A family of closed competitor sets is a good class when every member, inside almost every small ball, does no worse in Hausdorff measure than its cone competitor and its cup competitors - the two elementary comparison surfaces built by coning over the trace on a sphere and by gluing in a spherical cap. For any minimizing sequence of countably rectifiable sets drawn from such a class, the associated Hausdorff measures converge weak-star to a limit carried by a rectifiable set of density at least one, which delivers lower semicontinuity of area and a monotonicity formula without leaving the theory of Radon measures. Students learn how that single principle yields existence for the homotopy-spanning formulation of Plateau's problem and for sliding minimizers, and why minimizing over closed sets differs from minimizing over integer rectifiable currents.
Plateau's problem asks for a surface of least area among the sets that span a prescribed boundary. This paper treats the formulation in which one minimizes the n-dimensional Hausdorff measure over a …