Convex Integration, Isometric Extensions - ETH E

Diss. ETH No. 23133
Convex Integration,
Isometric Extensions &
Approximations of Curves
A thesis submitted to attain the degree of
Doctor of Sciences of ETH Zurich
(Dr. sc. ETH Zurich)
presented by
Micha Wasem
MSc Math. University of Fribourg
born February 28, 1986
citizen of
Schwarzenburg BE, Switzerland
accepted on the recommendation of
Prof. Dr. Norbert Hungerbühler, advisor
Prof. Dr. Camillo de Lellis, co-advisor
2016
Für meine Eltern
Für sie git’s nüt, wo’s nid git,
u aus wo’s git, git’s nid für ging,
sie nimmt’s wie’s chunnt u laht’s la gah.
– Büne Huber –
Zusammenfassung
In der vorliegenden Dissertation wird einerseits ein Fortsetzungsproblem für
isometrische Immersionen und Einbettungen in der Riemann’schen Geometrie untersucht und andererseits werden mit änhlichen Methoden zwei Approximationsresultate bewiesen – eines für Legendrekurven in dreidimensionalen
Kontaktmannigfaltigkeiten und das andere für Kurven mit vorgeschriebener
Krümmung in Euklidischen Räumen mit drei oder mehr Dimensionen.
Im ersten Kapitel befassen wir uns mit dem Fortsetzungsproblem: Den Ausgangspunkt bildet ein Resultat von Jacobowitz von 1974, welches notwendige
und hinreichende Bedingungen für die isometrische C 2 -Fortsetzbarkeit von
auf Hyperflächen vorgeschriebenen Isometrien liefert. Mit den von Nash entwickelten Methoden zur Konstruktion isometrischer Immersionen und Einbettungen der Klasse C 1 werden einseitige isometrische C 1 -Fortsetzungen
konstruiert, die auch dann existieren können, wenn einseitige C 2 -Fortsetzungen nicht zugelassen sind. Für die so konstruierten C 1 -Fortsetzungen weisen wir ein parametrisches, C 0 -dichtes h-Prinzip im Sinne von Gromov nach.
Die Konstruktion benötigt mindestens eine Kodimension.
In Kapitel 2 zeigen wir, dass im äquidimensionalen Fall im Allgemeinen keine isometrischen C 1 -Fortsetzungen existieren können. Durch Abschwächung
der Regularität beweisen wir aber mit einem ähnlichen Verfahren die Existenz von Lipschitz-Fortsetzungen, welche die Isometriebedingung fast überall
erfüllen. Das technische Rüstzeug für die Konstruktion der Fortsetzungen ist
in beiden Fällen die von Nash antizipierte – und schliesslich von Gromov
entwickelte – Methode der konvexen Integration. Die technische Schwierigkeit besteht darin, dass diese Methode im Allgemeinen auf der Hyperfläche,
wo die fortzusetzende Abbildung definiert ist, die Metrik nicht festlässt, was
aber mit Hilfe von Abschneidefunktionen behoben werden kann. Für die Konstruktion von äquidimensionalen Isometrien gilt es allerdings zu erwähnen,
dass es andere, allgemeinere Methoden gibt.
Das dritte Kapitel geht aus einer gemeinsamen Arbeit mit Thomas Mettler und meinem Doktorvater Norbert Hungerbühler hervor. Wir präsentieren
einen alternativen Beweis für ein wohlbekanntes Approximationsresultat aus
der Kontaktgeometrie. Dieser Beweis ermöglicht es, Legendrekurven erstmals
explizit zu konstruieren und zu visualisieren, die eine gegebene stetige Kurve
in einer 3-dimensionalen Kontaktmannigfaltigkeit beliebig gut approximieren.
Im letzten Kapitel beweisen wir, dass C 2 -Kurven mit vorgeschriebener Krümmung im n-dimensionalen Euklidischen Raum ein C 1 -dichtes h-Prinzip erfüllen, falls n > 3. Als Korollar dieses Resultates finden wir insbesondere eine Anwendung in der Knotentheorie: Wir beweisen die Existenz eines C 2 Knotens mit vorgeschriebener positiver Krümmung in jeder Isotopieklasse
und verallgemeinern damit ein Resultat von McAtee aus dem Jahr 2004,
welches die Existenz von C 2 -Knoten mit konstanter Krümmung in jeder Isotopieklasse garantiert.
Abstract
In this dissertation, we investigate on the one hand an extension problem for
isometric immersions and embeddings in Riemannian geometry and on the
other hand, using similar methods, we prove two approximation results: One
concerns Legendrian curves in contact 3-manifolds and the other is about
curves with prescribed curvature in Euclidean spaces of dimension at least 3.
The first chapter deals with isometric extensions: The starting point is a result by Jacobowitz from 1974, giving necessary and sufficient conditions for
the isometric C 2 -extendability of isometries that are prescribed on a hypersurface. Using Nash’s method for the construction of isometric immersions
and embeddings of class C 1 , we show the existence of one-sided isometric
C 1 -extensions which may exist even if one-sided classical extensions cannot
due to Jacobowitz’ result. Furthermore, we show that the so-constructed
extensions satisfy a C 0 -dense parametric h-principle in the sense of Gromov.
The construction requires at least one codimension.
In the second chapter, we show that isometric C 1 -extensions cannot generally exist in codimension 0. However, by relaxing the regularity, we show
with a similar method the existence of Lipschitz-extensions which satisfy the
isometry condition in a weak sense. The tool for the construction is in both
convex integration – a method anticipated by Nash and developed subsequently by Gromov. The main technical difficulty is that this method may
change the metric on the hypersurface, where the isometry one wishes to
extend is defined, but this can be circumvented using cut-off functions. Despite this, there are more general methods available for the construction of
different notions of weak isometries in codimension 0.
The third chapter stems from a project with Thomas Mettler and my advisor
Norbert Hungerbühler. We present an alternative proof of a well-known approximation theorem from contact geometry. This proof allows for an explicit
construction of Legendrian curves that approximate a given continuous curve
in a contact 3-manifold and delivers the first illustration of approximating
Legendrian curves.
In the final chapter, we establish a C 1 -dense h-principle for curves with prescribed curvature in Euclidean n-space provided n > 3. As a special case
of a corollary of this result, we obtain an application in knot theory and
prove that there exists a C 2 -knot of prescribed positive curvature in each
isotopy class. This generalizes a result by McAtee from 2004, which gives
the existence of C 2 -knots of constant curvature in every isotopy class.
Contents
Introduction
1
Preliminaries
8
1 Isometric C 1 -Extensions
1.1 Introduction . . . . . . . . . . . . . . . .
1.2 Obstructions and Adapted Subsolutions
1.3 Convex Integration . . . . . . . . . . . .
1.4 Iteration . . . . . . . . . . . . . . . . . .
1.5 h-Principle . . . . . . . . . . . . . . . . .
1.6 From Immersions to Embeddings . . . .
1.7 Global C 1 -Extensions . . . . . . . . . . .
1.8 Applications . . . . . . . . . . . . . . . .
2 Isometric Lipschitz-Extensions
2.1 Introduction . . . . . . . . . .
2.2 Obstructions . . . . . . . . . .
2.3 Subsolutions . . . . . . . . . .
2.4 Convex Integration . . . . . .
2.5 Iteration . . . . . . . . . . . .
2.6 Application . . . . . . . . . .
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3 Legendrian Approximation of Curves
3.1 Introduction . . . . . . . . . . . . . .
3.2 Legendrian Approximation in R3 . .
3.3 Gluing . . . . . . . . . . . . . . . . .
3.4 Examples . . . . . . . . . . . . . . .
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13
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48
48
51
53
54
58
66
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68
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69
72
75
Contents
4 h-Principle for Curves with Prescribed Curvature
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Step . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3 Iteration and Application to Knot Theory . . . . . . . . . . .
77
77
78
81
Bibliography
83
Dank
91
Introduction
Are abstract Riemannian manifolds and submanifolds of Euclidean space the
same? This question dates back to 1854, when Bernhard Riemann defined
the notion of an abstract Riemannian manifold in [Rie54]. In order to answer
the question affirmatively, one needs to isometrically embed a Riemannian
manifold (M, g) of dimension n into some Euclidean space (Rq , g0 ), where
g0 = h·, ·i denotes the Euclidean standard inner product, i.e. one has to find
an embedding u : M → Rq satisfying
g = u∗ g0 .
(1)
In local coordinates, the problem (1) amounts to solve the following system
of fully nonlinear first oder partial differential equations:
gij = h∂i u, ∂j ui.
Here, the q components of u are unknown and there are sn = n(n + 1)/2
equations due to the symmetry of g. The number sn is usually referred to as
Janet dimension.
In 1871, Ludwig Schläfli conjectured the local solvability of (1) in the formally determined case q = sn in the article [Sch73] (published in 1873). The
Schläfli-Conjecture has been verified in the analytic setting by Janet (1926
[Jan26]), Cartan (1927 [Car27]) and Burstin (1931 [Bur31]). In the smooth
setting, the problem is still open and the best-known result is due to Greene
(1969 [Gre69]) who constructed a smooth local solution when q = sn +n. For
the surface case, a better result due to Poznyak (1973 [Poz73]) is available,
that requires q = 4 and still leaves the conjecture open for n = 2. At low
regularity, there is a counterexample due to Pogorelov (1971 [Pog71]) who
constructed a metric of class C 2,1 on the 2-disk that has no neighborhood
of the origin that can be isometrically C 2 -embedded into R3 . In 1954, John
–1–
Introduction
Nash proved the surprising existence of C 1 -solutions to (1) in the formally
overdetermined case where q = n + 2. Gromov writes in [Gro00, p. 128]:
“This is sheer madness from a hard-minded analyst’s point of view as the [...]
components of u satisfy an overdetermined system [...], where one expects no
solutions at all!” Even more surprisingly, Nash showed that his C 1 -solutions
are very flexible and form a dense set in the space of short immersions (see
the theorem below for precise statements). The techniques he introduced to
tackle the problem were completely new as well. In [RS10, p. 394], Gromov
said: “At first, I looked at [...] Nashs paper and thought it was just nonsense. [...] it could not be true. But then I read it, and it was incredible. It
could not be true, but it was true [...]. He had a tremendous analytic power
combined with geometric intuition. This was a fantastic discovery for me:
How the world may be different from what you think!”
Building upon the work by Nash, Kuiper extended the result to q = n + 1
in [Kui55]. A further refinement has been obtained by Conti, de Lellis and
Székelyhidi in 2012 [CDS12], where the authors improved Kuiper’s result
from C 1 towards C 1,α provided α < 1/(1 + 2sn ). The so obtained Hölder
exponent α < 1/7 for surfaces has been recently improved by de Lellis and
Inauen towards α < 1/5 ([Ina15]). In [Gro86], Gromov constructed Lipschitzmaps that solve (1) almost everywhere in the equidimensional case q = n.
In order to find a global solution to (1), one first needs to know whether
every differentiable n-manifold can be embedded into some Euclidean space.
This question was answered by Whitney in 1936. He showed in [Whi36]
that every differentiable n-manifold can be embedded in R2n+1 and lowered
the dimension requirement in 1944 to 2n [Whi44]. In order to state the
known global results, we need to introduce some terminology: An immersion
u : M → Rq is called short, provided the equality in (1) is replaced by >
in the sense of quadratic forms, i.e. if g − u∗ g0 is positive definite. Note
that for compact manifolds, every immersion can be turned into a short one
by a suitable rescaling. The first global solution to the isometric embedding
problem is due to Nash and (with lower codimension requirement) to Kuiper:
Theorem (Nash 1954 [Nas54], Kuiper 1955 [Kui55])
Let u : (M n , g) → (Rq , g0 ) be a short immersion or embedding. Then, if
q > n, u can be uniformly approximated by C 1 -isometric immersions or
embeddings.
–2–
Introduction
For this statement, there is a refinement for compact manifolds due to Conti,
de Lellis and Székelyhidi (2012 [CDS12]) as well, where the C 1 regularity is
improved towards C 1,α , where α < 1/(1+2(n+1)sn ). It is known that such a
statement can not hold in class C 2 and Borisov showed in 1959 (see Theorem
1.4), that isometric C 1,α -embeddings of the standard 2-sphere into R3 are
rigid, if α > 2/3. It is still an open problem to determine the optimal regularity in the Nash-Kuiper theorem (see [Yau93, p. 8, Problem 27]). Allowing
for high codimension, Källén showed in [Käl78] that one can obtain isometric
C 1,α -immersions (-embeddings) of compact manifolds for any α < 1.
The Nash-Kuiper theorem has some very counterintuitive consequences: For
example, the hyperbolic plane H2 admits an isometric C 1 -embedding into R3 ,
which is in sharp contrast to Hilbert’s Theorem (1901) implying that such an
embedding can not be of class C 2 . The Nash-Kuiper theorem also implies the
breakdown of the aforementioned rigidity of isometric embeddings S 2 → R3 .
According to the density statement of the theorem, the standard sphere S 2
admits an isometric embedding into an arbitrary small portion of R3 . Another example is the flat torus R2 /Z2 that can be isometrically C 1 -embedded
into R3 . Again, such an embedding can not be of class C 2 and the lowest
known target dimension for the existence of a smooth isometric embedding
of R2 /Z2 into Euclidean space is 5 [Gro86, p. 298].
It took more than 50 years until the first image of such a counterintuitive
isometric embedding was produced. In 2012, Borelli, Jabrane, Lazarus and
Thibert produced in a beautiful work a picture of an isometric embedding of
R2 /Z2 into R3 (see Figure 1 and the references [BJLT12] and [BJLT13]).
We would like to mention furthermore that Nash proved a global isometric embedding result for high codimension and high regularity (i.e. C k for
k = 3, . . . , ∞ in [Nas56] and for the analytic case in [Nas66]), but we will
only focus on the low regularity case in the present work.
There is an extension problem related to (1): Let ι : Σ ,→ M be a hypersurface and let f : Σ → (Rq , g0 ) be a smooth isometric immersion (embedding).
Here, Σ is equipped with the metric ι∗ g and we will identify ι(Σ) and Σ.
When does f admit an extension to an isometric immersion (embedding)
–3–
Introduction
Figure 1: Isometric C 1 -embedding of the flat torus in R3
v : U → Rq satisfying
v ∗ g0 = g
(2)
v|Σ = f,
where U ⊂ M is a neighborhood of Σ? This question was first considered by
Jacobowitz in 1974 for the case of high codimension and high regularity in
[Jac74]. Jacobowitz derived a necessary condition on the second fundamental
forms of ι : Σ ,→ M and f : Σ → Rq respectively for isometric C 2 -extensions
to exist and showed that this condition is almost sufficient to prove local existence in the analytic and smooth categories requiring the same conditions
on the dimension (q = sn and q = sn + n) as in the respective local existence
theorems by Janet, Cartan, Burstin and Greene.
If Σ is a curve in R2 , q = 3 and f ∈ C 5 , the author gave an explicit geometric
construction for such isometric extensions in his Master Thesis (see [Was11]
and also the Master Thesis by Gilles Angelsberg [Ang03]). Figure 2 shows
an isometric extension that has been obtained in this way. In this example,
49
Σ = (0, 10
) × {0} ⊂ R2 and the map
f : (0, 49
) → R3 , f (t) := (6 − t) cos(4t), (6 − t) sin(4t), t2
10
has been extended to (0, 49
) × (−ε, ε) for some ε > 0.
10
–4–
Introduction
Figure 2: Isometric extension
In the present work, we will focus on the low regularity and low codimension
case of problem (2). Using a length comparison argument we will show
that Jacobowitz’ obstruction to local isometric C 2 -extendability is also an
obstruction to local isometric C 1 - and Lipschitz-extendability. However, if
Ω denotes the restriction of the neighborhood U in (2) to one side of Σ
only, we will prove the following existence theorem for one-sided isometric
C 1 -extensions under very mild hypotheses on Σ and f in codimension 1 (see
Theorem 1.2 for a more precise statement):
Theorem (C 1 -Extensions)
Let u : Ω̄ → Rn+1 be an extension of f that is short on Ω̄ \ Σ and isometric
on Ω̄ ∩ Σ. Then for every ε > 0, there exists a C 1 -immersion v : Ω̄ → Rn+1
satisfying v ∗ g0 = g, v|B = f and ku − vkC 0 (Ω̄) < ε. If u is an embedding, v
can be chosen to be an embedding as well.
This theorem provides an analogue of the Nash-Kuiper theorem for isometric
extensions. Together with a global and a parametric variant, it will be the
content of the first chapter.
In the second chapter we will discuss (2) in codimension 0 and our main
–5–
Introduction
Figure 3: A trefoil knot (black) and an approximating Legendrian curve
result is the following analogue of the previous theorem (cf. Theorem 2.3):
Theorem (Lipschitz-Extensions)
Let u : Ω̄ → Rn be an extension of f that is short on Ω̄ \ Σ and isometric
on Ω̄ ∩ Σ. Then for every ε > 0, there exists a Lipschitz map v : Ω̄ → Rn
a.e.
satisfying v|B = f , v ∗ g0 = g and ku − vkC 0 (Ω̄) < ε.
Both theorems make use of a method called convex integration, a tool developed by Gromov in [Gro73] for which the proof of the Nash-Kuiper theorem
was a precursor. The strategy consists in reducing the metric defect of an
immersion successively while controlling the C 1 - respectively the C 0,1 -norm
during the process.
The last two chapters deal with two other applications of convex integration:
In Chapter 3, we give an alternative and explicit proof of the well-known
Theorem 3.1 which roughly states as follows:
Theorem (Legendrian Approximation)
Every continuous curve in a contact 3-manifold can be C 0 -approximated by
a Legendrian curve.
Our proof delivers the first illustrations of Legendrian approximations for
given curves (see Figure 3).
–6–
Introduction
In Chapter 4 we show that curves in Rn with prescribed curvature satisfy
a C 1 -dense h-principle provided n > 3. Here, the curvature of a curve γ is
denoted by kγ and I denotes a compact interval. Our main result is Theorem
4.1:
Theorem (h-Principle for Curves with Prescribed Curvature)
Let γ0 ∈ C 2 (I, Rn ), n > 3 be an immersed (embedded) curve. Then for every
k ∈ C ∞ (I) satisfying k > kγ0 and every ε > 0, there exists a regular homotopy
(an isotopy) γs ∈ C 2 (I, Rn ), s ∈ [0, 1] such that kγ0 − γs kC 1 (I) < ε for all s
and such that kγ1 = k.
The statement about the isotopy becomes relevant only when γ0 is a knot in
R3 . Then, the approximating curve will be isotopic to γ0 . As a special case of
this result, we recover in particular the main result of [MG07] that provides
the existence of C 2 -knots of constant curvature in each isotopy class.
–7–
Preliminaries
Notations and Conventions
By a manifold M , we always understand a second countable, locally Euclidean topological Hausdorff space equipped with a smooth structure. We
denote by Γ(E) the module of smooth sections of a vector bundle E → M .
A Riemannian manifold (M, g) is a manifold together with a smooth metric
g. The Euclidean inner product will be denoted by either g0 or h·, ·i, the
outer product of two vectors a and b will be denoted by a ⊗ b := abT and we
will use the notation a b := 12 (a ⊗ b + b ⊗ a). The symbol ∧ stands for the
wedge-product, ~ for concatenation and we will tacitly use the identification
S1 ∼
= R/2πZ. Furthermore, we will make repeated use of the usual C k -norms
X
kf kC k (Ω) := sup
|∂ α f (x)|,
x∈Ω
|α|6k
where k ∈ N, Ω ⊂ Rn and α = (α1 , . . . , αn ) is a multi index.
Convex Integration
The main idea of convex integration is captured in the following lemma (see
[Spr10, Prop. 2.11, p. 28] for a reference):
Lemma (Fundamental Lemma of Convex Integration Theory)
Let R ⊂ Rn be a path-connected set and let p be contained in the interior of
the convex hull of R. Then there exist a continuous loop γ : S 1 → R having
p as its barycenter, i.e.
I
1
γ(t) dt = p.
2π S 1
We will discuss an elementary application that illustrates the way we will use
–8–
Preliminaries
(0, 1)
R
Figure 4: Illustration of R
convex integration throughout this thesis: Suppose
f : [0, 2π] → R2 , t 7→ (0, t)
is given and the problem consists in finding an approximation g : [0, 2π] → R2
of f that satisfies g 0 ∈ R, where
R := (a, b) ∈ R2 , |b| 6 ε|a|
and ε > 0 is a small fixed number. In other words, we intend to find a
solution g to the differential relation g 0 ∈ R such that kg − f kC 0 ([0,2π]) is as
small as desired. A priori, there is seemingly little hope to find such a g,
since f ([0, 1]) is a vertical line and the condition g 0 ∈ R forces g([0, 1]) to
have almost horizontal tangents. But the set R is ample, i.e. its convex hull
equals all of R2 and contains therefore the point (0, 1) ≡ f 0 (t). According to
the preceding lemma, we can find a loop γ : S 1 → R with barycenter (0, 1),
for example γ(s) := 2(ε−1 cos s, cos2 s). We claim now that the map
Z t
g(t) :=
γ(λs) ds
0
solves our problem, if λ 0. Since f (0) = g(0) and f 0 and g 0 agree in
average on shorter and shorter intervals as λ → ∞, f and g tend to become
close. Indeed g 0 ∈ R holds by construction and kg − f kC 0 ([0,2π]) = O(λ−1 )
follows from
1 4
.
|g(t) − f (t)| 6
sin(λt),
sin(2λt)
2λ ε
Since γ has to touch the origin, it is clear that g cannot be an immersion.
This is no longer true if we consider an analogous problem in 3-dimensions,
where f is replaced by fe(t) = (0, 0, t) and R by
n
o
√
e := (a, b, c) ∈ R3 , |c| 6 ε a2 + b2 .
R
–9–
Preliminaries
Figure 5: Illustration of f and g for the choices ε =
1
10
and λ = 50
e \ {0}, e.g. γ(s) := ε−1 (cos s, sin s, ε) and
Here we can choose γ : S 1 → R
the solution will be an embedded helix. In similar fashion, one can show
that the solutions to the differential relation g 0 ∈ R lie C 0 -dense in the space
of continuous plane curves. This is a manifestation of a so called C 0 -dense
h-principle, which will be explained below. A direct generalization of this
simple example leads to the explicit construction of Legendrian curves in
contact 3-manifolds, that approximate given continuous curves (see Chapter 3). In all of our proofs however, the philosophy consists in perturbing
a map in some C k -topology in order to achieve a desired change of order k+1.
Here is another example that involves no approximation: For each function
k ∈ C 0 ([0, 2π]), there is a unique (up to rigid motions) curve γ : [0, 2π] → R2
that is parametrized by arc length having signed curvature k, i.e. hJ γ̇, γ̈i = k,
where J = ( 01 −1
0 ). What conditions do we have to impose on k in order to
ensure that γ will be a closed curve? If γ should be parametrized by its arc
length, we require γ̇ ∈ S 1 , i.e. γ̇ = (cos ϑ, sin ϑ)T and a direct computation
shows that k = ϑ̇. Of course, γ will be closed if and only if the barycenter of
γ̇ is zero, i.e.
Z 2π 1
cos ϑ(t)
dt = 0
sin ϑ(t)
2π 0
and there are infinitely many choices for ϑ, as the origin is contained in the
convex hull of S 1 ⊂ R2 . But for example, if we want the curvature of γ
to be k(t) = a cos(t), then we find that it will be closed if and only if the
amplitude a is a zero of the Bessel function J0 (see Figure 6). In fact, the
– 10 –
Preliminaries
Figure 6: Closed curves with curvature a cos t, where a runs through the first
four zeros of J0 .
distance between the end-points of γ is in this case given by 2π|J0 (a)| as
follows from
Z 2π 1
cos(a sin t)
J0 (a)
dt =
.
sin(a sin t)
0
2π 0
The so-constructed curves are exactly the closed 2-minimal plane curves with
winding number zero in the sense of [CDVV95].
h-Principle
A differential relation R is a condition imposed on the partial derivatives of
some unknown function. A genuine solution is any function that satisfies R.
