Existence and compactness of minimizers of the Yamabe problem on manifolds with boundary
Let \((M,g)\) be a Riemannian manifold with boundary. In [J. Differ. Geom. 35, No. 1, 21--84 (1992; Zbl 0771.53017)], \textit{P. Escobar} showed that, if the dimension of \(M\) is greater than or equal to three, \(M\) can be endowed with a metric \(\tilde g\) conformal to \(g\) having constant scalar curvature and minimal boundary. The total scalar curvature plus total mean curvature functional \(E\), defined on the space of Riemannian metrics of a compact manifold with boundary, arises in connection with this generalization of the Yamabe problem for manifolds with boundary. The critical points of \(E\) on a given conformal class of metrics (subject to various volume and area constraints) are metrics of constant scalar curvature and constant mean curvature on the boundary. In the present paper, the author considers the set of all metrics conformal to \(g\) and satisfying \(aV + bA =1\), where \(a\) and \(b\) are positive real numbers, \(V\) and \(A\) are the volume of \(M\) and area of the boundary of \(M\) respectively. He proves the existence of minimizers of the functional \(E\) restricted to the above set and when the infimum of \(E\) on that set is stricly less than the corresponding quantity on the standard Euclidean half-sphere. This allows to deform \(g\) conformally to a metric having constant scalar curvature \(R\) and constant mean curvature \(H\) on the boundary and such that \(R\) and \(H\) are subject to the relation \(bR =2nHa\). Moreover, the author establishes compactness results for the set of all minimizers when the base metric is allowed to vary on a small \(C^3\) neighbourhood in the set of Riemannian metrics.
scientific article; zbMATH DE number 2109260