Quaternionic Calabi–Yau Problem

• Quaternionic Calabi–Yau problem is a generalization of Yau’s Calabi conjecture to quaternionic geometry, involving the quaternionic Monge–Ampère equation on HKT manifolds.
• It employs advanced techniques like the continuity method, Moser iteration, and Evans–Krylov theory to establish existence, uniqueness, and regularity of solutions.
• The problem has significant implications for defining canonical metrics in hypercomplex geometry and advancing pluripotential methods in non-Kähler settings.

The quaternionic Calabi–Yau problem is the quaternionic analogue of the celebrated Calabi–Yau problem in complex Kähler geometry. The central object is the quaternionic Monge–Ampère equation posed on a compact hypercomplex manifold equipped with an HKT (hyperKähler with torsion) or hyperKähler metric. The problem seeks to find a smooth potential modifying the HKT form to realize a prescribed volume form, generalizing Yau’s solution of the Calabi conjecture to the quaternionic setting and elucidating the structure of canonical metrics in quaternionic geometry.

1. Geometric Foundations

A hypercomplex manifold is a smooth manifold $M$ of real dimension $4n$ endowed with three integrable almost-complex structures $I$, $J$, $K$ satisfying the quaternionic relations $IJ = -JI = K$, $JK = -KJ = I$, $KI = -IK = J$. The Obata connection provides a unique torsion-free connection preserving $I$, $J$, $K$. A Riemannian metric $g$ is called hyperhermitian if it is Hermitian with respect to all three structures; the associated $(2,0)$-form with respect to $I$ is $\Omega = \omega_J - i\,\omega_K$.

An HKT structure requires $\partial\Omega = 0$, where $\partial$ is the $(1,0)$-Dolbeault operator for $(M,I)$. When $d\Omega = 0$, the metric is \textit{hyperKähler}; otherwise, if $\partial\Omega=0$ but not necessarily $d\Omega = 0$, the structure is \textit{hyperKähler with torsion} (HKT). The triviality of the canonical bundle $K_{(M,I)}$ in the holomorphic category plays the role of the Calabi–Yau condition in the quaternionic setting (Dinew et al., 2021).

2. Quaternionic Monge–Ampère Equation

The quaternionic Monge–Ampère operator arises by considering the twisted differential $\partial_J := J^{-1} \partial J$ and the second-order operator $\partial \partial_J \phi$ acting on smooth real-valued functions $\phi$. The problem is to solve the nonlinear PDE

$(\Omega+\partial\partial_J \phi)^n = e^F \Omega^n$

on a compact HKT or hyperKähler manifold $(M, I, J, K, g)$, given a smooth function $F$ and normalization ensuring agreement of the total quaternionic volume: $\int_M e^F\,\Omega^n = \int_M \Omega^n.$ This equation generalizes to allow right-hand side densities given by nowhere-vanishing holomorphic forms (Dinew et al., 2021, Alesker et al., 2015). The operator’s ellipticity and strict positivity are maintained by the requirement that $\Omega + \partial\partial_J \phi > 0$ as a $(2,0)$-form.

3. Existence, Uniqueness, and Regularity Results

The existence and uniqueness of smooth solutions to the quaternionic Calabi–Yau problem on compact manifolds were initially conjectured by Alesker and Verbitsky for HKT manifolds with holomorphically trivial canonical bundle. The problem was first solved under the assumption of a flat hyperKähler metric using the continuity method and detailed a priori estimates for the operator (Alesker, 2011). The flatness assumption provided global parallel coordinates and simplified the second-order estimates.

The main advance eliminating the requirement of local flatness was achieved by Dinew and Sroka, who provided a full nonlinear existence and uniqueness theorem in the case of compact hyperKähler manifolds of arbitrary dimension. Their strategy involved working in Obata-geodesic coordinates to suppress $J$-derivative terms, adapting Moser iteration and pluripotential techniques for $C^0$ bounds, establishing gradient and second-order estimates inspired by Błocki’s and Pogorelov-type arguments, and invoking the Evans–Krylov theorem for higher regularity (Dinew et al., 2021). Uniqueness up to an additive constant is established via maximum principle arguments (Alesker et al., 2015).

