A Classification of Complete Self-shrinkers

Abstract

Let $X:M^n\to {\mathbb{R}}^{n+1}$ be an $n$-dimensional complete self-shrinker. We obtain a complete classification of complete self-shrinkers with positive constant scalar curvature. More precisely, we prove that the round sphere $S^n(\sqrt{n})$ and the standard generalized cylinder $S^k(\sqrt{k})\times {\mathbb{R}}^{n-k}$ for $2\le k\le n-1$ are the only complete self-shrinkers with positive constant scalar curvature. The key difficulty is to characterize the residual case in Cheng-Li-Wei \cite{CLW}: $R>0$, $S<1$, and ${\sup }_{M}S=1$, where $R$ and $S$ denote the scalar curvature and the squared norm of the second fundamental form, respectively. For this case, it seems a hard task that the generalized maximum principle yields a useful information. In order to overcome this substantial difficulty, our key ingredient is to get a uniform positive lower bound for the Bakry-Émery Ricci curvature so that we can make use of the comparison theorem of Wei-Wylie \cite{WeiWylie} to conclude that the Gaussian volume is finite. Furthermore, the gap theorems on $S$ are given.