Remark on a diameter bound for complete manifolds with positive Bakry-\'{E}mery Ricci curvature

TL;DR

This paper presents a new upper diameter bound for complete manifolds with positive Bakry-Émery Ricci curvature, improving previous estimates and applying to Ricci solitons and geometric inequalities.

Contribution

It introduces an improved diameter estimate under specific curvature and potential function bounds, with applications to Ricci solitons and geometric inequalities.

Findings

New upper diameter bound for manifolds with positive Bakry-Émery Ricci curvature.

Application of the bound to compact Ricci solitons and scalar curvature.

Provision of conditions for Ricci solitons to satisfy Hitchin-Thorpe inequality.

Abstract

In this paper, we shall give a new upper diameter estimate for complete Riemannian manifolds in the case that the Bakry-\'Emery Ricci curvature has a positive lower bound and the norm of the potential function has an upper bound. Our diameter estimate improves previous ones obtained by Wei and Wylie (J. Differential Geom. 83, 377--405, 2009) and Limoncu (Math. Z. 271, 715--722, 2012). As an application, we shall give an upper diameter bound for compact Ricci solitons in terms of the maximum value of the scalar curvature. By using such a diameter bound, we shall provide some new sufficient conditions for four-dimensional compact Ricci solitons to satisfy the Hitchin-Thorpe inequality.