Discrete Yamabe problem for polyhedral surfaces

Abstract

We introduce a new discretization of the Gaussian curvature on piecewise at surfaces. As the prime new feature the curvature is scaled by the factor 1/r2 upon scaling the metric globally with the factor r. We develop a variational principle to tackle the corresponding discrete uniformisation theorem – we show that each piecewise at surface is discrete conformally equivalent to one with constant discrete Gaussian curvature. This surface is in general not unique. We demonstrate uniqueness for particular cases and disprove it in general by providing explicit counterexamples. Special attention is paid to dealing with change of combinatorics.