Algebraic and Geometric aspects of CMC surfaces

Algebraic and Geometric aspects of
CMC surfaces
Laurent Hauswirth
31 Mai 2014
Minimal surfaces in R3 de genre zero
Theorem (Colding, Collin, Meeks, Minicozzi, Perez, Ros,
Rosenberg)
A minimal surface properly embedded in R3 with genus
zero is a plane, a catenoid, a helicoid or a Riemann’s type
example.
Remark Minimal surfaces properly embedded of genus 0
are foliated by constant curvature curves.
Minimal surfaces in R3 de genre zero
picture : Matthias Weber
Compact minimal annuli in R3
• Conformal immersion parametrized by third coordinate
X : {(x, y) ∈ R2 ; C1 ≤ y ≤ C2 and 0 ≤ x ≤ τ } → R3
X(x, y) = (G(x, y), y) and X(x + τ, y) = X(x, y)
x → X(x, C1 ) and x → X(x, C2 ) are circles
• Metric ds2 = cosh2 ω|dz|2
• n3 =< N, e3 >= tanh ω
• Minimal surface equation : ∆ω = 0
Compact minimal annuli in R3-Shiffmann ’56
• Minimal surface equation : ∆ω = 0
• v = cosh2 ω∂x (k(y = t)) = ωxy − tanh ωωx ωy is a Jacobi field
• Jacobi operator
Lv = cosh
−2
2|∇ω|2
ω ∆v +
v =0
cosh2 ω
• v = 0 on the boundary and v = 0 describe at least four
nodal domain on the annulus by four vertex theorem
• Index L ≤ 1 imply v = 0 and the annulus is foliated by
circles.
Compact minimal annuli in R3
• v = ωxy − tanh ωωx ωy Jacobi field
• Minimal surface : X + tvN + O(t2 )
• Metric : ω(t) = ω + tu + O(t2 )
• KDV equation :
dω
dt
= u = ωzzz − 2ωz2
CMC sphere in R3
Hopf : An immersed CMC topological sphere is round
Alexandrov : An embedded compact CMC surface is a
sphere (hence round).
Open problem : In any tubular neighborhood of a unit
sphere, does there exists an immersed H = 1 surface ?
CMC Tori in R3
H = 1/2 constant mean curvature Torus conformally
immersed :
X : C\Γ → R3
Induced metric : ds2 = e2ω |dz|2
Holomorphic quadratic differential :
1
Q = hXzz , Ni(dz)2 = (dz)2
4
Gauss equation :
∆0 ω + sinh ω cosh ω = 0
CMC immersed Tori in R3
¨
picture : Felix Jakob Knoppel
Curve theory and vortex filament flow
Consider a closed curve γ0 : S1 → R3 parametrized by arc
length and integrate the flow given by its curvature along
the binormal :
dγ
= γ 0 ∧ γ 00 = kγ (t)B(t) and γ(0) = γ0
dt
Define the Complex curvature :
Z
ψ(t, s) = k(t, s) exp i τ (t, s)ds
Equation (NLS) for vortex filament flow :
1
idψ
+ ψss + |ψ|2 ψ = 0
dt
2
Vortex filament flow
¨
picture : Felix Jakob Knoppel
Vortex filament flow
¨
picture : Felix Jakob Knoppel
Soul conjecture of Pinkall
Let γ : S1 −→ R3 be an embedded curve and Tγ () an
-tubular neighborhood of the curve γ.
Does there exists for a constant H > 0 large enough a
constant mean curvature H torus immersed in Tγ ().
-The set of finite type closed curve solution of the vortex
filament flow are dense in the set of embedded curves of
R3
-When H → ∞ a CMC H sequence of tori converge to a
curve of finite type solution of the vortex filament flow.
CMC Annuli in R3
picture : Nick Schmitt
Minimal annuli properly embedded in S2 × R
There is a two-parameter family of annuli :
-Flat cylinder : a geodesic Γ product with R : Γ × R
-Helicoid : Foliated by horizontal geodesics turning with
constant speed around two vertical axis.
