Remark on a diameter bound for complete Riemannian manifolds with positive Bakry-Émery Ricci curvature

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• 作者：Homare Tadano^a ; ^b ; ^{h-tadano@cr.math.sci.osaka-u.ac.jp" class="auth_mail" title="E-mail the corresponding author} ; ^{homare.tadano@usc.es" class="auth_mail" title="E-mail the corresponding author}
• 关键词：primary ; 53C21 ; secondary ; 53C20 ; 53C25
• 刊名：Differential Geometry and its Applications
• 出版年：2016
• 出版时间：February 2016
• 年：2016
• 卷：44
• 期：Complete
• 页码：136-143
• 全文大小：272 K

文摘

In this paper, we shall give a new upper diameter estimate for complete Riemannian manifolds in the case that the Bakry–Émery 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 (2009) [12] and Limoncu (2012) [8]. As an application, we shall give an upper diameter bound for compact shrinking 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 shrinking Ricci solitons to satisfy the Hitchin–Thorpe inequality.