A formal solution is obtained by replacing the derivatives by new independent
variables. In this way one obtains an underlying algebraic relation, whose
solvability is a necessary condition for the solvability of R. The differential
relation R is said to satisfy the h-principle, if every formal solution can be
deformed into a genuine solution via a homotopy of formal solutions. If there
exists in addition a genuine solution C k -close to each formal solution, we say
that R satifies a C k -dense h-principle. If two genuine solutions, which are
homotopic though formal solutions, are actually homotopic through genuine
solutions, the h-principle is called parametric. The h-principle may have different notions of formal solutions which will be explained in Chapter 1 and 4.
– 11 –
Preliminaries
For more elementary examples and a very accessible introduction to convex
integration theory, we recommend the lecture notes by Vincent Borrelli1 as a
reference, where the reader may also find many very appealing illustrations.
For many more examples in various areas of mathematics, where convex integration was successfully used, we refer the reader to [Gro86], [EM02], [Spr10]
or [DjS12], where also the concept of h-principle is explained in greater detail.
1
http://math.univ-lyon1.fr/~borrelli/Recherche.html, February 2016
– 12 –
Chapter 1
Isometric C 1-Extensions
1.1
Introduction
The goal of this chapter is to obtain an analogue of the Nash-Kuiper theorem
for one-sided isometric extensions and to verify that these extensions satisfy
a C 0 -dense parametric h-principle in the sense of Gromov.
In local coordinates (2) can be reformulated as follows: Equip an open ball
in Rn centered at zero with an appropriate metric g and let the isometric
immersion f : B → Rn+1 be prescribed on B which is the intersection of
the closure of the ball with Rn−1 × {0}. The intersections of the ball with
Rn−1 × R>0 and Rn−1 × R60 are then called one-sided neighborhoods of B.
The image of a one-sided neighborhood of B under the inverse of a local
chart will be called a one-sided neighborhood (of Σ) as well (see Figure 1.1).
Definition 1.1
Let Ω be a one-sided neighborhood of B. A C ∞ -immersion u : Ω̄ → Rn+1 is
called subsolution adapted to (f, g), whenever u|B = f and g − u∗ g0 > 0 in
the sense of quadratic forms with equality on B only.
We are now ready to present the main theorem of this chapter:
Theorem 1.2 (C 1 -Extensions)
Let u : Ω̄ → Rn+1 be a subsolution adapted to (f, g). Then for every ε > 0,
there exists a C 1 -immersion v : Ω̄ → Rn+1 satisfying v ∗ g0 = g, v|B = f and
ku−vkC 0 (Ω̄) < ε. Moreover, the maps u and v are homotopic within the space
– 13 –
1.1. Introduction
M
Ω̄
Σ
B
U
Figure 1.1: One-sided neighborhoods
of subsolutions adapted to (f, g), and if u is an embedding we can choose v
to be an embedding as well.
As a corollary of Theorem 1.2 we obtain the following statement about isometric extensions of the standard inclusion ι : S 1 ,→ R2 × {0} ⊂ R3 to
isometric embeddings S 2 → R3 (here S 1 is the equator of S 2 ).
Corollary 1.3 (Wild Extensions on S 2 )
There are infinitely many isometric C 1 -embeddings v : S 2 → R3 satisfying
v|S 1 = ι.
Observe that this corollary is in sharp contrast to the following uniqueness
theorem (see [Bor58a, Bor58b, Bor59a, Bor59b, Bor60, CDS12]):
Theorem 1.4 (Borisov)
If α > 2/3, the standard inclusion S 2 ,→ R3 is – up to reflection across the
plane containing ι(S 1 ) – the only isometric C 1,α -extension of ι to S 2 .
Remark 1.5
The proof of Theorem 1.4 relies on the conservation of a weak form of Gaussian curvature for C 1,α -immersions whenever α > 2/3 (see [CDS12] and the
discussion in [DjS12, p. 369]). As already mentioned in the introduction, the
existence and determination of a threshold α between rigid and wild solutions is still an open problem (see for instance [GA13], [CDS12] and [Yau93,
Problem 27, p. 8]), but it has been conjectured to be α = 1/3, 1/2 or 2/3
(see for example [LP15]).
– 14 –
1.2. Obstructions and Adapted Subsolutions
Outline of the Chapter
In Section 1.2, we will present an obstruction to isometric C 1 -extendability
(Proposition 1.8) and the construction of adapted subsolutions (Proposition
1.9). In order to convert adapted subsolutions into isometric extensions, the
strategy consists in writing the metric defect of an adapted subsolution as a
sum of primitive metrics and add successively error terms. This uses a corrugation that is presented in Section 1.3. In Section 1.4 we construct one-sided
isometric C 1 -extensions. These extensions satisfy a C 0 -dense parametric hprinciple, which is the content of Section 1.5. In Section 1.6, we show how one
obtains embedded isometric extensions and finally, Section 1.7 indicates how
to obtain global results from the local ones by a partition of unity argument.
We will formulate all our results for the codimension 1 case q = n + 1, but
they might as well be carried on to higher codimension (see Remark 1.14).
1.2
Obstructions and Adapted Subsolutions
Obstructions
In this section, we will show that Jacobowitz’ necessary condition for the
existence of isometric C 2 -extensions is in fact an obstruction to isometric
C 1 -extensions. Recall that ι : Σ ,→ M is a hypersurface of an n-dimensional
Riemannian manifold (M, g) and f : Σ → Rn+1 is an isometric immersion
we seek to extend to a neighborhood U ⊂ M of a point in Σ.
Let A ∈ Γ(S2 (T ∗ Σ)⊗N Σ) denote the second fundamental form of ι : Σ ,→ M ,
and h(X, Y ) := g(ν, A(X, Y )), where ν ∈ Γ(N Σ) is a unique (up to sign) unit
vector field. Let further Ā ∈ Γ(S2 (T ∗ Σ) ⊗ f ∗ N Σ̄) be the second fundamental
form of Σ̄ := f (Σ) in Rq . In [Jac74], Jacobowitz showed that if u ∈ C 2 (U, Rq )
solves (2), then there exists a unit vector field ν̄ ∈ Γ(f ∗ N Σ̄) such that
h(X, Y ) = hν̄, Ā(X, Y )i
for all vector fields X, Y ∈ Γ(T (Σ ∩ U )). In particular,
|h(X, Y )|g 6 |Ā(X, Y )|
and we get as a corollary:
– 15 –
(1.1)
1.2. Obstructions and Adapted Subsolutions
Corollary 1.6 (C 2 -Obstruction)
If there exists a unit vector v ∈ Tp Σ such that |h(v, v)|g > |Ā(v, v)|, no
isometric extension u ∈ C 2 (U, Rq ) can exist.
In order to prove the obstruction to isometric C 1 -extendability, we need the
following lemma that seems to appear for the first time in [Haa47, Theorem
5] but we will give an independent proof for the convenience of the reader:
Lemma 1.7
Let γ : [0, ε) → (M, g) be a unit speed C 4 -curve with γ(0) = p and let kg
denote the geodesic curvature of γ in p. Then the geodesic distance from p
to γ(t) satisfies
d(p, γ(t)) = t −
kg2 3
t + O(t4 ) for t → 0.
24
Proof. In order to fix notation, the Levi-Civita connection of g is denoted by
∇ and we will invoke the Einstein summation convention (summation over
repeated indices). Pick geodesic normal coordinates centered at p (see e.g.
[KN96]). Then
Z
t
g (γ̇(s), ∂r (s)) ds,
d(p, γ(t)) =
0
where ∂r (s) := (d expp )γ̄(s) γ̄(s) |γ̄(s)|−1
is the radial vector field and
g
γ̄(t) := exp−1
p (γ(t)).
Using the Gauss-Lemma we find
Z t
d(p, γ(t)) =
g (γ̇(s), ∂r (s)) ds
0
Z t γ̄(s)
˙
g (d expp )γ̄(s) (γ̄(s)),
(d expp )γ̄(s) |γ̄(s)|
=
ds
g
0
Z t γ̄(s)
˙
=
g γ̄(s),
ds
|γ̄(s)|g
0
Z t
d
=
|γ̄(s)|g ds
0 ds
= |γ̄(t)|g .
Consider the expansion
˙
γ̄(t) = γ̄(0)t
+
...
γ̄¨ (0) 2 γ̄ (0) 3
t +
t + O(t4 ) for t → 0.
2!
3!
– 16 –
1.2. Obstructions and Adapted Subsolutions
Fix the coordinates such that γ̇ = γ̄˙ i ∂i and such that in zero γ̇(0) = ∂1 |t=0 .
Since
(d expp )0 : T0 (Tp M ) ∼
= Tp M → Tp M
is the identity, we find
Since ∇γ̇ γ̇|t=0 = ∇∂1 γ̇|t=0
˙
γ̄(0)
= γ̇(0).
and ∇2γ̇ γ̇ t=0 = ∇2∂1 γ̇ t=0 , we obtain
(1.2)
∇∂1 γ̄˙ i ∂i = γ̄¨ k ∂k + γ̄˙ i Γk1i ∂k
and since the Christoffel symbols vanish in p, we obtain
∇γ̇ γ̇|t=0 = γ̄¨ (0).
(1.3)
For the second covariant derivative we compute
∇2∂1 γ̇ = ∇∂1 γ̄¨ k + γ̄˙ i Γk1i ∂k
...
k
i k
i
k
¨
˙
= γ̄ + γ̄ Γ1i + γ̄ ∂1 Γ1i ∂k + γ̄¨ k + γ̄˙ i Γk1i Γl1k ∂l .
Evaluating in zero gives
...
∇2γ̇ γ̇ t=0 = γ̄ (0) + γ̄˙ i ∂1 Γk1i ∂k t=0 .
Using the following identity that holds in normal coordinates (see [LZ95]
equation (6))
1 i
i
(0) + Rkjl
(0)
∂l Γijk (0) = − Rjkl
3
and γ̇(0) = ∂1 |t=0 , we obtain
...
...
1 k
k
∇2γ̇ γ̇ t=0 = γ̄ (0) −
R111 (0) + R111
(0) ∂k |t=0 = γ̄ (0),
(1.4)
3
where the last equality follows from the antisymmetry of the curvature tensor
in the last two slots. Invoking (1.2), (1.3) and (1.4) it follows that (for t → 0)
1
1
1
d(p, γ(t))2 = t2 + g(∇γ̇ γ̇, γ̇)|t=0 t3 + kg2 t4 + g(γ̇, ∇2γ̇ γ̇)t=0 t4 + O(t5 )
2
4
3
and since γ is parametrized by arc length, g(∇γ̇ γ̇, γ̇)|t=0 = 0 and
kg2 = − g(γ̇, ∇2γ̇ γ̇)t=0 .
This implies
kg2 4
t + O(t5 ).
12
√
The desired result follows from invoking 1 + x = 1 + 12 x + O(x2 ) for x → 0.
d(p, γ(t))2 = t2 −
– 17 –
1.2. Obstructions and Adapted Subsolutions
Proposition 1.8 (C 1 -Obstruction)
If there exists a unit vector v ∈ Tp Σ such that |h(v, v)|g > |Ā(v, v)|, no
isometric extension u ∈ C 1 (U, Rq ) can exist.
Proof. We argue by contradiction. Suppose there exists such an extension u
and let γ : [0, ε) → Σ ∩ U be a geodesic with γ(0) = p and γ̇(0) = v such
that for every fixed t ∈ [0, ε), dM (p, γ(t)) is realized by a minimizing geodesic
σ : [0, 1] → U . Let p̄ = f (p) and γ̄ = f ◦ γ. Observe that
Z 1
Z 1
d
|σ̇(t)|g dt = dM (p, γ(t)),
|u(σ(1)) − u(σ(0))| 6
dt (u ◦ σ)(t) dt =
0
0
hence |γ̄(t) − p̄| 6 dM (p, γ(t)). Since we have
Σ
kg (p) = (∇M
γ̇ γ̇)(0) = (∇γ̇ γ̇)(0) + A(v, v),
we find
|h(v, v)|2g 3
dM (p, γ(t)) = t −
t + O(t4 ) for t → 0.
24
This together with an analogous computation of the geodesic curvature of γ̄
in p̄ gives
|γ̄(t) − p̄| − dM (p, γ(t)) =
1
|h(v, v)|2g − |Ā(v, v)|2 t3 + O(t4 ) for t → 0
24
contradicting |γ̄(t) − p̄| 6 dM (p, γ(t)).
Observe that the foregoing Proposition does not exclude the existence of
a C 1 -solution to (2) on a one-sided neighborhood of Σ (the part of U that
doesn’t contain the geodesic segment σ in M that relies p and γ(t)). However
such a “one-sided” isometric extension cannot be of class C 2 since equation
(1.1) is a pointwise statement that applies to points in Σ.
Subsolutions
We now present a sufficient condition for the existence of adapted subsolutions.
Proposition 1.9
Let f : Σ → Rn+1 be an isometric immersion. Suppose there exists a unit
normal field ν̄ ∈ Γ(f ∗ N Σ̄) such that h(·, ·) − hĀ(·, ·), ν̄i is positive definite.
Then around every p ∈ Σ, there exists a subsolution adapted to (f, g). This
adapted subsolution can be chosen to be an embedding.
– 18 –
1.2. Obstructions and Adapted Subsolutions
Proof. Choose a submanifold chart around p ∈ Σ as in Definition 1.1 and let
ρ : B → Σ be a parametrization. Consider the maps
ψ : B × [0, ε] → M
(x, t) 7→ expρ(x) (−tν(x))
u : B × [0, ε] → Rn+1
(x, t) 7→ (f ◦ ρ)(x) − s(t)ν̄(x),
where s ∈ C ∞ (R>0 , R) satisfies s(0) = 0, s0 (0) = 1, s00 (0) < 0 and ν ∈ Γ(N Σ)
is the unit vector field such that hA(·, ·), νi = h(·, ·). We claim that the map
u is a subsolution adapted to (f ◦ ψ, ψ ∗ g). It is clear from the definition, that
u|B = f ◦ ψ|B and we need to show that ψ ∗ g − u∗ g0 > 0 with equality on B
only. We think of (M, g) as being smoothly and isometrically embedded in
some Euclidean space in order to consider the Taylor expansion of ψ around
t = 0 (see e.g. [MMASC14]):
1e
ψ(x, t) = ρ(x) − tν(x) + A(ν(x),
ν(x))t2 + O(t3 ),
2
e is the second fundamental form of M with respect to that embedwhere A
ding. The coefficients of the pullback metric of ψ are
h∂i ψ, ∂j ψi = h∂i ρ, ∂j ρi − th∂i ρ, ∂j νi − th∂j ρ, ∂i νi + O(t2 )
e ν)i − th∂i ν, νi + O(t2 ) = O(t2 )
h∂i ψ, ∂t ψi = h∂i ρ, νi − th∂i ρ, A(ν,
e ν)i + O(t2 ) = |ν|2 + O(t2 ),
h∂t ψ, ∂t ψi = |ν|2 − 2thν, A(ν,
and the ones for u
h∂i u, ∂j ui = h∂i (f ◦ ρ), ∂j (f ◦ ρ)i − s(t)h∂i (f ◦ ρ), ∂j ν̄i − s(t)h∂j (f ◦ ρ), ∂i ν̄i
h∂i u, ∂t ui = 0
h∂t u, ∂t ui = s0 (t)2 |ν̄|2 .
A direct computation shows that
ψ ∗ g − u ∗ g0 =
2thij − 2s(t)hĀ(∂i , ∂j ), ν̄i ij
0
0
1 − s0 (t)2
!
+ O(t2 ),
which is positive definite if and only if 2t hij − hĀ(∂i , ∂j ), ν̄i + O(t2 ) is
positive definite, which is the case for small t > 0 by assumption. The
statement regarding embeddings follows immediately from the compactness
of B × [0, ε], the fact that exp is a local diffeomorphism and an appropriate
choice of ε > 0.
– 19 –
1.3. Convex Integration
Example 1.10 (Adapted Subsolutions despite the C 1 -Obstruction)
The Euclidean metric of Rn in polar coordinates (r, ϕ2 , . . . ϕn ) reads
dr ⊗ dr + r
2
n
X
i,j=2
gij dϕi ⊗ dϕj ,
where the gij do not depend on r. We now equip Rn with a new metric
ĝ = dr ⊗ dr + Ψ(r)
n
X
i,j=2
gij dϕi ⊗ dϕj ,
where Ψ is a smooth real valued function satisfying Ψ(1) = 1 and Ψ0 (1) > 2.
The map f : S n−1 ,→ Rn × {0} ⊂ Rn+1 is an isometric embedding of
S n−1 ⊂ (Rn , ĝ) into (Rn+1 , g0 ). Let ∂i := ∂ϕi for i ∈ {2, . . . , n} and hĝ
denote the scalar second fundamental form of S n−1 in Rn . Since −∂r is a
unit normal vector field on S n−1 , we find that
1
hĝij = −dr(∇ĝ∂i ∂j ) = −dr(Γkij ∂k ) = −Γrij = Ψ0 (1)gij
2
and hence hĝij − hgij0 = 21 (Ψ0 (1) − 2)gij > 0 in the sense of quadratic forms.
Choosing v := ∂i for any i ∈ {2, . . . , n}, the C 1 -obstruction shows that there
exists no ĝ-isometric C 1 -extension of f to a neighborhood of S n−1 but according to the previous Proposition, we can construct a subsolution adapted to
(f, ĝ) on a one-sided neighborhood of S n−1 of the form {1 6 |x| < 1+ε} ⊂ Rn
for some ε > 0.
1.3
Convex Integration
The goal of this section is to turn adapted subsolutions into one-sided isometric C 1 -extensions. The construction of these extensions is based on the
method of Nash [Nas54], Kuiper [Kui55], Conti, de Lellis and Székelyhidi
[CDS12]. We start with a subsolution adapted to (f, g), u : Ω̄ → Rn+1 and
decompose the metric defect g − u∗ g0 into a sum of primitive metrics (see
[GA13, p. 202, Lemma 1] for an explanation) as
(g − u∗ g0 )x =
m
X
k=1
a2k (x)νk ⊗ νk ,
where a2k are nonnegative smooth functions on Ω̄ \ B that extend continuously to B and vanish on B, νk ∈ S n−1 and m ∈ N is a finite number but at
– 20 –
1.3. Convex Integration
most m0 6 m terms in the above sum are non-zero for fixed x (see [GA13]),
where m0 depends on the dimension n only.
A stage then consists of m steps, of which each aims at adding one primitive
metric a2 ν ⊗ ν. Fix orthonormal coordinates in the target so that the the
metric u∗ g0 can be written as ∇uT ∇u, where ∇u = (∂j ui )ij . For a specific
unit vector ν ∈ S n−1 and a smooth nonnegative function a ∈ C ∞ (Ω̄), we aim
at finding v : Ω̄ → Rn+1 satisfying ∇v T ∇v ≈ ∇uT ∇u + a2 ν ⊗ ν. Nash solved
this problem using the Nash Twist:
a(x)
(cos(λhx, νi)β1 (x) + sin(λhx, νi)β2 (x)) ,
(1.5)
λ
where βi are mutually orthogonal unit normal fields, requiring thus 2 codimensions. We will explain this ansatz in more detail below. The improvement
to codimension 1 has first been achieved by Kuiper [Kui55] with the use of
a different ansatz (strain).
v(x) = u(x) +
We will use a corrugation introduced by Conti, de Lellis and Székelyhidi
[CDS12] (see equation (1.16)) and modify it slightly in order to achieve the
desired metric change within the class of adapted subsolutions. First, we give
a geometric motivation for the choice of the corrugation that follows [GA13].
Choose vectors
−1
ξe := ∇u · ∇uT ∇u
· ν,
ζe := ? (∂1 u ∧ ∂2 u ∧ . . . ∧ ∂n u) ,
where ? denotes the Hodge star with respect to the usual metric and orientation in Rn+1 . Let
ξe
ζe
ξ :=
,
ζ :=
.
(1.6)
e2
e ξ|
e
|ξ|
|ζ||
We use an ansatz of the form
1
v(x) = u(x) + (Γ1 (x, λhx, νi)ξ(x) + Γ2 (x, λhx, νi)ζ(x)) ,
λ
where Γ ∈ C ∞ (Ω̄ × S 1 , R2 ), (x, t) 7→ Γ(x, t) is a family of loops still to be
constructed. The differential of v reads
1
∇v = ∇u + ∂t Γ1 ξ ⊗ ν + ∂t Γ2 ζ ⊗ ν + E,
λ
where E := ξ∇x Γ1 + ζ∇x Γ2 + Γ1 ∇ξ + Γ2 ∇ζ and therefore
∇v T ∇v = ∇uT ∇u +
1
2∂t Γ1 + (∂t Γ1 )2 + (∂t Γ2 )2 ν ⊗ ν + r,
e2
|ξ|
– 21 –
(1.7)
1.3. Convex Integration
Sr21 (0)
Sr12 (−1)
Figure 1.2: The left image illustrates the Nash Twist and the right one, the
codimension 1 corrugation.
with
r=
2
1
Sym ∇uT E + ∂t Γ1 ν E T ξ + ∂t Γ2 ν E T ζ + 2 E T E ,
λ
λ
where Sym(A) := 21 (A + AT ) and a b := Sym(a ⊗ b).
In order to equate the coefficient of ν ⊗ ν in (1.7) and a2 , ∂t Γ needs to satisfy
e 2 a2 . Since we require Γ to be
the circle equation (∂t Γ1 + 1)2 + ∂t Γ22 = 1 + |ξ|
H
2π-periodic, we also need S 1 ∂t Γ dt = 0. Note that in codimension 2, where
ξ and ζ can be replaced by mutually orthogonal normal vectors say β1 and
β2 , a similar ansatz leads to the circle equation ∂t Γ21 + ∂t Γ22 = a2 which is
fulfilled by the choice ∂t Γ(x, t) := a(x)(− sin(t), cos(t)). This explains the
Nash Twist (1.5). We look for a 2π-periodic map ∂t Γ(x, ·) that takes values
in a circle parametrized by
√
cos(f (s)b(t))
1
2
(s, t) 7→ 1 + s
−
,
sin(f (s)b(t))
0
e and f (s) and b(t) are still to be chosen. In order to satisfy
where s := |ξ|a
the periodicity condition, we require
I √
1
cos(f (s)b(t))
1
2
1+s
−
dt = 0.
sin(f (s)b(t))
0
2π S 1
The second component should be zero independently of s when integrated.
This forces b to be 2π-periodic and antisymmetric with respect to π. The
simplest choice is b(t) := sin t. For the first component, we aim at finding a
– 22 –
1.3. Convex Integration
function f such that
1
J0 (f (s)) :=
2π
Z
0
2π
cos(f (s) sin t) dt = √
1
.
1 + s2
Note that J0 is the zeroth Bessel function of the first kind.
Lemma 1.11 (Existence of f )
There exists a function f ∈ C ∞ (R) such that J0 (f (s)) =
satisfying
√
2 + s2
0 < f 0 (s) 6
.
1 + s2
√ 1
1+s2
=: w(s)
(1.8)
Proof. Local existence around zero (see [CDS12]): Consider the function
1
F (s, r) := J0 (r 2 ) − √
1
.
1 + s2
Direct computations involving the series expansion of J0 yield F (0, 0) = 0
and ∂r F (0, 0) = − 14 . The implicit function Theorem applies and we get some
δ > 0 and a function h ∈ C ∞ ((−δ, δ)) with h(0) = 0 such that F (s, h(s)) = 0.
