Analysis of weighted Laplacian and applications to Ricci solitons
type: Item
instance of: scholarly article
title: Analysis of weighted Laplacian and applications to Ricci solitons
zbMATH Open document ID: 1245.53039
author: Ovidiu Munteanu
author: Jiaping Wang
published in: Communications in Analysis and Geometry
publication date: 2012-07-04T00:00:00Z
full work available at URL: https://arxiv.org/abs/1112.3027
reviewed by: Laurent Guillopé
Mathematics Subject Classification ID: 53C21
Mathematics Subject Classification ID: 58J50
Mathematics Subject Classification ID: 53C44
Mathematics Subject Classification ID: 53C24
zbMATH DE Number: 6052980
zbMATH Keywords: gradient Ricci soliton
zbMATH Keywords: Bakry-Emery curvature
zbMATH Keywords: Ricci curvature
zbMATH Keywords: Poincaré inequality
zbMATH Keywords: rigidity
zbMATH Keywords: Laplacian spectrum
MaRDI profile type: Publication
arXiv ID: 1112.3027
Wikidata QID: Q125934263
DOI: 10.4310/CAG.2012.V20.N1.A3
review text: A gradient Ricci soliton \((M,g,f)\) is a Riemannian manifold \((M,g)\) with a function \(f\) on \(M\) such that \(\text{Ric}_g +\text{Hess}(f)=\lambda g\) for some scalar \(\lambda\): this class of manifolds, which extends Einstein manifolds, appears in Ricci flow analysis both as a self-similar solution and as a singularity model. By rescaling the metric, the scalar \(\lambda\) can be taken equal to \(-1/2,0\) or \( 1/2\), the soliton being called expanding, steady or shrinking resp.NEWLINENEWLINENEWLINEFor an expanding soliton with lower bound \(S_g\geq(1-n)/2\) of the scalar curvature \(S_g\), the authors prove that \(M\), if with more than one end, is isometric to the product \(G\times N\) where \(N\) is a compact Einstein manifold and \(G\) is the Gaussian expanding Ricci soliton \((\mathbb R, dx^2,x^2/4)\). This result can be compared to \textit{J. Cheeger} and \textit{D. Gromoll}'s split theorem for non-negative Ricci curvature [J. Differ. Geom. 6, 119--128 (1971; Zbl 0223.53033)], Li-Wang's wrapped product theorem for Riemannian manifolds with Ricci curvature bounded below and the authors' result for steady Ricci solitons [Commun. Anal. Geom. 19, No. 3, 451--486 (2011; Zbl 1246.53056)].NEWLINENEWLINENEWLINEFurthermore, the authors prove a linear growth rate for the geodesic ball volume on any non-compact shrinking gradient Ricci soliton: \(\text{Vol}\, B(m,r)\geq c r\); the upper bound \(\text{Vol}\,B(m,r)\leq C r^n\), with \(n=\dim M\), has been proved by \textit{H. D. Cao} and \textit{D. Zhou} [J. Differ. Geom. 85, No. 2, 175--186 (2010; Zbl 1246.53051)]. These two bounds correspond to the bounds (upper by Bishop, lower by Yau) for Riemannian manifolds with \(\text{Ric}_g \geq 0\).NEWLINENEWLINENEWLINEThese geometrical results are consequences of analytical results on smooth metric spaces \((M,g,\text{e}^{-f}d\text{vol}_g)\) with deformed Laplace-Beltrami operator \(\Delta_{g,f}=\Delta_g-\langle\nabla_gf,\nabla_g\rangle\) and Bakry-Émery curvature \(\text…
description: scientific article; zbMATH DE number 6052980