Rigidity Theorems for the Weyl Problem of Convex Surfaces in Hyperbolic 3-Space

Abstract

In this paper, we study the rigidity of non-compact convex sets in hyperbolic 3-space. We prove that any intrinsic isometry between the boundaries of two non-compact closed convex subsets in hyperbolic 3-space extends to a global isometry of the ambient space, provided that their ideal boundaries are circle-type closed sets with countably many connected components. Moreover, the same conclusion holds if the ideal boundaries consist of a circle-type closed set with finitely many connected components together with a set of one-dimensional Hausdorff measure zero. This result generalizes a recent rigidity theorem of Luo, Luo, and Rao by allowing the ideal boundaries to contain disk components. As a direct consequence, we establish a uniqueness result concerning the Weyl problem for convex surfaces in hyperbolic 3-space, as proposed by Luo and Wu. In particular, our approach provides an alternative proof of the discrete Schwarz lemma. The proof uses Pogorelov's rigidity theorem for compact convex bodies in $\mathbb{R}^3$, the Pogorelov map, and properties of locally convex surfaces in $\mathbb{R}^3$.