HKT Ricci flow maximal-existence-time cone conjecture

Establish that for every compact HKT manifold $(M^{4n},I,J,K,g_0,\\Omega_0)$, the maximal smooth solution of the HKT Ricci flow exists precisely on the interval $[0,\\tau^*)$, where $\\tau^*:=\\sup\\{t\\ge 0:[\\Omega_0]_{qBC}-t c_1^{qBC}\\in\\mathcal P\\}$ and $\\mathcal P$ is the cone of quaternionic Bott–Chern classes containing a positive HKT form.

Background

The paper introduces an HKT-preserving Hermitian curvature flow, called HKT Ricci flow, and studies its short-time existence, curvature obstruction, scalar-curvature monotonicity, and several conditional long-time existence results. The flow evolves the HKT form by ∂tΩ=−Φ(g)\partial_t\Omega=-\Phi(g), where Φ(g)\Phi(g) represents the quaternionic analogue of Ricci curvature.

The conjecture identifies the maximal smooth existence time with the time at which the evolving quaternionic Bott–Chern cohomology class leaves the positive HKT cone. The authors prove only conditional extension criteria when either a metric-trace bound or a torsion bound is available, and establish the conjecture in quaternionic dimension one; the general higher-dimensional assertion therefore remains unresolved in the paper.

References

To begin we formulate a maximal existence time conjecture in analogy Kähler-Ricci flow criteria established by Tian-Zhang . A related conjecture for hyperHermitian metrics flowing by equation (\ref{f:Omegaflow}) appears in Conjecture 5.1. In the HKT setting, there is a natural notion of positive cone in analogy with the Kähler cone, and the conjecture predicts smooth existence of the HKT flow as long as the class remains in this cone:

— Hermitian curvature flow and HKT geometry  (2608.14358 - Brienza et al., 14 Aug 2026) in Conjecture 1.1 (Introduction); Conjecture 3.2, Section 3.1

\begin{conj} \label{c:coneconj} Let $(M{4n}, I, J, K, g_0, \Omega_0)$ be a compact HKT manifold. Let $\tau* := \sup{ t \geq 0\ |\ [\Omega_0]_{qBC} - t c_1{qBC} \in \mathcal P }$ The maximal smooth solution to (\ref{f:HKTflow}) exists on $[0, \tau*)$.\end{conj}

— Hermitian curvature flow and HKT geometry  (2608.14358 - Brienza et al., 14 Aug 2026) in Conjecture 3.2, Section 3.1