A Strong Stability Condition on Minimal Submanifolds

It is well known that the square of 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 generalises the above scenario. In particular, if a closed minimal submanifold Σ is strongly stable, then:

1. The distance function to Σ satisfies a convex property in a neighbourhood of Σ, which implies that Σ is the unique closed minimal submanifold in this neighbourhood, up to a dimensional constraint.
2. The mean curvature flow that starts with a closed submanifold in a C¹ neighborhood of Σ converges smoothly to Σ.

   Many examples, including several well-known calibrated submanifolds, are shown to satisfy this strong stability condition. This is based on joint work with Chung-Jun Tsai.