Conformal deformations of metrics on non-compact quotients of real hyperbolic space (Q2753161)

The article is devoted to conformal deformations of metrics on non-compact quotients of a real hyperbolic space. The hyberbolic space \(\mathbb{H}^n\) is realized as \(SO_0(n,1) /SO(n)\), where the group \(SO_0(n,1)\) consists of matrices preserving the bilinear form in \(\mathbb{R}^{n+1}\) of signature \((n,1)\).NEWLINENEWLINENEWLINETheorem 1.1. Let \(\gamma\) be a hyperbolic element of \(SO_0(n,1)\), \(n\geq 3\), and \(\Gamma:= \langle\gamma \rangle\). Let \(a,b,k,p\) be constants such that \(a\leq b< 0\), \(-{(n-1)^2 \over 4}<k <0\), \(p>1\), and let \(K\) be a Hölder continuous function on \(M:=\Gamma \setminus\mathbb{H}^n\). NEWLINENEWLINENEWLINEThen there exist \(\varepsilon >0\), \(\alpha>1\) such that if \(a\leq k\leq b\) outside a compact subset \(M_0\) of \(M\) and \(K\leq \varepsilon\) on \(M\) then equation (3) \(\Delta u+Ku^p=ku\) has a solution \(u\in C^2(M)\) such that NEWLINE\[NEWLINE\left( {k\over a}\right)^{1/(p-1)}\leq u\leq \left({k \over b}\right)^{1/(p-1)} e^\alpha.\tag{4}NEWLINE\]NEWLINE The problem of finding a positive solution \(u\in C^2(M)\) of (3) is equivalent to the following: Given a complete noncompact Riemannian manifold \(M\) of dimension \(n\geq 3\) with metric \(g\) and scalar curvature \(S\), find a complete metric \(\widetilde g\) on \(M\), conformal to \(g\), with a given function \(\widetilde S\) as its scalar curvature.