Structure theorems of the scalar curvature equation on subdomains of a compact Riemannian manifold
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.
scientific article; zbMATH DE number 1102917