Stability properties of complete self-shrinking surfaces in $\mathbb{R}^3$

Hilário Alencar; Gregório Silva Neto; Detang Zhou

Stability properties of complete self-shrinking surfaces in $\mathbb{R}^3$

Differential Geometry 2022-05-02 v2 Analysis of PDEs Spectral Theory

Authors: Hilário Alencar , Gregório Silva Neto , Detang Zhou

Abstract

This paper studies rigidity for immersed self-shrinkers of the mean curvature flow of surfaces in the three-dimensional Euclidean space $\mathbb{R}^3.$ We prove that an immersed self-shrinker with finite $L$ -index must be proper and of finite topology. As one of consequences, there is no stable two-dimensional self-shrinker in $\mathbb{R}^3$ without assuming properness. We conclude the paper by giving an affirmative answer to a question of Mantegazza.

Keywords

surfaces with constant mean curvature surfaces and embeddings isometric immersions

Cite

@article{arxiv.2106.09165,
  title  = {Stability properties of complete self-shrinking surfaces in $\mathbb{R}^3$},
  author = {Hilário Alencar and Gregório Silva Neto and Detang Zhou},
  journal= {arXiv preprint arXiv:2106.09165},
  year   = {2022}
}

Comments

We made a mistake in the proof of Lemma 2.4, we thanks PR. M\'arcio Batista (UFAL) for pointed us out this

Stability properties of complete self-shrinking surfaces in $\mathbb{R}^3$

Abstract

Keywords

Cite

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