Scalar curvature of self-shrinkers

Abstract

In this paper, we study scalar curvature of $n$-dimensional self-shrinkers in the Euclidean space $\mathbb R^{n+1}$. If the scalar curvature of an $n$-dimensional self-shrinker is constant, then we prove that the scalar curvature $R$ satisfies $R\leq n-1$. Furthermore, we classify $n$-dimensional complete self-shrinkers in $\mathbb R^{n+1}$ with non-negative constant scalar curvature. We also study $n$-dimensional complete self-shrinkers in $\mathbb R^{n+1}$ with constant squared norm $S$ of the second fundamental form. We partially resolve the conjecture on $n$-dimensional complete self-shrinkers in $\mathbb R^{n+1}$ with constant squared norm $S$ of the second fundamental form.