Moreover, since ∂s F (0, 0) = 0 and ∂s2 F (0, 0) = 1, we find h0 (0) = 0 and
h00 (0) = 4. Hence we get a smooth f (s) satisfying f 2 (s) = h(s). Direct
√
computations yield f (0) = 0 and f 0 (0) = 2.
In order to prove global existence, observe that w ∈ C ∞ (R) takes values in
(0, 1]. Since J0 : [0, µ] → [0, 1] is a bijection (µ being the smallest positive
zero of J0 ) and J00 doesn’t admit any zero on (0, µ], its inverse J0−1 is in
C ∞ ([0, 1)). Now set
f (s) := sgn s · J0−1 (w(s))
This function is smooth on R \ {0}. Since f corresponds around zero to the
function constructed by means of the implicit function Theorem, f ∈ C ∞ (R).
We will now prove an estimate, from which (1.8) follows. We will need the
following three estimates:
2x − 2
x−1
6 log x 6 √ , x > 1
x+1
x
|f (s)|
√
6 |J1 (f (s))|
2 1 + s2
x2 x4
x2
−
6 J2 (x) 6
8
96
8
– 23 –
(1.9)
(1.10)
(1.11)
1.3. Convex Integration
µ
f
−µ
Figure 1.3: The graph of the function f over the interval [−5, 5].
A proof of (1.9) can be found in [Top07]. We prove (1.10) using an integral
representation for Bessel functions, trigonometric identities and integration
by parts:
Z 2π
1
sin(f (s) sin t) sin tdt
J1 (f (s)) =
2π 0
Z
f (s) 2π
=
cos2 t cos(f (s) sin t)dt
2π 0
Z
Z
f (s) 2π
f (s) 2π
=
cos(2t) cos(f (s) sin t)dt +
cos(f (s) sin t)dt
4π 0
4π 0
f (s)
=
(J2 (f (s)) + J0 (f (s)))
2 1
f (s)
J2 (f (s)) + √
.
=
2
1 + s2
(1.12)
The identity (1.12) implies (1.10) since J2 is nonnegative on [−µ, µ]. In order
to prove (1.11) we use again an integral representation for J2 and integration
by parts to obtain
Z 2π
1
J2 (x) =
cos(2t) cos(x sin t)dt
2π 0
Z 2π
x
=
sin(2t) cos t sin(x sin t)dt
4π 0
Z
x2
x2 2π
6
| sin(2t) cos t sin t|dt = .
4π 0
8
For the other inequality of (1.11), we use the same expression for J2 and use
– 24 –
1.3. Convex Integration
3
sin x = x + R3 (x), where |R3 (x)| 6 |x|3! . It follows that
Z 2π
x
sin(2t) cos t (x sin t + R3 (x sin t)) dt
J2 (x) =
4π 0
Z 2π
x2
x
=
+
sin(2t) cos t R3 (x sin t)dt
8
4π 0
and hence
x2
x4
− J2 (x) 6
8
24π
Z
2π
| sin(2t) cos t sin3 t|dt =
0
x4
.
96
Using the definition of f implies
0 < f 0 (s) =
s
p
J1 (f (s)) (1 + s2 )3
and hence (1.10) implies
f 0 (s) 6
2s
.
f (s)(1 + s2 )
Multiplication with f and integration together with the second inequality of
(1.9) gives
s
p
2s2
|f (s)| 6 2 log(1 + s2 ) 6 √
.
(1.13)
1 + s2
We return to the identity (1.12) and use (1.11) together with
on [−µ, µ] to obtain
x2
16
6
x2
8
−
x4
96
16s
32s
√
√
6 f 0 (s) 6
2
2
2
f (s)(1 +
+ f (s) 1 + s )
f (s)(1 + s )(16 + f 2 (s) 1 + s2 )
(1.14)
Multiplication of the first inequality of (1.14) with f and (1.13) implies
s2 )(8
8|s|
1 2 0 6
(f ) (s)
(4 + s2 )(1 + s2 )
2
and hence integrating together with the first inequality of (1.9)
s
r
2
2
4s + 4
s2
.
|f (s)| > 2
log
>
4
3
s2 + 4
8 + 5s2
This lower bound on f can be plugged in the second inequality of (1.14) to
obtain
√
3
8 + 5s2
0
√
.
f (s) 6
2(1 + s2 ) 8 + s2 5 + 1 + s2
– 25 –
1.3. Convex Integration
√
3 √
From this estimate, (1.8) follows by using
8 + 5s2 6 8 + 4s2 (8 + 6s2 )
√
in the numerator and 5 + 1 + s2 > 6 in the denominator.
With this choice of f , let Γ : R2 → R2 ,
Z t √
cos(f (s) sin u)
1
2
Γ(s, t) :=
1+s
−
du.
sin(f (s) sin u)
0
0
Lemma 1.12 (Corrugation)
The function Γ is 2π-periodic in the second argument, hence Γ : R×S 1 → R2 ,
and it holds that
√
|∂t Γ(s, t)| 6 2|s|,
(1.15)
√
where the constant 2 is optimal.
Proof. For the periodicity we compute directly (see also [CDS12] for this
computation):
Z t+2π √
cos(f (s) sin u)
1
2
Γ(s, t + 2π) − Γ(s, t) =
1+s
−
du
sin(f (s) sin u)
0
t
Z 2π √
cos(f (s) sin u)
1
2
1+s
−
du
=
sin(f (s) sin u)
0
0
√
1 + s2 J0 (f (s)) − 1
0
= 2π
=
.
0
0
In order to prove (1.15), observe that since ∂t Γ(0, t) = 0, integrating in s
yields
Z s
∂t Γ(s, t) =
∂s ∂t Γ(r, t) dr,
0
√
and we need to show that |∂s ∂t Γ| is bounded by 2. We compute
s
cos(f (s) sin t)
∂s ∂t Γ(s, t) = √
+
1 + s2 sin(f (s) sin t)
√
− sin(f (s) sin t)
0
2
+ 1 + s f (s) sin t
,
cos(f (s) sin t)
hence using (1.8)
|∂s ∂t Γ(s, t)|2 6
Since ∂s ∂t Γ(0, π2 ) =
√
s2
+ (1 + s2 )[f 0 (s)]2 6 2.
1 + s2
2, the constant is optimal.
– 26 –
1.4. Iteration
Figure 1.4: u(S 1 ) and v(S 1 ) for the choices λ = 3, . . . , 7
Remark 1.13
Note that Conti, de Lellis and Székelyhidi use the same corrugation function,
but in [CDS12], Γ is shown to exist on [0, ε]×S 1 for some small number ε > 0
and estimates on the derivatives of all orders are provided. The difference
is that here Γ is shown to exist on all of R × S 1 , the estimate (1.15) holds
globally and we don’t need estimates on the higher order derivatives at the
price of getting lower regularity of the solution in the end.
Adding a primitive metric a2 ν ⊗ ν to ∇uT ∇u is hence done with the ansatz
i
1h
e λh·, νi)ξ + Γ2 (a|ξ|,
e λh·, νi)ζ
v =u+
Γ1 (a|ξ|,
(1.16)
λ
and we say that v is obtained from u by convex integration.
Remark 1.14
In higher codimension q > n + 1, one can always find (locally) a smooth
normal vector field (that plays the role of ζ), hence the same construction
can be be used in the case of higher codimension. For this reason we will
restrict ourselves to the codimension 1 case.
Example 1.15
If the corrugation above is applied to the map u : S 1 → R2 that sends S 1 to
a circle with radius 1/2 in R2 , ansatz (1.16) produces the picture shown in
Figure 1.4.
1.4
Iteration
Step
Since the ansatz (1.16) reaches the desired metric change only up to an error
term O(λ−1 ), the corrugation above is not suitable for adapted subsolutions
– 27 –
1.4. Iteration
since it possibly adds a metric defect on B, where the adapted subsolution is
already isometric. We will overcome this difficulty by replacing the primitive
metric we wish to add by a “cut-off” primitive metric that vanishes near
B. Adding this modified primitive metric can then be done while leaving
the initial map u unchanged near B. In order to do this, let ` > 0 and let
ηe` be a C ∞ -function defined on R such that ηe` vanishes on (−∞, `/2], is
monotonically increasing on [`/2, `] and constant with value 1 elsewhere. Let
then η` : Ω̄ → R, η` (x1 , . . . , xn ) = ηe` (xn ) and let
Ω̄j := {x ∈ Ω̄, dist(x, B) 6 j},
where dist denotes the Euclidean distance.
Ωcj
j
Ω̄j
Figure 1.5: Illustration of the definition of Ω̄j
Proposition 1.16 (k-th Step)
Let uk−1 ∈ C ∞ (Ω̄, Rn+1 ) be an immersion. Then for every ε > 0 and every
0 < δ < 1 there exists an immersion uk ∈ C ∞ (Ω̄, Rn+1 ) that agrees with uk−1
on Ω̄`/2 such that the following estimates hold, provided the real parameter
λk > 0 is suitably chosen:
kuk − uk−1 kC 0 (Ω̄) 6 ε,
√
|∇uk − ∇uk−1 | 6 2ak + O(λ−1
k )
T
T
∇uk ∇uk − ∇uk−1 ∇uk−1 + (1 − δ)η`2 a2k νk ⊗ νk 0 6
C (Ω̄)
(1.17)
(1.18)
δ2
.
2m
(1.19)
Proof. We use the ansatz
1
uk (x) = uk−1 (x)+
Γ1 (s, λk hx, νk i) ξk (x)+Γ2 (s, λk hx, νk i) ζk (x) , (1.20)
λk
where s := (1 − δ)1/2 η` (x)ak (x)|ξek (x)| and the vector fields ξk and ζk are
constructed as in (1.6) and ak is nonnegative. Observe that ak might fail
– 28 –
1.4. Iteration
to be smooth on B, but since η` ak is smooth, so is uk . Since Γ(0, t) = 0
and η` |Ω̄`/2 = 0, uk and uk−1 agree on Ω̄`/2 . From (1.20) we immediately get
|uk − uk−1 | 6 C/λk , where C depends on Ω̄ and k. Choosing λk adequately
proves (1.17). For the differential one gets
∇uk = ∇uk−1 + ∂t Γ1 ξk ⊗ νk + ∂t Γ2 ζk ⊗ νk +
1
Ek ,
λk
where
1
Ek := (1 − δ) /2 (∂s Γ1 ξk + ∂s Γ2 ζk ) ⊗ grad(η` ak |ξek |) + Γ1 ∇ξk + Γ2 ∇ζk .
Using the definition of ξk and ζk we obtain
|∂t Γ1 ξk ⊗ νk + ∂t Γ2 ζk ⊗ νk |2 6 | (∂t Γ1 ξk + ∂t Γ2 ζk ) ⊗ νk |2
6 |∂t Γ1 ξk + ∂t Γ2 ζk |2
6 |ξek |−2 |∂t Γ|2 .
This estimate together with (1.15) leads to the pointwise estimate
|∇uk − ∇uk−1 | 6 |∂t Γ1 ξk ⊗ νk + ∂t Γ2 ζk ⊗ νk | + λ−1
k |Ek |
6 |ξek |−1 |∂t Γ| + λ−1
k |Ek |
√
1/2
6 2(1 − δ) ak η` + λ−1
k |Ek |
√
6 2ak + λ−1
k |Ek |.
This proves (1.18). The pullback metric is given by
∇uTk ∇uk = ∇uTk−1 ∇uk−1 + (1 − δ)η`2 a2k νk ⊗ νk + rk ,
where
rk :=
2
2
1
Sym ∇uTk−1 Ek +
∂t Γ1 νk EkT ξk + ∂t Γ2 νk EkT ζk + 2 EkT Ek .
λk
λk
λk
Observe that |Ek | 6 C(ak + η` ), where the constant C depends on `, k and
Ω̄. This leads to the estimate
|rk | 6
C
(ak + η` + a2k + η`2 ),
λk
(1.21)
2
δ
where again, the constant depends on `, k and Ω̄. Hence krk kC 0 (Ω̄) 6 2m
provided λk is large enough. The computation of the pullback metric also
implies ∇uTk ∇uk > ∇uTk−1 ∇uk−1 + rk , hence uk is an immersion provided λk
is large enough.
– 29 –
1.4. Iteration
Stage
A stage consists in adding iteratively “cut-off” primitive metrics while controlling the C 1 -norm of the resulting maps. This is the key ingredient to
later obtain the convergence in C 1 (Ω̄, Rn+1 ). The process leaves the initial
map unchanged near B.
Proposition 1.17 (Stage)
Let u ∈ C ∞ (Ω̄, Rn+1 ) be a subsolution adapted to (f, g). For any ε > 0 there
exists a map u
e ∈ C ∞ (Ω̄, Rn+1 ) with the following properties:
ku − u
ekC 0 (Ω̄) 6 ε,
(1.22)
kg − ∇e
uT ∇e
ukC 0 (Ω̄) 6 ε,
(1.23)
1/2
k∇u − ∇e
ukC 0 (Ω̄) 6 Ckg − ∇uT ∇ukC 0 (Ω̄) .
(1.24)
Moreover, u
e is an adapted subsolution with respect to (f, g) provided ε > 0 is
small enough.
Proof. Choose ` > 0 such that
kg − ∇uT ∇ukC 0 (Ω̄` ) <
ε
2
(1.25)
and δ such that the following two conditions are met:
δ id < g − ∇uT ∇u |Ωc ,
`/2
n
√ o
−1
T
ε
δ < min 2 kg − ∇u ∇ukC 0 (Ω̄) , ε .
(1.26)
(1.27)
Now we use Proposition 1.16 iteratively and choose λk in each step such that
the following three conditions hold:
krk kC 0 (Ω̄) 6
kuk − uk−1 kC 0 (Ω̄) <
1
kEk kC 0 (Ω̄)
λk
6
δ2
,
2m
ε
,
m
1
kg
m
(1.28)
(1.29)
1/2
− ∇uT ∇ukC 0 (Ω̄) .
(1.30)
We start with the map u0 = u and will get after m steps the desired map
u
e := um . Using (1.29) we find
ke
u − ukC 0 (Ω̄) 6
m
X
k=1
kuk − uk−1 kC 0 (Ω̄) < ε.
– 30 –
1.4. Iteration
This proves (1.22). We have
g − ∇e
uT ∇e
u = g − ∇uT ∇u + ∇uT ∇u − ∇e
uT ∇e
u
m
X
=
a2k νk ⊗ νk + ∇uTk−1 ∇uk−1 − ∇uTk ∇uk
k=1
=
m
X
k=1
=
m
X
k=1
a2k νk ⊗ νk − (1 − δ)η`2 a2k νk ⊗ νk − rk
(1.31)
1 − (1 − δ)η`2 a2k νk ⊗ νk − rk .
Using (1.25), (1.27) and (1.28) we get on Ω̄` :
kg − ∇e
uT ∇e
ukC 0 (Ω̄` ) 6 kg − ∇uT ∇ukC 0 (Ω̄` ) +
δ2
ε ε
< + 6 ε.
2
2 2
On Ωc` we use (1.27), (1.28) and (1.31) to obtain
kg − ∇e
uT ∇e
ukC 0 (Ωc` ) 6 δkg − ∇uT ∇ukC 0 (Ω̄) +
δ2
< ε.
2
which proves (1.23). For the proof of (1.24), we use (1.18), (1.30) and the
uniform bound
m
X
T
∗
kg − ∇u ∇ukC 0 (Ω̄) > |(g − u g0 )x (νk , νk )| =
a2i (x)|hνk , νi i|2 > a2k (x)
i=1
to obtain (since at most m0 of the functions ak are non-zero for fixed x) the
uniform bound
m X
√
2|ak | + λ−1
kE
k
|∇e
u − ∇u| 6
0
k C (Ω̄)
k
k=1
√
1/2
1/2
2m0 kg − ∇uT ∇ukC 0 (Ω̄) + kg − ∇uT ∇ukC 0 (Ω̄)
√
1/2
6
2m0 + 1 kg − ∇uT ∇ukC 0 (Ω̄) .
6
We need to show that the new map u
e is again an adapted subsolution. Since
u
e|Ω̄`/2 = u0 |Ω̄`/2 we need to verify the shortness condition on Ωc`/2 only. We
use (1.26), (1.28) and (1.31):
m
X
g − ∇e
uT ∇e
u > δ g − ∇uT ∇u −
rk
k=1
> δ 2 id −
m
X
k=1
– 31 –
2
rk >
δ
id .
2
1.4. Iteration
By choosing ε > 0 small enough, we ensure that u
e is an immersion. Observe
that since g is positive definite on Ω̄, there exists a positive minimum µ of
the function Ω̄ × S n−1 → R, (x, v) 7→ (g)x (v, v) and hence g > µ id in the
sense of quadratic forms. Pick any v ∈ S n−1 . Then
|(∇e
u)x v|2 = gx (v, v) − (g − u
e∗ g0 )x (v, v) > µ − ε > 0,
whenever ε > 0 is small enough.
Corollary 1.18
Let u, u
e and ε > 0 be from the previous proposition. Then there is a homotopy
e
H : [0, 1]× Ω̄ → Rn+1 relying u and u
e within the space of subsolutions adapted
to (f, g) and we have the estimates
e − ukC 0 (Ω̄) 6 ε,
kH
∗
∗
(1.32)
∗
e ·) g0 − u g0 6 kg − u g0 kC 0 (Ω̄) + ε id .
H(τ,
(1.33)
Proof. Let η(τ ) := ηe1/2 (τ ) and consider the map H : [0, 1] × Ω̄ → Rn+1 given
by
1
(τ, x) 7→ uk−1 (x) +
Γ1 (s(τ ), λk hx, νk i) ξk (x) + Γ2 (s(τ ), λk hx, νk i) ζk (x) ,
λk
where s(τ ) := η(τ )(1 − δ)1/2 η` ak |ξek |. Then, H is a homotopy relating uk−1
and uk with the property that uk−1 ≡ H(τ, ·) ≡ uk on Ω̄`/2 . We must show
that whenever uk−1 is an adapted subsolution, so is H(τ, ·) for all τ . Observe
that H(τ, ·) adds the metric term
(1 − δ)η`2 η 2 (τ )a2k νk ⊗ νk + rk (τ )
to u∗k−1 g0 . Inequality (1.21) implies that (1.28) holds uniformly in τ provided
δ2
λk is large enough i.e. krk (τ )kC 0 (Ω̄) 6 2m
. It follows that
u∗k−1 g0 −
δ2
δ2
id 6 H(τ, ·)∗ g0 6 u∗k g0 +
id .
2m
2m
(1.34)
We choose δ in the proof of the previous proposition such that u∗ g0 > δ 2 id
holds additionally. Since u∗k g0 − u∗k−1 g0 > rk , we find inductively
δ2
∗
2
u k g0 > δ − k
id > 0,
2m
– 32 –
1.4. Iteration
δ2
id > δ 2 m−1
id > 0, showing that H(τ, ·) is never singular.
hence u∗k g0 − 2m
2m
We are left to show that g − H(τ, ·)∗ g0 > 0 on Ωc`/2 , which follows from the
following computation:
∗
g − H(τ, ·) g0 > g −
>
k
X
j=1
(1 − (1 − δ)η`2 )a2j νj ⊗ νj − rj +
m
X
+
j=k+1
>
m
X
j=1
δ2
−
id
2m
u∗k g0
a2j νj ⊗ νj −
δ2
id
2m
δ2
δa2j νj ⊗ νj − rj −
id
2m
δ 2 (m + 1)
> δ(g − u∗ g0 ) −
id
m
m−1
> δ2
id > 0.
2m
e relating u and u
The homotopy H
e is now obtained from the concatenation of
the homotopies relating uk−1 and uk for k = 1, . . . , m. From (1.29) we know
that
ε
kH(τ, ·) − uk−1 kC 0 (Ω̄) 6 .
m
e the uniform estimate (1.32). The inequality (1.34)
In particular we get for H
e ·)∗ g0 6 u
implies u∗ g0 − δ 2 id 6 H(τ,
e∗ g0 + δ 2 id and hence we get using (1.31)
and (1.27)
e ·)∗ g0 − u∗ g0 6 u
H(τ,
e∗ g0 − u∗ g0 + δ 2 id
m
X
6
(1 − δ)η`2 a2k νk ⊗ νk + δ 2 id
j=1
6 kg − u∗ g0 kC 0 (Ω̄) + ε id,
thus proving (1.33).
Passage to the Limit
Proposition 1.17 can be used iteratively with an adequate sequence (εk )k>1
to achieve the convergence in C 1 (Ω̄, Rn+1 ).
– 33 –
1.5. h-Principle
Theorem 1.19 (Iteration and C 0 -density)
Let u0 : Ω̄ → Rn+1 be a subsolution adapted to (f, g). For every ε > 0
there exists a sequence (uk )k∈N of adapted subsolutions uk ∈ C ∞ (Ω̄, Rn+1 )
converging to an isometric immersion u ∈ C 1 (Ω̄, Rn+1 ) such that u|B = u0 |B
and ku − u0 kC 0 (Ω̄) 6 ε.
Proof. Apply Proposition 1.17 iteratively to u0 with a sequence (εk )k>1 satP
P√
isfying
εk 6 ε,
εk < ∞ and choose each εk such that after the k-th
stage, the resulting map is again an adapted subsolution. Since for j > i we
have
kuj − ui kC 1 (Ω̄) 6 kuj − ui kC 0 (Ω̄) + k∇uj − ∇ui kC 0 (Ω̄)
∞
∞
X
X
6
kuk − uk−1 kC 0 (Ω̄) +
k∇uk − ∇uk−1 kC 0 (Ω̄)
6
k=i+1
∞
X
k=i+1
εk +
k=i+1
∞
X
√
k=i
i,j→∞
εk −→ 0,
(uk )k∈N is Cauchy in C 1 (Ω̄, Rn+1 ) and therefore admits a limit map
u : Ω̄ → Rn+1 .
Because of the convergence in C 1 (Ω̄, Rn+1 ), we obtain
u∗ g0 = lim u∗k g0 = g,
k→∞
which also implies that u is immersive. Moreover
ku − u0 kC 0 (Ω̄) 6
∞
X
k=1
kuk − uk−1 kC 0 (Ω̄) 6
∞
X
εk 6 ε.
k=1
The equality on B is clear since uk+1 |B = uk |B for all k ∈ N.
1.5
h-Principle
We will now show that there is a homotopy of subsolutions adapted to (f, g)
relying u and u0 from the previous Theorem. This implies that one-sided
isometric C 1 -extensions satisfy an h-principle.
Corollary 1.20
The maps u and u0 from the previous Theorem are homotopic within the
space of subsolutions adapted to (f, g).
– 34 –
1.5. h-Principle
2.0
1.5
1.0
..
.
u2
u1
0.5
3
4
1
2
u0
0.2
0.4
...1 −
0.6
1
2n
0.8
1.0
Figure 1.6: Construction of the homotopy H
Proof. Observe that Theorem 1.19 together with Corollary 1.18 delivers a
Cauchy sequence uk : Ω̄ → Rn+1 in C 1 (Ω̄, Rn+1 ) and homotopies hk relating uk−1 and uk within the space of subsolutions adapted to (f, g). Let
ck (τ, x) := uk (x) be the constant homotopy and define the following homotopies (~ denotes concatenation):
Hk := h1 ~ (h2 ~ · · · ~ (hk ~ ck )) .
