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

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28 Citations (Scopus)

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.

Original languageEnglish (US)
Pages (from-to)201-237
Number of pages37
JournalJournal of Differential Geometry
Volume87
Issue number2
StatePublished - Feb 2011

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Scalar Curvature
Flat Manifold
Three-dimensional
Curvature
Angle
Derivative
Circle packing
Einstein Manifold
Formulation
Total curvature
Riemannian Metric
Rigidity
Hilbert
Albert Einstein
Convexity
Euclidean
Circle
Intersection
Vary
Analogue

ASJC Scopus subject areas

  • Algebra and Number Theory
  • Analysis
  • Geometry and Topology

Cite this

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