Remarks on non-compact complete Ricci expanding solitons

TL;DR

This paper investigates the properties of non-compact gradient Ricci expanding solitons, revealing conditions for scalar curvature maxima, positivity, and exponential decay, thereby advancing understanding of their geometric structure.

Contribution

It provides new results on scalar curvature behavior and decay in non-compact Ricci expanding solitons with non-negative Ricci curvature.

Findings

Scalar curvature has at least one maximum point.

Scalar curvature is positive unless the manifold is Ricci flat.

Exponential decay of scalar curvature in $\

Abstract

In this paper, we study gradient Ricci expanding solitons $(X,g)$ satisfying $$ Rc=cg+D^2f, $$ where $Rc$ is the Ricci curvature, $c<0$ is a constant, and $D^2f$ is the Hessian of the potential function $f$ on $X$. We show that for a gradient expanding soliton $(X,g)$ with non-negative Ricci curvature, the scalar curvature $R$ has at least one maximum point on $X$, which is the only minimum point of the potential function $f$. Furthermore, $R>0$ on $X$ unless $(X,g)$ is Ricci flat. We also show that there is exponentially decay for scalar curvature for $\epsilon$-pinched complete non-compact expanding solitons.