Remarks on scalar curvature of Yamabe solitons

TL;DR

This paper investigates the scalar curvature properties of Yamabe solitons, establishing conditions under which they have constant scalar curvature and providing new insights into their geometric behavior.

Contribution

It proves that complete Yamabe solitons with certain conditions have constant scalar curvature and offers a new proof of the Kazdan-Warner condition.

Findings

Complete Yamabe solitons with non-positive Ricci curvature have constant scalar curvature.

Quadratic decay of Ricci curvature implies non-negative scalar curvature.

A new proof of the Kazdan-Warner condition is provided.

Abstract

In this paper, we consider the scalar curvature of Yamabe solitons. In particular we show that, with natural conditions and non positive Ricci curvature, any complete Yamabe soliton has constant scalar curvature, namely, it is a Yamabe metric. We also show that the quadratic decay at infinity of the Ricci curvature of a complete non-compact Yamabe soliton has non-negative scalar curvature. A new proof of Kazdan-Warner condition is also presented.