Regularity of Isoperimetric Regions that are Close to a Smooth Manifold

In this paper we prove a regularity theorem for isoperimetric regions T that are close in flat norm to an open bounded set B with smooth boundary in a smooth complete (possibly noncompact) n-dimensional Riemannian manifold () (the dimension n being arbitrary) with Ricci curvature bounded below and volume of balls uniformly bounded below with respect to its center by a positive constant. In fact we prove that under the above assumptions the boundary of T is smooth and is the normal graph of function u whose Holder norms are controlled by the volume of the symmetric difference . Moreover we allow the metric g to be variable and obtain a suitable regularity result for applications to the study of the isoperimetric profile.