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

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 Smoczykcite{MSS} and Ancari and Chengcite{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 Chengcite{AC}.