Remarks on scalar curvature of gradient Yamabe solitons with non-positive Ricci curvature

Abstract

The aim of this short note is the study of the scalar curvature of a complete gradient Yamabe solitons. In particular we show an integral inequality for a gradient Yamabe soliton and as a consequence we proved that under a linear growth of the potential function f the gradient Yamabe soliton has constant scalar curvature. Also, with natural conditions and non-positive Ricci curvature, any complete Yamabe soliton has constant scalar curvature, namely, it is a Yamabe metric and become rotationally symmetric.