|DATE||September 19 (Tue), 2017|
|INSTITUTE||National Taiwan University|
|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 $\Sigma$ is strongly stable, then:
1. The distance function to $\Sigma$ satisfies a convex property in a neighborhood
of $\Sigma$, which implies that $\Sigma$ 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 $\Sigma$ converges smoothly to $\Sigma$.
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.