A novel type of Sobolev-Poincar\'e inequality for submanifolds of Euclidean space

Abstract

For functions on generalised connected surfaces (of any dimensions) with boundary and mean curvature, we establish an oscillation estimate in which the mean curvature enters in a novel way. As application we prove an a priori estimate of the geodesic diameter of compact connected smooth immersions in terms of their boundary data and mean curvature. These results are developed in the framework of varifolds. For this purpose, we establish that the notion of indecomposability is the appropriate substitute for connectedness and that it has a strong regularising effect; we thus obtain a new natural class of varifolds to study. Finally, our development leads to a variety of questions that are of substance both in the smooth and the nonsmooth setting.Comment: 35 pages, no figure