Nonconvex constant mean curvature surfaces in generic Riemannian $3$-manifolds

Orateur: ZOLOTAREVA Tatiana
Localisation: École polytechnique, France
Type: Séminaire de géométrie
Site: Hors LAMA , IHP
Salle: 421
Date de début: 25/01/2016 - 14:00
Date de fin: 25/01/2016 - 14:00

In Euclidean $3$-space, Hopf's Theorem asserts that round spheres are the only topological spheres whose mean curvature is constant. In 1990, R. Ye proved the existence of embedded constant mean curvature hypersurfaces in Riemannian manifolds obtained by perturbing geodesic spheres centered near nondegenerate critical points of the scalar curvature function. In our result we prove the existence in ''generic" Riemannian $3$-manifolds of topological spheres that have large constant mean curvature but are not convex. These surfaces are obtained by perturbing the connected sums of two tangent geodesic spheres of small radii whose centers are lined up along a geodesic which passes through a critical point of the scalar curvature function with velocity equal to a unit eigenvector associated to a simple non-zero eigenvalue of the Hessian of the scalar curvature.