Uniqueness of fixed-base-point tangent flows

Establish uniqueness of the Type I tangent flow at each fixed base point of a finite-time noncollapsing Kähler–Ricci flow, independently of the subsequence and the sequence of times approaching the singular time.

Background

Theorem 8.5 proves subsequential convergence of fixed-base-point Type I blow-ups to singular shrinking Kähler–Ricci soliton spaces and shows that the limiting Nash entropy is independent of the subsequence. However, the theorem does not prove that the resulting soliton or tangent flow itself is unique.

The paper notes that uniqueness is known in special cases admitting a holomorphic approximation by an asymptotically conical Kähler–Ricci shrinker, but presents uniqueness in general as part of a conjectural picture.

References

In general, uniqueness is part of the conjectural picture in Conjecture 6.8 and Remark 6.9.

— Finite Time Noncollapsing Kähler-Ricci Flow: Gaussian Type Heat Kernel, Gromov-Hausdorff Limits  (2610.21824 - Zhang et al., 15 Oct 2026) in Remark 8.6(3), “remarkA”

The question of whether the blow-up limits based at Ricci vertices are tangent flows is raised in Section 1 and Conjecture 2.2, Remark 2.3.

— Finite Time Noncollapsing Kähler-Ricci Flow: Gaussian Type Heat Kernel, Gromov-Hausdorff Limits  (2610.21824 - Zhang et al., 15 Oct 2026) in Remark 8.7, “openremark”

We note that Theorem 1.3 establishes subsequential convergence and the normal analytic structure of the limit, while the uniqueness and optimal-degeneration assertions of Conjecture 1.3 remain open.

— Type I estimates for the Kähler-Ricci flow II  (2610.22941 - Jian et al., 16 Oct 2026) in Discussion following Theorem 1.3, Section 1.3, p. 5