Mean curvature flow and Riemannian submersions

TL;DR

This paper establishes conditions under which mean curvature flow commutes with Riemannian submersions and applies this to generate new examples of evolving submanifolds in various geometric settings.

Contribution

It provides a sufficient condition for the commutation of mean curvature flow with Riemannian submersions and constructs new examples of submanifold evolution in different geometries.

Findings

Mean curvature flow commutes with Riemannian submersions under certain conditions.

New examples of mean curvature flow evolution are constructed in spheres, complex projective spaces, Heisenberg group, and tangent sphere bundles.

The results extend understanding of geometric flows in complex and sub-Riemannian geometries.

Abstract

We give a sufficient condition ensuring that the mean curvature flow commutes with a Riemannian submersion and we use this result to create new examples of evolution by mean curvature flow. In particular we consider evolution of pinched submanifolds of the sphere, of the complex projective space, of the Heisenberg group and the tangent sphere bundle equipped with the Sasaki metric.