Lagrangian potential theory and a Lagrangian equation of MongeAmpère type
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 MongeAmpere 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 MongeAmpere operator is an explicit polynomial on ${\rm Sym}^2(TX)$ whose principle branch defines the space of Lagharmonics. Interestingly the operator depends only on the Laplacian and the SKEWHermitian 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, Lagplurihamonics, 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.
 Publication:

arXiv eprints
 Pub Date:
 December 2017
 DOI:
 10.48550/arXiv.1712.03525
 arXiv:
 arXiv:1712.03525
 Bibcode:
 2017arXiv171203525R
 Keywords:

 Mathematics  Differential Geometry;
 Mathematics  Complex Variables;
 58J32;
 53D05;
 53D12;
 32U99
 EPrint:
 Change of title