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Chung-Jun Tsai: Lectures

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 \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.

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