Rigidity of Area-Minimizing Free Boundary Surfaces in Mean Convex Three-Manifolds

Free Boundary Minimal Annuli in Convex Three-manifolds

We prove the existence of free boundary minimal annuli inside suitably convex subsets of three-dimensional Riemannian manifolds with nonnegative Ricci curvature − including strictly convex domains of the Euclidean space R.

Area-minimizing Projective Planes in Three-manifolds

On the Local Rigidity of Einstein Manifolds with Convex Boundary

Let (M, g) be a compact Einstein manifold with non-empty boundary ∂M . We prove that Killing fields at ∂M extend to Killings fields of (any) (M, g) provided ∂M is (weakly) convex and π1(M,∂M) = {e}. This gives a new proof of the classical infinitesimal rigidity of convex surfaces in Euclidean space and generalizes the result to Einstein metrics of any dimension.

Curvature Free Rigidity for Higher Rank Three-manifolds

We prove two rigidity results for complete Riemannian threemanifolds of higher rank. Complete three-manifolds have higher spherical rank if and only if they are spherical space forms. Complete finite volume threemanifolds have higher hyperbolic rank if and only if they are finite volume hyperbolic space forms.

Compactness of the Space of Embedded Minimal Surfaces with Free Boundary in Three-manifolds with Nonnegative Ricci Curvature and Convex Boundary

We prove a lower bound for the first Steklov eigenvalue of embedded minimal hypersurfaces with free boundary in a compact n-dimensional Riemannian manifold which has nonnegative Ricci curvature and strictly convex boundary. When n “ 3, this implies an apriori curvature estimate for these minimal surfaces in terms of the geometry of the ambient manifold and the topology of the minimal surface. A...