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Koebe conjecture and the Weyl problem for convex surfaces in hyperbolic 3-space

Feng Luo, Tianqi Wu

Abstract

We prove that the Koebe circle domain conjecture is equivalent to the Weyl type problem that every complete hyperbolic surface of genus zero is isometric to the boundary of the hyperbolic convex hull of the complement of a circle domain. It provides a new way to approach the Koebe's conjecture using convex geometry. Combining our result with the work of He-Schramm on the Koebe conjecture, one establishes that every simply connected non-compact polyhedral surface is discrete conformal to the complex plane or the open unit disk. The main tool we use is Schramm's transboundary extremal lengths.

Koebe conjecture and the Weyl problem for convex surfaces in hyperbolic 3-space

Abstract

We prove that the Koebe circle domain conjecture is equivalent to the Weyl type problem that every complete hyperbolic surface of genus zero is isometric to the boundary of the hyperbolic convex hull of the complement of a circle domain. It provides a new way to approach the Koebe's conjecture using convex geometry. Combining our result with the work of He-Schramm on the Koebe conjecture, one establishes that every simply connected non-compact polyhedral surface is discrete conformal to the complex plane or the open unit disk. The main tool we use is Schramm's transboundary extremal lengths.

Paper Structure

This paper contains 39 sections, 48 theorems, 113 equations, 8 figures.

Table of Contents

  1. Introduction
  2. The main result
  3. History and a generalized Weyl problem in $\mathbf{H}^3$
  4. Organization of the paper and acknowledgement
  5. Notations, Preliminaries, and Outline of the proof of Theorem \ref{['1.222']}
  6. Notations and Preliminaries
  7. Basic facts about convex surfaces and shortest distance projections
  8. Basic facts about area and conformal structures on surfaces of bounded curvature
  9. Outline of the proof of Theorem \ref{['1.222']} (a)
  10. Outline of the proof of Theorem \ref{['1.222']} (b)
  11. A proof of a special case of Theorem \ref{['1.222']}
  12. Shortest distance projections and area estimates on convex surfaces
  13. The Poincaré and Klein models of hyperbolic 3-space
  14. Shortest distance projections and area of hyperbolic convex surfaces
  15. Another estimate for the hyperbolic shortest distance projection
  16. ...and 24 more sections

Key Result

Theorem 1.1

(a) For any circle domain $U \subset \hat{\mathbb{C}}$ whose boundary contains at least three points, there exists a circle type closed set $Y \subset \mathbf S^2$ such that the Poincaré metric $(U, d^U)$ is isometric to $\partial C_P(Y)$. (b) For any circle type closed set $Y \subset \mathbf S^2$ w

Figures (8)

  • Figure 1: Lipschitz property of inversion
  • Figure 2: Lipschitz property of nearest point projection
  • Figure 3: Eestimate for shortest distance nearest projection
  • Figure 4: The rings $A_j$'s.
  • Figure 5: The annulus surface $S$ in $\partial C_P(W) \cup W$.
  • ...and 3 more figures

Theorems & Definitions (83)

  • Theorem 1.1
  • Corollary 1.2
  • Lemma 2.1
  • Corollary 2.2
  • Lemma 2.3
  • Theorem 3.1
  • proof
  • Theorem 4.1
  • Proposition 4.2
  • proof
  • ...and 73 more