The Harnack estimate for the modified Ricci flow on complete \(\mathbb{R}^ 2\).
type: Item
instance of: scholarly article
title: The Harnack estimate for the modified Ricci flow on complete \(\mathbb{R}^ 2\).
zbMATH Open document ID: 1046.53045
author: Shu-Cheng Chang
author: Shi-Kuo Hong
author: Chin-Tung Wu
published in: Rocky Mountain Journal of Mathematics
publication date: 2003-12-03T00:00:00Z
full work available at URL: CONT33-1.html
review text: The authors prove a trace Harnack inequality and a matrix Harnack inequality for the ''modified Ricci flow'' \[ {\partial g\over \partial t} = - {R \over 1+R } \] on \(({\mathbb R}^{2} ,e^{u(t)} g_{\mathbb{R}^{2}})\), where \(g_{\mathbb{R}^{2}}\) is the standard Euclidean metric on \({\mathbb R}^{2} \), and \(R\) is the scalar curvature. This flow was first proposed by S. T. Yau to study the uniformization theorem on complete noncompact surfaces. The Harnack quantity is found by considering expanding gradient solitons for the flow, following \textit{R. S. Hamilton} [J. Diff Geom. 37, 225-243 (1993; Zbl 0804.53023)]. The inequalities are then proved using methods of \textit{R. Hamilton} and \textit{B. Chow} [Commun. Pure Appl. Math. 45, No. 8, 1003--1014 (1992; Zbl 0785.53027)].
reviewed by: Christine Guenther
Mathematics Subject Classification ID: 53C44
Mathematics Subject Classification ID: 53C21
zbMATH DE Number: 2012049
zbMATH Keywords: Harnack inequality
zbMATH Keywords: modified Ricci flow
zbMATH Keywords: Harnack quantity
MaRDI profile type: Publication
OpenAlex ID: W1998390228
cites work: The yamabe flow on locally conformally flat manifolds with positive ricci curvature
cites work: Three-manifolds with positive Ricci curvature
cites work: Q3812872
cites work: The Harnack estimate for the Ricci flow
cites work: Ricci deformation of the metric on complete non-compact Riemannian manifolds
cites work: The Ricci flow on complete \(\mathbb{R}^ 2\)
DOI: 10.1216/RMJM/1181069987
description: scientific article; zbMATH DE number 2012049