A gap theorem of four-dimensional gradient shrinking solitons

TL;DR

This paper proves a gap theorem for four-dimensional gradient shrinking solitons, establishing conditions under which such solitons are flat or have Ricci curvature eigenvalues bounded below by a positive multiple of scalar curvature.

Contribution

It introduces new lower bounds for Ricci eigenvalues in four-dimensional gradient shrinking solitons under curvature pinching conditions, including a sharp bound of one-third of the scalar curvature.

Findings

Either the soliton is flat or Ricci eigenvalues satisfy a positive lower bound.

Established a sharp lower bound of 1/3 of scalar curvature for Ricci eigenvalues.

Provided conditions under which the soliton's geometry is constrained by curvature bounds.

Abstract

In this paper, we will prove a gap theorem for four-dimensional gradient shrinking soliton. More precisely, we will show that any complete four-dimensional gradient shrinking soliton with nonnegative and bounded Ricci curvature, satisfying a pinched Weyl curvature, either is flat, or $\lambda_1 + \lambda_2\ge c_0 R>0$ everywhere for some $c_0\approx 0.29167$, where $\{\lambda_i\}$ are the two least eigenvalues of Ricci curvature. Furthermore, we will show that $\lambda_1 + \lambda_2\ge \frac 13R>0$ under a better pinched Weyl tensor assumption. We point out that the lower bound $\frac 13R$ is sharp.