Diameter estimation of gradient $ρ$-Einstein solitons

Abstract

Our aim in this article is to give a lower bound of the diameter of a compact gradient ρ-Einstein soliton satisfying some given conditions. We have also deduced some conditions of the gradient ρ-Einstein soliton with bounded Ricci curvature to become non-shrinking and non-expanding. Further, we have proved that a complete non-compact gradient shrinking or expanding Schouten soliton with non-constant potential and a boundedness condition on scalar curvature must be non-parabolic.