Boundary regularity for the Ricci equation, geometric convergence, and   Gel'fand's inverse boundary problem
  This paper explores and ties together three themes. The first is to establish regularity of a metric tensor, on a manifold with boundary, on which there are given Ricci curvature bounds, on the manifold and its boundary, and a Lipschitz bound on the mean curvature of the boundary. The second is to establish geometric convergence of a (sub)sequence of manifolds with boundary with such geometrical bounds and also an upper bound on the diameter and a lower bound on injectivity and boundary injectivity radius, making use of the first part. The third theme involves the uniqueness and conditional stability of an inverse problem proposed by Gel'fand making essential use of the results of the first two parts. 
Michael T. Anderson, Atsushi Katsuda, Yaroslav Kurylev, Matti Lassas   and Michael E. Taylor