## Journal of Differential Geometry

- J. Differential Geom.
- Volume 90, Number 2 (2012), 303-327.

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

Yanir A. Rubinstein and Steve Zelditch

#### Abstract

The Cauchy problem for the homogeneous real/complex Monge–Ampère equation (HRMA/HCMA) arises from the initial value
problem for geodesics in the space of Kähler metrics equipped
with the Mabuchi metric. This Cauchy problem is believed to be
ill-posed and a basic problem is to characterize initial data of
(weak) solutions which exist up to time $T$. In this article, we use a
quantization method to construct a subsolution of the HCMA on
a general projective variety, and we conjecture that it solves the
equation for as long as the unique solution exists. The subsolution,
called the “quantum analytic continuation potential,” is obtained
by (i) Toeplitz quantizing the Hamiltonian flow determined
by the Cauchy data, (ii) analytically continuing the quantization,
and (iii) taking a certain logarithmic classical limit. We then prove
that in the case of torus invariant metrics (where the HCMA reduces
to the HRMA) the 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, "The Cauchy problem for the homogeneous
Monge–Ampère equation, II. Legendre transform," *Adv. Math.*, 228 (2011),
2989–3025, it is proved that the Legendre transform potential ceases to
solve the HCMA once it ceases to be smooth. The results here and
in the sequels in particular characterize the initial data of smooth
geodesic rays.

#### Article information

**Source**

J. Differential Geom., Volume 90, Number 2 (2012), 303-327.

**Dates**

First available in Project Euclid: 24 April 2012

**Permanent link to this document**

https://projecteuclid.org/euclid.jdg/1335230849

**Digital Object Identifier**

doi:10.4310/jdg/1335230849

**Mathematical Reviews number (MathSciNet)**

MR2899878

**Zentralblatt MATH identifier**

1250.32036

#### Citation

Rubinstein, Yanir A.; Zelditch, Steve. The Cauchy problem for the homogeneous Monge-Ampère equation, I. Toeplitz quantization. J. Differential Geom. 90 (2012), no. 2, 303--327. doi:10.4310/jdg/1335230849. https://projecteuclid.org/euclid.jdg/1335230849