### Geometry-Topology Seminar

Thursday, February 2, 2017 - 4:30pm

#### Antoine Song

Princeton Univeristy

Location

University of Pennsylvania

DRL 4C8

A theorem of Calabi and Cao asserts that a closed geodesic of least length in a two-sphere with nonnegative curvature is simple. We will show that a higher dimensional version of this result holds without any restriction on the curvature: in a closed (n+1)-manifold with n between 2 and 6, a least area closed immersed minimal hypersurface exists and is always embedded. As a corollary of this result, we can prove the following 3-dimensional version of a theorem due to Toponogov. In a closed 3-manifold with scalar curvature at least 6 and not isometric to the round sphere, there exists an embedded closed minimal surface of area less than 4\pi.