For a connected planar (genus-zero) surface with a complete path metric of curvature at least −1-1, the hyperbolic Weyl problem asks whether the surface is isometric to the boundary of a convex set in hyperbolic 3-space whose ideal boundary at infinity is a circle-type closed set (a compact subset of the sphere at infinity each of whose connected components is a closed round disk or a point), and whether this realization is unique up to ambient isometry. Recent work of Luo–Wu, Luo–Luo–Rao, and Zhao settles both existence and uniqueness only under a countability restriction on the topological ends (equivalently, on the components of the ideal boundary). The open core is the case of uncountably many ends: existence there is equivalent, by a theorem of Luo and Wu, to the classical Koebe circle domain conjecture for planar domains with uncountably many boundary components, and uniqueness requires extending the known rigidity theorems to circle-type ideal boundaries with uncountably many point components. Whether the countability hypothesis is optimal is also open.

Background

The classical Weyl problem asks whether every Riemannian metric of positive Gaussian curvature on the 2-sphere can be realized as the intrinsic metric of a convex surface in Euclidean 3-space; it was answered affirmatively by Alexandrov and Pogorelov. Pogorelov's rigidity theorem strengthens this: if two compact convex bodies in R3\mathbb{R}^{3} (or in hyperbolic 3-space H3\mathbb{H}^{3}, via the Pogorelov map — Pogorelov's construction that transports an isometry between convex surfaces in constant-curvature H3\mathbb{H}^{3} or S3\mathbb{S}^{3} to an isometry between convex surfaces in Euclidean R3\mathbb{R}^{3}, where Euclidean rigidity applies) have boundaries that are isometric in their intrinsic path metrics, the isometry extends to an ambient isometry, so the realization is unique.

For non-compact convex sets in H3\mathbb{H}^{3} the naive extension fails. A construction attributed to Thurston takes the hyperbolic convex hull CP(J)C_{P}(J) — the smallest closed convex subset of H3\mathbb{H}^{3} whose closure at infinity contains JJ — of a Jordan curve JJ on the sphere at infinity ∂H3\partial\mathbb{H}^{3}: the boundary ∂CP(J)\partial C_{P}(J) (often called the Thurston dome of JJ) is intrinsically isometric to two copies of the hyperbolic plane H2\mathbb{H}^{2} regardless of JJ, but CP(J1)C_{P}(J_{1}) and CP(J2)C_{P}(J_{2}) are globally isometric only if the curves differ by a Möbius transformation (a conformal self-map of the Riemann sphere; the Möbius transformations are exactly the extensions to ∂H3\partial\mathbb{H}^{3} of the isometries of H3\mathbb{H}^{3}), e.g. a square versus a non-square rectangle. Hence some restriction on the behavior at infinity is necessary for rigidity.

The right framework singles out circle-type data. A planar (genus-zero) surface is a surface homeomorphic to a connected domain in the 2-sphere, and a topological end of such a non-compact surface is an equivalence class of nested complementary compact subsets — intuitively, a distinct way of escaping to infinity, one per boundary component of the domain. A circle domain is an open connected subset of the Riemann sphere C^=∂H3\hat{\mathbb{C}}=\partial\mathbb{H}^{3} each of whose boundary components is a round circle or a point; a compact set is a circle-type closed set if its complement is a circle domain, i.e. each of its connected components is a closed round disk or a point. A theorem of Thurston states that for any closed set X⊂∂H3X\subset\partial\mathbb{H}^{3} with at least three points, the boundary of the hyperbolic convex hull ∂CP(X)\partial C_{P}(X) (the metric double, in the 2-dimensional case) is a complete hyperbolic surface of genus zero. Conversely, the hyperbolic Weyl problem, posed by Luo and Wu, asks for the reverse direction: every genus-zero complete hyperbolic surface — more generally, every connected planar surface with a complete path metric of curvature at least −1-1 (the curvature bound understood in the sense of Alexandrov's comparison geometry for length spaces) — should be isometric to such a convex hull boundary with circle-type ideal boundary, uniquely up to isometry of H3\mathbb{H}^{3} (equivalently, up to Möbius transformations of the ideal set).

