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A Rigidity Theorem for Convex Sets in Hyperbolic 3-Space

Feng Luo, Yanwen Luo, Zhenghao Rao

Abstract

Pogorelov's rigidity theorem states that a compact convex body in the hyperbolic 3-space is determined up to isometry by the intrinsic path metric on its boundary. The main result of this paper addresses a rigidity problem for non-compact closed convex 3-dimensional subsets in hyperbolic 3-space. We show that the intrinsic path metric on the boundary determines a closed convex set up to isometry, provided that the set of limit points of the convex set at infinity of the hyperbolic 3-space has vanishing 1-dimensional Hausdorff measure, i.e., zero length. Furthermore, this zero-length condition is optimal. This can be considered as an analogue of the Painlevé removability theorem in complex analysis, which states that sets of zero length are removable for bounded holomorphic functions. As a corollary, we show that if the underlying complex structure of a connected polyhedral surface is of parabolic type, then it is discrete conformal, unique up to scaling, to a complete flat surface marked with a discrete subset. The proof uses Pogorelov's rigidity theorem for compact convex bodies in $\mathbb R^3$, the Pogorelov map, and the Tabor--Tabor theorem on the extension of locally convex functions.

A Rigidity Theorem for Convex Sets in Hyperbolic 3-Space

Abstract

Pogorelov's rigidity theorem states that a compact convex body in the hyperbolic 3-space is determined up to isometry by the intrinsic path metric on its boundary. The main result of this paper addresses a rigidity problem for non-compact closed convex 3-dimensional subsets in hyperbolic 3-space. We show that the intrinsic path metric on the boundary determines a closed convex set up to isometry, provided that the set of limit points of the convex set at infinity of the hyperbolic 3-space has vanishing 1-dimensional Hausdorff measure, i.e., zero length. Furthermore, this zero-length condition is optimal. This can be considered as an analogue of the Painlevé removability theorem in complex analysis, which states that sets of zero length are removable for bounded holomorphic functions. As a corollary, we show that if the underlying complex structure of a connected polyhedral surface is of parabolic type, then it is discrete conformal, unique up to scaling, to a complete flat surface marked with a discrete subset. The proof uses Pogorelov's rigidity theorem for compact convex bodies in , the Pogorelov map, and the Tabor--Tabor theorem on the extension of locally convex functions.
Paper Structure (12 sections, 25 theorems, 84 equations, 9 figures)

This paper contains 12 sections, 25 theorems, 84 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Basic Properties of the Pogorelov map
  4. The Pogorelov map on Minkowski spaces
  5. The Pogorelov map on the hyperboloid $\mathcal{H}^n$
  6. Congruency-preserving property
  7. Star-shapedness and local convexity
  8. Two extension results
  9. Proof to Theorem \ref{['main2']}
  10. Proof of Theorem \ref{['main2']} in case (i): both $X_1$ and $X_2$ are $3$-dimensional
  11. Proof of Theorem \ref{['main2']} in case (ii): both $X_1$ and $X_2$ are 2-dimensional
  12. Proof of Theorem \ref{['main2']} in case (iii): exactly one of $X_1$ and $X_2$ is 3-dimensional

Key Result

Theorem 1.1

If $P$ and $Q$ are two compact convex bodies in the 3-dimensional Euclidean space $\mathbb R^3$ whose boundaries are isometric with respect to the intrinsic path metrics, then any isometry between the boundaries of $P$ and $Q$ extends to an isometry of $\mathbb R^3$.

Figures (9)

  • Figure 1: Thurston's isometry $T$ between domes and the Riemann mapping $R$ between the domains below domes.
  • Figure 2: The discrete Riemann mapping $f:V \to W$ is induced by the isometric map between the domes over $\Omega-V$ and $\mathbb{D}-W$.
  • Figure 3: Length cross-ratio, the empty-disk condition, and convex hulls.
  • Figure 4: A Delaunay triangulation in $\mathbb C$ and the convex hull of its vertex set.
  • Figure 5: A circle packing on $\mathbb C$.
  • ...and 4 more figures

Theorems & Definitions (41)

  • Theorem 1.1: Pogorelov
  • Theorem 1.2
  • Corollary 1.3
  • Corollary 1.4
  • Theorem 1.5: Schramm, schramm
  • Corollary 1.6
  • Lemma 2.1
  • proof
  • Definition 3.1: Pogorelov
  • Lemma 3.2
  • ...and 31 more