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Blow Up of Compact Mean Curvature Flow Solutions with Bounded Mean Curvature

Zichang Liu

Abstract

In 1994, Velázquez constructed a countable family of complete hypersurfaces flowing in $\mathbb{R}^{2N}$ $(N\geq 4)$ by mean curvature, each of which develops a type II singularity at the origin in finite time. Later Guo and Sesum showed that for a non-empty subset of Velázquez's solutions, the mean curvature blows up near the origin, at a rate smaller than that of the second fundamental form; recently Stolarski proved another subset of these solutions has bounded mean curvature up to the singular time. In this paper, we follow their arguments to construct compact mean curvature flow solutions in $\mathbb{R}^n$ $(n\geq 8)$ with bounded mean curvature.

Blow Up of Compact Mean Curvature Flow Solutions with Bounded Mean Curvature

Abstract

In 1994, Velázquez constructed a countable family of complete hypersurfaces flowing in by mean curvature, each of which develops a type II singularity at the origin in finite time. Later Guo and Sesum showed that for a non-empty subset of Velázquez's solutions, the mean curvature blows up near the origin, at a rate smaller than that of the second fundamental form; recently Stolarski proved another subset of these solutions has bounded mean curvature up to the singular time. In this paper, we follow their arguments to construct compact mean curvature flow solutions in with bounded mean curvature.
Paper Structure (8 sections, 37 theorems, 521 equations)

This paper contains 8 sections, 37 theorems, 521 equations.

Table of Contents

  1. Introduction
  2. Minimal Hypersurfaces Tangent to Lawson's Cones at Infinity
  3. Admissible Flow
  4. Construction of Velázquez's Solutions
  5. $C^0$ Estimates
  6. Smooth Estimates and Determination of the Constant $\Lambda$
  7. Vanishing Theorems of Parabolic Equations on Lawson's Cones and the Related Minimal Hypersurfaces
  8. Boundedness of the Mean Curvature

Key Result

Theorem 1.1

(vel) Let $N\geq 4$, $l\geq 2$ be integers. For $t_0<0$, $|t_0|\ll 1$, there exists a family of $O(N)\times O(N)$-invariant mean curvature flow solutions $\{\Sigma_{l}^{2N-1}(t)\}_{t_0\leq t<0}$ in $\mathbb{R}^{2N}$ such that

Theorems & Definitions (38)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Lemma 2.1
  • Proposition 3.1
  • Lemma 4.1
  • Proposition 4.2
  • Proposition 4.3
  • Lemma 4.4
  • ...and 28 more