Higher-dimensional Gromov–Hausdorff continuity through Kähler–Ricci surgery

Prove Gromov–Hausdorff continuity through Kähler–Ricci flow surgeries in complex dimensions greater than two, namely that the incoming and outgoing flows converge to the same metric space at the singular time.

Background

The analytic minimal model program predicts that the metric evolution remains continuous, in the Gromov–Hausdorff sense, across the discrete birational transformations induced by Kähler–Ricci flow. The paper notes that this continuity has been established for complex surfaces and in certain highly symmetric higher-dimensional settings.

The general higher-dimensional continuity problem is explicitly identified as unresolved and is part of the broader conjectural picture for Kähler–Ricci flow through singularities.

References

In higher dimensions, this continuity is still conjectural, though it is known in certain situations where $\omega$ is invariant under a large symmetry group .

— Kähler-Ricci Tangent Flows in the Analytic Minimal Model Program  (2608.19152 - Chen et al., 19 Aug 2026) in Section 1, subsection “Background and local models”