Combinatorial scalar curvature and rigidity of ball packings (original) (raw)
Related papers
2009
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.
Rigidity of polyhedral surfaces, II
Geometry & Topology, 2009
This paper investigates several global rigidity issues for polyhedral surfaces including inversive distance circle packings. Inversive distance circle packings are polyhedral surfaces introduced by P. Bowers and K. Stephenson in [2] as a generalization of Andreev-Thurston's circle packing. They conjectured that inversive distance circle packings are rigid. Using a recent work of R. Guo [9] on variational principle associated to the inversive distance circle packing, we prove rigidity conjecture of Bowers-Stephenson in this paper. We also show that each polyhedral metric on a triangulated surface is determined by various discrete curvatures introduced in [12], verifying a conjecture in . As a consequence, we show that the discrete Laplacian operator determines a Euclidean polyhedral metric up to scaling.
Some Progress in Conformal Geometry
Symmetry, Integrability and Geometry: Methods and Applications, 2007
In this paper we describe our current research in the theory of partial differential equations in conformal geometry. We introduce a bubble tree structure to study the degeneration of a class of Yamabe metrics on Bach flat manifolds satisfying some global conformal bounds on compact manifolds of dimension 4. As applications, we establish a gap theorem, a finiteness theorem for diffeomorphism type for this class, and diameter bound of the σ 2-metrics in a class of conformal 4-manifolds. For conformally compact Einstein metrics we introduce an eigenfunction compactification. As a consequence we obtain some topological constraints in terms of renormalized volumes.
Duality structures and discrete conformal variations of piecewise constant curvature surfaces
2014
Source: arXiv CITATIONS 2 READS 26 2 authors, including: Abstract. A piecewise constant curvature manifold is a triangulated manifold that is assigned a geometry by specifying lengths of edges and stipulating that for a chosen background geometry (Euclidean, hyperbolic, or spherical), each simplex has an isometric embedding into the background geometry with the chosen edge lengths. Additional structure is defined either by giving a geometric structure to the Poincaré dual of the triangulation or by assigning a discrete metric, a way of assigning length to oriented edges. This notion leads to a notion of discrete conformal structure, generalizing the discrete conformal structures based on circle packings and their generalizations studied by Thurston and others. We define and analyze conformal variations of piecewise constant curvature 2-manifolds, giving particular attention to the variation of angles. We give formulas for the derivatives of angles in each background geometry, which yield formulas for the derivatives of curvatures. Our formulas allow us to identify particular curvature functionals associated with conformal variations. Finally, we provide a complete classification of discrete conformal structures in each of the background geometries.
Discrete conformal maps and ideal hyperbolic polyhedra
Geometry & Topology, 2015
We establish a connection between two previously unrelated topics: a particular discrete version of conformal geometry for triangulated surfaces, and the geometry of ideal polyhedra in hyperbolic three-space. Two triangulated surfaces are considered discretely conformally equivalent if the edge lengths are related by scale factors associated with the vertices. This simple definition leads to a surprisingly rich theory featuring Möbius invariance, the definition of discrete conformal maps as circumcirclepreserving piecewise projective maps, and two variational principles. We show how literally the same theory can be reinterpreted to address the problem of constructing an ideal hyperbolic polyhedron with prescribed intrinsic metric. This synthesis enables us to derive a companion theory of discrete conformal maps for hyperbolic triangulations. It also shows how the definitions of discrete conformality considered here are closely related to the established definition of discrete conformality in terms of circle packings.
Simplices modelled on spaces of constant curvature
J. Comput. Geom., 2019
We give non-degeneracy criteria for Riemannian simplices based on simplices in spaces of constant sectional curvature. It extends previous work on Riemannian simplices, where we developed Riemannian simplices with respect to Euclidean reference simplices. The criteria we give in this article are in terms of quality measures for spaces of constant curvature that we develop here. We see that simplices in spaces that have nearly constant curvature, are already nondegenerate under very weak quality demands. This is of importance because it allows for sampling of Riemannian manifolds based on anisotropy of the manifold and not (absolute) curvature.
Riemannian simplices and triangulations
Geometriae Dedicata, 2015
We study a natural intrinsic definition of geometric simplices in Riemannian manifolds of arbitrary finite dimension, and exploit these simplices to obtain criteria for triangulating compact Riemannian manifolds. These geometric simplices are defined using Karcher means. Given a finite set of vertices in a convex set on the manifold, the point that minimises the weighted sum of squared distances to the vertices is the Karcher mean relative to the weights. Using barycentric coordinates as the weights, we obtain a smooth map from the standard Euclidean simplex to the manifold. A Riemannian simplex is defined as the image of this barycentric coordinate map. In this work we articulate criteria that guarantee that the barycentric coordinate map is a smooth embedding. If it is not, we say the Riemannian simplex is degenerate. Quality measures for the "thickness" or "fatness" of Euclidean simplices can be adapted to apply to these Riemannian simplices. For manifolds of dimension 2, the simplex is non-degenerate if it has a positive quality measure, as in the Euclidean case. However, when the dimension is greater than two, non-degeneracy can be guaranteed only when the quality exceeds a positive bound that depends on the size of the simplex and local bounds on the absolute values of the sectional curvatures of the manifold. An analysis of the geometry of non-degenerate Riemannian simplices leads to conditions which guarantee that a simplicial complex is homeomorphic to the manifold.
Circle packings in the approximation of conformal mappings
Bulletin of the American Mathematical Society, 1990
Connections between circle packings and analytic functions were first suggested by William Thurston [T2], who conjectured that the conformai mapping of a simply connected plane domain Q to the unit disc A could be approximated by manipulating hexagonal circle configurations lying in Q. The conjecture was confirmed by Rodin and Sullivan [RS]. Their proof relies heavily on the hexagonal combinatorics of the circle configurations, a restriction not suggested by the underlying intuition. The purpose of this note is to announce that Thurston's conjecture is true with much weaker combinatoric hypotheses and to outline the proof. The main lines of argument are those developed by Rodin and Sullivan, but the proof is independent. The deepest part of their work-a uniqueness result of Sullivan's which depends on Mostow rigidity-is replaced here by probabilistic arguments. We work in the setting of hyperbolic geometry and make use of the discrete Schwarz-Pick lemma proven in [BS] to understand the behavior of circle configurations. We analyze how curvature distributes itself around a packing as successive differential changes are made to boundary circles, ultimately modelling this process as a random walk. The proof that a certain limiting random walk is recurrent replaces the uniqueness result of Sullivan. Details will appear elsewhere. The author gratefully acknowledges support of the National Science Foundation and the Tennessee Science Alliance. STATEMENT OF THE MAIN RESULT Let P denote a finite collection of circles in the plane having mutually disjoint interiors. Connect centers of tangent circles with euclidean line segments. If a triangulation of a simply connected closed region of the plane results, we say that P is a circle packing.