A Note on Discrete Einstein Metrics

Abstract

In this note, we prove that the space of all admissible piecewise linear metrics parameterized by the square of length on a triangulated manifold is a convex cone. We further study Regge’s Einstein-Hilbert action and give a more reasonable definition of discrete Einstein metric than the former version. Finally, we introduce a discrete Ricci flow for three dimensional triangulated manifolds, which is closely related to the existence of discrete Einstein metrics.