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 $\Sigma$ 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 $1\leq k\leq m-1$.