June 7, 2018
TITLE: A strong stability condition on minimal submanifolds
ABSTRACT: It is well known that the distance function to a totally geodesic submanifold of a negatively curved ambient manifold is a convex function. We identify a strong stability condition on minimal submanifolds that generalizes the above scenario. In particular, if a closed minimal submanifold is strongly stable, then:
1. The distance function to satisfies a convex property in a neighborhood of , which implies that is the unique closed minimal submanifold in this neighborhood, up to a dimensional constraint.
2. The mean curvature flow that starts with a closed submanifold in a C^1 neighborhood of converges smoothly to .
Many examples, including several well-known types of calibrated submanifolds, are shown to satisfy this strong stability condition. This is based on joint work with Mu-Tao Wang.