Let (M,I,J,K,g_0) be a compact HKT manifold, let Omega_0 be the HKT form of the initial metric, and let T_max denote the maximal existence time of the unnormalized HKT-Ricci flow starting from g_0. Define tau^(Omega_0):=supt≥ 0:[Omega_0]-t,c_1^BC(M) contains a positive HKT form. The conjecture asserts that T_max=tau^(Omega_0); equivalently, the flow exists on [0,tau^(Omega_0)) and cannot be continued past tau^(Omega_0) when tau^(Omega_0)<∞.

HKT-Ricci-flow maximal-time conjecture

Let $(M,I,J,K,g_0)$ be a compact HKT manifold, let $\Omega_0$ be the HKT form of the initial metric, and let $T_{\max}$ denote the maximal existence time of the unnormalized HKT-Ricci flow starting from $g_0$ . Define $\tau^*(\Omega_0):=\sup\{t\ge 0:[\Omega_0]-t\,c_1^{\mathrm{BC}}(M)\text{ contains a positive HKT form}\}$ . The conjecture asserts that $T_{\max}=\tau^*(\Omega_0)$ ; equivalently, the flow exists on $[0,\tau^*(\Omega_0))$ and cannot be continued past $\tau^*(\Omega_0)$ when $\tau^*(\Omega_0)<\infty$ .

References

Primary source

arXiv

Additional references

HKT geometry on locally conformally hyperkähler manifolds — arXiv — Giovanni Gentili, Luigi Vezzoni

Posted 16 hours ago · edited 16 hours ago

Progress summary

Refreshed 16 hours ago

Claimed progress

A recent paper proves the predicted flow-time formula for compact locally conformally hyperkähler manifolds, but the conjecture for all compact HKT manifolds remains open.

The conjecture predicts that the HKT flow on a compact manifold exists precisely up to the cohomologically defined time $\tau^{*}$ . The general statement is not proved; the latest result covers only the locally conformally hyperkähler subclass.

Known results

In quaternionic dimension $n=1$ , the conjecture is proved: hyperKähler manifolds have immortal flow, while quaternionic Hopf surfaces admit immortal normalized flow.
For general compact HKT manifolds, only conditional resolutions under additional a priori estimates are reported.
The hyperKähler case is supported by long-time-flow and quaternionic Monge–Ampère results.

October 6, 2026 locally conformally hyperkähler result

Giovanni Gentili and Luigi Vezzoni report a proof of the conjectured maximal-time formula in the compact locally conformally hyperkähler setting and construct nontrivial solitons on quaternionic Hopf surfaces. This is substantive progress, not a proof for arbitrary compact HKT manifolds.

Current status (as of October 2026): The conjecture is proved in quaternionic dimension $n=1$ and in the compact locally conformally hyperkähler subclass, while the general compact HKT case remains open.

Sources

No solutions have been posted yet.