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

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