The point of this talk is to present a Lagrangian potential theory, which is in many ways analogous to classical pluripotential theory in complex analysis. I will also introduce a new Lagrangian differential operator of Monge-Ampère type. This ideas are new even in ℂn. However, they apply quite generally, perhaps most importantly to symplectic manifolds equipped with a Gromov metric.
The Lagrangian Monge-Ampère operator is an explicit polynomial on Sym2(TX) whose principle branch defines the space of Lag- harmonics. Interestingly this operator depends only on the Laplacian and the SKEW-Hermitian part of the Hessian if the function. 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. We shall also look at the notions of Lagrangian plurisubharmonic and harmonic functions, Lagrangian convex manifolds and boundaries, and an analogue of the Levi problem. Parallels of this Lagrangian potential theory with standard (complex) pluripotential theory are emphasized.
This video is part of Harvard University‘s conference JDG 2017.
