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 S 0 . We fix a constant V and we consider all surfaces S with boundary γ such that S and S 0 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 of 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 R 3 , solutions exist for all volumes and are geodesic balls. Benoˆ ıt Daniel Constant mean curvature surfaces
Classical examples of minimal surfaces in R 3 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 R 3 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 R 3 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 R 3 is a round sphere. Idea of proof: for CMC surfaces in R 3 , there exists a holomorphic quadratic differential Q whose zeroes are the umbilical points of the surface: it is the Hopf differential, on 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 S 3 ( κ ) and hyperbolic spaces H 3 ( κ ). Benoˆ ıt Daniel Constant mean curvature surfaces
The Alexandrov theorem Theorem (Alexandrov, 1956) Any compact embedded CMC surface in R 3 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 H 3 ( 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 R 3 . This conjecture was disproved: Wente (1986) and then Abresch (1987) indeed constructed CMC tori in R 3 , and Kapouleas (1995) constructed CMC surfaces of any genus g � 2 in R 3 . None of these surfaces is of course embedded, by Alexandrov’s theorem. Benoˆ ıt Daniel Constant mean curvature surfaces
Embedded CMC tori in S 3 Theorem (Brendle, 2012; Lawson’s conjecture) The Clifford torus is the only embedded minimal torus in S 3 . Theorem (Andrews-Li, 2012; Pinkall and Sterling’s conjecture) Any embedded CMC torus in S 3 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 R 2 , constant sectional curvature spheres S 2 ( κ ), hyperbolic spaces H 2 ( 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 R 3 , constant sectional curvature sphere S 3 ( κ ), hyperbolic space H 3 ( κ ). Benoˆ ıt Daniel Constant mean curvature surfaces
If dim G = 4, then ˆ M belongs to a two-parameter family denoted by E 3 ( κ, τ ). κ < 0 κ = 0 κ > 0 H 2 ( κ ) × R R 3 S 2 ( κ ) × R τ = 0 S 3 ( κ/ 4), � PSL 2 ( R ) τ � = 0 Nil 3 Berger spheres These manifolds admit a Riemannian fibration ̟ : E 3 ( κ, τ ) → M 2 ( κ ) where M 2 ( 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 Sol 3 . Benoˆ ıt Daniel Constant mean curvature surfaces
Isometries of E 3 ( κ, τ ) The fibers of the fibration ̟ are geodesics called vertical geodesics. Any isometry of E 3 ( κ, τ ) induces an isometry of M 2 ( κ ). In particular, translations along fibers are isometries called vertical translations, around any vertical geodesic there exists a one-parameter family of rotations. Remark: rotations by angle π around horizontal geodesics are isometries, in H 2 ( κ ) × R and S 2 ( 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 E 3 ( κ, τ ) In E 3 ( 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 H 2 � − κ/ 4, there is no rotational CMC H sphere, if H 2 > − κ/ 4, there exists a unique one up to ambient isometries. The Hopf theorem was extended to E 3 ( κ, τ ). Theorem (Abresch-Rosenberg, 2004) Any immersed CMC sphere in E 3 ( κ, τ ) is rotational. Idea: construction of a holomorphic quadratic differential for CMC surfaces in E 3 ( κ, τ ). Benoˆ ıt Daniel Constant mean curvature surfaces
Compact embedded CMC surfaces in E 3 ( κ, τ ) The Alexandrov theorem was extended to H 2 ( κ ) × R and a hemisphere of S 2 ( κ ) times R . Theorem (Hsiang-Hsiang, 1989) Any compact embedded CMC surface in H 2 ( κ ) × R or in a hemisphere of S 2 ( κ ) times R is a rotational sphere. Idea: apply the Alexandrov reflection with respect to “vertical planes”. This problem is still open in E 3 ( κ, τ ) for τ � = 0. Benoˆ ıt Daniel Constant mean curvature surfaces
Generalisation to Sol 3 Sol 3 = R 3 endowed with the metric e 2 x 3 d x 2 1 + e − 2 x 3 d x 2 2 + d x 2 3 . Motivation : it is the unique “Thurston geometry” where the problem is open. Main difficulties. There is no one-parameter family of rotations in Sol 3 , 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 Sol 3 Theorem (D.-Mira, Meeks) For every H > 0 there exists a unique immersed CMC H sphere in Sol 3 (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 Sol 3 : I ( v ) = Volume(Ω)= v Area( ∂ Ω) . inf By results of Pittet on the isoperimetric profile of homogeneous manifolds, we have log v � I ( v ) � c 2 v c 1 v for large v. log v Benoˆ ıt Daniel Constant mean curvature surfaces
Idea of proof: existence Moreover, the isoperimetric problem admits a solution for any volume v (since Sol 3 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 ( S H ) H> 0 where S H 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 one negative eigenvalue. Benoˆ ıt Daniel Constant mean curvature surfaces
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