Classification of spherical quadrilaterals

Definitions and mathematical statement

A spherical quadrilateral is an oriented surface homeomorphic to a closed disk, with four distinct boundary vertices $v_1,v_2,v_3,v_4$ in cyclic order, equipped with a metric of Gaussian curvature $1$ away from the vertices. Each boundary arc between successive vertices is geodesic, and the interior angle at $v_j$ is $\pi\alpha_j$, with $\alpha_j>0$. The metric is smooth in the interior and along the open sides; the vertices have the prescribed corner singularities.

Two such quadrilaterals are identified by an isometry preserving the ordered vertices. The underlying conformal quadrilateral has modulus $m>0$: it is conformally equivalent to a Euclidean rectangle of width $1$ and height $m$, with vertices matched in the specified cyclic order.

Classify spherical quadrilaterals for arbitrary positive angle data. Determine which quadruples $(\alpha_1,\ldots,\alpha_4)$ are realizable, which moduli $m$ occur for each realizable quadruple, and the isometry classes for fixed angles and modulus, including their possible multiplicity. The area identity \[ \operatorname{Area}=\pi (\alpha_1+\alpha_2+\alpha_3+\alpha_4-2) \] gives a necessary positivity condition, but the problem requires the full restrictions. Angles larger than $\pi$ are allowed. The metric need not come from a simply embedded region of the standard sphere; its developing map may overlap. Restricting to convex spherical quadrilaterals would omit the main general cases.

Short English statement

Classify curvature-one quadrilaterals by their angles, conformal modulus, and possible nonuniqueness.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 3700011 (AMR-036-0011), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Eremenko - My favorite unsolved problems, Possible shapes of spherical quadrilaterals (2025 update). Original source indexed by AMR-036-0011. https://www.math.purdue.edu/~eremenko/uns1.html.
• S3 A. Eremenko, Possible shapes of spherical quadrilaterals. Author-maintained primary problem note. https://www.math.purdue.edu/~eremenko/dvi/quadr.pdf.