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Remarks on mean curvature flow solitons in warped products

Giulio Colombo, Luciano Mari, Marco Rigoli

Abstract

We study some properties of mean curvature flow solitons in general Riemannian manifolds and in warped products, with emphasis on constant curvature and Schwarzschild type spaces. We focus on splitting and rigidity results under various geometric conditions, ranging from the stability of the soliton to the fact that the image of its Gauss map be contained in suitable regions of the sphere. We also investigate the case of entire graphs.

Remarks on mean curvature flow solitons in warped products

Abstract

We study some properties of mean curvature flow solitons in general Riemannian manifolds and in warped products, with emphasis on constant curvature and Schwarzschild type spaces. We focus on splitting and rigidity results under various geometric conditions, ranging from the stability of the soliton to the fact that the image of its Gauss map be contained in suitable regions of the sphere. We also investigate the case of entire graphs.

Paper Structure

This paper contains 4 sections, 29 theorems, 152 equations.

Table of Contents

  1. Introduction
  2. Invariance by a $1$-parameter group of isometries
  3. Stability
  4. MCF graphs

Key Result

Theorem 1

Let $\psi:M^m\to\mathbb R^{m+1}$ be an oriented, connected, complete proper self-shrinker whose spherical Gauss map $\nu$ has image contained in a closed hemisphere of $\mathbb S^m$. Then either $\psi$ is totally geodesic (that is, $\psi(M)$ is an affine hyperplane) or $\psi$ is a cylinder over some

Theorems & Definitions (67)

  • Definition 1.1
  • Example 1
  • Example 2
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Remark 1
  • Corollary 4
  • Theorem 5
  • Corollary 6
  • ...and 57 more