On Euler characteristic and Hitchin-Thorpe inequality for four-dimensional compact Ricci solitons

TL;DR

This paper explores the geometry of 4-dimensional compact gradient Ricci solitons, establishing conditions under which they satisfy the Hitchin-Thorpe inequality and providing volume estimates.

Contribution

It proves that under an upper bound on the potential function's range, such solitons must satisfy the Hitchin-Thorpe inequality, linking Ricci solitons to classical geometric inequalities.

Findings

Ricci solitons satisfy Hitchin-Thorpe inequality under certain bounds

Volume estimates for 4-dimensional compact Ricci solitons

Conditions linking potential function bounds to geometric inequalities

Abstract

In this article, we investigate the geometry of $4$-dimensional compact gradient Ricci solitons. We prove that, under an upper bound condition on the range of the potential function, a $4$-dimensional compact gradient Ricci soliton must satisfy the classical Hitchin-Thorpe inequality. In addition, some volume estimates are also obtained.