Index Estimate of Self-Shrinkers in $\mathbb{R}^3$ with Asymptotically Conical Ends

TL;DR

This paper develops a method using Gaussian harmonic forms to analyze the Morse index of self-shrinkers in -dimensional space with asymptotically conical ends, extending previous index estimates.

Contribution

It introduces a new approach to estimate the Morse index of self-shrinkers with asymptotically conical ends using Gaussian harmonic forms.

Findings

Morse index 3d 84;(2g + r - 1)/3

Constructed Gaussian harmonic forms detecting asymptotically conical ends

Extended previous index estimates for self-shrinkers

Abstract

We construct Gaussian Harmonic forms of finite Gaussian weighted $L^2$-norm on non-compact surfaces that detect each asymptotically conical end. As an application we prove an extension of the index estimates of self-shrinkers in $[11]$ under the existence of such ends. We show that the Morse index of a self-shrinker is greater or equal to $\frac{2g+r-1}{3}$, where $r$ is the number of asymptotically conical ends.