We will show that
H(τ, x) :=

 lim Hk (τ, x), whenever τ ∈ [0, 1)
k→∞

u(x), if τ = 1
is the desired homotopy between u and u
e. We first show that H is continuous
1
6 τ < 1 − 21k . Since
in τ = 1. For fixed τ ∈ [0, 1), choose k such that 1 − 2k−1
k → ∞ as τ → 1, we find
k
k
b2 τ c
b2 τ c
|H(τ, x) − H(1, x)| 6 H(τ, x) − H
, x + H
, x − u(x)
k
k
2
2
6 |H(τ, x) − uk−1 (x)| + |uk−1 (x) − u(x)|
k→∞
6 εk + kuk−1 − ukC 0 (Ω̄) −→ 0
uniformly in x, where the last inequality follows from (1.32). Similarly, we
– 35 –
1.5. h-Principle
prove continuity of H∗ g0 in τ = 1.
k
∗ b2 τ c
∗
∗
+
|H(τ, x) g0 − gx | 6 H(τ, x) g0 − H
,
x
g
0
2k
k
∗
b2 τ c
,
x
+ H
g
−
g
0
x
2k
6|H(τ, x)∗ g0 − (u∗k−1 g0 )x | + |(u∗k−1 g0 )x − gx |
k→∞
6εk−1 + εk + ku∗k−1 g0 − gkC 0 (Ω̄) −→ 0
uniformly in x, where the last inequality follows from (1.33).
Corollary 1.20 together with Theorem 1.19 means in the language of Gromov
[Gro86], Eliashberg and Mishashev [EM02], that the one-sided isometric C 1 extensions satisfy a C 0 -dense h-principle. The next Proposition shows that
this C 0 -dense h-principle is also parametric, i.e. whenever two isometric extensions u are homotopic within the space of adapted subsolutions, then
there is a homotopy of solutions relying them. We start with a homotopy
H : [0, 1] × Ω̄ → Rn+1 , H(τ, ·) =: uτ , where u0 and u1 are isometric C 1 extensions and uτ is a subsolution adapted to (f, g) for τ ∈ (0, 1), that is,
u0 and u1 are isometric C 1 -extensions that can be deformed into each other
via adapted subsolutions. The goal is to show that there is a homotopy that
carries u0 to u1 in the space of C 1 -isometric extensions.
Proposition 1.21 (Parametric Stage)
Let uτ ∈ C ∞ (Ω̄, Rn+1 ) be defined as above. For any ε > 0 there exists
e ·) := u
a homotopy H(τ,
eτ ∈ C ∞ (Ω̄, Rn+1 ) such that we have the following
estimates uniformly in τ :
kuτ − u
eτ kC 0 (Ω̄) 6 ε,
kg − (e
uτ )∗ g0 kC 0 (Ω̄) 6 ε,
(1.35)
(1.36)
1/2
k∇uτ − ∇e
uτ kC 0 (Ω̄) 6 Ckg − (∇uτ )T ∇uτ kC 0 (Ω̄) .
(1.37)
Moreover u
e0 and u
e1 are isometric C 1 -extensions and u
eτ is a subsolution
adapted to (f, g) for τ ∈ (0, 1) provided ε > 0 is small enough.
P
τ 2
Proof. We can decompose g−(uτ )∗ g0 = m
k=1 (ak ) νk ⊗νk and use Proposition
1.16 with aτk instead of ak and replace the function η` by a new function
Θ` (τ, x) := η` (x)η` (τ )η` (1 − τ ). This is done because otherwise we cannot
have an estimate corresponding to (1.26) (see (1.39)). With these choices,
– 36 –
1.5. h-Principle
we can obtain the same estimates as in Proposition 1.16: Choose ` > 0 such
that
|g − (∇uτ )T ∇uτ | < 2ε
(1.38)
holds on [0, 1] × Ω̄ \ Θ−1
` (1) and δ such that
δ id < g − (∇uτ )T ∇uτ |[0,1]×Ω̄\Θ−1 (0) ,
`
)
( −1
√
ε
, ε .
max kg − (∇uτ )T ∇uτ kC 0 (Ω̄)
δ < min
2 τ ∈[0,1]
(1.39)
(1.40)
e ·) =: u
We iterate Proposition 1.16 to get after m steps a homotopy H(τ,
eτ
with ke
uτ − uτ k < ε uniformly in τ . This shows (1.35) and is done exactly as
in the non-parametric case. The computation (1.31) is replaced by
τ T
τ
g − (∇e
u ) ∇e
u =
where krkτ kC 0 (Ω̄) 6
δ2
2m
m
X
k=1
1 − (1 − δ)Θ2` (aτk )2 νk ⊗ νk − rkτ ,
(1.41)
can be achieved as in Proposition 1.16. Using (1.38)
and (1.40) we get on [0, 1] × Ω̄ \ Θ−1
` (1):
|g − (∇e
uτ )T ∇e
uτ | 6 |g − (∇uτ )T ∇uτ | +
ε ε
δ2
< + 6 ε.
2
2 2
On Θ−1
` (1) we use (1.40) and (1.41) to obtain
δ2
< ε,
2
which proves (1.36). The proof of (1.37) is obtained exactly as in the nonparametric case by choosing λk in each step large enough:
m X
√ τ
1
1/2
τ
τ
τ
|∇e
u − ∇u | 6
2|ak | + kEk kC 0 (Ω̄) 6 Ckg − (∇uτ )T ∇uτ kC 0 (Ω̄) .
λk
k=1
|g − (∇e
uτ )T ∇e
uτ | 6 δkg − (∇uτ )T ∇uτ kC 0 (Ω̄) +
We are left to show that uτ is an adapted subsolution for τ ∈ (0, 1) and a
solution for τ = 0, 1. Since u
eτ ≡ uτ on Θ−1
eτ is
` (0) we must only show that u
short on Θ−1
` ((0, 1]). We compute
g − (∇e
uτ )T ∇e
uτ > δ(g − (∇uτ )T ∇uτ ) −
m
X
k=1
rkτ > δ 2 id −
δ2
id > 0.
2
The rest of the proof is exactly the same as in the non-parametric case.
Theorem 1.19 and its proof can be taken over word by word for the parametric
case and delivers the desired homotopy.
– 37 –
1.6. From Immersions to Embeddings
1.6
From Immersions to Embeddings
This section follows [GA13] very closely. We show that if the adapted subsolution is an embedding (see Proposition 1.9), we can arrange the isometric
extension to be an embedding as well. Since Ω̄ is compact and all the maps
throughout the iteration are immersive, we just need to check the injectivity.
Step and Stage
First we show that adding a primitive metric works in the class of embeddings. Taylor’s formula implies that v(y) − v(x) = (∇v)x (y − x) + O(|y − x|2 ).
Write h := x − y. Then
|v(y) − v(x)|2 = (∇v T ∇v)x h, h + O(|h|3 )
= (∇uT ∇u)x h, h + (1 − δ)η`2 (x)a2 (x)|hν, hi|2 +
+ O λ−1 |h|2 + O(|h|3 )
>|u(y) − u(x)|2 + O λ−1 |h|2 + O(|h|3 )
=|u(y) − u(x)|2 1 + O λ−1 + O(|h|) ,
hence there exists µ > 0 and λ large enough, such that
|v(y) − v(x)| > |u(y) − u(x)|/2,
whenever |h| < µ. The uniform convergence of ku − vkC 0 (Ω̄) → 0 as λ → ∞
implies that for any δ > 0 there exists λ large enough such that
|v(y) − v(x)| 6 |v(y) − u(y)| + |u(y) − u(x)| + |u(x) − v(x)|
6 |u(y) − u(x)| + δ.
This in turn implies the uniform convergence
|v(y) − v(x)| λ→∞
→ 1
|u(y) − u(x)|
on Λµ := {(x, y) ∈ Ω̄ × Ω̄, |y − x| > µ}. Hence |v(y) − v(x)| > |u(y) − u(x)|/2
whenever |y − x| > µ and λ large enough. This shows that after a step
(and in particular after a stage), we can get an embedding provided u is an
embedding.
– 38 –
1.7. Global C 1 -Extensions
Passage to the Limit
Now we show that the map u obtained in Theorem 1.19 is an isometric
embedding provided u0 is an embedding. We have
|u(y) − u(x)|2 = h(u∗ g0 )x h, hi + o(|h|2 )
= h(u∗ g0 − u∗0 g0 )x h, hi + h(u∗0 g0 )x h, hi + o(|h|2 )
m
X
=
a2k (x) h(νk ⊗ νk )x h, hi + h(u∗0 g0 )x h, hi + o(|h|2 )
>
k=1
h(u∗0 g0 )x h, hi
+ o(|h|2 )
= |u0 (x) − u0 (y)|2 (1 + o(1)) .
As before, we get the existence of a number µ > 0 such that
|u0 (x) − u0 (y)|
2
provided |h| < µ. Note that this calculation holds for any u being an isometric extension and is thus independent of the choice of ε in the Theorem.
If u0 is an embedding, we have
|u(x) − u(y)| >
min |u0 (x) − u0 (y)| > m
e > 0.
(x,y)∈Λµ
Now choose ε := m/4.
e
The Theorem delivers a map u with
ku − u0 kC 0 (Ω̄) <
m
e
,
4
and we conclude that
m
e
6 |u(x) − u(y)|.
2
On Λµ we thus find |u(x) − u(y)| > |u0 (x) − u0 (y)|/2 which proves that u
is an isometric embedding. This together with Theorem 1.19 and Corollary
1.20 completes the proof of Theorem 1.2.
|u0 (x) − u0 (y)| −
1.7
Global C 1-Extensions
In this section, “global” has to be understood in the sense that we want to
construct solutions to (2) on a neighborhood of Σ (and not only of a point
in Σ). In order to fix the setting, let (M, g) be an oriented connected and
compact Riemannian n-manifold and Σ an oriented connected and compact
codimension 1 submanifold of M with trivial normal bundle. Let further
f : Σ → Rn+1 be an isometric immersion.
– 39 –
1.7. Global C 1 -Extensions
Norms on Manifolds
Let A := {ϕj : Uj → Vj ⊂ Rn }j be a finite atlas of M such that the Uj
are diffeomorphic to open balls in Rn and such that the maps ϕj extend to
diffeomorphisms ϕj : Ūj → V̄j . The inverse maps are denoted by ψj . Let A
furthermore be the completion of a submanifold atlas of Σ in the sense that
a coordinate neighborhood Ū belongs either to the submanifold atlas (i.e.
maps Ū ∩ Σ to Rn × {0}) or doesn’t intersect Σ.
Definition 1.22
For f ∈ C 1 (M ), ω ∈ Γ(T ∗ M ) and g ∈ Γ(S2 (T ∗ M )) we define the following
norms:
kf kC 0 (M ) = max |f (p)| = max kψj∗ f kC 0 (V̄j ) ,
j
p∈M
kωkC 0 (M ) :=
max kψj∗ ωkC 0 (V̄j )
j
kgkC 0 (M ) := max kψj∗ gkC 0 (V̄j ) ,
j
kf kC 1 (M ) := kf kC 0 (M ) + kdf kC 0 (M ) .
Using these definitions we obtain
Lemma 1.23
Let f ∈ C 0 (M ), ω ∈ Γ(T ∗ M ) and g ∈ Γ(S2 (T ∗ M )) be compactly supported
in Ūk . Then there exists a constant C only depending on the atlas such that
the following estimates hold
kf kC 0 (M ) = kf ◦ ψk kC 0 (V̄k ) ,
kωkC 0 (M ) 6 Ckψk∗ ωkC 0 (V̄k ) ,
kgkC 0 (M ) 6 Ckψk∗ gkC 0 (V̄k ) .
Proof. Unwinding the definitions gives the first equality. For the second and
third estimate, one finds
kωkC 0 (M ) = max max sup |ω((dψj )x (v))|
j
x∈V̄j v∈S n−1
= max max sup |ψk∗ ω(∇(ϕk ◦ ψj )x (v))|
j
x∈V̄j v∈S n−1
∗
∇(ϕk ◦ψj )x (v) = max max sup |∇(ϕk ◦ ψj )x (v)| ψk ω |∇(ϕk ◦ψj )x (v)| j
x∈V̄j v∈S n−1
6 max k∇(ϕk ◦ ψj )kC 0 (V̄j ) kψk∗ ωkC 0 (V̄k ) =: C1 kψk∗ ωkC 0 (V̄k )
k,j
– 40 –
1.7. Global C 1 -Extensions
kgkC 0 (M ) = max max sup |g((dψj )x (v), (dψj )x (v))|
j
x∈V̄j v∈S n−1
= max max sup |ψk∗ g(∇(ϕk ◦ ψj )x (v), ∇(ϕk ◦ ψj )x (v))|
j
x∈V̄j v∈S n−1
6 max k∇(ϕk ◦ ψj )k2C 0 (V̄j ) kψk∗ gkC 0 (V̄k ) =: C2 kψk∗ gkC 0 (V̄k )
k,j
Choosing C := max{C1 , C2 } proves the claim.
One-sided Neighborhoods
We need to introduce an equivalence relation on M to get an adequate notion
of one-sided neighborhood: Pick a submanifold chart ϕ : Ū → V̄ . The points
p, q ∈ Ū are equivalent whenever the n-th coordinate of ϕ(p) and ϕ(q) has
the same sign. Points of coordinate neighborhoods that don’t intersect Σ are
equivalent. The transitive closure of this equivalence relation is denoted by
∼ and defines then an equivalence relation on M that doesn’t depend on the
choice of A.
Lemma 1.24
It holds that 2 6 # (M/∼ ) 6 3.
Proof. The points in Σ are not equivalent to the points in M \Σ. This proves
the first inequality. On the other hand each point (path-connectedness) is
equivalent to a point in a submanifold chart. In a submanifold chart, ∼ has
trivially 3 equivalence classes, since the n-th component of the coordinate
expression of a point is positive, negative or zero.
Definition 1.25
Let Ū be a closed neighborhood of Σ in M . If ∼ has three equivalence classes,
Ū can be divided into three parts U+ , Ū ∩ Σ and U− according to ∼. We call
Ū± one-sided neighborhoods of Σ.
If ∼ has only two equivalence classes, we restrict our considerations to neighborhoods that split into three equivalence classes and use the definition
above. This is always possible since the normal bundle of Σ is trivial and
diffeomorphic to a tubular neighborhood of Σ in M via Ψ : Σ × (−ε, ε) → M .
Cut-Off
Let Φ := Ψ−1 , π : Σ × (−ε, ε) → (−ε, ε) be the projection onto the second
factor and let Ω̄ be a one-sided neighborhood of Σ that has a nonempty
– 41 –
1.7. Global C 1 -Extensions
intersection with Ψ(Σ × [0, ε)) and set η` : Ω̄ → [0, 1]
(
(e
η` ◦ π ◦ Φ)(p), whenever p ∈ Ψ(Σ × [0, ε))
p 7→
1 elsewhere
and fix the following notation: Ω̄β := Ω̄ ∩ Ψ(Σ × [0, β]). Observe that η`
vanishes on Ω̄`/2 and equals one on Ωc` .
Step
Let {ςj ∈ Cc∞ (Uj )}j be a partition of unity subordinate to A in the sense
P 2
that
ς ≡ 1.
j
Definition 1.26
Let Ω̄ be a one-sided neighborhood of Σ. A subsolution adapted to (f, g) is
a smooth immersion u : Ω̄ → Rn+1 satisfying u|Σ = f and g − u∗ g0 > 0 in
the sense of quadratic forms with equality on Σ only.
The metric defect of an adapted subsolution can be written as
X
g − u∗ g0 =
j
ςj2 (g − u∗ g0 ).
With the notation ςˆj := ςj ◦ ψj , one gets
ςj2 (g − u∗ g0 ) = ϕ∗j ςˆj2 (ψj∗ g − (u ◦ ψj )∗ g0 )
and with the use of a decomposition into primitive metrics for
ψj∗ g
∗
− (u ◦ ψj ) g0 =
m
X
k=1
a2k,j νkj ⊗ νkj ,
the metric defect can be finally written as
X
g − u∗ g0 =
ϕ∗j (ˆ
ςj ak,j )2 νkj ⊗ νkj .
k,j
Note that m depends on j in general but we will suppress this dependence
since we can choose the maximal m over all charts.
Proposition 1.27 (Global step)
Let u ∈ C ∞ (Ω̄, Rn+1 ) be an immersion. For every ε > 0 there exists an
– 42 –
1.7. Global C 1 -Extensions
immersion v ∈ C ∞ (Ω̄, Rn+1 ) that agrees with u on Ω̄`/2 such that the following
estimates hold:
kv − ukC 0 (Ω̄) 6
ε
,
#Am
(1.42)
1/2
kdv − dukC 0 (Ω̄) 6 Ckg − u∗ g0 kC 0 (Ω̄) ,
(1.43)
δ 2 (g − u∗ g0 )
ςj ak,j · η` ◦ ψj )2 νkj ⊗ νkj 6
v ∗ g0 − u∗ g0 + ϕ∗j (1 − δ)(ˆ
. (1.44)
2#Am
Proof. We use the ansatz
1
j
j
v(p) = u(p) +
Γ1 s, λk,j hx, νk i ξk,j (x) + Γ2 s, λk,j hx, νk i ζk,j (x) ,
λk,j
where s := (1 − δ)1/2 η` (ψj (x))ˆ
ςj (x)ak,j (x)|ξek,j (x)|, x = ϕj (p) and the vector
fields ξk,j and ζk,j are constructed as in the local step with the map u ◦ ψj
instead of u. Observe that the map v −u is compactly supported in Ūj . Since
in this case
C
kv − ukC 0 (Ω̄) 6
kΓ1 ξk,j + Γ2 ζk,j kC 0 (V̄j ) ,
λk,j
we can choose λk,j large enough to get (1.42). For the pullback one finds
v ∗ g0 = ϕ∗j (v ◦ ψj )∗ g0
= ϕ∗j (u ◦ ψj )∗ g0 + (1 − δ)(ˆ
ςj ak,j · η` ◦ ψj )2 νkj ⊗ νkj + rk,j
= u∗ g0 + ϕ∗j (1 − δ)(ˆ
ςj ak,j · η` ◦ ψj )2 νkj ⊗ νkj + ϕ∗j rk,j .
In view of the local step and since kϕ∗j rk,j kC 0 (Ω̄) 6 Ckrk,j kC 0 (V̄j ) by Lemma
1.23 we can choose λk,j large enough such that
ϕ∗j rk,j |Ωc 6
`/2
δ
(g − u∗ g)|Ωc
`/2
2#Am
(1.45)
in the sense of quadratic forms. Since ϕ∗j rk,j |Ω̄`/2 ≡ 0, this proves (1.44).
Observe that dv − du is also compactly supported in Ūj , hence one gets
kdv − dukC 0 (Ω̄) 6 Ck∇((v − u) ◦ ψj )kC 0 (V̄j )
6 Ckak,j ςˆj kC 0 (V̄j ) + O(λ−1
k,j )
(1.46)
1/2
6 Ckg − u∗0 g0 kC 0 (Ω̄)
as in the local step. The maps u and v agree on Ω̄`/2 by construction and
are immersions (similar argument as in the local step).
– 43 –
1.7. Global C 1 -Extensions
Stage
To prove the global stage, we use Proposition 1.27 #Am times to prove the
same estimates as in the local case.
Proposition 1.28 (Global Stage)
Let u ∈ C ∞ (Ω̄, Rn+1 ) be a subsolution adapted to (f, g). For any ε > 0 there
exists a map u
e ∈ C ∞ (Ω̄, Rn+1 ) with the following properties:
ku − u
ekC 0 (Ω̄) 6 ε,
(1.47)
∗
kg − u
e g0 kC 0 (Ω̄) 6 ε,
(1.48)
1/2
kdu − de
ukC 0 (Ω̄) 6 Ckg − u∗ g0 kC 0 (Ω̄) .
(1.49)
Moreover, u
e is a subsolution adapted to (f, g) provided ε > 0 is small enough.
Proof. Choose ` > 0 and δ > 0 such that
kg − u∗ g0 kC 0 (Ω̄` ) < 2ε ,
(1.50)
δ < min
nε
2
√ o
−1
∗
kg − u g0 kC 0 (Ω̄) , ε .
(1.51)
Using Proposition 1.27 iteratively we find after #Am steps a new map
u
e := u#Am such that ke
u − ukC 0 (Ω̄) 6 ε, provided the frequencies λk,j in
each step are chosen appropriately. Moreover
g−u
e∗ g0 = g − u∗ g0 − (e
u∗ g0 − u∗ g0 )
X
=
ϕ∗j (ˆ
ςj ak,j )2 νkj ⊗ νkj −
k,j
−
=
ϕ∗j
(1 − δ)(ˆ
ςj ak,j · η` ◦
ψj )2 νkj
⊗
νkj
−
ϕ∗j rk,j
X
ϕ∗j (1 − (1 − δ)(η` ◦ ψj )2 )(ˆ
ςj ak,j )2 νkj ⊗ νkj − ϕ∗j rk,j .
k,j
On Ω̄` this yields
g−u
e∗ g0 6
X
k,j
ϕ∗j (ˆ
ςj ak,j )2 νkj ⊗ νkj − ϕ∗j rk,j
6 g − u∗ g0 −
X
ϕ∗j rk,j
k,j
and therefore using (1.50) and (1.51):
kg − u
e∗ g0 kC 0 (Ω̄` ) 6 kg − u∗ g0 kC 0 (Ω̄` ) +
– 44 –
δ2
6 ε.
2
1.8. Applications
On Ωc` we find
g−u
e∗ g0 6
X δ ϕ∗j (ˆ
ςj ak,j )2 νkj ⊗ νkj − ϕ∗j rk,j
k,j
6 δ(g − u∗ g0 ) −
X
ϕ∗j rk,j
k,j
and hence kg − u
e∗ g0 kC 0 (Ωc` ) 6 δkg − u∗ g0 kC 0 (Ω̄) +
For the shortness we find on Ωc`/2 using (1.45):
g−u
e∗ g0 >
δ2
2
6 ε. This proves (1.48).
X δ ϕ∗j (ˆ
ςj ak,j )2 νkj ⊗ νkj − ϕ∗j rk,j
k,j
> δ(g − u∗ g0 ) −
X
k,j
δ
ϕ∗j rk,j > δ(g − u∗ g0 ) − (g − u∗ g0 )
2
δ
> (g − u∗ g0 ) > 0.
2
The estimate (1.49) then follows from (1.46) and an appropriate choice of
frequencies λk,j in each step (as in the local case). The rest of the argument
is similar to the one in the local case.
Theorem 1.19 and its proof can be taken over word by word to the global
case.
1.8
Applications
We can now prove Corollary 1.3:
Proof. Whenever ε > 0 is small enough, the image of the map


cos ϑ cos ϕ
Φε : [− π2 , π2 ] × [0, 2π] → R3 , (ϑ, ϕ) 7→ 1 − ε sin2 ϑ  cos ϑ sin ϕ 
sin ϑ
is a 2-dimensional submanifold S of R3 . One can check that S is diffeomorphic to S 2 (S 2 corresponds to the choice ε = 0) and that with the induced
metric of R3 we have gS 2 − gS > 0 with equality on the equator only. We
can apply now the global variant of Theorem 1.19 to each of the hemispheres
S±2 and get sequences of maps (u±
k )k∈N . Observe that for each k, the two
maps give rise to an embedding uk : S 2 → R3 since an open neighborhood
– 45 –
1.8. Applications
of the equator remains unchanged after k steps. The C 1 -convergence of the
sequences implies that also the limit maps
v ± := lim u±
k
k→∞
give rise to an isometric C 1 -embedding v : S 2 → R3 which extends the
standard inclusion S 2 ⊃ S 1 ,→ R3 .
Example 1.29 (Dirichlet Problem)
Let Ω = D2 , g be a Riemannian metric on Ω and let f : ∂Ω → R2 × {0} ⊂ R3
be a smooth isometric embedding. One can consider the Dirichlet Problem
for maps u ∈ C 1 (Ω̄, R3 ) given by
(
∇uT ∇u = g in Ω
(1.52)
u = f on ∂Ω.
There is a rigidity theorem for solutions to this problem: In order to state
it, recall that (Ω̄, g) is called a smooth positive disk, if g has positive Gauss
curvature Kg > 0.
Theorem 1.30 (Hong, [Hon99])
Let (Ω̄, g) be a smooth positive disk with positive geodesic curvature along ∂Ω.
