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High codimension mean curvature flow of spacelike-convex submanifolds with one spacelike codimension

Ben Andrews, Qiyu Zhou

Abstract

In the pseudo-Euclidean space $\R^{n+1,k}$, we consider the mean curvature flow %$F: §^n \times [0, T) \to \R^{n+1,k}$ of $n$-dimensional spacelike submanifolds with spacelike codimension one and arbitrary timelike codimension $k$. We show that if the initial submanifold is compact and \emph{spacelike-convex} (the acceleration along every geodesic is strictly spacelike), then natural quantities measuring curvature pinching and noncollapsing are preserved under the flow. Moreover, we prove an analogue of the Huisken and Gage-Hamilton theorems in this setting, which states that the mean curvature flow deforms any such submanifold to a point in finite time, and that the solution is asymptotic to a shrinking sphere in a maximally spacelike affine subspace $\R^{n+1,0}\subset \R^{n+1,k}$.

High codimension mean curvature flow of spacelike-convex submanifolds with one spacelike codimension

Abstract

In the pseudo-Euclidean space , we consider the mean curvature flow % of -dimensional spacelike submanifolds with spacelike codimension one and arbitrary timelike codimension . We show that if the initial submanifold is compact and \emph{spacelike-convex} (the acceleration along every geodesic is strictly spacelike), then natural quantities measuring curvature pinching and noncollapsing are preserved under the flow. Moreover, we prove an analogue of the Huisken and Gage-Hamilton theorems in this setting, which states that the mean curvature flow deforms any such submanifold to a point in finite time, and that the solution is asymptotic to a shrinking sphere in a maximally spacelike affine subspace .
Paper Structure (23 sections, 44 theorems, 244 equations, 5 figures)

This paper contains 23 sections, 44 theorems, 244 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Pseudo-Euclidean space
  4. High codimension bundle machinery
  5. Connection and curvature
  6. Pull back bundles
  7. Time dependent immersion and subbundles
  8. Evolution equations
  9. spacelike-convex submanifolds of one spacelike codimension
  10. Gauss map parameterisations
  11. Other properties
  12. Curvature pinching
  13. Noncollapsing
  14. Formulation
  15. Geometric implications
  16. ...and 8 more sections

Key Result

Proposition 1.1

Suppose that ${\Sigma}$ is compact and $F: {\Sigma}^n \times I \to {\mathbb R}^{n+1,k}$ is a solution to MCF with $F(.,0)$ spacelike and spacelike-convex. Then the inward pinching ratio $\alpha$ is non-decreasing in $t$, and the outward pinching ratio $\beta$ is non-increasing in $t$.

Figures (5)

  • Figure 1: An illustration of pinching when $N_x{\Sigma} \cong {\mathbb R}^{1,1}$, the second fundamental form $h(v,v)$ lies in the red shaded region for all $v\in T{\Sigma}, |v|^2 = 1$, where $\nu \perp H(x)$ is a unit timelike basis in $N_x{\Sigma}$.
  • Figure 2: Noncollapsing illustration in the direction of $\widehat{H}(x)$, where the blue and orange hyperboloids represent the interior and exterior touching pseudosphere at $x$ respectively.
  • Figure :
  • Figure :
  • Figure :

Theorems & Definitions (86)

  • Proposition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 2.1: Causal character
  • Definition 2.2
  • Definition 2.3
  • Lemma 2.4
  • proof
  • Definition 3.1
  • Proposition 3.2
  • ...and 76 more