Weakly differentiable functions on varifolds

In this talk a theory of weakly differentiable functions on rectifiable varifolds with locally bounded first variation will be presented. The concept proposed here is defined by means of integration by parts identities for certain compositions with smooth functions. Results include a variety of Sobolev Poincaré type embeddings, embeddings into spaces of continuous and sometimes Hölder continuous functions, pointwise differentiability results both of approximate and integral type as well as coarea formulae.

As applications the finiteness of the geodesic distance associated to varifolds with suitable summability of the mean curvature and a characterisation of curvature varifolds are obtained.