Classification of Elliptic Affine Spheres and Hemispheres

Definitions and mathematical statement

Scope and missing definitions. The supplied record concerns elliptic affine spheres, hyperbolic affine spheres, completeness, and a proposed affine-hemisphere condition. It does not define these notions or specify the dimension, ambient setting, regularity, or global hypotheses on the hypersurfaces. In particular, it does not state which metric or other notion determines completeness, nor does it give the affine-hemisphere condition. These choices remain unresolved rather than being imposed here.

Research question, as supplied. Why are complete elliptic affine spheres much rarer than hyperbolic affine spheres, and can replacing completeness by the affine-hemisphere condition provide a meaningful classification?

Limits of the formulation. The expression "much rarer" has no specified mathematical meaning in the record: no comparison of classes, parameter spaces, or existence criteria is stated. The requested classification likewise has no specified equivalence relation, prescribed data, or proposed conclusion. Thus the supplied question describes a research direction, but does not determine a precise quantified conjecture or classification problem. A precise statement requires the definitions and hypotheses above, together with an explicit classification target; these cannot be recovered from the supplied text alone.

Short English statement

The question asks why complete elliptic affine spheres appear more restrictive than hyperbolic affine spheres, and whether an affine-hemisphere condition leads to a useful classification. The record leaves completeness, the hemisphere condition, the comparison, and the classification target unspecified, so a precise conjecture remains unresolved.

Sources

• Supplied local UnsolvedMath registry record, permanent ID 30003064, source problem code catalogue entry, titled "Classification of Elliptic Affine Spheres and Hemispheres," in the Oberwolfach Reports Open Problems collection. The supplied proposed year is 2016; no individual proposer is identified.
• Supplied source citation: Asymptotic Geometric Analysis (2016), Oberwolfach Reports, pp. 507--565, DOI 10.4171/owr/2016/11. https://doi.org/10.4171/owr/2016/11. The background identifies Oberwolfach Reports as reports of mathematical workshops and their open-problem discussions.
• Supplied literature source labeled "affine hemispheres": https://www.weizmann.ac.il/math/klartag/sites/math.klartag/files/uploads/hemi_paper.pdf. The record supplies no further bibliographic details or specific passage defining the intended condition.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open and states that affine hemispheres are proposed as a way to study the scarcity of complete elliptic affine spheres, while the record contains no classification theorem resolving the broader existence and classification question. This is supplied dated curation derived from the OWR material, not independently verified here; time-sensitive status and cited claims require independent verification.