Then there is a unique (up to rigid motions) smooth isometric embedding
u : Ω̄ → R3 such that u(∂Ω) is a planar curve.
The global variant of Theorem 1.2 implies that whenever there exists a subsolution u0 adapted to (f, g), then for any ε > 0, there exists a C 1 -solution u
to (1.52) such that ku − u0 kC 0 (Ω̄) < ε. In order to produce adapted subsolutions, we can use Hong’s theorem: Fix a positive disk (Ω̄, g) that satisfies the
assumptions from Hong’s theorem and consider a perturbed metric ge that is
C 2 -close to g and such that g − ge > 0 with equality on ∂Ω only. This can be
achieved by setting ge = ϕg, where ϕ : Ω̄ → R is a smooth function such that
ϕ|∂Ω ≡ 1, 0 < ϕ|Ω < 1 and such that k1 − ϕkC 2 (Ω̄) is very small. It follows
from the estimate
kKg − KgekC 0 (Ω̄) 6 Ckg − gekC 2 (Ω̄)
that (Ω, ge) also satisfies the assumptions from Hong’s theorem. The corresponding isometric embedding u
e : Ω̄ → R3 is then a subsolution adapted to
(f, g), where f = u
e|∂Ω .
– 46 –
1.8. Applications
Example 1.31 (Coin through Paper Hole)
Consider the map γa (t) = Ca (cos t, a sin t)T , where a > 0 and Ca is chosen,
such that
Z
2π
0
|γ̇a (t)| dt = 2π.
A reparametrization of γa by arc length delivers an isometric embedding
fa : S 1 → R2 ⊂ R3 .
Let ε > 0 and consider the maps α : S 1 × [0, ε] → R2 and β : S 1 × [0, ε] → R3
given by α(r, t) := (1 + r + r2 )(cos t, sin t)T and β(r, t) := fa (t) + re3 , where
{e1 , e2 , e3 } denotes the standard basis of R3 . These maps have the property
that
4r(1 + r)
0
∗
∗
α gR2 − β gR3 =
0
r(1 + r)(2 + r + r2 )
and hence the map β ◦ α−1 is subsolution adapted to (fa , gR2 ). Theorem 1.19
delivers an isometric extension of fa for every a which can be interpreted as
follows: A direct computation shows that
lim diamR2 (fa (S 1 )) = π,
a→0
hence one can cut a circle of radius 1 out of a sheet of paper and push an
idealized coin of diameter < π through the hole when deforming the paper
accordingly.
– 47 –
Chapter 2
Isometric Lipschitz-Extensions
2.1
Introduction
In this chapter we analyze problem (2) in the equidimensional case, i.e. if
q = n. That is, for a given smooth isometric immersion f : Σ → Rn of
a hypersurface Σ ⊂ (M n , g) into Rn equipped with the standard Euclidean
metric g0 , we will look for a map v : U → Rn satisfying
v ∗ g0 = g
v|Σ = f,
(2.1)
where U ⊂ M is a neighborhood of a point in Σ. We have already seen
in Chapter 1 that there is a curvature-obstruction to the existence of C 2 and C 1 -solutions to (2.1). In this chapter we will prove along the very same
lines that this obstruction carries over to the Lipschitz case (Proposition 2.6).
However, restricting the neighborhood U to one side of Σ only – as it is done
in the codimension one case – one can hope to construct one-sided solutions
of low regularity, where curvature does not exist. The type of flexibility the
C 1 -extensions of Chapter 1 enjoy in codimension one is not expected in the
equidimensional case since the classical Liouville Theorem (see for example
[Cia05, p. 30-31]) implies that any two images of isometric C 1 -immersions
into Euclidean space are congruent, so even if g is flat, there will not be
C 1 -solutions to (2.1) in general. In this case, it seems natural to relax the
regularity and consider piecewise C 1 -maps (think of folding a piece of paper).
Indeed, Dacorogna, Marcellini and Paolini ([DMP10], [DMP08]) constructed
piecewise isometric C 1 -immersions from a flat square into R2 mapping the
boundary to a single point.
– 48 –
2.1. Introduction
It is remarkable that for general metrics, there is a curvature obstruction
even at low regularity in the equidimensional case. We will show that there
are no differentiable isometric extensions if g is not flat (Proposition 2.5).
Hence we will further relax the regularity and focus on Lipschitz-maps instead. Problem (2.1) is thus replaced by the problem of finding a Lipschitz
map v : U → Rn that satisfies
v ∗ g0 = g
a.e.
v|Σ = f,
(2.2)
where a.e. refers to the measure on M induced by g. This setting provides
enough flexibility to circumvent the curvature obstruction and the one given
by Liouville’s theorem.
In [MŠ98] and [MŠ03], Müller and Šverák constructed solutions to the following related Dirichlet problem, where Ω is a bounded domain in Rn :
v ∗ g0 = id Ln -a.e. in Ω
v = ϕ on ∂Ω,
(2.3)
where Ln denotes the n-dimensional Lebesgue measure. They require the
boundary datum ϕ to be a short map, which is not the case for f in (2.2).
As mentioned before, maps satisfying (2.2) may collapse entire submanifolds
to single points, hence this definition does not reflect a truly geometric notion of isometry. A more natural notion of isometry (which is equivalent if
v ∈ C 1 ) arises if one requires an isometry to be a map that preserves the
length of every rectifiable curve (see [KSS14]). Our results do not extend to
this stronger setting. See also [Pet10] for an even stronger notion of isometry.
The construction of isometric maps in [KSS14] is based on a Baire-Category
approach and produces residuality results. We will use a weak variant of the
Nash-type convex integration iteration scheme from Chapter 1, which only
allows us to prove a density result.
Recall from Chapter 1, that the strategy consisted in starting with a subsolution u : (M n , g) → Rq and adding primitive metrics by using corrugations
(q = n + 1) or the Nash Twist (q > n + 2) at all scales to correct the metric
error successively while controlling the C 1 -norm of the perturbed maps during the process. This led to the convergence in C 1 . The basic building block
– 49 –
2.1. Introduction
here was a family of loops with average zero in a suitable (q − n)-dimensional
sphere. The method does not apply to the equidimensional case, since here,
the corresponding sphere is degenerate and consists of two isolated points.
However, the points define an interval I ⊂ R and one may choose a loop in
I that is concentrated on the two boundary points (see Lemmata 2.10 and
2.11 for precise statements). Using this loop we obtain a corrugation function
that leads to weaker estimates than the corresponding ones in codimension
one, but still ensure the required Lipschitz-regularity of a solution to (2.2).
In local coordinates, the setting for the problem (2.2) can be reformulated
as follows: Consider an n-polytope (P, g) in Rn with an appropriate metric
g such that the origin is contained in P̊ and let the isometric immersion
f : B → Rn be prescribed on B := P ∩ (Rn−1 × {0}).
Definition 2.1
The sets P ∩ (Rn−1 × R>0 ) and P ∩ (Rn−1 × R60 ) are called one-sided neighborhoods of B.
Let Ω̄ be a one-sided neighborhood of B. A map u belongs to Cp∞ (Ω̄, Rn ) if
there is a finite simplicial decomposition of Ω̄ into non-degenerate n-simplices
such that the restriction of u onto each of the simplices is smooth. Similarly,
a map u belongs to the space Aff p (Ω̄, Rn ) if there is a finite simplicial decomposition of Ω̄ into non-degenerate n-simplices such that the restriction of u
onto each of the simplices is affine.
Definition 2.2
Let Ω̄ be a one-sided neighborhood of B. A map u ∈ Cp∞ (Ω̄, Rn ) ∩ C 0 (Ω̄, Rn )
is called subsolution adapted to (f, g), if u|B = f and g − u∗ g0 > 0 in the
sense of quadratic forms with equality on B only (i.e. g − u∗ g0 is positive
definite on Ω̄ \ B and zero on B).
The main result of the present chapter is the following equidimensional analogue of Theorem 1.2
Theorem 2.3 (Lipschitz-Extensions)
Let u : Ω̄ → Rn be a subsolution adapted to (f, g). Then for every ε > 0,
there exists a Lipschitz map v : Ω̄ → Rn satisfying v|B = f , v ∗ g0 = g Ln -a.e.
and ku − vkC 0 (Ω̄) < ε.
– 50 –
2.2. Obstructions
We obtain a corollary regarding isometric extensions of the standard inclusion
ι : S 1 ,→ D̄2 ⊂ R2
to maps S 2 → D̄2 , where D̄2 denotes the closed two-dimensional unit disk
and S 1 the equator of S 2 . Here µS 2 denotes the standard measure on S 2 .
Corollary 2.4 (Isometric Collapse of S 2 )
There exist infinitely many Lipschitz-maps v : S 2 → D̄2 satisfying v|S 1 = ι
and v ∗ g0 = gS 2 µS 2 -a.e.
This corollary shows the strong interaction between codimension and regularity: The standard inclusion S 1 ,→ R2 × {0} ,→ R3 can be extended to an
isometric immersion v ∈ C 1,α (S 2 , R3 ) in a unique way (up to reflection across
the R2 × {0}-plane) provided α > 2/3 (see Theorem 1.4), but infinitely many
isometric extensions v ∈ C 1 (S 2 , R3 ) exist (Corollary 1.3). We will also show
that no isometric extension v ∈ C 1 (S 2 , R2 ) can exist. More precisely, we will
show that the maps v from Corollary 2.4 cannot be locally C 1 nor locally
injective, hence the restriction to Lipschitz maps is not excessive.
2.2
Obstructions
We will first show that curvature is an obstruction to merely differentiable
(not necessarily C 1 ) isometric immersions in the equidimensional case which
leads to the choice of the Lipschitz-regularity in (2.2).
Proposition 2.5
An n-dimensional Riemannian manifold (M n , g) can be locally isometrically
embedded by a differentiable map into (Rn , g0 ) if and only if g is flat and in
this case, the map is in fact of class C ∞ .
Proof. The if part of the statement is a classical theorem in differential geometry (see for example Theorem 3.1 in [Tay06]). A local isometric immersion f
is locally a distance preserving homeomorphism by the differentiable local inversion theorem (see [Ray02]) and the condition f ∗ g0 = g. Such a map must
be a local C ∞ -isometric diffeomorphism by Theorem 2.1 in [Tay06] since g
is smooth by assumption. Hence the statement is a simple consequence of
the classical fact that the Riemannian curvature tensor is preserved under
C ∞ -isometries.
– 51 –
2.2. Obstructions
The following proposition is a Lipschitz variant of Proposition 1.8. We will
use the notation introduced in Chapter 1.
Proposition 2.6 (Lipschitz-Obstruction)
If there exists a unit vector v ∈ Tp Σ such that |h(v, v)|g > |Ā(v, v)|, no
isometric Lipschitz-extension u : U → Rn can exist.
Proof. We argue by contradiction. Suppose u exists and let (for ε > 0 small
enough) γ : [0, ε) → Σ ∩ U be a geodesic with γ(0) = p and γ̇(0) = v such
that dM (p, γ(t)) can be realized by a minimizing geodesic σ : [0, 1] → U for
all t.
Claim: For every δ > 0 there exists a curve c : [0, 1] → U joining p and γ(t)
such that u is differentiable in c(t) L1 -a.e. in [0, 1] and such that
Z 1
|ċ(t)|g dt < dM (p, γ(t)) + δ.
0
Proof of the claim: Let D ⊂ U be the set where u is differentiable,
P := [0, 1] × Bρn−1 (0)
and consider the map
z : P → U, (s, x) 7→ expσ(s) (s(s − 1)x),
where x ∈ Bρn−1 (0) ⊂ σ̇(s)⊥ ∈ Tσ(s) U . If ρ > 0 is small enough, the restriction of z onto (0, 1) × Bρn−1 (0) is a diffeomorphism onto its image, hence
e := z −1 (D) has full Ln -measure in P. We obtain
D
Z
Z 1
Z
Z
χDe |∂s z|g ds dx
|∂s z|g ds dx =
|∂s z|g ds dx =
Bρn−1 (0)
e
P∩D
P
hence for Ln−1 -a.e. x it holds that
Z 1
Z
χDe (s, x)|∂s z(s, x)|g ds =
0
0
Since
Z
lim
x→0
0
0
1
|∂s z(s, x)|g ds.
1
|∂s z(s, x)|g ds = dM (p, γ(t)),
we can choose an appropriate x such that c(t) := z(t, x) has the desired
properties. This finishes the proof of the claim.
Let p̄ = f (p) and γ̄ = f ◦ γ. Observe that
Z 1
Z 1
d
|u(σ(1)) − u(σ(0))| 6
|σ̇(t)|g dt < dM (p, γ(t)) + δ.
dt (u ◦ σ)(t) dt =
0
0
– 52 –
2.3. Subsolutions
We conclude that |γ̄(t)− p̄| 6 dM (p, γ(t)), since δ was arbitrary. The opposite
inequality |γ̄(t) − p̄| > dM (p, γ(t)) that delivers the contradiction is obtained
exactly in the same way as in the proof of Proposition 1.8.
2.3
Subsolutions
For the construction of subsolutions, we refer to Proposition 1.9 which takes
over to the equidimensional case.
Approximation of Subsolutions by Piecewise Affine Maps
We need to introduce the notion of an adapted piecewise affine subsolution
and therefore the following approximation result (see [Sai79] for a proof):
Proposition 2.7
Let u ∈ Cp∞ (Ω̄, Rn ) ∩ C 0 (Ω̄, Rn ). Then for every ε > 0, there exists a map
v ∈ Aff p (Ω̄, Rn ) ∩ C 0 (Ω̄, Rn ) such that
ku − vkW 1,∞ (Ω̄) < ε.
Remark 2.8
Note that if an adapted subsolution u is approximated by a piecewise affine
map v, we cannot ensure that u|B = v|B since this would require u|B to be
piecewise affine already. To circumvent this problem, let
Ω̄` := {x ∈ Ω̄, dist(x, B) 6 `}
and define η` ∈ C ∞ (Ω̄, [0, 1]) to be
(
0, if x ∈ Ω̄`/2
η` (x) :=
1, if x ∈ Ωc` := x ∈ Ω̄, dist(x, B) > `
and 0 < η` < 1 elsewhere (as in Chapter 1).
Ωcℓ
ℓ
Ω̄ℓ
– 53 –
2.4. Convex Integration
Definition 2.9
For fixed ` > 0, decompose Ω̄`/2 and the closure of its complement in Ω̄
separately into non-degenerate simplices, approximate u by v in the sense of
Proposition 2.7 and finally replace v by w := u + η`/2 (v − u). This map has
the property that w ≡ u on Ω̄`/4 and the restriction of w to Ωc`/2 is a piecewise
affine map. If such a map has in addition the property that g − ∇wT ∇w > 0
Ln -a.e., but ∇wT ∇w = g on B, we will call it a piecewise affine subsolution
adapted to (f, g).
The estimates
kw − ukC 0 (Ω̄) 6 ku − vkC 0 (Ω̄)
k∇w − ∇ukL∞ (Ω̄) 6 k∇η`/2 kC 0 (Ω̄) ku − vkC 0 (Ω̄) + k∇u − ∇vkL∞ (Ω̄)
ensure that we can approximate an adapted subsolution by an adapted piecewise affine subsolution in W 1,∞ (Ω̄).
2.4
Convex Integration
We start with a piecewise affine subsolution u : Ω̄ → Rn adapted to (f, g),
that is piecewise affine on Ωc`/2 and decompose the metric defect into a sum
of primitive metrics as
∗
(g − u g0 )x =
m
X
k=1
a2k (x)νk ⊗ νk ,
where the a2k are nonnegative on Ω̄ \ B, zero on B, belong to Cp∞ (Ω̄, Rn ) and
extend continuously to B. The sum is locally finite with at most m0 6 m
terms being nonzero for a fixed point x and νk ∈ S n−1 are fixed unit vectors. We intend to correct this metric defect by successively adding primitive metrics, i.e. metric terms of the form a2 ν ⊗ ν. Adding a primitive
metric is done in a step and a stage consists then of m steps, where the
number m ∈ N is finite but may change from stage to stage. Fix orthonormal coordinates in the target so that the the metric u∗ g0 can be written as
∇uT ∇u, where ∇u = (∂j ui )ij . For a specific unit vector ν ∈ S n−1 and a
nonnegative function a ∈ C ∞ (Ω̄) we aim at finding v : Ω̄ → Rq satisfying
∇v T ∇v ≈ ∇uT ∇u + a2 ν ⊗ ν. Recall that Nash solved this problem using the
Nash Twist
1
v(x) = u(x) +
N1 (a(x), λhx, νi)β1 (x) + N2 (a(x), λhx, νi)β2 (x) , (2.4)
λ
– 54 –
2.4. Convex Integration
where N(s, t) := s(− sin t, cos t) satisfies the circle equation ∂t N21 + ∂t N22 = s2
and βi are mutually orthogonal unit normal fields, requiring thus codimension
at least two (q > n + 2). Recall that in Chapter 1, we used a family of loops
satisfying the circle equation (∂t Γ1 + 1)2 + ∂t Γ22 = 1 + s2 in the codimension
one case. Observe that both, N and Γ satisfy the average condition
I
I
∂t N dt = 0 and
∂t Γ dt = 0
(2.5)
S1
S1
and – as a crucial ingredient to control the C 1 -norm during the iteration –
the following C 1 -estimates:
|∂t N(s, t)| = |s|
√
|∂t Γ(s, t)| 6 2|s|
(2.6)
for all (t, s) ∈ R × S 1 . In the equidimensional case, where there is no normal
e ξ|
e −2 . A similar ansatz like (2.4) or
vector at all, let ξe := ∇u−T · ν and ξ := ξ|
(1.16) is
1
e
λhx, νi)ξ,
(2.7)
v(x) = u(x) + L(a(x)|ξ(x)|,
λ
where L : R × S 1 → R2 , (s, t) 7→ L(s, t) is a smooth family of loops still to be
constructed. Direct computations show that
∇v = ∇u + ∂t Lξ ⊗ ν + O(λ−1 )
and therefore by the definition of ξ:
∇v T ∇v = ∇uT ∇u +
1
(2∂t L + ∂t L2 )ν ⊗ ν + O(λ−1 ).
e2
|ξ|
In order to obtain ∇v T ∇v = ∇uT ∇u + a2 ν ⊗ ν + O(λ−1 ), ∂t L needs to satisfy
e 2 a2 =: 1 + s2 . Since the circle here
the circle equation (1 + ∂t L)2 = 1 + |ξ|
is zero-dimensional, it consists of two isolated points and there is no smooth
map ∂t L to that circle being zero in average. We will circumvent this problem
by replacing the equality in the circle equation by a pointwise and an average
inequality (see Lemma 2.10 and Figure 2.4). This idea is due to Székelyhidi.
Note that the definition of ξ in (2.7) requires u to be immersive. If ∇u does
not have full rank, we will choose ξ to be a unit vector field in ker ∇uT
which will lead to the circle equation ∂t L2 = s2 (see Lemma 2.11).
– 55 –
2.4. Convex Integration
Lemma 2.10 (Regular Corrugation)
For every ε > 0 and c > 0 there exists L ∈ C ∞ ([0, c] × S 1 ), (s, t) 7→ L(s, t)
satisfying the following conditions:
1
2π
(1 + ∂t L)2 6 1 + s2
s2 − ∂t L2 dt < ε
I
1
∂t L dt = 0
2π S 1
I
S1
(2.8)
(2.9)
(2.10)
Proof. Consider the 2π-periodic extension of p̂s : [0, 2π] → [−1, 1], where
s ∈ [0, c]:

i
h π
−1, x ∈ π 1 + √ 1
√ 1
,
3
−
2
2
1+s2
1+s2
p̂s (t) :=
 1 else.
For 0 < ε < 12 fixed, choose 0 < δ < επ/(2(1+c2 )) and let ϕδ denote the usual
symmetric standard mollifier. The function p : R×S 1 → R, (s, t) 7→ ϕδ ∗ p̂s (t)
is smooth and a direct computation using Fubini’s Theorem implies
I
1
1
p(s, t) dt = √
.
2π S 1
1 + s2
Since p2 (s, ·) equals one on a domain of measure at least 2π − 4δ on each
period we get
I
2δ
1
ε
p2 (s, t) dt > 1 −
>1−
.
2π S 1
π
1 + s2
The function L : R × S 1 → R
Z t √
2
L(s, t) :=
1 + s · p(s, u) − 1 du
0
then has all the desired properties.
Lemma 2.11 (Singular Corrugation)
e ∈ C ∞ ([0, c] × S 1 ), (s, t) 7→ L(s,
e t)
For every ε > 0 and c > 0 there exists L
satisfying the following conditions:
e 2 6 s2
∂t L
1
e2 dt < ε
s2 − ∂ t L
2π S 1
I
1
e dt = 0
∂t L
2π S 1
(2.11)
I
– 56 –
(2.12)
(2.13)
2.4. Convex Integration
Sr21 (0)
Sr12 (−1)
Sr01 (0)
Sr02 (−1)
Figure 2.1: From left to right and from top to bottom the picture illustrates
e and L
the Nash Twist, the corrugation from Chapter 1 and the maps L
from the Lipschitz case. The radii of the spheres are given by r1 = |s| and
√
r2 = 1 + s 2 .
Proof. Consider the convolution of the 2π-periodic extension of the map
(
−s, t ∈ [ π2 , 3π
]
2
[0, 2π] 3 t 7→
s else
with ϕδ , where δ < επ/(2c2 ). This convolution gives rise to a map p(s, t).
The map
Z t
e
L(s, t) :=
p(s, u) du
0
then has all the desired properties.
e analogous
The conditions (2.9) and (2.12) oppose a C 1 -estimate for L and L
to (2.6), hence the C 1 -norms of the maps cannot be controlled during the
iteration, but we will use a suitable L2 -estimate due to Székelyhidi using an
identity relating the first derivatives to the trace of the metric defect and
integration by parts later on (see Proposition 2.14).
– 57 –
2.5. Iteration
2.5
Iteration
Let ` > 0 and let u be a piecewise affine subsolution adapted to (f, g) in the
sense of Definition 2.9 and decompose the metric defect as
∗
g − u g0 =
m
X
k=1
a2k νk ⊗ νk .
Since u is already isometric on B, we will add a “cut-off” error η`2 (g − u∗ g0 )
as in Chapter 1.
k-th Step
Let uk−1 be a piecewise affine subsolution adapted to (f, g) s.t. uk−1 is
piecewise affine on Ωc`/2 . We introduce a map Θk ∈ C ∞ (Ω̄, [0, 1]) in the
k-th step that is associated to uk−1 as follows: Let {Si }i be the simplicial
decomposition of Ω̄ according to uk−1 and let Uk be an open neighborhood
of
[
Kk := Ω̄ \ S̊i
i
Ukc
and set Θk = 1 on
and Θk = 0 on Kk . Observe that Uk can be chosen to
have arbitrary small Lebesgue-measure. Let 0 < δ < 1 (the exact value will
be determined later). We will now discuss the step for a simplex S ⊂ Ωc`/2 .
The restriction of uk−1 to S is an affine function. If ∇uk−1 is regular on S,
let
ξek := ∇uk−1 (∇uTk−1 ∇uk−1 )−1 · νk
ξk := ξek |ξek |−2
1
sk := (1 − δ) /2 Θk η` ak |ξek |
Lk (x) := L (sk , λk hx, νk i) .
and define uk := uk−1 + λ1k Lk ξk . We have uk ∈ C ∞ (S, Rn ) and kuk−1 −uk kC 0 (S)
can be made arbitrarily small provided the free parameter λk is large enough.
