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

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.

arXiv commentWe made a mistake in the proof of Lemma 2.4, we thanks PR. Márcio Batista (UFAL) for pointed us out this