Quaternionic plurisubharmonicity and convexity expansion

Core idea

This dependency-ordered index develops the concepts surrounding quaternionic plurisubharmonic functions, their applications to convex-body valuations and HKT geometry, and the dimension-sensitive octonionic extension. Click any term to expand it inline.

Classical potential theory

• Harmonic function (H)
• Subharmonic function (SH)
• Plurisubharmonic function (PSH)
• Pluriharmonic function (PH)
• Exact H/SH/PSH/PH relationships
• Levi form
• Strictly plurisubharmonic function
• Complex Monge–Ampère operator

Quaternionic foundations

• Quaternion division algebra
• Quaternionic vector space
• Hyperhermitian form and matrix
• Moore determinant
• Mixed discriminant
• Cauchy–Fueter operators
• Quaternionic Hessian

Quaternionic pluripotential theory

• Quaternionic plurisubharmonic function
• Strictly quaternionic plurisubharmonic function
• Quaternionic Monge–Ampère measure
• Mixed quaternionic Monge–Ampère measure
• Continuity of quaternionic Monge–Ampère measures
• Quaternionic Błocki formula
• Quaternionic Monge–Ampère equation
• Strictly quaternionically pseudoconvex domain
• Dirichlet theorem for the quaternionic Monge–Ampère equation

Convex bodies and valuations

• Convex body
• Hausdorff distance
• Support function of a convex body
• Valuation on convex bodies
• Mixed volume
• Pluripotential construction of convex-body valuations

The bridge is structural: a support function is convex and hence quaternionic PSH; the max/min identity for support functions combines with the quaternionic Błocki formula to give finite additivity; and continuity of Monge–Ampère measures converts Hausdorff convergence into continuity of the valuation.

Hypercomplex and HKT geometry

• Hypercomplex manifold
• Hyper-Hermitian manifold
• The del-J operator
• Quaternionic PSH function on a hypercomplex manifold
• HKT metric
• Local HKT potential theorem
• Hyperkähler manifold

Here the bridge is geometric: positivity of $\partial\partial_Ju$ is the curved analogue of positivity of the flat quaternionic Hessian, and strict positivity turns the potential into an HKT metric.

Octonionic plane and Spin(9)

• Octonion algebra
• Octonionic spin factor
• Determinant of a two-by-two octonionic Hermitian matrix
• Octonionic projective line
• Octonionic special linear group
• Spin representation of Spin(9)
• Affine octonionic line
• Octonionic Radon transform
• Octonionic Hessian
• Octonionic plurisubharmonic function
• Octonionic Monge–Ampère measure
• Octonionic pseudovolume

The octonionic theory in this index is deliberately confined to $\mathbb O^2\cong\mathbb R^{16}$. That is where octonionic line geometry, the Hermitian $2\times2$ determinant, and the real spin representation of $\operatorname{Spin}(9)$ fit together. It is not an automatic theory on arbitrary $\mathbb O^n$.

Research directions for later source-guided expansion

The following neighborhoods are related but are not silently treated as part of the 2006 survey:

• pluripotential theory on general quaternionic manifolds using Baston operators and line-bundle-valued potentials;
• the quaternionic Calabi problem on compact HKT manifolds;
• singular domains of definition, finite-energy classes, capacities, and quaternionic $m$-subharmonic functions;
• calibrated plurisubharmonicity as a framework encompassing additional geometric positivity notions;
• the classification and algebra structure of invariant valuations for quaternionic groups and exceptional groups acting transitively on spheres.

These topics should receive their own semantic audit and source manifest. In particular, general quaternionic-manifold PSH sections should not be merged with hypercomplex functions: their Baston-operator formulation has different geometric data.

References

1. Semyon Alesker, “Quaternionic plurisubharmonic functions and their applications to convexity,” 2016 revision. arXiv record.
2. Semyon Alesker, “Non-commutative linear algebra and plurisubharmonic functions of quaternionic variables,” revised 2024. arXiv record.
3. Semyon Alesker, “Valuations on convex sets, non-commutative determinants, and pluripotential theory,” Advances in Mathematics 195 (2005), 561–595. arXiv record.
4. Semyon Alesker and Misha Verbitsky, “Plurisubharmonic functions on hypercomplex manifolds and HKT-geometry,” Journal of Geometric Analysis 16 (2006), 375–399. arXiv record.
5. Semyon Alesker, “Plurisubharmonic functions on the octonionic plane and $\operatorname{Spin}(9)$-invariant valuations on convex sets,” Journal of Geometric Analysis 18 (2008), 651–686. arXiv record.