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Monge-Ampère measures on balanced polyhedral spaces

Ana María Botero, Enrica Mazzon, Léonard Pille-Schneider

Abstract

We study classes of convex functions on balanced polyhedral spaces and establish various structural properties, including a compactness theorem for polyhedrally plurisubharmonic functions. Using tropical intersection theory, we construct Monge--Ampère measures, first associated with piecewise affine functions, and then we extend it to polyhedrally plurisubharmonic functions. We investigate polyhedral Monge--Ampère equations on balanced polyhedral spaces via a variational approach, providing sufficient conditions for the existence of solutions as well as explicit counterexamples. Finally, we relate our framework to non-archimedean pluripotential theory and explore its connection with the non-archimedean Monge--Ampère equation.

Monge-Ampère measures on balanced polyhedral spaces

Abstract

We study classes of convex functions on balanced polyhedral spaces and establish various structural properties, including a compactness theorem for polyhedrally plurisubharmonic functions. Using tropical intersection theory, we construct Monge--Ampère measures, first associated with piecewise affine functions, and then we extend it to polyhedrally plurisubharmonic functions. We investigate polyhedral Monge--Ampère equations on balanced polyhedral spaces via a variational approach, providing sufficient conditions for the existence of solutions as well as explicit counterexamples. Finally, we relate our framework to non-archimedean pluripotential theory and explore its connection with the non-archimedean Monge--Ampère equation.
Paper Structure (39 sections, 68 theorems, 223 equations, 2 figures)

This paper contains 39 sections, 68 theorems, 223 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Convex functions on polyhedral spaces
  3. The polyhedral Monge--Ampère operator
  4. The polyhedral Monge--Ampère equation
  5. Relation to the non-archimedean Monge–Ampère equation
  6. Future research directions
  7. Preliminaries on polyhedral spaces
  8. Polyhedral complexes
  9. Compactifications of polyhedral spaces
  10. Cycles
  11. Intersection product
  12. Spaces of convex functions
  13. Polyhedral convexity
  14. Strict convexity
  15. Convex functions with growth condition
  16. ...and 24 more sections

Key Result

Theorem 1.1

The space $\operatorname{PSH}(\mathbf{X}, \gamma)/\mathbb{R}$ is compact for the topology of pointwise convergence.

Figures (2)

  • Figure 1: A polyhedrally smooth polyhedral space which is not smooth
  • Figure 2: The polytope $\Delta$, its normal fan $\Sigma$ and the tropicalization $\mathbf{X}$

Theorems & Definitions (198)

  • Theorem 1.1: Theorem \ref{['thm:comp']}
  • Theorem 1.2: Theorem \ref{['thm:PMA PCreg']}
  • Theorem 1.3: Proposition \ref{['prop:MA maxsol']}
  • Theorem 1.4: Theorem \ref{['theo:soldim1']}
  • Definition 2.1: Rational polyhedron
  • Definition 2.2: Rational polyhedral complex
  • Definition 2.3: Refinement
  • Definition 2.4: Rational polyhedral space
  • Remark 2.5
  • Definition 2.6: Restriction
  • ...and 188 more