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arXiv · 2609.30697

The second gap for self-shrinkers with constant norm of the second fundamental form

Pengpeng Cheng·Tongzhu Li
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Abstract

Let $X: M^{n}\to \mathbb{R}^{n+1}$ be a complete self-shrinker with constant squared norm of the second fundamental form $S$. In this paper, we prove that if $S\leq \frac{10}{7}$, then $S=1$ or $S=0$, and the self-shrinker is isometric to either a round sphere $\mathbb{S}^n(\sqrt{n})$ with the center at the origin, or a cylinder $\mathbb{S}^k(\sqrt{k})\times \mathbb{R}^{n-k},~~1\leq k\leq n-1$, or a plane through the origin.

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Pengpeng Cheng, Tongzhu Li. 2026-09-25. The second gap for self-shrinkers with constant norm of the second fundamental form. https://arxiv.org/abs/2609.30697

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