Compactification of a class of conformally flat 4-manifold
type: Item
instance of: scholarly article
title: Compactification of a class of conformally flat 4-manifold
zbMATH Open document ID: 0990.53026
author: Jie Qing
author: Paul C. Yang
author: Sun-Yung Alice Chang
published in: Inventiones Mathematicae
publication date: 2002-08-12T00:00:00Z
review text: Let \((M,g)\) be a complete non-compact \(4\)-manifold such that \(M=N\cup \{ \bigcup_{i=1}^{k}E_{i}\} \), where \((N,g)\) is a compact Riemannian manifold with boundary \(\partial N=\bigcup_{i=1}^{k}\partial E_{i}\), and each \(E_{i}\) is a conformally flat simple end of \(M\) (i.e. \((E_{i},g)=(\mathbb{R}^{4}\backslash B,\) \(e^{2\omega i}|dx|^{2})\) for some function \(\omega _{i}\), where \(B\) is the unit ball in \(\mathbb{R}^{4}\). Then we say \((M,g)\) is a complete 4-manifold with a finite number of conformally flat simple ends. According to the authors there are many examples of such manifolds, and they give some. This paper contains the detailed proof of three results that generalise Huber's result on complete surfaces of finite total curvature. Following the introduction, let us recall that in the theory of complete surfaces of finite total curvature, \textit{S. Cohn-Vossen} [Compos. Math. 2, 69-133 (1935; Zbl 0011.22501)] showed that if the Gauss curvature of a complete analytic metric is absolutely integrable, then \(\int KdA\leqq 2\pi \chi \), where \(\chi \) is the Euler number of the surface. \textit{A. Huber} [Comment. Math. Helv. 32, 13-72 (1957; Zbl 0080.15001)] extended this inequality to metrics with weaker regularity and proved that such surfaces can be conformally compactified by adjoining a finite number of points. For dimension four \textit{R. E. Greene} and \textit{H. Wu} [Acta Math. 137, 209-245 (1976; Zbl 0372.53019)] obtained a generalization of Cohn-Vossen's inequality to complete manifolds of positive sectional curvature outside a compact set. In a previous paper, the authors [Duke Math. J. 103, 523-544 (2000; Zbl 0971.53028)] proved the following result: suppose that \(e^{2\omega }|dx|^{2}\) on \(\mathbb{R}^{4}\) is a complete metric with its \(Q\)-curvature (Paneitz curvature) absolutely integrable, and suppose that its scalar curvature is negative at infinity, then \[ 1-\frac{1}{4\pi ^{4}}\int_{\mathbb{R}^{4}}Qe^{4\omega} dx=\lim_{r…
reviewed by: Ana Pereira do Vale
Mathematics Subject Classification ID: 53C20
Mathematics Subject Classification ID: 53A30
zbMATH DE Number: 1533361
zbMATH Keywords: conformal compactification
zbMATH Keywords: conformally flat
zbMATH Keywords: conformal metric
zbMATH Keywords: positive sectional curvature
zbMATH Keywords: Paneitz operator
zbMATH Keywords: Paneitz curvature
zbMATH Keywords: volume growth
zbMATH Keywords: conformally flat end
zbMATH Keywords: Kleinian groups
zbMATH Keywords: convex polyhedra
zbMATH Keywords: geometrically finite convex polyhedra
MaRDI profile type: Publication
full work available at URL: s002220000083
OpenAlex ID: W2070612742
Wikidata QID: Q126181007
DOI: 10.1007/S002220000083
description: scientific article; zbMATH DE number 1533361