On $n$-dimensional complete self-similar solutions to the mean curvature flow in ${\mathbb{R}}^{n+1}$ with nonnegative constant scalar curvature

Abstract

As is well known, self-similar solutions to the mean curvature flow, including self-shrinkers, translating solitons and self-expanders, arise naturally in the singularity analysis of the mean curvature flow. Recently, Guo \cite{Guo} proved that $n$-dimensional compact self-shrinkers in ${\mathbb{R}}^{n+1}$ with scalar curvature bounded from above or below by some constant are isometric to the round sphere ${\mathbb{S}}^n(\sqrt{n})$, which implies that $n$-dimensional compact self-shrinkers in ${\mathbb{R}}^{n+1}$ with constant scalar curvature are isometric to the round sphere ${\mathbb{S}}^n(\sqrt{n})$(see also \cite{Hui1}). Complete classifications of $n$-dimensional translating solitons in ${\mathbb{R}}^{n+1}$ with nonnegative constant scalar curvature and of $n$-dimensional self-expanders in ${\mathbb{R}}^{n+1}$ with nonnegative constant scalar curvature were given by Martín, Savas-Halilaj and Smoczyk\cite{MSS} and Ancari and Cheng\cite{AC}, respectively. In this paper we give complete classifications of $n$-dimensional complete self-shrinkers in ${\mathbb{R}}^{n+1}$ with nonnegative constant scalar curvature. We will also give alternative proofs of the classification theorems due to Martín, Savas-Halilaj and Smoczyk \cite{MSS} and Ancari and Cheng\cite{AC}.

15 pages. We noted that Theorem 1.8 in our paper has been proved earlier by Ancari and Cheng in a recent paper [AC]