Open Problems and Questions about Geodesics — Question 45 (Babenko)

Definitions and mathematical statement

The supplied question refers to a sphere with round metric $g_{\mathrm{rd}}$, a space $\mathcal G$ of metrics with conical singularities, a particular metric $g_{\mathrm{ex}}$ described as "the metric from the example above," and a quantity $\sigma(g)$. The referenced example and the definition of $\sigma$ are absent from the supplied record. The dimension of the sphere, the permitted conical singularities, and the topology or differentiable structure on $\mathcal G$ are also unspecified. Thus these symbols are placeholders for source data that must be recovered before the question has a fully determined mathematical meaning.

Question (Babenko; incomplete source formulation). With the missing data fixed as intended in the source, does there exist a continuous or smooth family \[ [0,1]\longrightarrow\mathcal G,\qquad t\longmapsto g_t, \] such that \[ g_0=g_{\mathrm{rd}},\qquad g_1=g_{\mathrm{ex}}, \] and $t\mapsto\sigma(g_t)$ is decreasing? Here $[0,1]$ records the interpolation between the two stated endpoint parameters. The source excerpt does not specify whether "decreasing" means \[ \sigma(g_t)\leq\sigma(g_s)\qquad(0\leq s<t\leq1) \] or the strict condition \[ \sigma(g_t)<\sigma(g_s)\qquad(0\leq s<t\leq1). \] Neither interpretation is selected in this draft. Likewise, continuity and smoothness remain alternative requested regularities whose meanings depend on the missing structure on $\mathcal G$; the round endpoint must be allowed in that space.

Short English statement

The question asks whether the round sphere can be joined to a particular example by a continuous or smooth family of metrics with conical singularities along which $\sigma$ decreases. The example, the meaning of $\sigma$, the family regularity, and the precise monotonicity requirement are not determined by the supplied excerpt.

Sources

• Local UnsolvedMath registry, permanent ID $800964$, source problem code catalogue entry, title "Open Problems and Questions about Geodesics --- Question 45 (Babenko)." The registry lists Burns and Matveev as proposers, gives the year 2019, and labels the problem open. This label is supplied registry metadata, not an independently verified assessment.
• Burns and Matveev, Open Problems and Questions about Geodesics (2019, as identified in the supplied record), Question 45: https://arxiv.org/abs/1308.5417. The supplied background attributes the question or its context to Babenko. It records extraction by arxiv_latex and an access date of 2026-10-16; that access information is supplied provenance, not a claim of independent source consultation. The preceding example and the definition of $\sigma$ were not included in the supplied record.