Consequences of strong stability of minimal submanifolds

Abstract

In this note we show that the recent dynamical stability result for small $C^1$-perturbations of strongly stable minimal submanifolds of C.-J. Tsai and M.-T. Wang directly extends to the enhanced Brakke flows of Ilmanen. We illustrate applications of this result, including a local uniqueness statement for strongly stable minimal submanifolds amongst stationary varifolds, and a mechanism to flow through some singularities of Lagrangian mean curvature flow which are proved to occur by Neves.