A comparison theorem for an elliptic operator
type: Item
instance of: scholarly article
title: A comparison theorem for an elliptic operator
zbMATH Open document ID: 0930.58012
DOI: 10.1023/A:1008698923961
author: Zhongmin Qian
published in: Potential Analysis
publication date: 2000-02-09T00:00:00Z
review text: Let \(\varrho(x)\) be the distance from a fixed reference point in a manifold. Using the Bakry-Emery technique, the author proves a comparison theorem similar to the upper bound for \(\Delta\varrho\) on a Riemannian manifold with lower bounded Ricci curvature. More precisely, consider a second-order elliptic differential operator \(L\) on a smooth manifold \(M\). The diffusion part of \(L\) gives rise to a Riemannian structure on \(M\), a corresponding carré du champ-operator \(\Gamma(f, g)={1\over 2}(L(fg)- fLg- gLf)\) and the Bakry-Emery curvature \(\Gamma_2(f,g)= {1\over 2}(L\Gamma(f,g)- \Gamma(Lf,g)- \Gamma(f, Lg))\). The author assumes that \(M\) is complete and that \(L\) satisfies the so-called curvature-dimension inequality \(\Gamma_2(f,f)\geq {1\over n}(Lf)^2- K\Gamma(f,f)\). In the case of the Laplace-Beltrami operator on an \(n\)-dimensional manifold, this is equivalent to a lower bound \(-K\) for the Ricci curvature. Under these assumptions, the author proves that on the complement of the cut-locus of the reference point the following comparison result holds: \[ L\varrho^2\leq n\Biggl\{1+ \sqrt{1+ {4K\varrho^2\over n}}\Biggr\}. \]
Mathematics Subject Classification ID: 58J05
Mathematics Subject Classification ID: 53C21
Mathematics Subject Classification ID: 58J60
zbMATH DE Number: 1160074
zbMATH Keywords: Bakry-Emery method
zbMATH Keywords: distance function
zbMATH Keywords: Ricci curvature
zbMATH Keywords: cut locus
zbMATH Keywords: comparison geometry
reviewed by: Walter Hoh
MaRDI profile type: Publication
full work available at URL: a:1008698923961
OpenAlex ID: W190780798
description: scientific article; zbMATH DE number 1160074