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

TL;DR

This paper investigates the stability and topological properties of immersed self-shrinking surfaces in three-dimensional space, proving that finite index implies properness and finite topology, and confirming the non-existence of stable non-proper self-shrinkers.

Contribution

It establishes a link between finite L-index and properness for self-shrinkers, and answers an open question about the stability of such surfaces in .

Findings

Finite L-index implies properness and finite topology.

No stable non-proper self-shrinker exists in .

Affirmative answer to Mantegazza's question.

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.