Mean curvature flows in manifolds of special holonomy

TL;DR

This paper investigates the uniqueness and stability of minimal submanifolds and mean curvature flows in special holonomy manifolds, demonstrating that certain zero sections are uniquely minimal and dynamically stable.

Contribution

It establishes the uniqueness and stability of zero sections as minimal submanifolds in specific special holonomy spaces, linking Ricci flatness and extrinsic geometry.

Findings

Zero sections are unique among compact minimal submanifolds.

Zero sections are dynamically stable under mean curvature flow.

The proof connects Ricci flatness with extrinsic geometry.

Abstract

We study the uniqueness of minimal submanifolds and the stability of the mean curvature flow in several well-known model spaces of manifolds of special holonomy. These include the Stenzel metric on the cotangent bundle of spheres, the Calabi metric on the cotangent bundle of complex projective spaces, and the Bryant--Salamon metrics on vector bundles over certain Einstein manifolds. In particular, we show that the zero sections, as calibrated submanifolds with respect to their respective ambient metrics, are unique among compact minimal submanifolds and are dynamically stable under the mean curvature flow. The proof relies on intricate interconnections of the Ricci flatness of the ambient space and the extrinsic geometry of the calibrated submanifolds.