Constructing barriers to minimal surfaces from polyhedral data

The results of geometric measure theory assert, under very general conditions, the existence of an area-minimizing surface spanning a given boundary. However, the arguments are compactness-based and do not indicate how one might actually find an area-minimizer.;The "Surface Evolver" computer program written by K. Brakke can produce triangulated polyhedral surfaces which very nearly minimize area in competition with polyhedral surfaces obtained by small displacements of the vertices.;This dissertation gives conditions, which the Evolver program could verify, that ensure the existence of a classical minimal submanifold closely approximated by the polyhedral "minimal candidate".;A central requirement is that two suitably separated stable polyhedral surfaces be produced when the Evolver is used to apply pressure to the minimal candidate, first on one side and then on the other.;It is shown how one may use such separated, pressurized polyhedral surfaces to produce nearby piecewise-smooth surfaces which are "barriers" to minimal surfaces. General existence theorems guarantee that there will be a smooth minimal submanifold trapped between the two barriers. (We need only know that such nearby barriers exist; it is not necessary to compute them explicitly.);In order to know that the barriers exist, one must have a reasonably equilateral triangulation on the pressurized surfaces; the Evolver can provide this. One also needs the triangulation size to be small in comparison to the overall geometry, as measured by a polyhedral approximation to "principal curvatures".;Under a "curvature stability" hypothesis, this method leads to a convergence proof for the Surface Evolver: as the triangulation grid-size decreases, the two barriers which trap the smooth minimal surface can be obtained arbitrarily close to the polyhedral minimal candidate.