A strong stability condition on minimal submanifolds and its implications

2017/10/01 by Tsai, Chung-Jun, Wang, Mu-Tao
#Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1710.00433

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 C¹ 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 C¹ 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.

Related

• Mean curvature flows in manifolds of special holonomy
• A Note on the Stability and Uniqueness for Solutions to the Minimal Surface System
• Uniqueness and Pseudolocality Theorems of the Mean Curvature Flow
• Dynamical Stability of Minimal Lagrangians in Kähler-Einstein Manifolds of Non-Positive Curvature
• Flowing maps to minimal surfaces: Existence and uniqueness of solutions