Application of the method of moving planes to conformally invariant equations
type: Item
instance of: scholarly article
title: Application of the method of moving planes to conformally invariant equations
zbMATH Open document ID: 1068.53023
author: Pengfei Guan
author: Chang-Shou Lin
author: Guofang Wang
published in: Mathematische Zeitschrift
publication date: 2005-02-11T00:00:00Z
review text: A basic question in Riemannian geometry is to determine to what extent curvature assumptions on the Riemannian metric \(g\) impose restrictions on the topology of \((M^n,g)\). Such a question is mainly motivated by the Gauss-Bonnet theorem for two-dimensional compact manifolds: \[ 2\pi\chi(M^2)=\int_{M^2}K\,d\mu\tag{1} \] where \(d\mu\) is the area element, \(K\) is the Gauss curvature, and \(\chi(M^2)\) is the Euler characteristic. Motivated by such a question A. Chang, M. Gursky and P. Yang asked the following: Given a compact Riemannian 4-manifold \((M^4,g_0)\) with both \(\int_{M^4} \sigma_2(A_0)\,dv_0\) and Yamabe invariant \(Y(g_0)=\inf_{g\in[g_0]} (\text{vol}(g))^{-1/2} \int_{M^4} R(g)\,dv_g\) positive, where \([g_0]\) denotes the conformal class of \(g_0\). Does there exist a conformal metric \(g=e^{2w}g_0\) with positive Ricci curvature Ric\(_g\)? Here \(A =\text{Ric}-{1\over 6}Rg\) denotes the Schouten tensor, \(\sigma_2(A)=-{1\over 2}| \text{Ric}| ^2+{1\over 6}R^2\) and \(R\) is the scalar curvature. In [\textit{S.-Y. A. Chang, M. J. Gursky}, and \textit{P. C. Yang}, Ann. Math. (2) 155, No. 3, 709--787 (2002; Zbl 1031.53062)], they gave a positive answer to the above question under the stronger assumption that there is a conformal metric \(g\) with \(\sigma_2(A)>0\). The authors of the paper under review prove a high dimensional version of the above result under the assumption that the manifold is locally conformally flat. Their proof uses the method of moving planes and the fundamental result of \textit{R. Schoen} and \textit{S.-T. Yau} [Invent. Math. 92, No. 1, 47--71 (1988; Zbl 0658.53038)] on developing maps for locally conformally manifolds. They also give a generalization of the above result for more general conformally invariant equations.
Mathematics Subject Classification ID: 53C20
Mathematics Subject Classification ID: 35J60
Mathematics Subject Classification ID: 58J05
zbMATH DE Number: 2133469
zbMATH Keywords: conformally invariant equations
zbMATH Keywords: moving planes method
MaRDI profile type: Publication
full work available at URL: s00209-003-0608-x
OpenAlex ID: W2160514810
DOI: 10.1007/S00209-003-0608-X
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description: scientific article; zbMATH DE number 2133469