For the Euclidean metric pulled back by uk , we find
∇uTk ∇uk = ∇uTk−1 ∇uk−1 +
1
2∂t Lk + ∂t L2k νk ⊗ νk + O λ−1
.
k
|ξek |2
If ∇uk−1 is singular, choose ξk ∈ ker(∇uTk−1 ) to be a unit vector, let
1
sk := (1 − δ) /2 Θk η` ak
ek (x) := L
e (sk , λk hx, νk i)
L
– 58 –
2.5. Iteration
ek ξk . As in the regular case, uk ∈ C ∞ (S, Rn ) and
and define uk := uk−1 + λ1k L
kuk−1 − uk kC 0 (S) can be made arbitrarily small. For the Euclidean metric
pulled back by uk we find
e2 νk ⊗ νk + O λ−1 .
∇uTk ∇uk = ∇uTk−1 ∇uk−1 + ∂t L
k
k
Stage
We approximate the resulting map after each step by an adapted piecewise
affine subsolution. This introduces a further subdivision of the simplicial
decomposition of Ω̄ after each step. In each step we will leave the map from
the foregoing step unchanged near B (due to the cut-off by η` ) and near
the (n − 1)-skeleton of the simplicial decomposition (thanks to Θk ). This
procedure does not allow for a pointwise control of the new metric error, but
we are still able to bound it in an L1 -sense:
Proposition 2.12 (Stage)
Let u be a subsolution adapted to (f, g). Then for any ε > 0, there exists a
piecewise affine subsolution u
e adapted to (f, g) satisfying
Z
Ω̄
Proof. Since
Z
Ω̄`
ku − u
ekC 0 (Ω̄) < ε
tr g − ∇e
uT ∇e
u dx < ε
(2.14)
(2.15)
`→0
tr(g − ∇uT ∇u) dx −→ 0,
there exists ` > 0 small enough such that after approximating u by a piecewise affine subsolution u
e0 adapted to (f, g) that is piecewise affine on Ωc`/2
we get
Z
ε
(2.16)
tr(g − ∇e
uT0 ∇e
u0 ) dx < .
7
Ω̄`
Choose (for ` now fixed) δ such that
δ id < (g − u
e∗0 g0 )|Ωc
Ln -a.e.
`/2
ε
δ<
7
Z
Ωc`
tr(g −
∇e
uT0 ∇e
u0 ) dx
(2.17)
!−1
.
(2.18)
We use the step iteratively to produce a sequence of maps starting with
u
e0 . After a step, say the k-th, we approximate the resulting map uk by an
– 59 –
2.5. Iteration
adapted piecewise affine subsolution u
ek that is piecewise affine on Ωc`/2 and
leave it unchanged on Ω̄`/2 .
After m steps, we set u
e := u
em . Choosing the free parameter λk sufficiently
large in each step together with suitable approximations by piecewise affine
subsolutions proves (2.14).
On each simplex (in Ωc`/2 ) from to the simplicial decomposition corresponding
to the map u
e we performed m steps and since after each step, the resulting
map was affine on that simplex, we used (depending on whether ∇e
uk−1 was
regular or singular) the “regular” or the “singular” k-th step. This splits
the set {1, . . . , m} into R and S corresponding to indices k where ∇e
uk−1 was
regular and singular respectively. A direct computation on a fixed simplex
shows
g − ∇e
uTm ∇e
um = g − ∇e
uT0 ∇e
u0 + ∇e
uT0 ∇e
u0 − ∇e
uTm ∇e
um
m
X
=
a2k νk ⊗ νk + ∇e
uTk−1 ∇e
uk−1 − ∇uTk ∇uk + E
k=1
X =
a2k − |ξek |−2 2∂t Lk + ∂t L2k νk ⊗ νk + O(λ−1
)
k
k∈R
+
X a2k
k∈S
−1
2
e
− ∂t Lk νk ⊗ νk + O(λk ) + E,
(2.19)
where
E :=
m
X
k=1
∇uTk ∇uk − ∇e
uTk ∇e
uk
satisfies kEkC 0 (Ω̄) < ε̂ and ε̂ > 0 will be fixed later (this is possible for every
ε̂ by the use of suitable approximations).
In order to prove that u
e is a piecewise affine subsolution adapted to (f, g),
first observe that every uk is piecewise smooth and continuous. This follows
from the infinite differentiability in the interior of every simplex and the fact
that uk agrees with uk−1 on Kk .
Now we prove shortness on Ωc`/2 : We use the computation (2.19) and the
– 60 –
2.5. Iteration
pointwise estimates (2.8), (2.11) and (2.17) to obtain Ln -a.e.
X g − ∇e
uTm ∇e
um =
a2k − |ξek |−2 2∂t Lk + ∂t L2k νk ⊗ νk + O(λ−1
)
k
k∈R
+
>
X k∈S
m
X
k=1
a2k
−1
2
e
− ∂t Lk νk ⊗ νk + O(λk ) + E
1 − (1 − δ)Θ2k η`2 a2k νk ⊗ νk + O(λ−1
k )+E
> δ(g − u
e∗0 g0 ) +
> δ 2 id +
m
X
k=1
m
X
k=1
O(λ−1
k ) − ε̂ id
O(λ−1
k ) − ε̂ id > 0
provided ε̂ is small enough and the frequencies λk are large enough. Note
that the pullback of the Euclidean metric is not defined on Km . In order to
prove (2.15), we use the following estimates on a simplex S in the regular
and singular case respectively:
Z ε
(2.20)
s2k − |ξek |−2 (2∂t Lk + ∂t L2k ) dx <
7m vol Ωc`/2
S
Z ε
e2k dx <
s2k − ∂t L
(2.21)
7m vol Ωc`/2
S
These estimates are direct consequences of (2.9) and (2.12) and the fact that
for every f ∈ C 0 (Ω̄ × S 1 )
Z
Z
I
1
λ→∞
f (x, t) dt dx.
f (x, λhx, νi) dx −→
Ω̄
Ω̄ 2π S 1
This is the content of Proposition 2.13 which will be stated after this proof.
Let
Z
Z
T
tr g − ∇e
um ∇e
um dx =
tr g − ∇e
uTm ∇e
um dx +
Ω̄
Ω̄`/2
|
+
{z
}
=:I1
N Z
X
i=1
|
Si
tr g − ∇e
uTm ∇e
um dx,
{z
=:I2
}
where N is the total number of simplices in the simplicial decomposition of
Ωc`/2 according to the map u
em . Since u
em and u
e0 agree on Ω̄`/2 , we use (2.16)
– 61 –
2.5. Iteration
to obtain I1 6 7ε . We use (2.19), (2.20), (2.21), ε̂ < 7ε (vol Ωc`/2 )−1 and a
P
ε
suitable choice of the λk to obtain “ O(λ−1
k ) < 7 ” for estimating I2 :
I2 6
N XZ X
i=1
+
k∈R
XZ k∈S
6
Si
Si
m Z
X
k=1
Ωc`/2
a2k
a2k − |ξek |−2 2∂t Lk + ∂t L2k
−
e2k
∂t L
dx +
dx +
2ε
7
(2.22)
(a2k − (1 − δ)Θ2k η`2 a2k ) dx +
3ε
3ε
=: I3 + .
7
7
Inequality (2.15) follows with the use of (2.16) and (2.18):
m Z
m Z
X
X
2
2 2
I3 6
(ak − (1 − δ)Θk η` ak ) dx +
(a2k − (1 − δ)Θk η`2 a2k ) dx
6
k=1 Ω̄`
m Z
X
Zk=1
6
Ω̄`
+
Ω̄`
k=1
a2k dx +
Z
m
X
k=1
δa2k dx +
Ωc` ∩Ukc
tr(g − ∇e
uT0 ∇e
u0 ) dx + δ
m
X
k=1
Z
Ωc`
Z
Ωc`
!
a2k dx
Ωc` ∩Uk
tr(g − ∇e
uT0 ∇e
u0 ) dx +
kak kL∞ (Ω̄) Ln (Uk )
The first two terms of the last line are bounded by 7ε and a suitable choice of
Uk ensures that Ln (Uk ) is small enough to make the same hold for the third
term as well.
Proposition 2.13
Let Ω ⊂ Rn be bounded, then for every f ∈ C 0 (Ω̄ × S 1 ) there holds
Z
Z
I
1
λ→∞
f (x, λhx, νi) dx −→
f (x, t) dt dx.
Ω̄
Ω̄ 2π S 1
Proof. The proof follows [Fré12, p. 13 - 15]. Split [0, 2π] into m intervals
of length 2π
. Let tk be the center of the k-th interval and let χk be its
m
characteristic function extended 2π-periodically onto R. Consider
fem (x, t) :=
m
X
f (x, tk )χk (t)
k=1
As m → ∞, fem converges uniformly to f . For ν ∈ Rn let
πν : Rn → R, x 7→ hx, νi
– 62 –
2.5. Iteration
and observe that as λ → ∞,
1
χk (λπν ( · )) +
2π
∗
I
χi (t) dt =
S1
1
in L∞ (Ω̄).
m
This is an immediate consequence of the following weak* convergence (in
L∞ (S 1 )) for any f ∈ C 0 (S 1 ) as λ → ∞:
I
∗ 1
f (λ · ) +
f (t) dt
2π S 1
(see for example [CD99, p. 33 - 37]). Hence we find
I
Z
Z
1
λ→∞
e
fem (x, t) dt dx
fm (x, λπν (x)) dx −→
Ω̄
Ω̄ 2π S 1
(2.23)
Now we write
Z I
1
f (x, t) dt dx 6 I1 + I2 + I3 ,
f (x, λπν (x)) dx −
2π S 1
Ω̄
where
Z e
I1 = f (x, λπν (x)) − fm (x, λπν (x)) dx
ZΩ̄ I
1
e
e
I2 = fm (x, λπν (x)) −
fm (x, t) dt dx
2π S 1
ZΩ̄ I 1
fem (x, t) − f (x, t) dt dx .
I3 = 2π S 1
Ω̄
Now fix ε > 0. The uniform convergence fem → f implies the existence of an
m such that I1 , I3 6 3ε . Using (2.23) we get I2 < 3ε by a suitable choice of λ.
This concludes the proof.
The following L2 -estimate due to Székelyhidi (see [GA13]) gives a way to
control the derivatives during the stage a posteriori:
Proposition 2.14 (Székelyhidi)
If ε > 0 from Proposition 2.12 is small enough, the maps u and u
e satisfy the
estimate
Z
2
k∇u − ∇e
ukL2 (Ω̄) 6 C
tr g − ∇uT ∇u dx.
(2.24)
Ω̄
Proof. Use the Definition of the Frobenius inner product to obtain
tr(g−∇e
uT ∇e
u) = tr(g−∇uT ∇u)−2 tr(∇uT (∇e
u −∇u))−|∇e
u −∇u|2 . (2.25)
– 63 –
2.5. Iteration
Denote by υ the outer unit normal vector of a simplex S. We then have the
following integration by parts formula (which holds since u is smooth in the
interior of S):
Z
Z
Z
T
tr(∇u (∇e
u − ∇u)) dx = − h∆u, u
e − ui dx+
tr(∇uT (e
u −u)⊗υ) dS,
S
S
∂S
|
{z
}
=:I
(2.26)
which is obtained from the following computation:
n Z
X
I=
∂i uk (∂i u
ek − ∂i uk ) dx
=
i,k=1 S
n X
i,k=1
=−
=−
−
Z
Z X
n
S k=1
Z
S
S
∂ii2 uk (e
uk
k
k
k
− u ) dx +
k
∆u (e
u − u ) dx +
h∆u, u
e − ui dx +
Z
∂S
Z
Z
∂S
∂S
k
k
k
i
∂i u (e
u − u )υ dS
tr(∇uT (e
u − u) ⊗ υ) dS
tr(∇uT (e
u − u) ⊗ υ) dS
Formulae (2.25) and (2.26) together with the shortness condition
g − ∇e
uT ∇e
u > 0 Ln -a.e.
imply
k∇e
u−
∇uk2L2 (Ω̄)
Z
6
Ω̄
tr(g − ∇uT ∇u) dx +
+ C k∆ukL∞ (Ω̄) + k∇ukL∞ (∂Ω) ke
u − ukC 0 (Ω̄) .
The last estimate is valid since u is Lipschitz and ∇u is therefore an L∞ function on ∂Ω. Now, if ε is small enough, we use (2.14) to get (2.24).
Passage to the Limit
We can now use Proposition 2.12 iteratively to prove our main result, Theorem 2.3:
Proof. Choose a sequence (εk )k such that
∞
X
εk 6 ε and such that
k=1
∞
X
√
k=1
– 64 –
εk < ∞.
2.5. Iteration
Since every adapted piecewise affine subsolution is also an adapted subsolution, we can apply the previous theorem iteratively starting with u0 := u.
The iteration gives rise to a sequence uk that is uniformly Lipschitz (short√
ness implies |∇uk | 6 tr g Ln -a.e.) and Cauchy in C 0 (Ω̄, Rn ) due to (2.14),
hence the limit map v is almost everywhere differentiable by Rademacher’s
Theorem and satisfies ku0 − vkC 0 (Ω̄) < ε by the choice of the εk . Because of
(2.15) we obtain furthermore
lim u∗k g0 = g
k→∞
Ln -a.e.
Choosing εk in each step so small that (2.24) holds, ensures that ∇uk is
Cauchy in L2 (Ω̄, Rn×n ) and converges (up to a subsequence) pointwise Ln -a.e.
to a limit map Λ. We have ∇v = Λ Ln -a.e., since Λ is the weak derivative of
v. Indeed, fix φ ∈ C0∞ (Ω̄). Using the uniform convergence of uk , integration
by parts and Hölder’s inequality we obtain
Z
Z
Z
Z
φ∇uk dx = − φΛ dx.
uk ⊗grad φ dx = − lim
v ⊗grad φ dx = lim
Ω̄
k→∞
k→∞
Ω̄
Ω̄
Ω̄
We are left to show that ∇v T ∇v = g Ln -a.e., which follows from
∇v T ∇v = ΛT Λ = lim ∇uTk
lim ∇uk = lim u∗k g0 = g Ln -a.e.
k→∞
k→∞
k→∞
Since uk |B = uk+1 |B for all k ∈ N, we find v|B = u|B .
Remark 2.15
The density statement from Theorem 2.3 can be reformulated as follows:
Let the space of subsolutions adapted to (f, g) equipped with the uniform
topology be denoted by Sub0 (f, g) and consider its C 0 -closure
Sub(f, g) = u : Ω̄ → Rn , u|B = f, g − ∇uT ∇u > 0 Ln -a.e.
and consider the functional F : Sub(f, g) → R given by
Z
F[u] :=
tr g − ∇uT ∇u dx.
Ω̄
Since Sub(f, g) consists of uniform Lipschitz functions, F is a well-defined
and non-negative upper semicontinuous functional and we obtain as a Corollary of Theorem 2.3
Corollary 2.16
The zero-set of F is dense in Sub(f, g).
– 65 –
2.6. Application
Remark 2.17 (Global Results)
The method presented here allows to obtain global results from the local
ones by a partition of unity argument. Here, “global” means that we want
to construct solutions to (2.2) on a neighborhood of Σ (and not only of a
point in Σ). The global result is obtained exactly as in Section 1.7, where
the step, the stage and the iteration are replaced by the step, the stage and
the iteration of the present chapter.
2.6
Application
We are now in position to prove Corollary 2.4:
Proof. We first construct a piecewise smooth map from the upper hemisphere
H+ to the disk D̄2 that is short everywhere but on the equator, where it is
isometric. The following considerations provide such a map. Consider the
maps
Φ ∈ C ∞ ([0, π2 ] × [0, 2π], R3 ) and Ψ ∈ C ∞ ([0, π2 ] × [0, 2π], R3 )
defined by


cos ϑ cos ϕ
Φ(ϑ, ϕ) :=  cos ϑ sin ϕ 
sin ϑ
cos ϑ cos ϕ
Ψ(ϑ, ϕ) := f (ϑ)
,
cos ϑ sin ϕ
where f : [0, π2 ] → [ 12 , 1] is defined by
(
ϑ 7→
ϑ2
2
1
2
− ϑ + 1 x ∈ [0, 1]
else
.
Observe that the composition Ψ◦Φ−1 gives rise to a modified projection from
H+ to D̄2 that is short everywhere but on the equator (where it is isometric).
Since the hemisphere is diffeomorphic to the disk, we can think of this map
as a map u : D̄2 → D̄2 that is short everywhere but on ∂D2 = S 1 ⊂ D̄2
with respect to a suitable metric in the source domain. Since one can inscribe two polygons P1 ⊂ P2 ⊂ D̄2 such that dist(∂D2 , P2 ) = `/4 and
dist(∂D2 , P1 ) = `/2 we can approximate u on P2 by a piecewise affine map
by Proposition 2.7 and get using Remark 2.8 a map v that is piecewise affine
– 66 –
2.6. Application
on P1 and corresponds to u on ∂D2 . Now we can proceed as in Proposition
2.12 to get a sequence of maps that converges to a isometric Lipschitz map
thanks to Theorem 2.3. Observe that one can ensure that the image of each
intermediate map is indeed D̄2 , since during the iteration, the maps are left
unchanged near B in every step and by choosing λ 1 large enough, the
claim follows from the C 0 -closeness of the maps during the iteration. Now
the same procedure can be performed for the lower hemisphere H− and the
resulting maps can be glued together along the equator S 1 in each step, since
the iteration doesn’t change the map in a neighborhood of S 1 . This concludes
the proof.
Remark 2.18
Proposition 2.5 shows that the map v constructed in Corollary 2.4 cannot
be C 1 or even merely differentiable, since the standard metric on S 2 has
Gaussian curvature 1. In this perspective, the Lipschitz regularity seems
to be optimal for equidimensional isometries. Moreover, it implies that the
singular set of the map v is dense in S 2 and v cannot even be locally injective.
We will now show that the Hausdorff-dimension of the singular set equals
one: In general, the set where the pullback of u constructed in Theorem 2.3
is undefined has Hausdorff-dimension n − 1 or is empty: Observe that the
k-th stage introduces a finite number Nk of simplices Si,k to the ones that are
present from the (k −1)-th step and the pullback of uk is in general undefined
on the (n − 1)-skeleton of the simplicial decomposition given by the {Si,k }i .
The singular set S of u can then be described as
S=
Nk
[[
∂Si,k .
k∈N i=1
It follows that dimH (S) = supk,i dimH ∂Si,k = n − 1, since the boundary of
an n-simplex has Hausdorff-dimension n − 1. If all Nk = 0, then the singular
set is of course empty.
– 67 –
Chapter 3
Legendrian Approximation
of Curves
3.1
Introduction
A contact structure on a 3-manifold M is a maximally non-integrable rank 2
subbundle ξ of the tangent bundle of M . If α is a 1-form on M whose kernel
is ξ, then ξ is a contact structure if and only if α ∧ dα 6= 0. A curve η in a
contact 3-manifold (M, ξ) is called Legendrian, whenever η ∗ α = 0 for some
(local) 1-form α defining ξ.
The purpose of this chapter is to give a detailed proof of the following statement which is often used in contact geometry and Legendrian knot theory.
Theorem 3.1 (Legendrian Approximation)
Any continuous map from a compact 1-manifold to a contact 3-manifold can
be approximated by a Legendrian curve in the C 0 -Whitney topology.
Whereas this theorem is a special case of Gromov’s h-principle for Legendrian immersions [Gro86], the curve-case can be treated by more elementary
techniques. Sketches of proofs of Theorem 3.1 have already appeared in the
literature, see for example [Etn05, p.6-7], [Gei06, p.40] or [Gei08, p.102]. Exploiting the fact that every contact 3-manifold is locally contactomorphic to
R3 equipped with the standard contact structure defined by α = dz − y dx,
Etnyre and Geiges indicate that either the front-projection (x, z) of a given
curve (x, y, z) can be approximated by a zig-zag-curve whose slope approximates the y-component of the curve or the Lagrangian projection (x, y) can
– 68 –
3.2. Legendrian Approximation in R3
be approximated by a curve whose area integral approximates the z component of the curve, which can be achieved by adding small negatively or
positively oriented loops.
Here, we give a different and analytically rigorous proof of Theorem 3.1
using convex integration. Our proof has the advantage of providing a constructive approximation. In particular, in the case of a continuous curve
in R3 equipped with the standard contact structure, we obtain an explicit
Legendrian curve given in terms of an elementary integral. For instance, we
obtain an explicit solution to the parallel parking problem in Example 3.5.
Example 3.6 shows how our technique recovers the zig-zag-curves and the
small loops in the front - respectively Lagrangian projections.
3.2
Legendrian Approximation in R3
We start by first treating the case where the contact manifold is R3 equipped
with the standard contact structure, that is, we aim to prove the following:
Proposition 3.2
Let υ ∈ C 0 ([0, 2π], R3 ). Then for every ε > 0, there exists a Legendrian curve
η ∈ C ∞ ([0, 2π], R3 ) such that kυ − ηkC 0 ([0,2π]) 6 ε.
Let the curve we wish to approximate be given by (x, y, z) ∈ C ∞ ([0, 2π], R3 ).
The regularity is no restriction due to a standard approximation argument using convolution. For every choice of smooth functions (a, c) ∈ C ∞ ([0, 2π], R2 )
satisfying ȧ 6= 0, we obtain a Legendrian curve η = (a, b, c) ∈ C ∞ ([0, 2π], R3 )
by defining b = ċ/ȧ. Therefore, if (ȧ(t), ċ(t)) lies in the set
Rt,ε := (u, v) ∈ R2 , |v − y(t)u| 6 ε min{|u|, |u|2 } ,
for every t ∈ [0, 2π], then kb − ykC 0 ([0,2π]) 6 ε. This condition can be achieved
by defining
Z t
(a(t), c(t)) := (x(0), z(0)) +
γ(u, nu) du,
0
∞
1
2
with γ ∈ C ([0, 2π] × S , R ) and n ∈ N, provided that γ(t, ·) ∈ Rt,ε .
Furthermore, if γ additionally satisfies
I
1
γ(t, s) ds = (ẋ(t), ż(t)),
2π S 1
– 69 –
3.2. Legendrian Approximation in R3
for all t ∈ [0, 2π], then – as we will show below – (a(t), c(t)) approaches
(x(t), z(t)) as n gets sufficiently large.
The set Rt,ε is ample, i.e., the interior of its convex hull is all of R2 . For
any given point (ẋ(t), ż(t)) ∈ R2 we will thus be able to find a loop in Rt,ε
having (ẋ(t), ż(t)) as its barycenter. This is the fundamental lemma of convex
integration already mentioned in the introduction (see [Spr10, Prop. 2.11,
p. 28]). In the particular case studied here we obtain an explicit formula for
γ:
Lemma 3.3
There exists a family of loops γ ∈ C ∞ ([0, 2π]×S 1 , R2 ) satisfying γ(t, ·) ∈ Rt,ε
and such that
I
1
γ(t, s) ds = (ẋ(t), ż(t)),
(3.1)
2π S 1
for all t ∈ [0, 2π].
Proof. The map γ := (γ1 , γ2 ), where
γ1 (t, s) := r cos s + ẋ(t)
and
2(ż(t) − y(t)ẋ(t))
γ1 (t, s)
γ2 (t, s) := γ1 (t, s) y(t) +
r2 + 2ẋ(t)2
satisfies (3.1) for every r > 0. If r is large enough one obtains γ(t, ·) ∈ Rt,ε ,
where r can be chosen independently of t by compactness of [0, 2π].
We now have:
Proof of Proposition 3.2. With the definitions above we obtain
b(t) :=
ċ(t)
2(ż(t) − y(t)ẋ(t))
= y(t) +
γ1 (t, nt).
