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A gap Theorem on closed self-shrinkers of mean curvature flow

Yuhang Zhao

TL;DR

The paper addresses the classification of compact self-shrinkers in mean curvature flow under a near-1 value for the squared norm of the second fundamental form. It introduces a gradient-estimate framework based on the Bochner formula and an auxiliary function to obtain sharp control of $|\nabla^{\Sigma} f|$ where $f=|X|^2$, under a pinching hypothesis. The main result shows that if $|\vec{II}|^2 \le 1 + \frac{1}{10\pi(n+2)}$, then the self-shrinker must be the round sphere $S^{n}(\sqrt{n})$, extending gap results to higher codimension and removing a lower-bound constraint in the compact case. This provides a rigorous gap theorem for compact self-shrinkers and reinforces extrinsic rigidity phenomena via gradient and energy-variation methods.

Abstract

In this paper, we prove a pinching theorem for $n-$dimensional closed self-shrinkers of the mean curvature flow. If the squared norm of the second fundamental form of a closed self-shrinker of arbitrary codimension satisfies: $ | \vec{\uppercase\expandafter{\romannumeral2}} |^2 \le 1 +\frac{1}{10 π(n+2)}$, then it must be the standard sphere $S^{n}(\sqrt{n})$. This result may provide some evidence for the open problem 13.76 in \cite{andrews2022extrinsic}.

A gap Theorem on closed self-shrinkers of mean curvature flow

TL;DR

The paper addresses the classification of compact self-shrinkers in mean curvature flow under a near-1 value for the squared norm of the second fundamental form. It introduces a gradient-estimate framework based on the Bochner formula and an auxiliary function to obtain sharp control of where , under a pinching hypothesis. The main result shows that if , then the self-shrinker must be the round sphere , extending gap results to higher codimension and removing a lower-bound constraint in the compact case. This provides a rigorous gap theorem for compact self-shrinkers and reinforces extrinsic rigidity phenomena via gradient and energy-variation methods.

Abstract

In this paper, we prove a pinching theorem for dimensional closed self-shrinkers of the mean curvature flow. If the squared norm of the second fundamental form of a closed self-shrinker of arbitrary codimension satisfies: , then it must be the standard sphere . This result may provide some evidence for the open problem 13.76 in \cite{andrews2022extrinsic}.

Paper Structure

This paper contains 4 sections, 7 theorems, 93 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proof of Theorem \ref{['maingradest']}
  4. Proof of Theorem \ref{['main']}

Key Result

Theorem 1.1

Let $X:\Sigma^{n} \to \mathbb{R}^{n+p} (n \ge 2)$ be an $n-$dimensional compact self-shrinker with the squared norm of the second fundamental satisfying then $X(\Sigma^{n})$ must be the standard sphere $S^{n}(\sqrt{n})$.

Theorems & Definitions (19)

  • Theorem 1.1
  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Theorem 1.2
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • ...and 9 more