Minimizers in polyhedral space

[摘要] Area minimizing surfaces and energy minimizing maps from surfaces into piecewise Euclidean pseudo 3-manifolds are studied. These spaces considered contain geometric singularities, having curvature concentrated at vertices and straight-line segments, and topological singularities, non-manifold points. We examine how these types of singularity affect area minimizing surfaces using calibration forms which also give information about foliations of those spaces by area minimizing surfaces. We extend the notion of foliation to allow for these singularities in the ambient space by defining a codimension one pseudo-foliation. Also the Poincare-Hopf index theorem tells us that a foliation of any compact connected surface other than a genus one torus will contain singularities. If we allow for this type of singularity (singular pseudo-foliation), we can place a piecewise Euclidean structure on the cone of any compact connected orientable boundaryless topological surface so that there is a space of such area minimizing piecewise planar pseudo-foliations containing S1 union a point. Conversely cones of non-connected compact surfaces will not admit pseudo-foliations. Dirichlet energy minimizing maps from one two-dimensional cone to any other two dimensional cone can map the cone point to the cone point. Thus a geometric singularity in the domain is mapped onto a geometric singularity in the range. However in an energy minimizing map from a Moebius band to a disc, we can show that the topological singularity of the map, where the image of a curve is a point, will not coincide with a positive curvature geometric singularity in the range. Topological collapsing is also demonstrated in limits of images under energy minimizing maps of planar domain surfaces into R 3 with Dirichlet boundary data. The limit of images of energy minimizing maps can be surface components connected by straight line segments. As the limit is approached the metric on the domain degenerates as its conformal class approaches the boundary of its moduli space. The straight-line segments are achieved in the Hausdorff set topology and in the general varifold topology. Also varifold compactness theorems were developed for this proof.

[发布机构] Rice University