Geometry of shrinking Ricci solitons

TL;DR

This paper studies the curvature properties of four-dimensional shrinking gradient Ricci solitons, establishing bounds and asymptotic behavior, and explores implications for the manifold's geometry, including conditions for being asymptotically conical.

Contribution

It provides new curvature estimates and asymptotic analysis for four-dimensional shrinking Ricci solitons, including bounds on the curvature operator and conditions for conical geometry.

Findings

Curvature operator satisfies | ext{Rm}| ≤ c S for bounded scalar curvature.

ext{Rm} is asymptotically nonnegative at infinity.

If scalar curvature tends to zero at infinity, the manifold is asymptotically conical.

Abstract

The main purpose of this paper is to investigate the curvature behavior of four dimensional shrinking gradient Ricci solitons. For such soliton $M$ with bounded scalar curvature $S$, it is shown that the curvature operator $\mathrm{Rm}$ of $M$ satisfies the estimate $|\mathrm{Rm}|\le c\,S$ for some constant $c$. Moreover, the curvature operator $\mathrm{Rm}$ is asymptotically nonnegative at infinity and admits a lower bound $\mathrm{Rm}\geq -c\,\left(\ln r\right)^{-1/4},$ where $r$ is the distance function to a fixed point in $M$. As application, we prove that if the scalar curvature converges to zero at infinity, then the manifold must be asymptotically conical. As a separate issue, a diameter upper bound for compact shrinking gradient Ricci solitons of arbitrary dimension is derived in terms of the injectivity radius.