Existence and Compactness of Minimizers of the Yamabe Problem on Manifolds with Boundary

Abstract

We show existence of minimizers of the Yamabe functional on a compact Riemannian manifold with boundary (M,g), of dimension n ≥ 3, restricted to the set of all metrics conformal to g and satisfying aV + bA = 1, where V and A are the volume of M and area of δM, respectively, when a and b are positive real numbers and when the infimum of the functional on that set is stricly less than the corresponding quantity on the standard Euclidean half-sphere. This shows that for such manifolds we can deform g conformally to obtain a metric with constant scalar curvature R and constant mean curvature h on the boundary which are related by bR = 2nha. These results are already known when (M,g) is locally conformally flat or when n ≥ 5 and δM is not umbilic. They extend for arbitrary positive a and b results known for the case when a = 1, b = 0, the case when a = 0, b = 1, and the case when b is small. We also show a compactness result for the set of all minimizers when the metric is allowed to vary on a small neighborhood of a given base metric satisfying the above condition.