-Onduloid : Rotational and periodic examples
-Riemman’s type examples (two parameter family of
periodic examples).
Theorem (-, Kilian,Schmidt) A properly embedded
minimal annulus in S2 × R is foliated by horizontal
constant curvature curves, therefore it is one example
beside flat cylinders, helicoids , a rotational examples or
Riemman’s type annuli.
Minimal annuli Alexandrov Embedded in S3
and CMC tori
-CMC tori are infinitely covered by CMC annuli (Choose
one period of the torus which remains closed)
-There is exactly two periods of the torus such that the
covering annulus is Alexandrov embedded.
Theorem (2012, Brendle for H=0, 2012 Andrews,Li for
H 6= 0)( 2013, -,Kilian,Schmidt) A CMC embedded torus
embedded in S3 is a rotational torus.
CMC embedded Tori in S3
picture : Nick Schmitt
Local geometric property
Consider ω : C → R solution of ∆0 ω + sinh ω cosh ω = 0
1 There exists one-parameter family of CMC H=1/2
isometric immersion. For λ ∈ S1 :
Yλ : C → R3 with metric ds2 = e2ω |dz|2
2 Family of Gauss map associate to the immersion :
Gλ : C → S2 harmonic map
3 The holomorphic quadratic Hopf differential of G is
Qλ =< (Gλ )z , (Gλ )z >S2 (dz)2
Qλ = λ−1 Q1
4 The Gauss map induce a minimal surface in S2 × R :
Z
p
2
Xλ : C → S × R with Xλ (z) = (Gλ (z), Re −2i Qλ )
5 The immersion CMC H = 1/2 given by Yλ is locally
isometric to a minimal surface in Y˜λ : Ω → S3 by solving
Gauss-Codazzi equation.
6 Minimal surface equation = HARMONIC map
Y˜λ : Ω → S3
Gλ : C → S2
( ∆G + |∇G|2 G = 0)
Remark
-Closing condition and Embeddness change with the
ambient space
R3 , S2 × R, S3
-The holomorphic quadratic differential Q has no the
same meaning. It is concerning the third coordinate or the
second fundamental form.
6-Dictionnary of the local immersion :
a- ω = 0 : γ × R —Cylinder—Clifford
b-ω depends only in one variable x or y :
Onduloid—–Delaunay—–Rotational embedded CMC in S3
Helicoid—–Nodoid—CMC Nodoid in S3
c-ω(x, y) = f (x)g(y) where f , g are elliptic functions.
Riemman annuli –Abresch family’s–”Villarceau” Family
Global geometric property
Theorem (Meeks,Rosenberg) A properly embedded
minimal annulus in S2 × R have bounded curvature
|K| ≤
where F3 =
flux.
R
γ
C
F3
< η, e3 > ds is the third coordinate of the
Proof of the theorem in S2 × R or in S3
1 - Properly embedded minimal annuli in S2 × R imply
bounded curvature |K| ≤ C
- Infinite covering of compact tori in S3 has bounded
curvature. It is an annulus which is mean convex
Alexandrov embedded
Mean Convex Alexandrov Embedded : X : M → S3
extends as an immersion to a 3-connected manifold N
with boundary ∂N = M with
1)The mean curvature with inward normal is non negative
2) The manifold N is complete with the induce metric
Proof of the theorem in S2 × R or in S3
2 Bounded curvature in S2 × R or ”covering torus” in S3
imply that the annulus is parabolic and can be
parametrized by a solution of
∆0 ω + sinh ω cosh ω = 0 with Qλ = λ−1 (dz)2
With closing condition
Xλ : C → S2 × R and Xλ (z + τ ) = Xλ (z)
3 Bounded curvature imply that the annulus has Finite
type
4 The space moduli of embedded annulus in S2 × R or
Alexandrov embedded annulus in S3 of finite type is path
connected and we can deform a such annulus up to a flat
example where ω = 0.
