Journal of Differential Geometry

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

David Glickenstein

Full-text: Open access

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.

Article information

Source
J. Differential Geom., Volume 87, Number 2 (2011), 201-238.

Dates
First available in Project Euclid: 4 May 2011

Permanent link to this document
https://projecteuclid.org/euclid.jdg/1304514973

Digital Object Identifier
doi:10.4310/jdg/1304514973

Mathematical Reviews number (MathSciNet)
MR2788656

Zentralblatt MATH identifier
1243.52014

Citation

Glickenstein, David. Discrete conformal variations and scalar curvature on piecewise flat two- and three-dimensional manifolds. J. Differential Geom. 87 (2011), no. 2, 201--238. doi:10.4310/jdg/1304514973. https://projecteuclid.org/euclid.jdg/1304514973


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