A strong stability condition on minimal submanifolds and its implications

Abstract

AbstractWe identify a strong stability condition on minimal submanifolds that implies uniqueness and dynamical stability properties. In particular, we prove a uniqueness theorem and a {\mathcal{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 {\mathcal{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 [24] which applies only to calibrated submanifolds of special holonomy ambient manifolds.