The Gauss-Bonnet-Chern mass of conformally flat manifolds
type: Item
instance of: scholarly article
zbMATH Open document ID: 1302.53038
author: Yuxin Ge
author: Guofang Wang
author: Jie Wu
publication date: 2014-10-21T00:00:00Z
full work available at URL: https://arxiv.org/abs/1212.3213
Mathematics Subject Classification ID: 53C20
Mathematics Subject Classification ID: 53C24
Mathematics Subject Classification ID: 53C25
Mathematics Subject Classification ID: 53C80
zbMATH DE Number: 6359518
zbMATH Keywords: Gauss-Bonnet-Chern mass
zbMATH Keywords: curvature
zbMATH Keywords: positive mass theorem
zbMATH Keywords: Penrose inequality
zbMATH Keywords: asymptotically flat conformally flat manifold
MaRDI profile type: Publication
OpenAlex ID: W2962777580
Wikidata QID: Q124853317
arXiv ID: 1212.3213
DOI: 10.1093/IMRN/RNT095
published in: IMRN. International Mathematics Research Notices
title: The Gauss-Bonnet-Chern mass of conformally flat manifolds
review text: The authors consider the \(k\)th Gauß-Bonnet-Chern mass \(m_{k}\), introduced in [the authors, Adv. Math. 266, 84--119 (2014; Zbl 1301.53032)] and constructed from the \(k\)th Gauß-Bonnet curvature \(L_{k}\) on an asymptotically flat conformally flat manifold \((M,g)=(\mathbb{R}^{n}\backslash\Omega,e^{-2u}\delta)\) (\(n>2k\) and \(\Omega\subset\mathbb{R}^{n}\) open) of decay order \(\tau > \frac{n-2k}{k+1}\) on which \(L_{k}\) is integrable. The main results are two positive mass theorems and a Penrose-type inequality. The positive mass theorems state that if \(M=\mathbb{R}^{n}\) and either \(L_{j}\geq 0\) for all \(j\leq k\) or \(k\) is even and \((-1)^{j}L_{j}\geq 0\) for all \(j\leq k\), then \(m_{k}\geq 0\) and equality holds iff \(M\) is flat. Furthermore, the Penrose-type inequality reads NEWLINE\[NEWLINEm_{k}\geq \left(\frac{|\partial\Omega|}{\omega_{n-1}} \right)^{\frac{n-2k}{n-1}}NEWLINE\]NEWLINE whenever \(\Omega\) is convex, \(\partial\Omega\) is minimal in \((M,g)\) and \(u\) is constant on \(\partial\Omega\), where \(\omega_{n-1}={\mathrm{Vol}}(S^{n-1})\); moreover, if \(k\geq 2\), then the inequality NEWLINE\[NEWLINEm_{k}\geq \left(\frac{\int_{\partial\Omega}R}{(n-1)(n-2)\omega_{n-1}} \right)^{\frac{n-2k}{n-3}}NEWLINE\]NEWLINE also holds, where \(R\) is the scalar curvature of \(\partial\Omega\) considered as a submanifold of \(\mathbb{R}^{n}\). These results are meant to support a conjecture posed by the authors in [loc. cit.] on the positivity of the Gauß-Bonnet-Chern mass.NEWLINENEWLINEThe structure of the paper is as follows: In Section 1 the Gauß-Bonnet curvature and Gauß-Bonnet-Chern mass are defined and the main results are stated and elaborated upon. Section 2 recasts the Gauß-Bonnet curvatures of locally conformally flat metrics in terms of symmetric functions and the Schouten tensor, which then simplifies the computations carried out in the following sections. The positive mass theorems are then proved in Section 3 and Penrose-type inequaliti…
reviewed by: Ahmad Afuni
description: scientific article; zbMATH DE number 6359518