A strong stability condition on minimal submanifolds and its implications

TL;DR

This paper introduces a strong stability condition for minimal submanifolds that guarantees uniqueness and long-term stability under mean curvature flow, extending previous results to a broader class of submanifolds.

Contribution

It establishes a new stability condition that ensures uniqueness and exponential convergence of mean curvature flow for minimal submanifolds, generalizing prior results.

Findings

Proves a uniqueness theorem for minimal submanifolds satisfying the stability condition.

Demonstrates exponential convergence of mean curvature flow to the minimal submanifold.

Extends previous stability results beyond calibrated submanifolds in special holonomy manifolds.

Abstract

We identify a strong stability condition on minimal submanifolds that implies uniqueness and dynamical stability properties. In particular, we prove a uniqueness theorem and a C^1 dynamical stability theorem of the mean curvature flow for minimal submanifolds that satisfy this condition. The latter theorem states that the mean curvature flow of any other submanifold in a C^1 neighborhood of such a minimal submanifold exists for all time, and converges exponentially to the minimal one. This extends our previous uniqueness and stability theorem [arXiv:1605.03645] which applies only to calibrated submanifolds of special holonomy ambient manifolds.