On the Weyl problem for complete surfaces in the hyperbolic and anti-de Sitter spaces

TL;DR

This paper explores generalizations of the Weyl problem for complete convex surfaces in hyperbolic, anti-de Sitter, Minkowski, and Half-pipe geometries, connecting classical geometry with modern open problems.

Contribution

It proposes three conjectural extensions of the Weyl problem to unbounded convex sets and surfaces, linking them to contemporary issues in geometry and complex analysis.

Findings

Descriptions of conjectural generalizations in hyperbolic and anti-de Sitter spaces.

Connections to Kleinian groups, circle packings, and grafting problems.

Collection of partial new results based on recent research.

Abstract

The classical Weyl problem (solved by Lewy, Alexandrov, Pogorelov, and others) asks whether any metric of curvature $K\geq 0$ on the sphere is induced on the boundary of a unique convex body in $\R^3$. The answer was extended to surfaces in hyperbolic space by Alexandrov in the 1950s, and a ``dual'' statement, describing convex bodies in terms of the third fundamental form of their boundary (e.g. their dihedral angles, for an ideal polyhedron) was later proved. We describe three conjectural generalizations of the Weyl problem in $\HH^3$ and its dual to unbounded convex subsets and convex surfaces, in ways that are relevant to contemporary geometry since a number of recent results and well-known open problems can be considered as special cases. One focus is on convex domain having a ``thin'' asymptotic boundary, for instance a quasicircle -- this part of the problem is strongly related to the theory of Kleinian groups. A second direction is towards convex subsets with a ``thick'' ideal boundary, for instance a disjoint union of disks -- here one find connections to problems in complex analysis, such as the Koebe circle domain conjecture. A third direction is towards complete, convex disks of infinite area in $\HH^3$ and surfaces in hyperbolic ends -- with connections to questions on circle packings or grafting on the hyperbolic disk. Similar statements are proposed in anti-de Sitter geometry, a Lorentzian cousin of hyperbolic geometry where interesting new phenomena can occur, and in Minkowski and Half-pipe geometry. We also collect some partial new results mostly based on recent works.