General split-signature Monge–Ampère integration

Integrate the constant-Hessian-determinant equation on the bounded Grassmannian domain for general split signature and identify the boundary behavior of the resulting solution as a barrier function for the bounded symmetric domain.

Background

For split signatures with p,q≥2, the paper replaces the non-convex vector light cone with the convex bounded symmetric domain D={Z∈R^{p×q}: I_p−ZZ^T≻0}. Theorem 11.26 reduces the condition det ∇²f=const for potentials of the form f(Z)=χ(det(I_p−ZZ^T)) to a transcendental equation involving the singular values of Z.

The authors explicitly leave unresolved both the integration of this equation and the characterization of the resulting solution near the boundary, where a barrier interpretation would connect the Monge–Ampère construction to information-geometric and optimization structures.