5 Study the possible deformation of flat cylinder to prove
isolated property of the embedded family into the family of
finite type annuli.
Space of CMC annuli via Integrable system
• Conformal immersion Xλ : C → S3 (orS2 × R)
• Periodic X(z + τ ) = X(z)
• Metric ds2 = e2ω |dz|2 (ords2 = cosh2 ω|dz|2 )
• Hopf differential Q =
λ−1
dz2
4
• ∆0 ω + sinh ω cosh ω = 0
Space of CMC annuli via Integrable system
• Consider a solution of Lu = ∆0 ω + sinh ω cosh ω = 0
where
ω : C → R is the metric.
•Linearisation of the equation and ALGEBRAIC family of
solutions u0 , u1 , u2 ..... :
∆0 ui + (cosh 2ω)ui = 0
Space of CMC annuli via Integrable
system-The hierarchy
• Deformation of the metric : ω(t) = ω0 + tu + ◦(t2 )
• Linearized sinh-Gordon operator
Lu = ∆0 u + (cosh 2ω)u = 0
•Hierarchy : Consider un a solution of Lun = 0 and find φ
solution of the system :
(φn )z = 4ωz un
(φn )¯z = −4 sinh ω cosh ωun
Then
un+1 = (un )zz − ωz φn satisfy Lun+1 = 0
Space of CMC annuli via Integrable
system-The hierarchy
• Consider a solution of Lu = ∆0 ω + sinh ω cosh ω = 0
where
ω : C → R is the metric.
•Linearisation of the equation and ALGEBRAIC family of
solutions u0 , u1 , u2 ..... :
∆0 ui + (cosh ω)ui = 0
•Find solution of sinh-Gordon equation
ω(t) = ω + tui + o(t2 ) which satisfy
∂ω
= ui .
dt
u0 = 0
u1 = ω z
u2 = ωzzz − 2ωz3 .......
Space of CMC annuli via Integrable system
•Family of immersions induced by metric ω(t) :
X(t) : C → S3 (or)S2 × R with X(t) = X + tξ + o(t2 )
• Integration of u1 = ωz is a translation in the surface.
Space of CMC annuli via Integrable system
• Integration of a KDV equation in the metric.
∂ω
= u2 = ωzzz − 2ωz3
dt
In the geometry of the surface of S2 × R, the variational
field is
< ξ, N >= λ∂x (kg )
where kg (x) is the curvature of the horizontal curve.
• u2 = a1 u1 imply ”Horizontal curve is a circle”
Space of CMC annuli via Integrable system• Finite type=KerL has finite dimension
g
X
ai ui = 0
i=1
Condition : A torus is compact or |ω| ≤ C on R × S1
• Integrating u0 , ....ug (finite type) define commuting flows
on the sinh-Gordon equation and immersions
X : C → S3 (or)S2 × R
Integrable system construct a differentiable group action
which act on the set of flow :
t = (z, t2 , t3 , ...tg ) ∈ Cg → ω(z, t2 , t3 , ...tg ) and X(z, t2 , t3 , ...tg )
ω(t + t0 ) = ω(t0 + t) and X(t + t0 ) = X(t0 + t)
• Long time existence for the deformations ω(t) and X(t)
• Immersions are not isolated and we have family
I(a) = {Family of surfaces with
g
X
ai ui = 0}
i=1
The set I(a) is called ISOSPECTRAL SET and has a
commutative group structure.
Space moduli of annuli M
• Let be ω with associate algebraic condition a.
How to solve the period problem for the immersion
X(z + τ ) = X(z) ?
Proposition : The closing condition is an isospectral
property
Definition : The space moduli of annuli is given by :
M = ∪{I(a); ∃τ (a) ∈ C with X(z + τ ) = X(z)}
Property The isospectral set I(a) is compact and
diffeomorphic to a (S1 )g for generic algebraic data a
( (CP1 )n × (S1 )g−2n ).
