1.2 (Choi) — Convex projective deformations from a CR structure

Definitions and mathematical statement

Let $M$ be a hyperbolic $3$-manifold admitting a CR structure, not assumed spherical. In real dimension $3$, a hypersurface-type CR structure consists of a rank-two distribution $H\subset TM$ and a complex structure $J:H\to H$, with the relevant integrability condition; no Levi nondegeneracy or strict pseudoconvexity hypothesis is printed. Spherical means locally CR equivalent to the standard CR $3$-sphere, and is expressly not required.

A real projective structure is an atlas into ${\mathbb{R}}{\mathbb{P}}^3$ with transition maps in $\operatorname{PGL}(4,{\mathbb{R}})$. A convex projective structure is described by suitable developing and holonomy data with convex developing image; proper versus strict convexity and global developing-map requirements must be fixed. Let $\mathcal D_{\mathrm{conv}}(M)$ denote the corresponding deformation space with the intended marking/equivalence conventions.

Choi asks whether the deformation theory of $\mathcal D_{\mathrm{conv}}(M)$ can be understood in terms of the CR structure on $M$. This asks for a precise construction or description relating the CR data to the projective deformations, not a supplied equality of two already defined moduli spaces. The source gives no map, parameterization, infinitesimal comparison or obstruction statement to prove. It proposes hyperbolic Coxeter $3$-orbifolds as a potentially easier setting for the same investigation. Completeness, compactness, volume and end/boundary conditions on $M$, as well as what CR data vary or remain fixed, are not specified.

Short English statement

Can a CR structure, even a nonspherical one, explain convex real projective deformations of a hyperbolic $3$-manifold? Investigate hyperbolic Coxeter $3$-orbifolds as the suggested easier case.

Sources

• Choi, attributed item 1.2 in Delp, Hoffoss and Manning, Problems In Groups, Geometry, and Three-Manifolds (2015), supplied PDF page 1. https://arxiv.org/pdf/1512.04620#page=1; source-status link https://arxiv.org/abs/1512.04620. Recorded access 2026-07-29; extraction source-tex checked against rendered PDF text, as reported by the supplied record.
• Local UnsolvedMath registry, data/unsolvedmath/problems.json, permanent ID $10900002$, exact source code catalogue entry. Its OPEN-TRIAGE summary mentions established surrounding holonomy openness/closedness work but supplies no exact additional reference or verified resolution of this question.