Mean curvature flow and Riemannian submersions

Giuseppe Pipoli

doi:10.1007/s10711-016-0157-1

Abstract

We prove that a sufficient condition ensuring that the mean curvature flow commutes with a Riemannian submersion is that the submersion has minimal fibers. We then lift some results taken from the literature (i.e., Andrews and Baker in J Differ Geom 85:357–395, 2010; Baker in The mean curvature flow of submanifolds of high codimension, 2011; Huisken in J Differ Geom 20:237–266, 1984; Math Z 195:205–219, 1987; Pipoli and Sinestrari in Mean curvature flow of pinched submanifolds of \({\mathbb {CP}}^{n}\)) to create new examples of evolution by mean curvature flow. In particular we consider the evolution of pinched submanifolds of the sphere, of the complex projective space, of the Heisenberg group and of the tangent sphere bundle equipped with the Sasaki metric.

Highlights

Let F0 : M → M, gbe a smooth immersion of an m-dimensional manifold into a Riemannian manifold M of dimension m + k, called ambient space
The first theorem proved in this paper explores the symmetries of the mean curvature flow and gives a sufficient condition ensuring that this flow commutes with a submersion
Since we deal with the mean curvature flow we want to understand how a submanifold of B is related to its lift to M: let π : (M, gM) → (B, gB) be a Riemannian submersion, and F : B → B an immersion. π −1(F(B)) is a submanifold of M of the same codimension of F(B)

Summary

Introduction

Let F0 : M → M, gbe a smooth immersion of an m-dimensional manifold into a Riemannian manifold M of dimension m + k, called ambient space. The main part of this paper is devoted to the applications to specific examples, where we obtain new convergence results for the mean curvature flow by lifting the known theorems for the base manifold to the ambient space. In Proposition 4.8 we study submanifolds of the tangent sphere bundle of the round sphere equipped with the Sasaki metric and prove the alternative between the convergence in finite time to an orbit and the convergence in infinite time to a minimal, but not totally geodesic, limit This result is obtained by lifting the classical result of Huisken [9] regarding pinched hypersurfaces of the sphere and its generalization [2] to arbitrary codimension. 3 first we show that the invariance of a submanifold with respect to a group of isometries of the ambient manifold is preserved by the mean curvature flow, Theorem 1.1 is proved.

Preliminaries

Symmetries of the mean curvature flow

Examples and applications