A note on the Kazdan-Warner type conditions
type: Item
instance of: scholarly article
title: A note on the Kazdan-Warner type conditions
zbMATH Open document ID: 0822.53026
author: Wenxiong Chen
author: Congming Li
published in: Journal of Differential Geometry
publication date: 1995-10-05T00:00:00Z
review text: We consider prescribing Gaussian curvature on \(S^ 2\). There are well- known Kazdan-Warner [\textit{J. L. Kazdan} and \textit{F. W. Warner}, Ann. Math., II. Ser. 99, 14-47 (1974; Zbl 0273.53034), ibid. 101, 317-331 (1975; Zbl 0297.53020)] and Bourguignon-Ezin [\textit{J. P. Bourguignon} and \textit{J. P. Ezin}, Trans. Am. Math. Soc. 301, 723-736 (1987; Zbl 0622.53023)] necessary conditions for a function \(K\) to be the Gaussian curvature of some pointwise conformal metric. Then are those necessary conditions also sufficient? This is a problem of common concern and has been open for a few years. In this paper, we answer the question negatively. First, we show that if \(K\) is rotationally symmetric and is monotone in the region where \(K > 0\), then the problem has no rotationally symmetric solution. Then we provide a family of functions \(K\) satisfying the Kazdan-Warner and Bourguignon-Ezin conditions, for which the problem has no solution at all. We also consider prescribing scalar curvature on \(S^ n\) for \(n \geq 3\). We prove the nonexistence of rotationally symmetric solutions for the above mentioned functions.
Mathematics Subject Classification ID: 53C20
Mathematics Subject Classification ID: 53C21
Mathematics Subject Classification ID: 58J60
zbMATH DE Number: 761164
zbMATH Keywords: nonlinear elliptic equations
zbMATH Keywords: method of moving planes
zbMATH Keywords: Kazdan-Warner conditions
zbMATH Keywords: prescribing Gaussian curvature
zbMATH Keywords: prescribing scalar curvature
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.4310/jdg/1214456217
OpenAlex ID: W1575131919
Wikidata QID: Q115178695
DOI: 10.4310/JDG/1214456217
description: scientific article; zbMATH DE number 761164