Structure theorems of the scalar curvature equation on subdomains of a compact Riemannian manifold
type: Item
instance of: scholarly article
title: Structure theorems of the scalar curvature equation on subdomains of a compact Riemannian manifold
zbMATH Open document ID: 0907.53030
author: Shin Kato
published in: Tsukuba Journal of Mathematics
publication date: 1999-03-04T00:00:00Z
review text: Suppose \((M,g)\) is a Riemannian manifold with \(\dim M=n\geq 3\), \(\Delta_g\) the Laplacian of \(g\), \(S_g\) the scalar curvature and \(L_g\) the conformal Laplacian of \(g\), that is, \[ L_g= -a_n\Delta_g+ S_g, \] where \(a_n= 4(n-1)/(n-2)\). Assume that \(u\) is a positive smooth function on \(M\), and define a conformal metric by \[ \widetilde{g}= u^{4/(n-2)}g. \] It follows that its scalar curvature is given by \[ S_{\widetilde{g}}= u^{-1} L_gu, \] where \(q= (n+2)/(n-2)= 4/(n-2)+1\). Therefore, there exists a smooth function \(f\) on \(M\) which can be expressed as the scalar curvature of some metric and which is pointwise conformal to \(g\) if and only if the problem \[ (f,M):\begin{cases} L_gu= fu^q,\\ u>0\end{cases} \quad\text{on }M \] has a smooth solution. The author studies the equation \((f,M)\) in the case when \((M,g)\) is a subdomain of a compact Riemannian manifold \((\overline{M}, \overline{g})\). The main aim of the paper is to study the case when \(\lambda_1 (L_{\overline{g}})>0\), \((M,g)\) is the complement \(\overline{M} \setminus \Sigma\) of a compact submanifold \(\Sigma\), and \(f\) is a nonpositive function.
Mathematics Subject Classification ID: 53C21
Mathematics Subject Classification ID: 35J30
zbMATH DE Number: 1102917
zbMATH Keywords: scalar curvature
zbMATH Keywords: compact Riemannian manifold
zbMATH Keywords: Martin boundary
zbMATH Keywords: Green function
zbMATH Keywords: supersolution
Wikidata QID: Q115233549
reviewed by: Themistocles M. Rassias
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.21099/tkbjm/1496163168
OpenAlex ID: W311914516
DOI: 10.21099/TKBJM/1496163168
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description: scientific article; zbMATH DE number 1102917