Discrete conformal variations and scalar curvature on piecewise flat two and three dimensional manifolds

TL;DR

This paper develops a framework for understanding how angles and scalar curvatures vary in piecewise flat manifolds under conformal changes, generalizing circle packing and enabling analysis of curvature functionals.

Contribution

It introduces formulas for angle derivatives and curvature variations in 2D and 3D piecewise flat manifolds, extending conformal variation concepts and enabling curvature analysis.

Findings

Derived explicit formulas for angle derivatives in 2D and 3D

Connected curvature variations to scalar curvature changes

Proved rigidity theorems for constant curvature and Einstein-like manifolds

Abstract

A piecewise flat manifold is a triangulated manifold given a geometry by specifying edge lengths (lengths of 1-simplices) and specifying that all simplices are Euclidean. We consider the variation of angles of piecewise flat manifolds as the geometry varies in a particular way, which we call a conformal variation. This variation generalizes variations within the class of circles with fixed intersection angles (such as circle packings) as well as other formulations of conformal variation of piecewise flat manifolds previously suggested. We describe the angle derivatives of the angles in two and three dimensional piecewise flat manifolds, giving rise to formulas for the derivatives of curvatures. The formulas for derivatives of curvature resemble the formulas for the change of scalar curvature under a conformal variation of Riemannian metric. They allow us to explicitly describe the variation of certain curvature functionals, including Regge's formulation of the Einstein-Hilbert functional (total scalar curvature), and to consider convexity of these functionals. They also allow us to prove rigidity theorems for certain analogues of constant curvature and Einstein manifolds in the piecewise flat setting.