Uniqueness and identification of the singular conifold Calabi–Yau metric

Determine whether the singular Calabi–Yau metric on the conifold obtained as the zero-Kähler-class limit of the Calabi–Yau metrics on the small resolution coincides, up to biholomorphism, with the singular Calabi–Yau metric constructed by Conlon–Rochon; equivalently, establish uniqueness of a metric on the conifold satisfying the stated conical-singularity and tangent-cone-at-infinity conditions.

Background

The paper constructs a family of complete Calabi–Yau metrics on the two small resolutions of the three-dimensional ordinary double point. As the Kähler class tends to zero, these metrics converge to a singular Calabi–Yau metric on the regular part of the conifold, with an isolated singularity modeled on the Stenzel metric and tangent cone at infinity equal to C × (C²/Z₂). Conlon–Rochon independently constructed a singular Calabi–Yau metric on the same conifold by degenerating smoothings rather than Kähler classes on small resolutions.

The authors expect the two singular metrics to agree, but do not establish this. Resolving the question would require a uniqueness theorem for Calabi–Yau metrics on the conifold with the specified local singularity, asymptotic geometry, and volume normalization, analogous to known uniqueness results for certain exotic Calabi–Yau metrics on C³.

References

While it seems natural to conjecture that the metric $\omega_{CY,0}$ coincides with the singular Conlon--Rochon metric on $Z$, the uniqueness of a metric on $Z$ satisfying the conditions of the above theorem remains an open question.

— Degenerations of exotic Calabi-Yau metrics through Atiyah's flop  (2609.08978 - Langlais et al., 8 Sep 2026) in Introduction, paragraph “Comparison with related constructions”; Section 6, subsection “Related questions,” paragraph “Relation to the Conlon–Rochon metrics”