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Discrete Yamabe problem for polyhedral surfaces

Hana Dal Poz Kouřimská

Abstract

We study a new discretization of the Gaussian curvature for polyhedral surfaces. This discrete Gaussian curvature is defined on each conical singularity of a polyhedral surface as the quotient of the angle defect and the area of the Voronoi cell corresponding to the singularity. We divide polyhedral surfaces into discrete conformal classes using a generalization of discrete conformal equivalence pioneered by Feng Luo. We subsequently show that, in every discrete conformal class, there exists a polyhedral surface with constant discrete Gaussian curvature. We also provide explicit examples to demonstrate that this surface is in general not unique.

Discrete Yamabe problem for polyhedral surfaces

Abstract

We study a new discretization of the Gaussian curvature for polyhedral surfaces. This discrete Gaussian curvature is defined on each conical singularity of a polyhedral surface as the quotient of the angle defect and the area of the Voronoi cell corresponding to the singularity. We divide polyhedral surfaces into discrete conformal classes using a generalization of discrete conformal equivalence pioneered by Feng Luo. We subsequently show that, in every discrete conformal class, there exists a polyhedral surface with constant discrete Gaussian curvature. We also provide explicit examples to demonstrate that this surface is in general not unique.

Paper Structure

This paper contains 27 sections, 30 theorems, 127 equations, 12 figures.

Table of Contents

  1. Introduction
  2. Fundamental definitions and results
  3. Tessellations of piecewise flat surfaces, discrete metric
  4. Voronoi tessellation
  5. Delaunay tessellation and triangulation
  6. Discrete metric
  7. Hyperbolic metrics, ideal tessellations, and Penner coordinates
  8. Ideal Delaunay tessellations and triangulations
  9. Penner coordinates
  10. From piecewise flat surfaces to decorated hyperbolic surfaces and back again
  11. Discrete conformal classes
  12. Counterexamples to uniqueness of metrics with constant curvature
  13. Tetrahedra with constant curvature
  14. Surfaces of genus two with two marked points and constant curvature
  15. Variational principles
  16. ...and 12 more sections

Key Result

Theorem 1.2

For every PL-metric $d$ on a marked surface $(S,V)$, there exists a discrete conformally equivalent PL-metric $\tilde{d}$ such that the piecewise flat surface $(S,V,\tilde{d})$ has constant discrete Gaussian curvature.

Figures (12)

  • Figure 1: A Delaunay and a non-Delaunay edge.
  • Figure 2: Penner coordinates of a decorated ideal hyperbolic triangle $ijk$, in the Poincaré disc model.
  • Figure 3:
  • Figure 4: A tetrahedron (left) and a division of areas in the two triangles (right).
  • Figure 5:
  • ...and 7 more figures

Theorems & Definitions (64)

  • Definition 1.1
  • Theorem 1.2: Discrete uniformization theorem
  • Proposition 2.1
  • Definition 2.2
  • Definition 2.4
  • Theorem 2.5: bo
  • Proposition 2.6: bo
  • Definition 2.7
  • Definition 2.8
  • Theorem 2.10: bo
  • ...and 54 more