Surveys in Differential Geometry

Volume 22 (2017)

Lagrangian potential theory and a Lagrangian equation of Monge–Ampère type

Pages: 217 – 257

DOI: https://dx.doi.org/10.4310/SDG.2017.v22.n1.a8

Authors

Harvey F. Reese (Department of Mathematics, Rice University, Houston, Texas, U.S.A.)

H. Blaine Lawson, Jr. (Department of Mathematics, Stony Brook University, Stony Brook, New York, U.S.A.)

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–Ampère type. This developement is new even in $\mathbf{C}^n$. However, it applies quite generally—perhaps most importantly to symplectic manifolds equipped with a Gromov metric.

The Lagrange operator is an explicit polynomial on $\mathrm{Sym}^2 (TX)$ 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. It 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 $\mathbf{C}^n$ there is another concept, Lag-pluriharmonics, 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.

Published 13 September 2018