  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.
