  This paper introduces first order Sobolev spaces on certain rectifiable
varifolds. These complete locally convex spaces are contained in the generally
nonlinear class of generalised weakly differentiable functions and share key
functional analytic properties with their Euclidean counterparts.
  Assuming the varifold to satisfy a uniform lower density bound and a
dimensionally critical summability condition on its mean curvature, the
following statements hold. Firstly, continuous and compact embeddings of
Sobolev spaces into Lebesgue spaces and spaces of continuous functions are
available. Secondly, the geodesic distance associated to the varifold is a
continuous, not necessarily H\"older continuous Sobolev function with bounded
derivative. Thirdly, if the varifold additionally has bounded mean curvature
and finite measure, the present Sobolev spaces are isomorphic to those
previously available for finite Radon measures yielding many new results for
those classes as well.
  Suitable versions of the embedding results obtained for Sobolev functions
hold in the larger class of generalised weakly differentiable functions.

Classical Analysis and ODEs;Analysis of PDEs;Differential Geometry