Constant mean curvature surfaces in homogeneous manifolds Beno t - - PowerPoint PPT Presentation

Constant mean curvature surfaces in homogeneous manifolds

Benoˆ ıt Daniel August 29, 2012

Benoˆ ıt Daniel Constant mean curvature surfaces

Constant mean curvature surfaces

Constant mean curvature (CMC) surfaces appear in variational problems. In a 3-dimensional Riemannian manifold ˆ M, we consider a fixed curve γ bounding a fixed compact embedded surface S0. We fix a constant V and we consider all surfaces S with boundary γ such that S and S0 bound a region whose (algebraic) volume is V . Among all these surfaces, we try to minimize the area of S. Solutions to this problem are CMC surfaces (with boundary).

Benoˆ ıt Daniel Constant mean curvature surfaces

Properties

Let S be a CMC surface in ˆ M (with or without boundary). Then every “small domain” of S is a solution to a problem of minimisation

• f area with fixed boundary and volume constraint.

When there is no volume constraint, we obtain minimal surfaces, i.e., surfaces with vanishing mean curvature. From now on we will only consider complete surfaces without boundary.

Benoˆ ıt Daniel Constant mean curvature surfaces

The isoperimetric problem

Let V > 0 be a fixed constant. Find all regions Ω of ˆ M such that Volume(Ω) = V and such that the area of ∂Ω is minimal. Solutions (if they exist) are bounded by one or several CMC surfaces. In R3, solutions exist for all volumes and are geodesic balls.

Benoˆ ıt Daniel Constant mean curvature surfaces

Classical examples of minimal surfaces in R3

helicoid catenoid

(images : Matthias Weber)

Existence : Lagrange, Meusnier, Euler (18th century). Uniqueness : Collin (1997) for the catenoid, Meeks-Rosenberg (2005) for the helicoid.

Benoˆ ıt Daniel Constant mean curvature surfaces

Classical examples of minimal surfaces in R3

Riemann’s minimal surface

(image : Matthias Weber)

Simply periodic minimal surface foliated by circles and lines. Uniqueness : Meeks-Perez-Ros (2007).

Benoˆ ıt Daniel Constant mean curvature surfaces

Classical examples of CMC surfaces in R3

Round sphere : H = 1/radius. Right cylinder : H = 1/(2 · radius). Unduloid. Nodoid.

Benoˆ ıt Daniel Constant mean curvature surfaces

The Hopf theorem

Theorem (Hopf, 1951) Any immersed constant mean curvature (CMC) sphere in R3 is a round sphere. Idea of proof: for CMC surfaces in R3, there exists a holomorphic quadratic differential Q whose zeroes are the umbilical points of the surface: it is the Hopf differential,

• n a CMC sphere, we then have Q ≡ 0, which means that the

sphere is totally umbilical, hence round. This result extends to constant curvature manifolds: spheres S3(κ) and hyperbolic spaces H3(κ).

Benoˆ ıt Daniel Constant mean curvature surfaces

The Alexandrov theorem

Theorem (Alexandrov, 1956) Any compact embedded CMC surface in R3 is a round sphere. Idea of proof: we use the “Alexandrov reflection” (moving planes) technique to show that the surface has a symmetry plane in every

• direction. This uses the maximum principle (CMC surfaces can be

locally described as graphs of solutions to an elliptic PDE). This result extends to hyperbolic spaces H3(κ) and constant curvature hemispheres.

Benoˆ ıt Daniel Constant mean curvature surfaces

Wente tori

Hopf conjectured that round spheres are the only compact CMC surfaces in R3. This conjecture was disproved: Wente (1986) and then Abresch (1987) indeed constructed CMC tori in R3, and Kapouleas (1995) constructed CMC surfaces of any genus g 2 in R3. None of these surfaces is of course embedded, by Alexandrov’s theorem.

Benoˆ ıt Daniel Constant mean curvature surfaces

Embedded CMC tori in S3

Theorem (Brendle, 2012; Lawson’s conjecture) The Clifford torus is the only embedded minimal torus in S3. Theorem (Andrews-Li, 2012; Pinkall and Sterling’s conjecture) Any embedded CMC torus in S3 is rotational.

