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 Mart\'{i}n, 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 Mart\'{i}n, Savas-Halilaj and Smoczyk \cite{MSS} and Ancari and Cheng\cite{AC}.