Diameter pinching in almost positive Ricci curvature
type: Item
instance of: scholarly article
title: Diameter pinching in almost positive Ricci curvature
zbMATH Open document ID: 1167.53033
DOI: 10.4171/CMH/160
author: Erwann Aubry
published in: Commentarii Mathematici Helvetici
publication date: 2009-04-21T00:00:00Z
full work available at URL: http://www.ems-ph.org/journals/show_pdf.php
review text: The author extends previous stability results of \textit{S. Ilias} [Ann. Inst. Fourier 43, No. 3, 843--863 (1993; Zbl 0783.53024)] to establish a diameter sphere theorem and its corresponding \(\lambda_1\) sphere theorem under \(L^p\) control of the curvature: Theorem. Let \(n\geq2\) be an integer, \(A>0\), and \(p>n\) be some reals. There exist positive constants \(C_i=C_i(p,n,A)\) such that any complete manifold which satisfies \[ \int_M({\underline{Ric}}-(n-1))^p_-\leq C_i\text{Vol}(M),\quad \int_M\bar\sigma_+^p<A\text{Vol}(M) \] and which in addition satisfies either of the following two conditions is homeomorphic to \(S^n\): (a) \(\text{Diam}(M)\geq\pi(1-C_1)\). (b) \(\lambda_1(M)\leq n(1+C_2)\).
reviewed by: Peter B. Gilkey
Mathematics Subject Classification ID: 53C20
zbMATH DE Number: 5545769
zbMATH Keywords: Ricci curvature
zbMATH Keywords: comparison theorems
zbMATH Keywords: integral bounds on the curvature
zbMATH Keywords: sphere theorems
MaRDI profile type: Publication
Wikidata QID: Q125363387
OpenAlex ID: W2030847752
description: scientific article; zbMATH DE number 5545769