Diameter estimation of gradient $\rho$-Einstein solitons

TL;DR

This paper establishes diameter bounds and curvature conditions for gradient -Einstein solitons, providing insights into their geometric properties and classifications under various curvature constraints.

Contribution

It introduces new diameter estimates and curvature conditions that determine when gradient -Einstein solitons are non-shrinking or non-expanding, advancing understanding of their structure.

Findings

Lower bounds on the diameter of compact gradient -Einstein solitons.

Conditions under which such solitons are non-shrinking or non-expanding.

Proof that certain non-compact solitons are non-parabolic under specified conditions.

Abstract

Our aim in this article is to give a lower bound of the diameter of a compact gradient $\rho$-Einstein soliton satisfying some given conditions. We have also deduced some conditions of the gradient $\rho$-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.