Derived Schläfli Formula for Spherical Cone Metrics

Definitions and mathematical statement

A spherical cone metric on the topological sphere $S^2$ is a metric of Gaussian curvature $+1$ away from finitely many marked points, with a conical singularity of total angle $\alpha_i>0$ at the $i$-th marked point. Let $k$ be the fixed number of such points, and set \[ A=2\pi(2-k)+\sum_{i=1}^{k}\alpha_i. \] In this curvature normalization, $A$ is the area given by the cone version of Gauss--Bonnet.

Conditional formulation of the recorded relation. Consider any differentiable family of realizable spherical cone metrics with fixed $k$, parametrized locally by $t$ in an open interval $I\subset{\mathbb{R}}$, with differentiable angles $\alpha_i(t)>0$. Assume that the construction called its Hopf lift to $S^3$ in the supplied record is defined along this family and has volume \[ V(t)=\frac{A(t)^2}{8},\qquad A(t)=2\pi(2-k)+\sum_{i=1}^{k}\alpha_i(t). \] The supplied relation, called the Schläfli formula in the record, is \[ 2\,\frac{\mathrm dV}{\mathrm dt}(t) =\frac{A(t)}{2}\sum_{i=1}^{k}\frac{\mathrm d\alpha_i}{\mathrm dt}(t) \qquad(t\in I), \] or, along such admissible deformations, \[ 2\,\mathrm dV=\frac{A}{2}\sum_{i=1}^{k}\mathrm d\alpha_i. \] The record does not specify the lifted metric, its normalization, the admissible angle configurations, or the deformation class. The formulation therefore assumes the stated volume formula and quantifies only over families for which the geometric construction exists; it does not assert that arbitrary independent angle variations are realizable. No limiting assertion at degenerations or vanishing cone angles is supplied.

Unresolved status of the intended problem. The supplied verification describes this relation as an already completed calculation from the stated volume formula, rather than an open problem. The registry nevertheless labels the entry open and its statement status unrecoverable. A separate supplied assessment says the extracted conjecture was truncated before its claim. These conflicting qualifications prevent identification of a distinct, precise open conjecture.

Short English statement

For a realizable differentiable family of spherical cone metrics with a fixed number of cone points, the record gives a volume formula for an unspecified Hopf lift and records the corresponding Schläfli relation. The supplied verification treats this as a completed calculation; the intended open problem and the precise lift construction remain unresolved.

Sources

• UnsolvedMath local registry, permanent ID $30002122$, source problem code catalogue entry, title: Derived Schläfli Formula for Spherical Cone Metrics. The supplied record retains the original statement for provenance, supplies no separate clean statement, labels the entry open, and marks the statement status as unrecoverable. No individual proposer is supplied; the recorded proposed year is 2012.
• Supplied report citation: Discrete Differential Geometry (2013), Oberwolfach Reports, pp. 2077--2137, DOI 10.4171/owr/2012/34. Source URL and literature-source link, labelled "OWR report": https://doi.org/10.4171/owr/2012/34. The supplied background describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions. No precise internal location for this relation is supplied.
• Supplied literature assessment dated 2026-08-21: status open; the extracted conjecture is described as truncated before its claim, preventing verification of a precise proposition. This is supplied dated curation, not an independently verified literature assessment. The background explicitly requests independent verification of time-sensitive status and cited claims.
• Supplied statement-verification note: the source derives the displayed relation directly from $V=A^2/8$, calls it the well-known Schläfli formula, and treats it as a completed calculation rather than an open problem. This source characterization is supplied metadata and has not been independently checked.