A Note on Discrete Einstein Metric

TL;DR

This paper explores the structure of discrete Einstein metrics on triangulated manifolds, proposing a convex cone framework, refining the Einstein metric definition, and introducing a discrete Ricci flow related to metric existence.

Contribution

It introduces a convex cone structure for piecewise linear metrics, refines the definition of discrete Einstein metrics, and proposes a discrete Ricci flow for 3D triangulated manifolds.

Findings

The space of admissible metrics forms a convex cone.

A more reasonable definition of discrete Einstein metric is provided.

A discrete Ricci flow for 3D triangulated manifolds is introduced.

Abstract

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