ȧ(t)
r2 + 2ẋ(t)2
We are left to show that |(a, c) − (x, z)| 6 ε provided n is large enough. This
follows from the following estimate
k(a, c) − (x, z)kC 0 ([0,2π])
4π 2
6
kγkC 1 ([0,2π]×S 1 ) .
n
(3.2)
The estimate is in fact a geometric property of the derivative and can be
interpreted as follows: Since (ȧ, ċ) and (ẋ, ż) coincide “in average” on shorter
– 70 –
3.2. Legendrian Approximation in R3
and shorter intervals as n gets bigger and bigger, (a, c) and (x, z) tend to
become close: Let
2πk 2π(k + 1)
nt
nt 2π
Ik :=
,
for k = 0, . . . ,
− 1 and J :=
,t .
n
n
2π
2π n
Then we can estimate D = |(a(t), c(t)) − (x(t), z(t))|:
Z t
Z t
D = γ(u, nu) du −
(ẋ, ż)(u) du
0
0
c−1 Z
bX
Z
Z 2π
1
γ(u, nu) du −
γ(u, v) dv du +
6
Ik
Ik 2π 0
k=0
Z
|γ(u, nu)| + kγkC 0 ([0,2π]×S 1 ) du
+
nt
2π
J
nt
b 2π
c−1 Z 2π Z
Z 2π
X
1
v
+
2kπ
1
+
6
γ
,
v
dv
−
γ(u,
v)
dv
du
n
n
2π
0
I
0
k
k=0
4π
kγkC 0 ([0,2π]×S 1 )
n
nt
b 2π
c−1 Z Z 2π X
1
v
+
2kπ
γ
6
, v − γ(u, v) dv du +
2π
n
Ik 0
k=0
+
4π
+
kγkC 0 ([0,2π]×S 1 )
n 2
nt 4π
4π
0 ([0,2π]×S 1 ) +
6
k∂
γk
kγkC 0 ([0,2π]×S 1 )
t
C
2π n2
n
4π
6
πk∂t γkC 0 ([0,2π]×S 1 ) + kγkC 0 ([0,2π]×S 1 ) .
n
By construction, the curve (a, b, c) is Legendrian and an approximation of
(x, y, z), provided n is large enough.
Next we show that we can approximate closed curves by closed Legendrian
curves.
Proposition 3.4
Let υ ∈ C 0 (S 1 , R3 ). Then for every ε > 0, there exists a Legendrian curve
η ∈ C ∞ (S 1 , R3 ) such that kυ − ηkC 0 (S 1 ) 6 ε.
Proof. Using standard regularization again, let the curve we wish to approximate be given by (x, y, z) ∈ C ∞ ([0, 2π], R3 ), where the values of (x, y, z) in 0
– 71 –
3.3. Gluing
and 2π agree to all orders. Define g(t) := γ12 (t, nt). Since kgkL1 ([0,2π]) = O(r2 )
as r → ∞, we can choose r > 0 large enough such that f := g/kgkL1 ([0,2π]) is
well-defined. With the notation
Z 2π
γ2 (u, nu) du,
I2 :=
0
we define η = (a, b, c) as follows:
Z t
(a(t), c(t)) := (x(0), z(0)) +
γ(u, nu) − (0, I2 f (u)) du,
(3.3)
0
ċ(t)
2(ż(t) − y(t)ẋ(t))
I2
b(t) :=
= y(t) + γ1 (t, nt)
.
−
ȧ(t)
r2 + 2ẋ(t)2
kgkL1 ([0,2π])
(3.4)
A straightforward computation shows that the values of (a, b, c) in 0 and 2π
agree to all orders, hence η ∈ C ∞ (S 1 , R3 ) and it is Legendre by construction.
2
Using (3.2) we obtain |I2 | 6 4πn kγ2 kC 1 ([0,2π]×S 1 ) , hence we find using (3.4) as
r → ∞:
1
kb − ykC 0 ([0,2π]) 6 kγ1 kC 0 ([0,2π]×S 1 ) 1 + kγkC 1 ([0,2π]×S 1 ) O(r−2 ).
n
For the remaining components we find find using (3.2) and (3.3) the uniform
bound
Z t
|I2 |
4π 2
kγkC 1 ([0,2π]×S 1 ) +
g(u) du
|(a(t), c(t)) − (x(t), z(t))| 6
n
kgkL1 ([0,2π]) 0
8π 2
6
kγkC 1 ([0,2π]×S 1 ) .
n
Choosing r large enough and n ∼ r2 concludes the proof.
3.3
Gluing
We show now how to glue together two local approximations of a curve Γ in
M on two intersecting coordinate neighborhoods. Let therefore Uσ and Uτ
in M be coordinate patches such that U = Uσ ∩ Uτ 6= ∅. Let Iσ and Iτ be
compact intervals such that I = Iσ ∩ Iτ contains an open neighborhood of
t = 0 (after shifting the variable t if necessary) and such that Γ(Iσ ) ⊂ Uσ ,
Γ(Iτ ) ⊂ Uτ . Assume without restriction that Γ is smooth and let (x, y, z)
– 72 –
3.3. Gluing
represent Γ on U . Suppose that (x, y, z) is approximated by Legendrian
curves σ : Iσ → R3 and τ : Iτ → R3 such that
kσ − (x, y, z)kC 0 (I) < ε2 ,
kτ − (x, y, z)kC 0 (I) < ε2
(3.5)
for some fixed 0 < ε < 21 . For r > 0, define R(r) to be the smallest number
such that B̄r (0) ⊂ convpR0,ε ∩ B̄R (0) . Note that R depends continuously
on r and if r > r0 := ε/ 1 + y(0)2 , then
R(r) =
rp
r
(1 + y(0)2 ) (1 + (|y(0)| + ε)2 ) =: w(y(0), ε).
ε
ε
(3.6)
Choose 0 < δ < ε2 such that [−δ, δ] ⊂ I and such that δk(x, y, z)kC 1 (I) 6 ε2
and define
p1 := (σ1 (−δ), σ3 (−δ)),
ṗ1 := (σ̇1 (−δ), σ̇3 (−δ)),
p2 := (τ1 (δ), τ3 (δ)),
ṗ2 := (τ̇1 (δ), τ̇3 (δ)).
From (3.5) and the choice of δ we obtain
ṗ1 , ṗ2 ∈ Cε := (u, v) ∈ R2 , |v − y(0)u| 6 ε|u|
and
p2 − p1
=: p ∈ Br̄ (0),
2δ
where r̄ = 2ε2 /δ. Since 3r̄ > r0 , we can express R(3r̄) by means of formula
(3.6). This will be used in computation (3.9). We construct a path
γ = (γ1 , γ2 ) : [−δ, δ] → Cε
as follows: For ρ < δ/2, let γ|[−δ,−δ+ρ] be a continuous path from ṗ1 to 0 and
let γ|[δ−ρ,δ] be a continuous path from 0 to ṗ2 . We construct γ such that the
quotient γ2 /γ1 is well-defined on [−δ, −δ + ρ] ∪ [δ − ρ, δ] and equals y(0) in
t = −δ + ρ and t = δ − ρ. Moreover, we require that
Z −δ+ρ
Z δ
δε
δε
|γ(t)| dt <
and
|γ(t)| dt < .
(3.7)
2
2
−δ
δ−ρ
On [−δ, −δ + ρ], such a path is for example given by
k
δ+t
t 7→ 1 −
ρ
!
σ̇1 (−δ)
y(0)σ̇1 (−δ) + (σ̇3 (−δ) − y(0)σ̇1 (−δ)) 1 −
– 73 –
δ+t
ρ
k
3.3. Gluing
provided k ∈ N is sufficiently large. We obtain
Z δ
Z −δ+ρ
1
γ(t) dt =: p̄ ∈ B3r̄ (0)
γ(t) dt −
2δp −
2(δ − ρ)
δ−ρ
−δ
and hence
p̄ ∈ int conv(BR(3r̄) (0) ∩ R0,ε ).
Using the fundamental lemma of convex integration we let γ|[−δ+ρ,δ−ρ] be a
continuous closed loop in BR(3r̄) (0) ∩ R0,ε based at 0 such that
1
2(δ − ρ)
δ−ρ
Z
γ(t) dt = p̄.
−δ+ρ
With these definitions we obtain
1
2δ
Z
δ
γ(t) = p.
−δ
Now we define η = (a, b, c) : [−δ, δ] → R3 by letting b(t) := ċ(t)/ȧ(t), where
Z t
γ(u)du.
(a, c)(t) := p1 +
−δ
The curve η is well-defined and Legendrian by construction. Moreover it
satisfies η(−δ) = σ(−δ) and η(δ) = τ (δ). Moreover, (a, c) and (σ1 , σ3 )
agree to first order in t = −δ and so do (a, c) and (τ1 , τ3 ) in t = δ. From
γ([−δ, δ]) ∈ Cε and the choice of δ we find
|b(t) − y(t)| 6 |b(t) − y(0)| + |y(t) − y(0)| 6 ε + δkykC 1 (I) < 2ε.
(3.8)
Using (3.5), (3.6), (3.7) and the choice of δ we obtain for the remaining
components the uniform bound
Z t
|(a, c)(t) − (x, z)(t)| 6 |p1 − (x, z)(−δ)| +
(|γ(u)| + |(ẋ, ż)(u)|) du
−δ
6 ε2 + δε +
Z
δ−ρ
−δ+ρ
|γ(u)| du + 2δk(x, z)kC 1 (I)
6 2ε + 2δR(3r̄)
2 !
1
.
6 ε 14 + 12 |y(0)| +
2
(3.9)
– 74 –
3.4. Examples
Finally, suppose υ is a continuous curve from a compact 1-manifold N (that
is, N is a compact interval or S 1 ) into a contact 3-manifold (M, ξ). We fix
some Riemannian metric g on M . Then it follows with the bounds (3.8), (3.9)
and the compactness of the domain of υ that for every ε > 0 there exists a
ξ-Legendrian curve η such that
sup dg (υ(t), η(t)) < ε,
t∈N
where dg denotes the metric on M induced by the Riemannian metric g.
In particular, every open neighborhood of υ ∈ C 0 (N, M ) – equipped with
the uniform topology – contains a Legendrian curve N → M . Since N is
assumed to be compact the uniform topology is the same as the Whitney
C 0 -topology, thus proving Theorem 3.1.
3.4
Examples
Example 3.5 (Parallel Parking)
The trajectory of a car moving in the plane can be thought of as a curve
[0, 2π] → S 1 × R2 .
Denoting by (ϕ, a, c) the natural coordinates on S 1 × R2 , the angle coordinate ϕ denotes the orientation of the car with respect to the a-axis and the
coordinates (a, c) the position of the car in the plane. Admissible motions of
the car are curves satisfying
ȧ sin ϕ = ċ cos ϕ.
The manifold S 1 × R2 together with the contact structure defined by the
kernel of the 1-form θ := sin ϕ da − cos ϕ dc is a contact 3-manifold. Indeed,
we have
θ ∧ dθ = − cos2 ϕ dϕ ∧ da ∧ dc − sin2 ϕ dϕ ∧ da ∧ dc = −dϕ ∧ da ∧ dc 6= 0.
Applying Theorem 3.1 with b = tan ϕ gives an explicit approximation of the
curve
t 7→ (x(t), y(t), z(t)) = (0, 0, t).
Lemma 3.3 gives the loop γ(t, s) = 2(r cos s, cos2 s) and hence the desired
Legendrian curve
(arccot(r sec(nt)), 2rt sinc(nt), t + t sinc(2nt)) ,
provided r is large enough and n ∼ r2 .
– 75 –
3.4. Examples
Figure 3.1: The front (top) and the Lagrangian projection (bottom) of the
Legendrian approximation of υ.
Example 3.6 (Legendrian Helix)
Suppose we want to find a Legendrian approximation with tolerance ε = 3/10
of the helix
υ : [0, 2π] → R3 , t 7→ (t, cos(5t), sin(5t)).
One can check that the choices n = 29 r2 and r = 30 are sufficient and the
approximating Legendrian curve is given by
3
sin(200t)
20
120
455
cos(5t) +
cos(5t) cos(200t)
b(t) =
451
451
459
1377
180
c(t) = sin(5t) +
sin(195t) +
sin(205t) +
sin(395t)+
5863
18491
35629
20
+
sin(405t).
4059
a(t) = t +
and produces the zig-zags and the small loops in its front and Lagrangian
projections (see Figure 3.1).
– 76 –
Chapter 4
h-Principle for Curves with
Prescribed Curvature
4.1
Introduction
In this chapter we will show that if n > 3, any given immersed C 2 -curve γ
in Rn with curvature kγ can be C 1 -approximated by an immersed C 2 -curve
with prescribed curvature k, provided k > kγ .
Let I = [0, b] ⊂ R be a compact interval. A curve γ ∈ C ` (I, Rn ), n > 3 is
called closed if it agrees in 0 and b to ` orders. The curvature of an immersed
curve γ ∈ C 2 (I, Rn ) is defined as kγ := |Ṫ|/|γ̇|, where T := γ̇/|γ̇|. For n = 3,
this formula reduces to the usual formula kγ = |γ̇ × γ̈|/|γ̇|3 . A homotopy
γs ∈ C ` (I, Rn ), s ∈ [0, 1], is called regular or an isotopy if γs is an immersion
or an embedding for all s. The main result of this chapter is the following
theorem:
Theorem 4.1 (h-Principle for Curves with Prescribed Curvature)
Let γ0 ∈ C 2 (I, Rn ), n > 3 be an immersed (embedded) curve. Then for every
k ∈ C ∞ (I) satisfying k > kγ0 and every ε > 0, there exists a regular homotopy
(an isotopy) γs ∈ C 2 (I, Rn ), s ∈ [0, 1] such that kγ0 − γs kC 1 (I) < ε for all s
and such that kγ1 = k.
We will start by observing that the sufficient condition k > kγ0 is almost
optimal in the sense that k > kγ0 is necessary for the statement to hold:
– 77 –
4.2. Step
Proposition 4.2
Let k ∈ C 0 (I) be non-negative, let γi ∈ C 2 (I, Rn ) be a sequence of immersions such that kγi = k and let γ ∈ C 2 (I, Rn ) be an immersion. If
lim kγ − γi kC 1 (I) = 0, then kγ 6 k.
i→∞
Proof. Let t0 ∈ I and let U be a relatively open interval containing t0 . From
the lower semicontinuity of the integral curvature functional (see [AR89,
Theorem 5.1.1, p. 120-121]) with respect to uniform convergence we obtain
Z
Z
Z
kγ (t) dt 6 lim inf
kγi (t) dt =
k(t) dt.
U
i→∞
U
U
By arbitrariness of U and continuity of kγ and k, it follows that kγ (t0 ) 6 k(t0 ).
The variant of Theorem 4.1 for closed curves (see Corollary 4.7) generalizes
a result due to McAtee [MG07], who proved that there exists a C 2 knot
of constant curvature in each isotopy class building upon the work of Koch
and Engelhardt [KE98], who gave the first explicit construction a non-planar
closed C 2 -curve of constant curvature. Ghomi proved in [Gho07] that curves
with constant curvature satisfy a relative C 1 -dense h-principle and obtained
in this way the existence of smooth knots with constant curvature in each
isotopy class that are C 1 -close to a given initial knot. Observe that we do not
achieve smooth curves with prescribed curvature, however, in our theorem,
the curvature need not be constant. Both, Ghomi’s result and Theorem 4.1
are – in the language of Gromov [Gro86] and Eliashberg, Mishachev [EM02]
– manifestations of a C 1 -dense h-principle.
We will prove Theorem 4.1 directly using a variant of convex integration à la
Nash and Kuiper. The strategy of the proof consists in reducing the curvature
defect k − kγ > 0 successively in steps while keeping careful control on the
C 2 -norm of the resulting maps during the process. Here we achieve a desired
C 2 -change by a C 1 -perturbation. In this respect, that strategy imitates the
strategy of Chapter 1, where we achieved a desired change of order one by a
C 0 -perturbation.
4.2
Step
As the main building block for the step, we make use of a function whose construction recycles the corrugation function previously constructed in Lemma
– 78 –
4.2. Step
1.12 from Chapter 1:
Lemma 4.3 (Building Block)
There exists C ∈ C ∞ (R × S 1 , R2 ), (s, t) 7→ C(s, t) satisfying
(1 + ∂tt C1 (s, t))2 + (∂tt C2 (s, t))2 = 1 + s2
√
|∂tt C(s, t)| 6 2|s|.
(4.1)
(4.2)
Proof. The function
C(s, t) :=
Z
t
0
t
Γ(s, u) du −
2π
Z
2π
Γ(s, u) du,
0
where Γ is the function from Lemma 1.12 has all the desired properties.
Proposition 4.4 (i-th Step)
Let γi−1 ∈ C ∞ (I, Rn ), n > 3 be an immersed (embedded) curve and let
k ∈ C ∞ (I) such that 0 < kγi−1 < k. Then there exists an immersed (embedded) curve γi ∈ C ∞ (I, Rn ) satisfying
kγi − γi−1 kC 1 (I) 6 ε,
√
1/2
kγ̈i − γ̈i−1 kC 0 (I) 6 2kγ̇i−1 k2C 0 (I) kk 2 − kγ2i−1 kC 0 (I) ,
2
kk −
kγ2i kC 0 (I)
(4.3)
(4.4)
6 ε,
(4.5)
0 < kγi < k.
(4.6)
Proof. First consider the map ϕ : I → J given by
Z t
t 7→
|γ̇(u)| du
0
and consider γ̄i−1 := γi−1 ◦ ϕ−1 ∈ C ∞ (J, Rn ) and k̄ := k ◦ ϕ−1 ∈ C ∞ (J),
where J = [0, ϕ(b)]. Let λ > 0, 0 < δ < 1, ξ = γ̄¨i−1 and ζ be a vector of
length |ξ| in (span{γ̄˙ i−1 , ξ})⊥ . Observe that ζ can be chosen to be closed if
γi−1 is closed. Indeed, the restriction of the (trivial) tangent bundle of Rn
onto the closed curve is trivial and γ̇ and ξ are two orthogonal sections in
this bundle. Hence there exist n − 2 additional linearly independent sections
that are orthogonal to span{γ̄˙ i−1 , ξ}. If n = 3, one can take ζ = γ̄˙ i−1 × ξ.
For
s
2
k̄ (t)
a(t) := (1 − δ)
−1 ,
|ξ(t)|2
– 79 –
4.2. Step
let
1
γ̄i (t) = γ̄i−1 (t) + 2 C1 (a(t), λt)ξ(t) + C2 (a(t), λt)ζ(t) .
λ
The first and the second derivative of γ̄i are given by
1
˙γ̄i = γ̄˙ i−1 +
∂t C1 (a, λ·)ξ + ∂t C2 (a, λ·)ζ + O(λ−2 )
λ
γ̄¨i = γ̄¨i−1 + ∂tt C1 (a, λ·)ξ + ∂tt C2 (a, λ·)ζ + O(λ−1 ).
(4.7)
This implies kγ̄i − γ̄i−1 kC 0 (J) = O(λ−2 ) and kγ̄i − γ̄i−1 kC 1 (J) = O(λ−1 ). Setting
γi = γ̄i ◦ ϕ we obtain (4.3) from choosing λ 1 and from
kγi − γi−1 kC 1 (I) 6 O(λ−1 ) + kϕ̇kC 0 (I) O(λ−1 ).
Since |ξ| = kγ̄i−1 , using (4.2) and a suitable choice of λ yields
|γ̄¨i − γ̄¨i−1 |2 = |ξ|2 |∂tt C(a, λ·)|2 + O(λ−1 )
6 2|ξ|2 a2 + O(λ−1 )
6 2(1 − δ)(k̄ 2 − |ξ|2 ) + O(λ−1 )
6 (2 − δ)(k̄ 2 − |ξ|2 )
and – since curvature is parameter-invariant – this implies the estimate (4.4)
√
1/2
kγ̈i − γ̈i−1 kC 0 (I) 6 2 − δkϕ̇k2C 0 (I) kk 2 − kγ2i−1 kC 0 (I) + kϕ̈kC 0 (I) O(λ−1 )
√
1/2
6 2kγ̇i−1 k2C 0 (I) kk 2 − kγ2i−1 kC 0 (I) .
Using (4.1) we obtain (4.5) and (4.6) from
k̄ 2 − kγ̄2i = k̄ 2 − |γ̄¨i − γ̄˙ i hγ̄˙ i , γ̄¨i i|2 + O(λ−1 )
= k̄ 2 − (1 + ∂tt C1 (a, λ·))2 + (∂tt C2 (a, λ·))2 |ξ|2 + O(λ−1 )
= k̄ 2 − (1 + a2 )|ξ|2 + O(λ−1 )
= δ(k̄ 2 − kγ̄2i−1 ) + O(λ−1 )
provided λ 1 and δ 1. We can ensure that γi is an embedding, since γi
and γi−1 can be made arbitrarily C 1 -close and the embeddings form an open
set in C 1 (I, Rn ). If γi−1 is closed, γi is closed if λ ∈ (2π/ϕ(b))N.
Remark 4.5
The foregoing proof might be simplified if n > 4. In this case, there exist
two mutually orthogonal vector fields ζ1 and ζ2 of length |ξ| such that
ζ1 , ζ2 ∈ (span{γ̄˙ i−1 , ξ})⊥ .
– 80 –
4.3. Iteration and Application to Knot Theory
Figure 4.1: “Top view” of γ̄i if γ̄i−1 is a unit circle, a(t) =
λ = 1, 2, . . . , 6.
√
8 and
If γi−1 is closed, ζ1 and ζ2 can also be chosen to be closed. Then the ansatz
(4.7) involving the rather complicated function C can be replaced by a modified Nash Twist
a(t)
γ̄i (t) = γ̄i−1 (t) + 2 cos(λt)ζ1 (t) + sin(λt)ζ2 (t) .
λ
4.3
Iteration and Application to Knot Theory
Using a C 2 -perturbation we may assume that the curvature kγ0 of the initial
curve γ0 in Theorem 4.1 is never zero. This is the content of the following
Lemma which is due to Ghomi [Gho07, Lemma 5.3], but we will give the
proof in our slightly different setting.
Lemma 4.6 (Ghomi)
Let γ ∈ C 2 (I, Rn ), n > 3 be an immersion and let k ∈ C 0 (I) satisfy k > kγ .
Then for every ε > 0 there exists an immersion γ̂ ∈ C 2 (I, Rn ) satisfying
k > kγ̂ > 0 and kγ − γ̂kC 2 (I) < ε.
Proof. Observe that the curvature of γ vanishes if and only if γ̇ and γ̈ are
linearly dependent. Consider therefore the subset
Λ ⊂ J 2 (I, Rn ) ∼
= I × Rn × Rn × Rn
of the 2-jet bundle J 2 (I, Rn ) containing all the elements having linearly dependent last two components. Note that dim Λ = 2n + 2. By Thom’s
transversality theorem, there exists a C 2 -dense set of curves γ̂ ∈ C 2 (I, Rn )
such that j 2 γ̂ is transversal to Λ. Since n > 3, we obtain
dim I + dim Λ = 2n + 3 < 3n + 1 = dim(J 2 (I, Rn ))
and we conclude that j 2 γ̂ and Λ do not intersect. By density we can choose
γ̂ in such a way that kγ − γ̂kC 2 (I) < ε. This inequality implies that we
– 81 –
4.3. Iteration and Application to Knot Theory
can also obtain kkγ − kγ̂ kC 0 (I) < ε and hence we can choose γ̂ satisfying
k > kγ̂ > 0.
Proof of Theorem 4.1. In view of Lemma 4.6 we can assume that γ0 has
non-vanishing curvature. Using standard regularization we can assume in
addition that γ0 ∈ C ∞ (I, Rn ), since the second derivative of the mollification
of γ0 can be made arbitrarily close to γ̈0 . After these modifications, γ0 is still
an immersion (embedding) since immersions and embeddings form an open
set in C 1 (I, Rn ) and we can use Proposition 4.4 iteratively starting with γ0
and a sequence εi → 0 such that
∞
X
εi = ε and
i=1
∞
X
√
i=1
εi < ∞.
