The Cauchy problem for the homogeneous Monge-Ampere equation, I. Toeplitz quantization

TL;DR

This paper explores a novel approach to solving the homogeneous Monge-Ampere equation using Toeplitz quantization of Hamiltonian flows, establishing a connection with Legendre transform potentials in the context of torus invariant metrics.

Contribution

It demonstrates that the quantum analytic continuation potential matches the Legendre transform potential for torus invariant metrics, providing a new method to solve the HRMA.

Findings

The quantum analytic continuation potential coincides with the Legendre transform potential for torus invariant metrics.

This potential solves the HRMA as long as it remains smooth.

The Legendre transform potential fails to solve the HRMA after a certain time.

Abstract

The Cauchy problem for the homogeneous (real and complex) Monge-Ampere equation (HRMA/HCMA) arises from the initial value problem for geodesics in the space of Kahler metrics. It is an ill-posed problem. We conjecture that, in its lifespan, the solution can be obtained by Toeplitz quantizing the Hamiltonian flow defined by the Cauchy data, analytically continuing the quantization, and then taking a kind of logarithmic classical limit. In this article, we prove that in the case of torus invariant metrics (where the HCMA reduces to the HRMA) this "quantum analytic continuation potential" coincides with the well-known Legendre transform potential, and hence solves the equation as long as it is smooth. In the sequel we prove that the Legendre transform potential ceases to solve the HRMA after that time.