• Find deformation of the algebraic condition a which
preserve the closing condition of the immersion X(z)
• Induce a deformation of a whole family of flow which are
contains in a differentiable compact manifolds I(a).
Remark
- Deformation isospectral= Integrate Uniformly bounded
variational field on the surface.
-Deformation of a=Integrate non bounded variational field
on the surface.
Proposition : Embeddness (or Alexandrov embeddness)
of annuli is an isospectral property.
Theorem : There is a path wich connect continuously I(a)
to the isospectral set of the flat solution ω = 0 preserving
closing condition and embeddness.
Definition The number g is the spectral genus
• g = 0 imply ω = 0
• g = 1 imply that translation in the surface depends only
in one parameter=rotational invariant.
• g = 2 contains family foliated by constant curvature
curves.
Spectral curve
• a(λ) ∈ C2g [λ] with 2g pairwise distinct roots which
satisfies the reality conditions
|a(0)| =
1
,
16
¯ −1 ) = a(λ) and a(λ) ∈ R− for all λ ∈ S1 .
λ2g a(λ
λg
Definition The spectral curve Σ of genus g associate to
the immersion X : C → S2 × R or S3 is defined by adding
(∞, 0) and (∞, ∞) as branch points in the
compactification of
Σ∗ = {(ν, λ) ∈ C2 ; ν 2 = λ−1 a(λ)}
Isospectral set
t
• Potential Pg = {ξλ ∈ sl2 (C); λg−1 ξ1/λ¯ = −ξλ }
−1
ξλ = λ
X
g
0 β−1
βn
n αn
+
λ
0 0
γn −αn
n=0
• Isospectral set : I(a) = {ξλ ∈ Pg ; det ξλ =
a(λ)
}
λ
• Exp[zξλ ] = Fλ (z)Bλ (z)
t
Fλ : C → ΛSU2 = {F1 λ¯ = Fλ−1 }
Bλ : C → Λ+ SL2 = {Bλ holomorphic on |λ| ≤ 1, B0 triangular}
Immersion
• Fλ : C → SU2 with Gλ (z) = Fλ (z)σ3 Fλ−1 (z) is harmonic
map
• CMC conformal immersion in S3
˜
(z) ∈ S3
X(z)
= Fλ1 (z)Fλ−1
2
2
with H = i λλ12 +λ
−λ1
Period problem
• The immersion is periodic iff Fλ (τ ) = ±Id.
• The eigenvalue of Fλ (τ ) is µ(λ) with
a µ : Σ \ {0, ∞} → C holomorphic and without zeroes
b µ has essential singularity.
√ −1
d ln µ − τ d λ extend holomorphically at λ = 0
√
d ln µ − τ¯d λ extend holomorphically at λ = ∞
c Under elliptic involution σ : (ν, λ) → (−ν, λ), we have
σ ∗ µ = µ−1 and µ = ±1 at zeroes of (λ − λ1 )(λ − λ2 )a(λ).
Double roots of a
Consider points where µ = ±1.
• Points λ = α0 on S1
• Points λ = α0 away from S1
Points λ = α0 on S1
a˜(λ) = (λ − α0 )2 a(λ)
µ
˜(λ) = µ(λ)
ξ˜λ = (λ − α0 )ξλ
˜ λ (z) = −Fλ (α0 z)
No change the frame F
Points λ = α0 away from S1
a˜(λ) = (λ − α0 )2 (1 − α
¯ 0 λ)2 a(λ)
µ
˜ = µ(λ)
Two different orbit
α β
˜
ξλ = (λ − α0 )(1 − α
¯ 0 λ)
γ −α
ξ˜λ =
(λ − α0 )(1 − α
¯ 0 λ)α
(λ − α0 )2 β
(1 − α
¯ 0 λ)2 γ
−(λ − α0 )(1 − α
¯ 0 λ)α
I(a) = I1 ∪ I2
Spectral genus one annulus and a Bubbleton

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