A geometric measure prescription problem via Monge-Ampère type variational methods

Many geometric measure prescription problems share a common Monge-Ampère type variational structure: one first constructs weak geometric solutions variationally and then studies their regularity through the associated Jacobian equation. The Aleksandrov problem on spherical domains, viewed as an optimal transport problem with a singular cost, is one such example. An Aleksandrov-type mass condition yields weak potentials and quantitative stay-away estimates.

In this talk, we illustrate this principle through the prescribed horospherical p-area measure problem in hyperbolic space. A constrained variational scheme produces weak horospherically convex solutions, while a local generated Jacobian formulation upgrades them to smooth ones. At the critical exponent, the main issue is loss of compactness through bubbling. A one-parameter max–min construction of subcritical solutions, combined with conformal blow-up analysis, rules out bubbling under suitable assumptions and yields a smooth solution of the hyperbolic Minkowski problem.