$\epsilon$-Regularity and Structure of 4-dimensional Shrinking Ricci Solitons

Abstract

A closed four dimensional manifold cannot possess a non-flat Ricci soliton metric with arbitrarily small $L^2$-norm of the curvature. In this paper, we localize this fact in the case of shrinking Ricci solitons by proving an $\epsilon$-regularity theorem, thus confirming a conjecture of Cheeger-Tian. As applications, we will also derive structural results concerning the degeneration of the metrics on a family of complete non-compact four dimensional shrinking Ricci solitons without a uniform entropy lower bound. In the appendix, we provide a detailed account of the equivariant good chopping theorem when collapsing with locally bounded curvature happens.

References in corpus (2)

• Geometry of shrinking Ricci solitons
• A note on the compactness theorem for 4d Ricci shrinkers

Cited by in corpus (1)

• Global $\epsilon$-regularity for 4-dimensional Ricci flow with integral scalar curvature bound