On Ricci solitons whose potential is convex

TL;DR

This paper investigates the geometric properties of Ricci solitons with convex or concave potentials, establishing conditions under which they are Ricci flat, split a line, or have limited scalar curvature critical points.

Contribution

It proves that certain complete gradient Ricci solitons with convex potentials are Ricci flat and split, and characterizes solitons with concave potentials and bounded Ricci curvature.

Findings

Complete gradient Ricci solitons with convex potential and finite weighted Dirichlet integral are Ricci flat.

Such solitons also isometrically split a line.

Gradient Ricci solitons with concave potential and bounded Ricci curvature are non-shrinking with at most one scalar curvature critical point.

Abstract

In this paper we consider the Ricci curvature of a Ricci soliton. In particular, we have showed that a complete gradient Ricci soliton with non-negative Ricci curvature possessing a non-constant convex potential function having finite weighted Dirichlet integral satisfying an integral condition is Ricci flat and also it isometrically splits a line. We have also proved that a gradient Ricci soliton with non-constant concave potential function and bounded Ricci curvature is non-shrinking and hence the scalar curvature has at most one critical point.