Lagrangian potential theory and a Lagrangian equation of Monge-Amp\`ere type

TL;DR

This paper develops a novel Lagrangian potential theory and Monge-Ampere type operator, extending classical pluripotential theory to symplectic manifolds, and solves associated Dirichlet problems.

Contribution

It introduces a new Lagrangian differential operator of Monge-Ampere type and establishes foundational theory including Lagrangian plurisubharmonic functions and convexity.

Findings

Defined the Lagrangian Monge-Ampere operator explicitly.

Solved the Dirichlet problem for the operator.

Established parallels with classical pluripotential theory.

Abstract

The purpose of this paper is to establish a Lagrangian potential theory, analogous to the classical pluripotential theory, and to define and study a Lagrangian differential operator of Monge-Ampere type. This development is new even in ${\bf C}^n$. However, it applies quite generally -- perhaps most importantly to symplectic manifolds equipped with a Gromov metric. The Lagrange Monge-Ampere operator is an explicit polynomial on ${\rm Sym}^2(TX)$ whose principle branch defines the space of Lag-harmonics. Interestingly the operator depends only on the Laplacian and the SKEW-Hermitian part of the Hessian. The Dirichlet problem for this operator is solved in both the homogeneous and inhomogeneous cases. The homogeneous case is also solved for each of the other branches. This paper also introduces and systematically studies the notions of Lagrangian plurisubharmonic and harmonic functions, and Lagrangian convexity. An analogue of the Levi Problem is proved. In ${\bf C}^n$ there is another concept, Lag-plurihamonics, which relate in several ways to the harmonics on any domain. Parallels of this Lagrangian potential theory with standard (complex) pluripotential theory are constantly emphasized.