Structure at infinity for shrinking Ricci solitons

Abstract

The paper mainly concerns the structure at infinity for complete gradient shrinking Ricci solitons. It is shown that for such a soliton with bounded curvature, if the round cylinder $\mathbb{R}\times \mathbb{S}^{n-1}/\Gamma$ occurs as a limit for a sequence of points going to infinity along an end, then the end is asymptotic to the same round cylinder at infinity. The result is then applied to obtain structural results at infinity for four dimensional gradient shrinking Ricci solitons. It is previously known that such solitons with scalar curvature approaching zero at infinity must be smoothly asymptotic to a cone. For the case that the scalar curvature is bounded from below by a positive constant, we conclude that along each end the soliton is asymptotic to a quotient of $\mathbb{R}\times \mathbb{S}^{3}$ or converges to a quotient of $\mathbb{R}^{2}\times \mathbb{S}^{2}$ along each integral curve of the gradient vector field of the potential function. For four dimensional Kahler Ricci solitons, stronger conclusion holds. Namely, they either are smoothly asymptotic to a cone or converge to a quotient of $\mathbb{R}^{2}\times \mathbb{S}^{2}$ at infinity.