Discrete Laplacians -- spherical and hyperbolic

Abstract

The discrete Laplacian on Euclidean triangulated surfaces is a well-established notion. We introduce discrete Laplacians on spherical and hyperbolic triangulated surfaces. On the one hand, our definitions are close to the Euclidean one in that the edge weights contain the cotangents of certain combinations of angles and are non-negative if and only if the triangulation is Delaunay. On the other hand, these discretizations are structure-preserving in several respects. We prove that the area of a convex polyhedron can be written in terms of the discrete spherical Laplacian of the support function, whose expression is the same as the area of a smooth convex body in terms of the usual spherical Laplacian. We show that the conformal factors of discrete conformal vector fields on a triangulated surface of curvature $k \in \{-1,1\}$ are $-2k$-eigenfunctions of our discrete Laplacians, exactly as in the smooth setting. The discrete conformality can be understood here both in the sense of the vertex scaling and in the sense of circle patterns. Finally, we connect the $-2k$-eigenfunctions to infinitesimal isometric deformations of a polyhedron inscribed into corresponding quadrics.