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

By David Glickenstein. 2009

123 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Computational Mechanics, Computer Graphics and Computer-Aided Design

View Paper: https://www.semanticscholar.org/paper/45bc7d16448beca916f7a23abe8a98fc4554dde8

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

1. Xu Xu
2. Huabin Ge
3. Feng Luo
4. Ren Guo
5. David Glickenstein
6. Xianfeng David Gu
7. Boris A. Springborn
8. Wenshuai Jiang

In Computational Mechanics

1. Xu Xu
2. Ren Guo
3. Feng Luo
4. Huabin Ge

In Computer Graphics and Computer-Aided Design

1. Ren Guo
2. Feng Luo
3. Xu Xu
4. Huabin Ge

Citations in the corpus

• Rigidity of inversive distance circle packings revisited (2017, methodology): “We refer to Glickenstein [18] for a nice geometric interpretation of the variational principle in [19].”
• Rigidity of polyhedral surfaces, II (2006, methodology): “A very nice geometric interpretation of the variational principle was given by Glickenstein [7].”
• Discrete quasi-Einstein metrics and combinatorial curvature flows in 3-dimension (2013, background): “In [10], Glickenstein considered the curvature Ci = Kiri.”
• A discrete uniformization theorem for polyhedral surfaces (2013, unknown): “Other different versions of discrete Riemann mapping theorem and discrete conformality have appeared in the work by Thurston [39], Cannon [12], Schramm [37], Glickenstein [18], Hersonsky [22], Mercat [31] and others.”
• Rigidity of Polyhedral Surfaces, III (2010, methodology): “A very nice geometric interpretation of the variational principle was given by Glickenstein [7].”