Geometric triangulations and discrete Laplacians on manifolds: An update

Abstract

This paper uses the technology of weighted triangulations to study discrete versions of the Laplacian on piecewise Euclidean manifolds. Given a collection of Euclidean simplices glued together along their boundary, a geometric structure on the Poincaré dual may be constructed by considering weights at the vertices. We show that this is equivalent to specifying sphere radii at vertices and generalized intersection angles at edges, or by specifying a certain way of dividing the edges. This geometric structure gives rise to a discrete Laplacian operator acting on functions on the vertices. We study these geometric structures in some detail, considering when dual volumes are nondegenerate, which corresponds to weighted Delaunay triangulations in dimension 2, and how one might find such nondegenerate weighted triangulations. Finally, we talk briefly about the possibilities of discrete Riemannian manifolds.