Monge-Ampere equations and moduli spaces of manifolds of circular type

TL;DR

This paper studies manifolds of circular type characterized by solutions to the Monge-Ampère equation, exploring their moduli spaces, invariants, and new characterizations of circular domains and the unit ball.

Contribution

It introduces a normal form for manifolds of circular type, defines a new holomorphic invariant, and provides new characterizations of circular domains and the unit ball.

Findings

Existence of a unique normal form for each equivalence class.

The normalizing maps form a finite-dimensional real manifold.

New characterizations of circular domains and the unit ball.

Abstract

A (bounded) manifold of circular type is a complex manifold M of dimension n admitting a (bounded) exhaustive real function u, defined on M minus a point x_o, so that: a) it is a smooth solution on $M\setminus {x_o}$ to the Monge-Amp\`ere equation $(d d^c u)^n = 0$; b) x_o is a singular point for u of logarithmic type and e^u extends smoothly on the blow up of M at x_o; c) $d d^c (e^u) >0$ at any point of $M\setminus {x_o}$. This class of manifolds naturally includes all smoothly bounded, strictly linearly convex domains and all smoothly bounded, strongly pseudoconvex circular domains of $\bC^n$. The moduli spaces of bounded manifolds of circular type are studied. In particular, for each biholomorphic equivalence class of them it is proved the existence of an essentially unique manifold in normal form. It is also shown that the class of normalizing maps for an n-dimensional manifold M is a new holomorphic invariant with the following property: it is parameterized by the points of a finite dimensional real manifold of dimension n^2 when M is a (non-convex) circular domain while it is of dimension $n^2 + 2 n$ when M is a strictly convex domain. New characterizations of the circular domains and of the unit ball are also obtained.