Diameter estimates and Hitchin-Thorpe inequality for four-dimensional compact Quasi-Einstein manifolds

TL;DR

This paper investigates diameter bounds and geometric inequalities for four-dimensional compact Quasi-Einstein manifolds, extending known results and relating potential function oscillation to manifold geometry.

Contribution

It introduces new diameter estimates and Hitchin-Thorpe inequality conditions specific to four-dimensional compact Quasi-Einstein manifolds.

Findings

Derived lower bounds for diameter based on potential function oscillation.

Established diameter conditions ensuring Hitchin-Thorpe inequality in four dimensions.

Extended diameter estimates from smooth metric measure spaces to Quasi-Einstein manifolds.

Abstract

We study compact $m$-quasi-Einstein manifolds and derive geometric estimates relating the oscillation of the potential function to the diameter of the manifold. We obtain lower bounds for the diameter in terms of the oscillation of the potential function. As an application in dimension four, we derive diameter conditions ensuring that compact $m$-quasi-Einstein manifolds satisfy the Hitchin--Thorpe inequality. Our results extend diameter estimates in smooth metric measure spaces and are consistent with known bounds in the limiting case corresponding to Ricci solitons. Finally, we provide a volume estimate involving the oscillation.