Rigidity and gap results for low index properly immersed self--shrinkers in $\mathbb{R}^{m+1}$

TL;DR

This paper classifies properly immersed self-shrinkers in Euclidean space with low Morse index, showing hyperplanes have index 1 and cylinders have index at least m+2, with precise characterizations.

Contribution

It establishes a rigidity and gap theorem for the Morse index of properly immersed self-shrinkers, identifying hyperplanes and cylinders as the only low-index examples.

Findings

Hyperplanes have Morse index 1.

Non-hyperplane self-shrinkers have index at least m+2.

Cylinders achieve the minimal index m+2 among non-hyperplanes.

Abstract

In this paper we show that the only properly immersed self--shrinkers $\Sigma$ in $\mathbb{R}^{m+1}$ with Morse index $1$ are the hyperplanes through the origin. Moreover, we prove that if $\Sigma$ is not a hyperplane through the origin then the index jumps and it is at least $m+2$, with equality if and only if $\Sigma$ is a cylinder $\mathbb{R}^{m-k}\times \mathbb{S}^{k}(\sqrt{k})$ for some $1\leq k\leq m-1$.