Conformal metrics with prescribed mean curvature on the boundary (Q1815288)

Given a compact \(n\)-dimensional manifold \((M,g)\) with boundary \(\partial M\) and a function \(f\) on \(\partial M\), does \(M\) carry a metric \(\tilde g\) that is conformally related to \(g\), such that \(\tilde g\) has zero scalar curvature and \(\partial M\) has mean curvature \(f\) with respect to \(\tilde g\)? This paper gives a partial answer to this question when the Sobolev quotient \(Q(M,\partial M)\) is finite and nonnegative. The conformal invariant \(Q(M,\partial M)\) is positive (zero) iff \(M\) carries a conformally related metric of positive (zero) scalar curvature and zero mean curvature on the boundary. If \(Q(M,\partial M)>0\), \(n=3\) and \(f\) is positive somewhere, such a \(\tilde g\) exists provided \((M,g)\) is not conformally related to the ball in \({\mathbb R}^3\). If \(n>5\) and \(\partial M\) has a nonumbilic point, the author gives a condition on \(f\) sufficient for the existence of \(\tilde g\). If \(n\geq 3\), \(M\) is locally conformally flat, \(\partial M\) is umbilic and \(M\) is not conformally related to the \(n\)-dimensional Euclidean ball, there is another sufficient condition on \(f\) for the existence of \(\tilde g\); in dimensions \(n=4,5\) it is not necessary to assert that \(M\) is locally conformally flat. If \(Q(M,\partial M)=0\) and \(n\geq 3\), the author gives a necessary and sufficient condition for the existence of \(\tilde g\). This paper improves the author's results in [Ann. Math., II. Ser. 136, 1-50 (1992; Zbl 0766.53033)].