Further, in the HKT nilmanifold context with abelian hypercomplex structures, the problem reduces (under torus symmetry) to explicit solvable PDEs on lower-dimensional tori, and methods such as the ABP estimate, Laplacian bounds, and Evans–Krylov regularity ensure existence of smooth, invariant potentials (Gentili et al., 2020).

4. Analytical Techniques and Estimates

A cornerstone is the uniform $C^0$ a priori estimate for potential solutions, initially established in the flat setting using adaptations of comparison principles and the ABP (Alexandrov–Bakelman–Pucci) inequality, as well as the construction of quaternionic Gauduchon forms (Alesker et al., 2011, Alesker et al., 2015). The $C^0$ estimate is a decisive ingredient in the continuity method, and its extension to general HKT backgrounds removed several prior restrictions.

Second-order estimates (Laplacian estimates) historically relied on the existence of a flat hyperKähler background but were subsequently generalized through coordinate computations exploiting curvature identities and the quaternionic symmetries of $J$. Gentili–Vezzoni provided a simplified derivation of the classical second-order estimate, showing each curvature–trace term vanishes individually in geodesic coordinates, leveraging the key commutation and curvature identities for hyperkähler metrics (Gentili et al., 13 Dec 2025). Once $C^2$ bounds are obtained, the Evans–Krylov theory yields $C^{2,\alpha}$-regularity, and bootstrapping provides smoothness of solutions.

5. Parabolic Flows and the Geometric Approach

An alternative to the continuity method is provided by the parabolic quaternionic Monge–Ampère flow: $$\frac{\partial \phi}{\partial t} = \log\frac{(\Omega+\partial\partial_J\phi)^n}{\Omega^n} - F,$$ with initial condition $\phi(\cdot,0)=0$, and normalized so that $\int_M \phi\,\Omega^n=0$ at all times. This flow is shown to have unique, long-time smooth solutions whose potentials converge to a solution of the static quaternionic Monge–Ampère equation in compact hyperKähler manifolds (Bedulli et al., 2023, Bedulli et al., 2021).

Maximum principle techniques yield uniform $C^0$ bounds along the flow, while parabolic versions of the second-order estimates and the Evans–Krylov theorem provide higher regularity. A decreasing quaternionic analogue of the Mabuchi functional can be constructed, and its properties ensure convergence of the flow to the unique potential solving the elliptic problem. This parabolic framework provides an independent existence and uniqueness proof and yields insights into the structure of canonical HKT metrics.

6. Special Cases and Examples

• Flat space/quaternionic tori: Early solutions for the quaternionic Monge–Ampère equation were carried out for flat models on quaternionic space $\mathbb{H}^n$.
• Nilmanifolds: On 8-dimensional abelian hypercomplex nilmanifolds, torus-invariant solutions are constructed for the reduced PDE with explicit form, utilizing the continuity method and fact that the local geometry is modeled on flat $\mathbb{H}^2$ (Gentili et al., 2020).
• K3 surfaces, Hilbert schemes, generalized Kummer varieties: The methods of Dinew and Sroka apply to any compact hyperKähler manifold, not requiring local flatness.

7. Open Problems and Future Directions

While existence and regularity are now established for compact hyperKähler manifolds with trivial canonical bundle, the fully general quaternionic Calabi–Yau problem on HKT manifolds without local flatness or further assumptions remains open. The key analytic challenge lies in carrying out second-order and higher a priori estimates in the absence of curvature vanishing. Recent advances in $C^0$ estimates for general compact HKT-manifolds (Alesker et al., 2015) provide foundational steps. Novel geometric flows—such as the quaternionic Chern–Ricci flow—have been formulated analogously to the complex setting, with conjectured properties toward producing canonical HKT metrics and deepening the understanding of metrics in quaternionic geometry (Bedulli et al., 2023).

A plausible implication is that further progress in the extension of second-order and regularity theory to arbitrary HKT backgrounds, and the construction of new canonical metrics or moduli spaces, would mirror developments in Kähler geometry and pluripotential theory, but must contend with fundamentally new quaternionic phenomena.