On the finiteness of the Morse index of self-shrinkers

TL;DR

This paper establishes conditions under which complete properly embedded self-shrinkers in Euclidean space have finite Morse index, linking geometric end behavior and genus to stability properties.

Contribution

It provides a sufficient condition for finite Morse index based on asymptotic geometry and genus of self-shrinkers, advancing understanding of their stability.

Findings

Self-shrinkers with asymptotically conical or cylindrical ends have finite Morse index.

Self-shrinkers in $\,\mathbb{R}^3$ with finite genus have finite Morse index.

The results connect geometric end behavior and topological complexity to stability properties.

Abstract

In this paper, we present a sufficient condition for finite Morse index of complete properly self-shrinkers. We prove that a complete properly embedded self-shrinker in $\mathbb{R}^{n+1}$ with finite asymptotically conical ends or asymptotically cylindrical ends must have finite Morse index. Moreover, as an application of this result, we show that a complete properly embedded self-shrinker in $\mathbb{R}^3$ with finite genus has finite Morse index.