On existence of the prescribing k-curvature problem on manifolds with boundary

Abstract

In this paper, we study the problem of conformally deforming a metric to a prescribed kth order symmetric function of the eigenvalues of the Schouten tensor on compact Riemannian manifolds with totally geodesic boundary. We prove the solvability of the problem and the compactness of the solution set for the case k ≥ n/2, provided the conformal class admits a k-admissible metric. These results have been proved by Gursky and Viaclovsky, Trudinger and Wang for the manifolds without boundary, and by Jin et al. and S. Chen for the locally conformally flat manifolds with boundary.