Benoˆ ıt Daniel Constant mean curvature surfaces

Homogeneous manifolds

A Riemannian manifold ˆ M is called homogeneous if, for every pair (x, y) of points of ˆ M, there exists an isometry ϕ of ˆ M such that y = ϕ(x). In dimension 2, a Riemannian homogeneous manifold has constant curvature, so there only exist three types of simply connected homogeneous manifolds: Euclidean space R2, constant sectional curvature spheres S2(κ), hyperbolic spaces H2(κ).

Benoˆ ıt Daniel Constant mean curvature surfaces

3-dimensional homogeneous manifolds

Let ˆ M a simply connected 3-dimensional Riemannian homogeneous manifold and G its isometry group. If dim G = 6, then ˆ M has constant sectional curvature: Euclidean space R3, constant sectional curvature sphere S3(κ), hyperbolic space H3(κ).

Benoˆ ıt Daniel Constant mean curvature surfaces

If dim G = 4, then ˆ M belongs to a two-parameter family denoted by E3(κ, τ). κ < 0 κ = 0 κ > 0 τ = 0 H2(κ) × R R3 S2(κ) × R τ = 0

• PSL2(R)

Nil3 S3(κ/4), Berger spheres These manifolds admit a Riemannian fibration ̟ : E3(κ, τ) → M2(κ) where M2(κ) is the simply connected constant curvature κ surface, τ is the bundle curvature (τ = 0 if and only if the fibration is a product fibration). If dim G = 3, then ˆ M is isometric to a Lie group endowed with a left invariant metric. Among Lie groups in this class we will particularly consider Sol3.

Benoˆ ıt Daniel Constant mean curvature surfaces

Isometries of E3(κ, τ)

The fibers of the fibration ̟ are geodesics called vertical geodesics. Any isometry of E3(κ, τ) induces an isometry of M2(κ). In particular, translations along fibers are isometries called vertical translations, around any vertical geodesic there exists a one-parameter family

• f rotations.

Remark: rotations by angle π around horizontal geodesics are isometries, in H2(κ) × R and S2(κ) × R, reflections with respect to totally geodesic surfaces (vertical or horizontal) are isometries, if τ = 0, all isometries preserve orientation.

Benoˆ ıt Daniel Constant mean curvature surfaces

CMC spheres in E3(κ, τ)

In E3(κ, τ), there exists a a one-parameter family of rotations around any vertical geodesic, which allows to construct rotational CMC spheres and to classify them (study of an ODE): if H2 −κ/4, there is no rotational CMC H sphere, if H2 > −κ/4, there exists a unique one up to ambient isometries. The Hopf theorem was extended to E3(κ, τ). Theorem (Abresch-Rosenberg, 2004) Any immersed CMC sphere in E3(κ, τ) is rotational. Idea: construction of a holomorphic quadratic differential for CMC surfaces in E3(κ, τ).

Benoˆ ıt Daniel Constant mean curvature surfaces

Compact embedded CMC surfaces in E3(κ, τ)

The Alexandrov theorem was extended to H2(κ) × R and a hemisphere of S2(κ) times R. Theorem (Hsiang-Hsiang, 1989) Any compact embedded CMC surface in H2(κ) × R or in a hemisphere

• f S2(κ) times R is a rotational sphere.

Idea: apply the Alexandrov reflection with respect to “vertical planes”. This problem is still open in E3(κ, τ) for τ = 0.

Benoˆ ıt Daniel Constant mean curvature surfaces

Generalisation to Sol3

Sol3 = R3 endowed with the metric e2x3dx2

1 + e−2x3dx2 2 + dx2 3.

Motivation : it is the unique “Thurston geometry” where the problem is open. Main difficulties. There is no one-parameter family of rotations in Sol3, which does not give a way to explicitely compute CMC spheres. More generally, we do not know a priori for which real numbers H > 0 there exists a CMC H sphere. There is no holomorphic quadratic differential of Abresch-Rosenberg-type.

Benoˆ ıt Daniel Constant mean curvature surfaces

Generalisation to Sol3

Theorem (D.-Mira, Meeks) For every H > 0 there exists a unique immersed CMC H sphere in Sol3 (up to translations). This sphere is moreover embedded.

Benoˆ ıt Daniel Constant mean curvature surfaces

Idea of proof: existence

We first consider the isoperimetric profile of Sol3: I(v) = inf

Volume(Ω)=v Area(∂Ω).

