Volume estimates and the asymptotic behavior of expanding gradient Ricci solitons

TL;DR

This paper investigates the volume growth and asymptotic geometry of expanding gradient Ricci solitons, establishing conditions under which they resemble Euclidean space at infinity.

Contribution

It provides new volume estimates and characterizes the tangent cone at infinity for certain expanding gradient Ricci solitons.

Findings

Establishes asymptotic volume ratio behavior of non-steady gradient Ricci solitons.

Provides local volume ratio estimates under specific curvature decay conditions.

Shows that certain expanding solitons have Euclidean space as their tangent cone at infinity.

Abstract

We study the asymptotic volume ratio of non-steady gradient Ricci solitons. Moreover, a local estimate of the volume ratio is obtained for expanding solitons which satisfy $\lim_{dist(O,x)\rightarrow\infty} |Sect|\cdot dist(O,x)^2=0$. Therefore, for such a soliton, we can show that it must have $\mathbb{R}^n$ as one of its tangent cone at infinity. (Here we assume that the soliton is simply connected at infinity, has only one end and $n\geq 3$.)