Ricci curvature, diameter and optimal volume bound
type: Item
instance of: scholarly article
title: Ricci curvature, diameter and optimal volume bound
zbMATH Open document ID: 0831.53027
author: Jyh-Yang Wu
published in: Journal of the Mathematical Society of Japan
publication date: 1996-02-12T00:00:00Z
review text: Let \(M\) be an \(n\)-dimensional Riemannian manifold with Ricci curvature \(\text{Ric}(M) \geq (n - 1)k\) and the diameter \(d(M) \leq D\). It is known that \(M\) has volume bounded above by the volume \(\widetilde {v}_k(D)\) of a \(D\)-ball in a simply connected space form \(M^n_k\). This may be restated as follows: for the class \(M_k\) of all closed Riemann \(n\)-manifolds with \(\text{Ric} (M) \geq (n - 1)k\) and \(d(M) = \pi\) -- herewith \(k\) is naturally smaller than or equal to 1 -- the volume defines a function on \(M_k\), \(\text{vol}: M_k \to \mathbb{R}^+\), with range in the interval \((0, \widetilde{v}_k (\pi))\). To the inquiry if the upper bound \(\widetilde {v}_k(\pi)\) is optimal for \(k = 1/4\) or 1, it is shown that given any \(\Lambda \geq k(n - 1)\), there exist a function \(\Phi_\Lambda : (-\infty, 1) \to \mathbb{R}^+\) and a number \(\delta_\Lambda > 0\) such that (1) \(\Phi_\Lambda (k) \geq \widetilde{v}_k (\pi)\) and equality holds if and only if \(k = 1/4\). (2) \(\text{vol}(M) \leq \Phi_\Lambda (k)\) for any \(M \in M^\Lambda_k\), where \(M^\Lambda_k\) is a subclass of \(M_k\) defined by \(\{M \in M_k \mid \text{Ric}(M) \leq \Lambda\}\). (3) If \(k = 1/4\), then \(M \in M^\Lambda_k\) is diffeomorphic to \(\mathbb{R} P^n\) provided that \(\text{vol} (M) \geq \widetilde {v}_k (\pi) - \delta_\Lambda\).
Mathematics Subject Classification ID: 53C21
Mathematics Subject Classification ID: 53C23
zbMATH DE Number: 800613
zbMATH Keywords: diameter
zbMATH Keywords: bounded Ricci curvature
zbMATH Keywords: bounded volume
reviewed by: Tanjiro Okubo
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2969/jmsj/04730537
OpenAlex ID: W2096979673
DOI: 10.2969/JMSJ/04730537
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description: scientific article; zbMATH DE number 800613