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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