The existence side is tied to complex analysis. Koebe's circle domain conjecture (1908) asserts that every planar domain is conformally equivalent to a circle domain; He and Schramm proved it for domains with at most countably many boundary components, which remains the best general result. Luo and Wu proved that the Koebe conjecture is in fact equivalent to the existence part of the hyperbolic Weyl problem (their tool is Schramm's transboundary extremal length — a variant of the classical conformal invariant in which the admissible curve families are allowed to cross the ideal boundary), so existence for surfaces with countably many topological ends follows from He–Schramm. On the uniqueness side, Luo–Luo–Rao established rigidity when the ideal boundary has vanishing 1-dimensional Hausdorff measure (measure-zero point sets), and Zhao recently extended rigidity to circle-type ideal boundaries with countably many components, or finitely many components together with a zero-measure remainder. Together these results resolve the hyperbolic-surface form of the conjecture for surfaces with countably many topological ends (Zhao's Theorem 1.6), and the uniqueness part of the stronger curvature-≥−1\geq -1 conjecture for countably many ends. Two structural facts delimit the remaining case: a circle-type closed set can have at most countably many disk components (each disk contains a point with rational coordinates), so beyond the countable regime the ideal boundary has uncountably many point components; and when the union of these point components has zero length, the Luo–Luo–Rao theorem already applies, so the genuinely unresolved uniqueness case concerns point components of positive total 1-dimensional Hausdorff measure.

Problem Statement

Resolve the Luo–Wu Weyl conjecture for convex surfaces in hyperbolic 3-space in the regime not covered by the countability theorems, i.e. for genus-zero surfaces with uncountably many topological ends. The underlying conjecture, in its strongest form (Conjecture 3 of Luo–Wu 2024, restated as Conjecture 1 of Zhao 2026), asserts: if (S,d)(S,d) is a connected planar surface with a complete path metric of curvature at least −1-1, then SS is isometric to a complete convex surface YY in H3\mathbb{H}^{3} bounding a convex set XX whose closure meets the sphere at infinity ∂H3\partial\mathbb{H}^{3} in a circle-type closed set, and YY is unique up to isometry of H3\mathbb{H}^{3}. The sharply posed surviving core, as identified in the source, has two linked parts, both open:

(a) Existence for uncountably many ends: prove or refute that every genus-zero complete hyperbolic surface with uncountably many topological ends is isometric to ∂CP(X)\partial C_{P}(X), the boundary of the hyperbolic convex hull of a circle-type closed set X⊂∂H3X\subset\partial\mathbb{H}^{3}. By Luo–Wu's Theorem 1.1 this is equivalent to the Koebe circle domain conjecture for planar domains with uncountably many boundary components: every domain in the Riemann sphere is conformally equivalent to a circle domain.

(b) Uniqueness/rigidity for uncountably many components: prove or refute that for two circle-type closed sets X1,X2⊂∂H3X_{1},X_{2}\subset\partial\mathbb{H}^{3} with uncountably many components (necessarily point components), any intrinsic isometry between the convex hull boundaries ∂CP(X1)\partial C_{P}(X_{1}) and ∂CP(X2)\partial C_{P}(X_{2}) forces X1X_{1} and X2X_{2} to differ by a Möbius transformation — i.e. extend Zhao's Theorem 1.1 beyond countably many components (the uncovered case is point components whose union has positive 1-dimensional Hausdorff measure).

Both parts constitute one scientific objective, the completion of the Weyl problem in H3\mathbb{H}^{3} for all genus-zero surfaces in the hyperbolic case of curvature identically −1-1; the existence half of the strongest curvature-≥−1\geq -1 form for complete path metrics whose curvature is not identically −1-1 is a separate component that is not addressed by (a) or (b). The source also records the open question whether the countability hypothesis on the ends (equivalently, on the ideal-boundary components) is optimal: does there exist a strictly weaker condition on the ideal boundary, compatible with Thurston's counterexample, that still guarantees rigidity?

References

  1. Feng Luo and Tianqi Wu, "The Koebe conjecture and the Weyl problem for convex surfaces in hyperbolic 3-space", Advances in Mathematics 458 (2024), Paper No. 109969, 56 pp. DOI: 10.1016/j.aim.2024.109969, https://doi.org/10.1016/j.aim.2024.109969; arXiv:1910.08001, https://arxiv.org/abs/1910.08001
  2. Xinrong Zhao, "Rigidity Theorems for the Weyl Problem of Convex Surfaces in Hyperbolic 3-Space", arXiv:2606.15402 (2026). DOI: 10.48550/arXiv.2606.15402, https://arxiv.org/abs/2606.15402
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