A gap theorem of four-dimensional gradient shrinking solitons

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 ${\lambda }_1+{\lambda }_2\ge \frac{1}{3}R>0$ under a better pinched Weyl tensor assumption. We point out that the lower bound $\frac{1}{3}R$ is sharp.

11 pages

References in corpus (2)

• Geometry of shrinking Ricci solitons
• Gradient shrinking Ricci solitons of half harmonic Weyl curvature

Cited by in corpus (1)

• Four-dimensional complete gradient shrinking Ricci solitons