By results of Pittet on the isoperimetric profile of homogeneous manifolds, we have c1v log v I(v) c2v log v for large v.

Benoˆ ıt Daniel Constant mean curvature surfaces

Idea of proof: existence

Moreover, the isoperimetric problem admits a solution for any volume v (since Sol3 is homogeneous), the existence of reflections in two directions allows to use Alexandrov reflection to show that isoperimetric surfaces (= boundaries of isoperimetric domains) are diffeomorphic to spheres (Rosenberg), I admits at every v left and right derivatives I′

−(v) and I′ +(v)

and there exist isoperimetric surfaces of mean curvature I′

−(v)/2

and I′

+(v)/2 respectively.

From that we deduce that inf{H > 0 | ∃ an isoperimetric sphere of mean curvature H} = 0.

Benoˆ ıt Daniel Constant mean curvature surfaces

Idea of proof: existence

We start with a solution to the isoperimetric problem for a very small volume and we deform it by an implicit function argument and by means of a curvature estimate and Meeks’ height estimate: we obtain a smooth family (SH)H>0 where SH is a CMC H sphere for every H > 0. Remark: we do not know if all these spheres are isoperimetric but we can prove they all have index 1, i.e., their Jacobi operator has exactly

• ne negative eigenvalue.

Benoˆ ıt Daniel Constant mean curvature surfaces

Idea of proof: uniqueness

We first show that the Gauss map G of the CMC H sphere we constructed before (which has index 1), is a diffeomorphism. Sphere whose Gauss map would not be a diffeomorphism.

Benoˆ ıt Daniel Constant mean curvature surfaces

Idea of proof: uniqueness

Theorem (Alencar-do Carmo-Tribuzy) Let Qdz2 be a complex quadratic differential on ¯

• C. Assume that

around every point z0 ∈ ¯ C we have |Q¯

z| f|Q| where f is a

real-valued continuous function around z0 (“Cauchy-Riemann inequality”). Then Q ≡ 0. Consequently, for a given H > 0, to prove uniqueness of CMC H spheres, we will try to find a quadratic differential that satisfies the Cauchy-Riemann inequality for every CMC H immersion. Then such a differential will vanish on CMC H spheres.

Benoˆ ıt Daniel Constant mean curvature surfaces

Idea of proof: uniqueness

Given a CMC H conformal immersion with Gauss map g, we will look for a quadratic differential of the form Q(g)dz2 with Q(g) = L(g)g2

z + M(g)gz¯

gz where L, M : ¯ C → C are to be determined. We compute that Q¯

z

= g¯

zg2 z(Lq + 2LA) + gz|gz|2(L¯ q + 2LB + M ¯

B) +gz|g¯

z|2(Mq + MA) + ¯

gz|gz|2(M¯

q + MB + M ¯

A). where A and B are certain (explicit) functions appearing in the Gauss map equation. Next we can choose M so that Mq + MA = 0, M¯

q + MB + M ¯

A = 0.

Benoˆ ıt Daniel Constant mean curvature surfaces

Idea of proof: uniqueness

If moreover L : ¯ C → C is a global solution to (Lq + 2LA)¯ L = (L¯

q + 2LB + M ¯

B) ¯ M, (1) then Q(g) satisfies the Cauchy-Riemann inequality for every Gauss map g. Let G be the Gauss map of the CMC H sphere whose we proved the existence (recall that G is a diffeomorphism). Then, if such Q exists, then by the theorem of Alencar-do Carmo-Tribuzy we must have Q(G) ≡ 0, which implies that L(G(z)) = −M(G(z)) ¯ Gz(z) Gz(z) . It turns out that this formula allows to define L satisfying (1). Hence this defines Q(g) satisfying the Cauchy-Riemann inequality for every Gauss map g.

Benoˆ ıt Daniel Constant mean curvature surfaces

Questions

Let O ∈ Sol3 and for every H > 0 let SH be the unique CMC H sphere centered at O.

1 Is (SH)H>0 a foliation of Sol3 \ {O}? 2 Are all spheres SH isoperimetric? We only know they have index

• ne. Yes to question 1 implies yes to question 2.

Benoˆ ıt Daniel Constant mean curvature surfaces