(4.8)
This implies the uniform estimate
kγ̇i kC 0 (I) 6 kγ̇0 kC 0 (I) +
i
X
`=1
kγ̇` − γ̇`−1 kC 0 (I)
6 kγ̇0 kC 0 (I) + ε
and hence we find for j > i using (4.3), (4.4) and (4.8):
kγj − γi kC 2 (I) 6
6
j
X
`=i+1
∞
X
`=i+1
kγ` − γ`−1 kC 2 (I)
ε` +
√
2
2(kγ̇0 kC 0 (I) + ε)
∞
X
√
`=i+1
j,i→∞
ε`−1 −→ 0.
The sequence (γi )i∈N is thus Cauchy in C 2 (I, Rn ) and the estimate (4.5)
implies that the limit map γ
e has curvature k. Using (4.3) and (4.8) we find
that the homotopy γs = (1 − s)γ + se
γ satisfies
kγs − γkC 1 (I) 6 ske
γ − γkC 1 (I) 6 ε.
This concludes the proof since immersions and embeddings form an open set
in C 1 (I, Rn ).
Since Proposition 4.4 may produce closed curves out of closed curves, we
obtain as an application of Theorem 4.1:
Corollary 4.7
Let γ0 ∈ C 2 (S 1 , R3 ) be a knot. Then for any positive k ∈ C ∞ (S 1 ), there
exists a knot γ1 ∈ C 2 (S 1 , R3 ) in the same isotopy class with curvature k.
– 82 –
4.3. Iteration and Application to Knot Theory
Proof. Note that γ0 and c · γ0 for c > 0 are isotopic and it holds that
kc·γ0 = kγ0 /c. Hence we can choose c in such a way that
kkc·γ0 kC 0 (S 1 ) < min1 k(t)
t∈S
and apply Theorem 4.1.
Remark 4.8
We recover the main result of [MG07] by choosing k in Corollary 4.7 to be
constant.
– 83 –
Bibliography
[Ang03]
G. Angelsberg, The Cauchy Problem for Isometries, Master’s
thesis, University of Fribourg, 2003.
[AR89]
A. D. Alexandrov and Yu. G. Reshetnyak, General theory of
irregular curves, Mathematics and its Applications (Soviet Series), vol. 29, Kluwer Academic Publishers Group, Dordrecht,
1989, Translated from the Russian by L. Ya. Yuzina. MR
1117220 (92h:53003)
[BJLT12]
Vincent Borrelli, Saı̈d Jabrane, Francis Lazarus, and Boris
Thibert, Flat Tori in Three-dimensional Space and Convex Integration, Proc. Natl. Acad. Sci. USA 109 (2012), no. 19, 7218–
7223. MR 2935570
[BJLT13]
, Isometric Embeddings of the Square Flat Torus in
Ambient Space, Ensaios Matemáticos [Mathematical Surveys],
vol. 24, Sociedade Brasileira de Matemática, Rio de Janeiro,
2013. MR 3154503
[Bor58a]
Ju. F. Borisov, The Parallel Translation on a Smooth Surface.
I, Vestnik Leningrad. Univ. 13 (1958), no. 7, 160–171. MR
0104277 (21 #3032)
[Bor58b]
, The Parallel Translation on a Smooth Surface. II, Vestnik Leningrad. Univ. 13 (1958), no. 19, 45–54. MR 0104278 (21
#3033)
[Bor59a]
, On the Connection Bewteen the Spatial Form
of Smooth Surfaces and their Intrinsic Geometry, Vestnik
Leningrad. Univ. 14 (1959), no. 13, 20–26. MR 0116295 (22
#7090)
– 84 –
Bibliography
[Bor59b]
, The Parallel Translation on a Smooth Surface. III,
Vestnik Leningrad. Univ. 14 (1959), no. 1, 34–50. MR 0104279
(21 #3034)
[Bor60]
, On the Question of Parallel Displacement on a Smooth
Surface and the Connection of Space Forms of Smooth Surfaces
with their Intrinsic Geometries., Vestnik Leningrad. Univ. 15
(1960), no. 19, 127–129. MR 0131225 (24 #A1078)
[Bur31]
C. Burstin, Ein Beitrag zum Problem der Einbettung der Riemann’schen Räume in Euklidischen Räumen, Mathematicheskii Sbornik 38 (1931), 74–85.
[Car27]
É. Cartan, Sur la possibilité de plonger un espace riemannien
donné dans un espace euclidien, Annales de la société polonaise
de mathématique (1927), 1–7.
[CD99]
Doina Cioranescu and Patrizia Donato, An Introduction to Homogenization, Oxford Lecture Series in Mathematics and its
Applications, vol. 17, The Clarendon Press, Oxford University
Press, New York, 1999. MR 1765047 (2001j:35019)
[CDS12]
Sergio Conti, Camillo De Lellis, and László Székelyhidi, hprinciple and Rigidity for C 1,α Isometric Embeddings., Nonlinear partial differential equations. The Abel symposium 2010.
Proceedings of the Abel symposium, Oslo, Norway, September
28–October 2, 2010, Berlin: Springer, 2012, pp. 83–116 (English).
[CDVV95]
Bang-Yen Chen, Franki Dillen, Leopold Verstraelen, and Luc
Vrancken, Compact hypersurfaces determined by a spectral variational principle, Kyushu J. Math. 49 (1995), no. 1, 103–121.
MR 1339701 (96h:53067)
[Cia05]
Philippe G. Ciarlet, An Introduction to Differential Geometry with Applications to Elasticity, Springer, Dordrecht,
2005, Reprinted from J. Elasticity 78/79 (2005), no. 1-3
[MR2196098]. MR 2312300 (2007m:74004)
[DjS12]
Camillo De Lellis and László jun. Székelyhidi, The h-principle
and the Equations of Fluid Dynamics., Bull. Am. Math. Soc.,
New Ser. 49 (2012), no. 3, 347–375 (English).
– 85 –
Bibliography
[DMP08]
B. Dacorogna, P. Marcellini, and E. Paolini, Lipschitzcontinuous Local Isometric Immersions: Rigid Maps and
Origami, J. Math. Pures Appl. (9) 90 (2008), no. 1, 66–81.
MR 2435215 (2009c:26026)
[DMP10]
Bernard Dacorogna, Paolo Marcellini, and Emanuele Paolini,
Origami and Partial Differential Equations, Notices Amer.
Math. Soc. 57 (2010), no. 5, 598–606. MR 2664038
[EM02]
Y. Eliashberg and N. Mishachev, Introduction to the hprinciple, Graduate Studies in Mathematics, vol. 48, American Mathematical Society, Providence, RI, 2002. MR 1909245
(2003g:53164)
[Etn05]
John B. Etnyre, Legendrian and Transversal Knots, Handbook
of knot theory, Elsevier B. V., Amsterdam, 2005, pp. 105–185.
MR 2179261 (2006j:57050)
[Fré12]
Emmanuel Frénod, Two-scale Convergence, CEMRACS’11:
Multiscale coupling of complex models in scientific computing,
ESAIM Proc., vol. 38, EDP Sci., Les Ulis, 2012, pp. 1–35. MR
3006533
[GA13]
et al. Giovanni Alberti (ed.), HCDTE Lecture Notes. Part ii.
Nonlinear Hyperbolic PDEs, Dispersive and Transport equations, American Institute of Mathematical Sciences, 2013.
[Gei06]
Hansjörg Geiges, Contact Geometry, Handbook of differential
geometry. Vol. II, Elsevier/North-Holland, Amsterdam, 2006,
pp. 315–382. MR 2194671 (2007c:53123)
[Gei08]
, An introduction to Contact Topology, Cambridge Studies in Advanced Mathematics, vol. 109, Cambridge University
Press, Cambridge, 2008. MR 2397738 (2008m:57064)
[Gho07]
Mohammad Ghomi, h-principles for Curves and Knots of
Constant Curvature, Geom. Dedicata 127 (2007), 19–35. MR
2338512 (2008j:53074)
[Gre69]
R.E. Greene, Isometric Embedings., Bull. Am. Math. Soc. 75
(1969), 1308–1310 (English).
– 86 –
Bibliography
[Gro73]
M. L. Gromov, Convex integration of differential relations. I,
Izv. Akad. Nauk SSSR Ser. Mat. 37 (1973), 329–343. MR
0413206 (54 #1323)
[Gro86]
Mikhael Gromov, Partial Differential Relations, Ergebnisse der
Mathematik und ihrer Grenzgebiete (3), vol. 9, Springer-Verlag,
Berlin, 1986. MR 864505 (90a:58201)
[Gro00]
Misha Gromov, Spaces and questions, Geom. Funct. Anal.
(2000), no. Special Volume, Part I, 118–161, GAFA 2000 (Tel
Aviv, 1999). MR 1826251 (2002e:53056)
[Haa47]
J. Haantjes, Distance Geometry. Curvature in Abstract Metric Spaces, Nederl. Akad. Wetensch., Proc. 50 (1947), 496–
508=Indagationes Math. 9, 302–314 (1947). MR 0021325
(9,53g)
[Hon99]
Jiaxing Hong, Darboux equations and isometric embedding of
Riemannian manifolds with nonnegative curvature in R3 , Chinese Ann. Math. Ser. B 20 (1999), no. 2, 123–136, A Chinese
summary appears in Chinese Ann. Math. Ser. A 20 (1999), no.
2, 267. MR 1699138 (2000d:53060)
[Ina15]
Dominik Inauen, A Nash-Kuiper Theorem for C 1,1/5−ε Embeddings of Surfaces in 3 Dimensions, Master’s thesis, University
of Zurich, 2015.
[Jac74]
H. Jacobowitz, Extending Isometric Embeddings, J. Differential
Geometry 9 (1974), 291–307. MR 0377773 (51 #13942)
[Jan26]
M. Janet, Sur la possibilité de plonger un espace riemannien
donné dans un espace euclidien, Annales de la société polonaise
de mathématique (1926), 38–43.
[Käl78]
Anders Källén, Isometric Embedding of a Smooth Compact
Manifold with a Metric of Low Regularity, Ark. Mat. 16 (1978),
no. 1, 29–50. MR 499136 (80c:53064)
[KE98]
Richard Koch and Christoph Engelhardt, Closed Space Curves
of Constant Curvature Consisting of Arcs of Circular Helices, J.
Geom. Graph. 2 (1998), no. 1, 17–31. MR 1647587 (99m:53006)
– 87 –
Bibliography
[KN96]
Shoshichi Kobayashi and Katsumi Nomizu, Foundations of Differential Geometry. Vol. I, Wiley Classics Library, John Wiley
& Sons Inc., New York, 1996, Reprint of the 1963 original, A
Wiley-Interscience Publication. MR 1393940 (97c:53001a)
[KSS14]
B. Kirchheim, E. Spadaro, and L. Szekelyhidi, Jr, Equidimensional Isometric Maps, ArXiv e-prints (2014).
[Kui55]
Nicolaas H. Kuiper, On C 1 -Isometric Imbeddings. I, II, Nederl.
Akad. Wetensch. Proc. Ser. A. 58 = Indag. Math. 17 (1955),
545–556, 683–689. MR 0075640 (17,782c)
[LP15]
M. Lewicka and M. R. Pakzad, Convex Integration for the
Monge-Ampère Equation in Two Dimensions, ArXiv e-prints
(2015).
[LZ95]
M. Liao and W. A. Zheng, Radial Part of Brownian Motion
on a Riemannian Manifold, Ann. Probab. 23 (1995), no. 1,
173–177. MR 1330766 (96j:58182)
[MG07]
Jenelle Marie McAtee Ganatra, Knots of Constant Curvature,
J. Knot Theory Ramifications 16 (2007), no. 4, 461–470. MR
2327737 (2008e:57005)
[MMASC14] M. G. Monera, A. Montesinos-Amilibia, and E. SanabriaCodesal, The Taylor Expansion of the Exponential Map and
Geometric Applications, Rev. R. Acad. Cienc. Exactas Fı́s.
Nat. Ser. A Math. RACSAM 108 (2014), no. 2, 881–906. MR
3249982
[MŠ98]
Stefan Müller and Vladimir Šverák, Unexpected Solutions of
First and Second Order Partial Differential Equations, Proceedings of the International Congress of Mathematicians, Vol.
II (Berlin, 1998), no. Extra Vol. II, 1998, pp. 691–702 (electronic). MR 1648117 (99j:35040)
[MŠ03]
S. Müller and V. Šverák, Convex Integration for Lipschitz Mappings and Counterexamples to Regularity, Ann. of Math. (2)
157 (2003), no. 3, 715–742. MR 1983780 (2005i:35028)
[Nas54]
John Nash, C 1 Isometric Imbeddings, Ann. of Math. (2) 60
(1954), 383–396. MR 0065993 (16,515e)
– 88 –
Bibliography
[Nas56]
, The Imbedding Problem for Riemannian Manifolds,
Ann. of Math. (2) 63 (1956), 20–63. MR 0075639 (17,782b)
[Nas66]
J. Nash, Analyticity of the solutions of implicit function problems with analytic data, Ann. of Math. (2) 84 (1966), 345–355.
MR 0205266 (34 #5099)
[Pet10]
A. Petrunin, Intrinsic Isometries in Euclidean Space, Algebra
i Analiz 22 (2010), no. 5, 140–153. MR 2828830 (2012e:54037)
[Pog71]
A. V. Pogorelov, An example of a two-dimensional Riemannian
metric that does not admit a local realization in E3 , Dokl. Akad.
Nauk SSSR 198 (1971), 42–43. MR 0286034 (44 #3250)
[Poz73]
E G Poznyak, Isometric immersions of two-dimensional riemannian metrics in euclidean space, Russian Mathematical
Surveys 28 (1973), no. 4, 47.
[Ray02]
Jean Saint Raymond, Local Inversion for Differentiable Functions and the Darboux property, Mathematika 49 (2002), no. 12, 141–158 (2004). MR 2059050 (2005c:26015)
[Rie54]
Bernhard Riemann, Über die Hypothesen, welche der Geometrie zugrunde liegen, Gesammelte mathematische Werke, wissenschaftlicher Nachlass und Nachträge (Raghavan Narasimhan
and Heinrich Weber, eds.), Springer, Berlin, 1854, pp. 304–319.
[RS10]
Martin Raussen and Christian Skau, Interview with Mikhail
Gromov, Notices Amer. Math. Soc. 57 (2010), no. 3, 391–403.
MR 2643447
[Sai79]
R. Saigal, On Piecewise Linear Approximations to Smooth
Mappings, Math. Oper. Res. 4 (1979), no. 2, 153–161. MR
543927 (80j:55002)
[Sch73]
L. Schlaefli, Nota alla memoria del signor beltrami “sugli spazı́i
di curvatura costante”, Annali di Mathematica Pura ed Applicata 2 (1871-1873), 176–193.
[Spr10]
David Spring, Convex Integration Theory, Modern Birkhäuser
Classics, Birkhäuser/Springer Basel AG, Basel, 2010, Solutions
to the h-principle in Geometry and Topology, Reprint of the
1998 edition [MR1488424]. MR 3024860
– 89 –
Bibliography
[Tay06]
Michael Taylor, Existence and Regularity of Isometries, Trans.
Amer. Math. Soc. 358 (2006), no. 6, 2415–2423 (electronic).
MR 2204038 (2007e:53035)
[Top07]
Flemming Topsøe, Some Bounds for the Logarithmic Function,
137–151. MR 2349596 (2009a:41019)
[Was11]
Micha Wasem, Origami & Paperfolding, Master’s thesis, University of Fribourg, 2011.
[Whi36]
Hassler Whitney, Differentiable manifolds, Ann. of Math. (2)
37 (1936), no. 3, 645–680. MR 1503303
[Whi44]
, The self-intersections of a smooth n-manifold in 2nspace, Ann. of Math. (2) 45 (1944), 220–246. MR 0010274
(5,273g)
[Yau93]
Shing-Tung Yau, Open Problems in Geometry, Differential geometry: partial differential equations on manifolds (Los Angeles, CA, 1990), Proc. Sympos. Pure Math., vol. 54, Amer. Math.
Soc., Providence, RI, 1993, pp. 1–28. MR 1216573 (94k:53001)
– 90 –
Dank
Ich möchte meinem Doktorvater Norbert Hungerbühler meinen herzlichen
Dank aussprechen. Vor etwa 10 Jahren habe ich unter seiner Obhut meine Reise in die endlosen Weiten der Mathematik angetreten: Ich besuchte
bei ihm die Kurse Analysis I-IV, verfasste unter seiner Leitung zwei Seminararbeiten und später reihte sich meine Masterarbeit und schliesslich die
vorliegende Arbeit hinzu. Es ist ein grosses Privileg, die Möglichkeit zu erhalten, Mathematik auf eigene Faust zu entdecken und schliesslich etwas ganz
Kleines selbst beizutragen. Zum Verfassen einer solchen Arbeit gehört aber
ein ganzer Lebensabschnitt – umso mehr stellt das Betreuen eines Doktoranden eine anspruchsvolle und facettenreiche Aufgabe dar, die Norbert Hungerbühler meisterhaft bewältigt hat: Trotz seines vollen Terminkalenders hat
er sich immer Zeit für mich genommen und hatte stets ein offenes Ohr für all
meine Fragen und Anliegen. Seine scharfsinnigen Ideen und die sorgfältigen
Erklärungen haben mir in Momenten der Orientierungslosigkeit immer wieder einen Weg gewiesen. So wurden die Besuche bei ihm im Büro zu kleinen
Oasen auf dem Weg eines Suchenden und ich habe Norbert über all die
Jahre als hervorragenden Mathematiker kennen- und schätzen gelernt. Was
mir aber vor Allem in Erinnerung bleiben wird, ist sein vorbildlicher Umgang
mit Menschen, seine grosse Geduld, sein Verständnis für allerlei kleinere oder
grössere Sorgen und sein respektvoller Umgang mit seinen Zöglingen. Durch
seine wohlwollende Art und seine Familien- und Menschenfreundlichkeit ist
Norbert auch für Themen ausserhalb der Mathematik ein Ansprechpartner
geworden. Für all die Zeit und die Energie, die Norbert in mich investiert
hat, möchte ich ihm von ganzem Herzen danken.
Ein herzlicher Dank gebührt auch Camillo de Lellis, der sich freundlicherweise bereiterklärt hat, das Koreferat zu übernehmen und diese Dissertation
sorgfältig durchzulesen. Seine herausragenden Arbeiten waren eine Quelle der
Inspiration und stellten gleichzeitig auch den Ausgangspunkt der vorliegen– 91 –
Dank
den Arbeit dar.
Ich möchte mich bei Marc Burger, Ana Cannas da Silva, Michael Struwe und
Peter Thurnheer für die angenehme Zusammenarbeit in der Lehre bedanken
und bei Hanspeter Scherbel und Patricia Malzacher-Lienhard für die stets
freundliche und positive Atmosphäre bei den administrativen Tätigkeiten.
Weiter danke ich Ruth Kellerhals für Ihre Einladung nach Freiburg und für
die Informationen zu Ludwig Schläfli. Merci à Vincent Borrelli (et tout le
groupe Hévéa) pour ton superbe tore plat et merci de m’avoir invité à Lyon
pour un exposé.
Ein weiterer Dank gebührt allen Mitgliedern der Gruppe 6 für die Unterstützung, die ich als Gruppenorganisator erhalten habe. Speziell zu erwähnen
sind hier Jonas und Igor, die Jungs aus meiner Generation; die Leute aus
F27: Berit, Felix, Kathrin und speziell meine Bürokollegen Martin, Luis und
Christian und schliesslich meine akademischen Geschwister Kathi, Andreas
und Thomas. Andreas’ buchstäblich endloser Enthusiasmus für die Mathematik hat mir geholfen, immer wieder Begeisterung zu finden. Dafür, für
seine Freundschaft und für alle Gespräche über andere wichtige Dinge wie
z.B. Eishockey danke ich ihm herzlich. Und Thomas danke ich dafür, ein
wichtiger Wegbegleiter geworden zu sein – ohne ihn hätte ich mich gar nicht
erst an das Unterfangen Dissertation herangewagt und ich freue mich, dass
er an einer wichtigen Stelle in mein Leben getreten und zu einem Freund geworden ist – geile Schnitzer! Weiter danke ich Theo für seine Unterstützung,
seine Hilfe beim Vorbereiten von Vorträgen und die Plaudereien im Büro
oder im bQm; Yann für die gemeinsame und leider viel zu kurze Zeit an der
ETH und für all den Blödsinn, den wir gelabert haben; Alexander für die
Diskussionen im Türrahmen und den Fernwehzürchern Alain, Alan, Bidu,
Bobby und Roberto aka Matthias für die schöne Zeit. Ich möchte mich auch
bei Nicolas für seine Freundschaft bedanken und für die zahllosen Diskussionen über Sinn und Unsinn, über Freude und Frust in der Mathematik und
über die Wege und Umwege des Seins.
Ein grosser Dank gebührt meinem Freund und Mitbewohner der Lost in
music WG Gianluca. Ich danke ihm für seinen Mut für die Revolutionen
im Leben, für seinen Humor, seine philosophische Tiefe und sein vielseitiges
kreatives Talent, welches ein Quell der Ablenkung und Inspiration war und
ist. Unsere zahlreichen Projekte haben den Alltag mit vielen Freuden erhellt
– 92 –
Dank
– an dieser Stelle sei beispielsweise das FIFA99-Buch erwähnt – welches aber
nur die Spitze des Eisbergs darstellt – merci Nöttu! Weiter danke ich meinen langjährigen Freunden und Wegbegleitern vom “Starm”: Julian, Lukas,
Philipp und Silvan: Sie alle spielen schon seit dem Kindesalter Hauptrollen
in der Geschichte meines Lebens und ich bin glücklich und dankbar, sie an
meiner Seite zu wissen. Ihre Vielseitigkeit, ihre Tiefe und ihr Humor haben
mich in allen Phasen meines Lebens erfreut, bereichert und geprägt. Trotz
der geografischen Distanz hatten sie für mich immer offene Türen, Ohren
und Herzen. Ein spezieller Dank geht an Borski aka Woronin: Ich danke ihm
für sein Interesse an meiner Arbeit, für all die Einträge im Logbuch unseres
Lebens und für die legendär gewordenen Olten-Treffen, in denen wir immer
wieder mit den Wogen des Lebens gekämpft und sie schliesslich geglättet
haben. Dadadanke!
Ich danke den Jungs vom Tambourenverein Murten, speziell Bäschtu, Kusi,
Nicku, Slude, Stefu, Ramon und Vinci für die tolle Stimmung und dafür,
mich nach all der Zeit nicht vergessen zu haben. Ich freue mich, bald wieder
mit ihnen Musik zu machen. An dieser Stelle danke ich auch noch dem HC
Fribourg-Gottéron: Merci pour la magnifique sasion 2012/2013!
Zu guter Letzt möchte ich mich bei meiner Familie bedanken: Meinen Eltern
für die mir geschenkte Liebe, für ihre bedingungslose Unterstützung zu jedem
Zeitpunkt meines Lebens und für das Beispiel, das sie mir waren und sind.
Auch wenn mein Vater diese Zeilen nicht mehr lesen kann, danke ich ihm
für alles, was er mir auf meine Reise mitgegeben hat. Meinem Bruder Lukas
danke ich für seine Freundschaft und dafür, mir immer ein Vorbild gewesen
zu sein. Ich habe von kaum einem Menschen mehr gelernt als von ihm. Ich
danke auch meiner neuen kleinen Familie: Josi für ihre Liebe, ihre Wärme, für
die opferbereite Unterstützung während der letzten Jahre und dafür, nie den
Kopf hängen zu lassen. Meiner kleinen Lenia – die viele Stunden hat auf ihren
Daddy verzichten müssen – danke ich für ihr Wesen, ihre Unbekümmertheit,
ihre Liebe und ihr Lachen.
– 93 –

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