Escape obstructions for a monotone round-to-Calabi-Croke path

Definitions and mathematical statement

Babenko's Question $8.7.1$ asks for a continuous or smooth family $(g_t)_{0\leq t\leq1}$ of sphere metrics allowing conical singularities, with $g_0$ round, $g_1$ the preceding example, and $\sigma(g_t)$ decreasing. The aid reconstructs the missing context as the Calabi--Croke metric, the double of an equilateral Euclidean triangle, and the scale-invariant systolic area \[ \sigma(g)=\frac{\operatorname{Area}(S^2,g)}{\operatorname{sys}(g)^2}, \] where $\operatorname{sys}(g)$ is the shortest nonconstant closed-geodesic length under the chosen conical-geodesic convention. This is a geodesic systole on $S^2$, not the length of a homotopically nontrivial loop. The precise topology on conical metrics and the word decreasing, strict versus nonincreasing, are not defined in the extracted sentence.

For the round metric $\sigma=1/\pi$. For the reconstructed Calabi--Croke endpoint, with three cone angles $2\pi/3$, the aid uses $\sigma=1/(2\sqrt3)$. After constant-area normalization $\operatorname{Area}=4\pi$, the question becomes a path with nondecreasing systole from \[ \operatorname{sys}(g_0)=2\pi\quad\text{to}\quad \operatorname{sys}(g_1)=\sqrt{8\pi\sqrt3}. \] Strict decrease of $\sigma$ would correspond to strict increase of the normalized systole.

The reported smooth-end obstruction assumes that $g_t$ is an ordinary smooth metric for $t$ in some initial interval and is $C^2$-continuous there. If $\sigma(g_t)$ is nonincreasing from the round endpoint, then on a smaller initial interval \[ \sigma(g_t)=1/\pi, \] and all these metrics are Zoll: their geodesics are closed with a common period in the stated equality case. Thus a strictly decreasing path starting immediately at the round metric is impossible in this smooth-through-the-end category. This local plateau does not rule out a nonincreasing conical path which later leaves that class, and no uniform plateau duration is specified.

The aid also reports that no continuous nonincreasing path can reach the Calabi--Croke endpoint if all interior metrics are smooth spheres of revolution, using the corresponding rotational systolic obstruction. This conclusion requires a path topology for which the relevant area and systole comparison passes to the endpoint; that topology and the exact rotational theorem are not provided. It is a restriction on the proposed ansatz, not a resolution for arbitrary conical metrics.

The canonical record ends "A simpler version of this question is," without the simpler question. Its missing statement is not invented. The original general conical-path question is reported open despite these partial obstructions.

Short English statement

The reconstructed question seeks a nonincreasing systolic-area path from the round sphere to the doubled equilateral triangle. The aid forces a Zoll plateau near the round endpoint if the path initially remains smooth and $C^2$-continuous, and reports an obstruction to keeping all intermediate metrics smooth and rotationally symmetric. General conical paths remain open.

Sources

• Local UnsolvedMath registry, permanent ID $20001918$, source code catalogue entry; canonical collection aim-geometry-notes.json, notes[255]. Source item $8.7$, actual question $8.7.1$, attributed to Babenko.
• AIM workshop "Geodesics," Burns--Matveev report "Open problems and questions about geodesics": https://aimath.org/pastworkshops/geodesicsproblems.pdf. The supplied verification reports inspecting Section $8.7$, pages 18--19, and adjacent records to reconstruct the endpoint and invariant. The exact extraction is truncated before a simpler version. This session did not open the PDF.
• Supplied research attempt $1$, partial_result, problem status at run "open," rigor label "proved," novelty confidence "low." Registry status partially_solved, classification PARTIAL-PROGRESS, updated 14 August 2026. Original remarks and literature are empty; no full authoritative references for the Zoll/local or rotational obstruction, proposer-field value, or proposed year are supplied.