Octonionic Monge-Ampère Operator

• The octonionic Monge-Ampère operator is a non-associative, non-commutative differential operator defined using 2×2 octonionic Hermitian matrices.
• It forms the basis of a novel pluripotential theory for octonionic plurisubharmonic functions via innovative analytic extensions and determinant concepts.
• Applications include solving the Dirichlet problem and establishing octonionic Calabi–Yau theorems on 16-dimensional manifolds, addressing challenges from octonion structure.

The octonionic Monge-Ampère operator is the non-associative, non-commutative analogue of the complex and quaternionic Monge-Ampère operators, defined on functions of two octonionic variables. This operator, and its associated equation, serve as the foundation for a pluripotential theory on octonionic spaces, analogous to those on complex and quaternionic manifolds, but with significant technical differences due to the structural properties of the octonions 𝕆. Two complementary developments have emerged in recent years: the analytic pluripotential approach for plurisubharmonic functions on open domains in 𝕆² (Wang, 13 Jan 2026), and the global geometric study on 16-dimensional manifolds with special affine and holonomy structure, notably the "octonionic Calabi-Yau theorem" and related Monge-Ampère equation (Alesker et al., 2022).

1. Algebraic Foundation: Octonionic Hessian, Determinant, and Operator

Given $x = (x_1, x_2) \in \mathbb{O}^2$, each $x_\alpha$ is expressed in real coordinates $x_{\alpha p}, p = 0, ..., 7$ using a standard octonion basis $e_0=1, ..., e_7$. The Dirac–Fueter operators are defined as

$$\bar{\partial}_\alpha = \sum_{p=0}^7 e_p \frac{\partial}{\partial x_{\alpha p}}, \quad \partial_\alpha = \sum_{p=0}^7 \frac{\partial}{\partial x_{\alpha p}}\, \bar{e}_p.$$

For real-valued, twice differentiable $u$, the octonionic Hessian is the $2\times2$ Hermitian matrix

$$\operatorname{Hess}_\mathbb{O}(u) = \begin{pmatrix} \bar{\partial}_1 \partial_1 u & \bar{\partial}_1 \partial_2 u \ \bar{\partial}_2 \partial_1 u & \bar{\partial}_2 \partial_2 u \end{pmatrix} \in \mathcal{H}_2(\mathbb{O}),$$

where $\mathcal{H}_2(\mathbb{O})$ denotes 2×2 octonionic Hermitian matrices.

The determinant for $$A = (a_{\bar\alpha \beta}) \in \mathcal{H}_2(\mathbb{O})$$ is

$$\det A = a_{\bar{1}1} a_{\bar{2}2} - |a_{\bar{1}2}|^2.$$

Given $A, B \in \mathcal{H}_2(\mathbb{O})$, the mixed determinant is defined by

$$\det(A, B) = \frac{1}{2}\bigl( a_{\bar{1}1} b_{\bar{2}2} + a_{\bar{2}2} b_{\bar{1}1} - 2 \Re(a_{\bar{1}2} b_{\bar{2}1}) \bigr).$$

The octonionic Monge-Ampère operator is defined, for $u$ in a suitable function class (e.g., $C^2$ and octonionic plurisubharmonic), as the measure $$\operatorname{MA}_\mathbb{O}(u) := \det\bigl[\operatorname{Hess}_\mathbb{O}(u)\bigr]$$ (Wang, 13 Jan 2026, Alesker et al., 2022).

2. Plurisubharmonicity and Extension to Radon Measures

A function $u$ is called octonionic plurisubharmonic (OPSH) on an open set $\Omega \subset \mathbb{O}^2$ if its restriction to every affine right-octonionic line is subharmonic as a function of eight real variables. For $u \in USC(\Omega) \cap L^1_{\text{loc}}(\Omega)$, the octonionic Monge–Ampère operator and mixed Monge–Ampère operator are extended to continuous OPSH functions as Radon measures, using approximation and an extension of Alesker's result (Wang, 13 Jan 2026).

3. Analytical and Pluripotential Theory: Key Theorems and Properties

Several core results are established for OPSH functions and the octonionic Monge–Ampère operator (Wang, 13 Jan 2026):

• Comparison Principle (Theorem 1.1): For OPSH $u, v \in C(\overline{\Omega})$, $$\int_{\{u<v\}} \det[\operatorname{Hess}_\mathbb{O}(u)] \geq \int_{\{u<v\}} \det[\operatorname{Hess}_\mathbb{O}(v)]$$.
• Integration by Parts (Lemma 3.1): For suitable boundary data, $$\int_{\Omega} v\,\det[\operatorname{Hess}_\mathbb{O}(u), \omega]$$ can be related to a boundary integral and a symmetrized interior term using the alternativity and weak associativity properties: $\Re((ab)c) = \Re(a(bc))$ for $a, b, c \in \mathbb{O}$.
• Quasicontinuity (Theorem 1.3): Any locally bounded OPSH function is continuous outside an open set of arbitrarily small capacity, defined via infima of integrals of the octonionic Monge–Ampère operator.
• Capacity Theory: The notion of octonionic capacity $C(K, \Omega)$ is developed, and relative extremal functions are characterized analogous to the complex case.

