Geometric triangulations and discrete Laplacians on manifolds

TL;DR

This paper explores discrete Laplacians on manifolds using triangulation techniques, analyzing their properties and algorithms, and extending graph Laplacian concepts to piecewise Euclidean manifolds.

Contribution

It introduces a detailed study of regular triangulations and their flip algorithms, extending weighted Laplacian concepts from graphs to manifolds.

Findings

Analysis of weighted and regular triangulations

Development of flip algorithms for triangulations

Extension of graph Laplacian to manifold setting

Abstract

This paper uses the technology of weighted and regular triangulations to study discrete versions of the Laplacian on piecewise Euclidean manifolds. Regular triangulations are studied in some detail, including flip algorithms. The Laplacian is then studied as an operator on functions of the vertices as a generalized weighted Laplacian on graphs.