Scalar curvature of self-shrinkers

TL;DR

This paper investigates the scalar curvature of n-dimensional self-shrinkers in Euclidean space, establishing bounds, classifications, and partial resolutions of conjectures related to constant scalar curvature and second fundamental form.

Contribution

It provides new bounds and classifications for self-shrinkers with constant scalar curvature and addresses conjectures concerning the squared norm of the second fundamental form.

Findings

Scalar curvature of self-shrinkers with positive constant R satisfies 0<R≤n-1.

Complete self-shrinkers with non-negative constant scalar curvature are classified.

Partial resolution of the conjecture on constant squared norm of the second fundamental form S.

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 a positive constant, then we prove that the scalar curvature $R$ satisfies $0<R\leq n-1$. 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 of the second fundamental form $S$. 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.