Consequences of strong stability of minimal submanifolds

TL;DR

This paper extends a recent stability result for minimal submanifolds to Brakke flows, providing applications such as local uniqueness and a method to navigate certain singularities in Lagrangian mean curvature flow.

Contribution

It generalizes the dynamical stability of strongly stable minimal submanifolds to enhanced Brakke flows, with new applications in geometric analysis.

Findings

Stability result extends to enhanced Brakke flows.

Establishes local uniqueness among stationary varifolds.

Provides a mechanism to flow through certain singularities.

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.