These results are crucial for the development of octonionic pluripotential theory, addressing the main obstruction posed by non-associativity of 𝕆.

4. The Dirichlet Problem and Regularity Theory

The Dirichlet problem for the homogeneous octonionic Monge–Ampère equation in the unit ball $B^2 \subset \mathbb{O}^2$ is addressed using the Perron–Bremermann envelope construction. The solution $u = \Psi_{B^2, \varphi}$ for continuous boundary data $\varphi$ lies in $C(\overline{B^2})$ and is maximal in $B^2$ (Wang, 13 Jan 2026).

A weighted automorphism $T_a$ of $B^2$, constructed using Cayley transforms, Heisenberg translations, and dilations—$$T_a = C^{-1} \circ D_{\delta_a} \circ \tau_{\zeta_a} \circ C$$—serves to correct for the fact that general automorphisms do not preserve OPSH functions. The weighted transformation

$$\left(T_a^* u\right)(x) := | \Psi_a(x) |^{-6} u(T_a(x)),\quad \Psi_a(x) := 2(1 + [T_a(x)]_2)^{-1} (1 + x_2),$$

restores plurisubharmonicity under pullback (Wang, 13 Jan 2026).

By an adaptation of Bedford–Taylor’s method, if $\varphi$ is $C^2$ on the boundary, then the envelope solution is $C_{\text{loc}}^{1,1}$ inside $B^2$. The proof proceeds via second-order difference-quotient estimates (using the convexity property under automorphisms), smoothing and averaging arguments showing that the Laplacian is locally bounded, and then classical elliptic estimates (Wang, 13 Jan 2026).

5. Global Geometry: Octonionic Kähler and Calabi-Yau Structures

On a 16-dimensional Spin(9)-affine manifold $M$, a Riemannian metric is octonionic Hermitian if, in each chart, its restriction to right-octonionic lines is a multiple of the Euclidean metric. Octonionic Kähler metrics arise when the Hermitian form $T_{\bar{i}j}(x)$ satisfies a set of first-order linear PDEs equivalent to being a Hessian of some real potential (Alesker et al., 2022).

The global octonionic Monge–Ampère equation for a smooth octonionic Kähler background $G_0$ and $f \in C^\infty(M)$ is

$$\det \bigl( G_0 + \operatorname{Hess}_\mathbb{O}(\phi) \bigr) = e^f\, \det(G_0).$$

This equation is elliptic (by Sylvester’s criterion), its linearization is a Fredholm operator of index zero, and the operator is equivariant under $GL_2(\mathbb{O})$ coordinate changes (Alesker et al., 2022).

The existence and uniqueness theorem for the octonionic Monge–Ampère equation is proved by the continuity method, following the classical scheme of the complex Calabi–Yau and quaternionic analogues:

1. The map $$\phi \mapsto \frac{\det(G_0 + \operatorname{Hess}_\mathbb{O} \phi)}{\det G_0}$$ is a local diffeomorphism.
2. Uniqueness up to constants is ensured by the maximum principle.
3. A priori estimates—$C^0$ via a Moser iteration adapted to octonionic integrals, gradient via integration by parts and Sobolev techniques, Laplacian via Chern–Lu-type inequalities.
4. The closedness follows from compactness and higher regularity via Evans–Krylov and Schauder bootstrapping (Alesker et al., 2022).

6. Structural and Algebraic Constraints

A central obstacle throughout is the non-associativity of the octonions, which prevents the definition of determinants for Hermitian matrices of size greater than $2 \times 2$. However, key determinant constructions—and crucially, all integration by parts and Sylvester-type positivity arguments—are valid because $\mathbb{O}$ is alternative (every subalgebra generated by two elements is associative) and satisfies the Moufang identities (Alesker et al., 2022, Wang, 13 Jan 2026).

The determinant in dimension two is well-defined, and all real parts and divergences commute appropriately, allowing the extension of pluripotential arguments. Beyond two variables, the lack of a higher-dimensional determinant is a major barrier to generalization.

7. Comparative Perspective and Implications

The octonionic Monge–Ampère operator occupies a terminal point in the hierarchy of Monge–Ampère-type nonlinear PDEs, corresponding to the real, complex (Calabi–Yau), quaternionic (HKT), and octonionic settings. Each stage in this sequence is characterized by increasing algebraic complexity:

• In the complex case, the theory uses the $dd^c$ operator, standard determinant, and classical pluripotential methods.
• The quaternionic case invokes the Moore determinant on Hermitian matrices and benefits from associativity, allowing for full generalization beyond $2 \times 2$.
• The octonionic case, due to alternativity and existence of a determinant only for $2 \times 2$ matrices, necessitates new techniques to address both local and global equations, with contemporary work limited to 16-dimensional manifolds with special geometry (Alesker et al., 2022, Wang, 13 Jan 2026).

These advances underpin ongoing research in octonionic and exceptional geometry, highlighting both the analytic depth and the rigidity introduced by the structure of the octonions.