A Rigidity Theorem for Convex Sets in Hyperbolic 3-Space

TL;DR

This paper extends Pogorelov's rigidity theorem to certain non-compact convex sets in hyperbolic 3-space, showing boundary metrics determine the sets up to isometry under a zero-length limit set condition.

Contribution

It establishes a new rigidity result for non-compact convex sets in hyperbolic 3-space with zero-length limit sets, generalizing classical compact case results.

Findings

Boundary metric determines convex set up to isometry under zero-length limit set

Zero-length condition is proven to be optimal

Connection to complex analysis via Painlevé removability theorem

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 Painlev\'e 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.