Giorgio Patrizio

Parabolic exhaustions for strictly convex domains.

Using results of L. Lempert, we are able to construct parabolic exhaustions for strictly convex domains D⊂ℂm with center at any given point of D. Using the theory of parabolic spaces and the geometric properties of these exhaustions, we can characterize the strictly convex domains biholomorphic to a circular domain and in particular to the ball in ℂm.

The pluricomplex Poisson kernel for strongly convex domains

Let D be a bounded strongly convex domain in the complex space of dimension n. For a fixed point p epsilon partial derivative D, we consider the solution of a homogeneous complex Monge-Ampere equation with a simple pole at p. We prove that such a solution enjoys many properties of the classical Poisson kernel in the unit disc and thus deserves to be called the pluricomplex Poisson kernel of D with pole at p. In particular we discuss extremality properties (such as a generalization of the classical Phragmen-Lindelof theorem), relations with the pluricomplex Green function of D, uniqueness in terms of the associated foliation and boundary behaviors. Finally, using such a kernel we obtain explicit reproducing formulas for plurisubharmonic functions.

Holomorphic curvature of Finsler metrics and complex geodesics

In his famous 1981 paper, Lempert proved that given a point in a strongly convex domain the complex geodesics (i.e., the extremal disks) for the Kobayashi metric passing through that point provide a very useful fibration of the domain. In this paper we address the question whether, given a smooth complex Finsler metric on a complex manifoldM, it is possible to find purely differential geometric properties of the metric ensuring the existence of such a fibration in complex geodesies ofM. We first discuss at some length the notion of holomorphic sectional curvature for a complex Finsler metric; then, using the differential equation of complex geodesies we obtained in [AP], we show that for every pair (p;v) ∈T M, withv ≠ 0, there is a (only a segment if the metric is not complete) complex geodesic passing throughp tangent tov iff the Finsler metric is Kähler, has constant holomorphic sectional curvature −4, and its curvature tensor satisfies a specific simmetry condition—which are the differential geometric conditions we were after. Finally, we show that a complex Finsler metric of constant holomorphic sectional curvature −4 satisfying the given simmetry condition on the curvature is necessarily the Kobayashi metric.

Holomorphic Dynamical Systems

This chapter is a survey on local dynamics of holomorphic maps in one and several complex variables, discussing in particular normal forms and the structure of local stable sets in the non-hyperbolic case, and including several proofs and a large bibliography.

Kähler Finsler Manifolds of Constant Holomorphic Curvature

0. Introduction The classification of simply connected Kähler manifolds of constant holomorphic curvature is a classical result. According to the classification, up to biholomorphic isometry there are only three possibilities: C endowed with the euclidean metric, P(C) endowed with (a suitable costant multiple of) the Fubini-Study metric, and the unit ball B in C endowed with (a suitable costant multiple of) the hyperbolic metric. In recent years questions coming from geometric function theory, and in particular the study of invariant metrics of complex manifolds, suggested to investigate the geometry of complex Finsler (rather than Hermitian) metrics with constant holomorphic curvature, satisfying some natural Kähler condition (agreeing with the usual one in the case of Hermitian metrics) and whose curvature has symmetries enjoyed by the function theoretic examples. In [AP1], and then in [AP2], among other results it was shown that these hypotheses are equivalent to the existence of geodesic complex curves. Since complex Finsler metrics have been considered for quite some time (we recall among other contributions [Ri] who possibly introduced them, [Ru], [K] who indicated the right setting for their study, [Ro], [F], [P]) it is natural to ask whether, at least under natural geometric assumptions, it is possible to obtain a satisfactory classification. Examples show that one should not expect a short list of models. In fact the strongly convex domains in C with their Kobayashi metric provide an infinite dimensional family of not equivalent (neither holomorphically nor isometrically) complex (weakly) Kähler Finsler manifolds of constant negative holomorphic curvature. Furthermore it is easy to endow C with infinite non isometric flat complex Kähler Finsler metrics (the strongly pseudoconvex Minkowski metrics). On the other hand, no example is known of non Hermitian complex Kähler Finsler manifold of positive constant holomorphic curvature. The difference of availability of examples seems to hint that there is a different situation according to the sign of the curvature, in striking contrast with the Hermitian situation. Indeed there are difficulties which do not allow one to extend easily the techniques of the Hermitian case — and even in the real case the classification of constant curvature Finsler manifolds is not clearly established. Finally, the relationship between complex and real geometry is not as effective as in the Hermitian situation. In this paper, using heavily the results of [AP2] and the previous work on the subject by the authors (in particular [AP1]), we address the classification problem and we are able to clarify the situation completely in the non-negative case and make some substantial progress in the negative one. Our work shows that the examples gave the right feeling about the problem. Namely, up to biholomorphic isometries, if some natural symmetry of the curvature is assumed the only complex Kähler Finsler manifold of positive constant holomorphic curvature is P(C) endowed with (a suitable constant multiple of) the Fubini-Study metric, and the only simply connected flat ones are C endowed with strongly pseudoconvex Minkowski metrics. For the negative case we are able to give sufficient conditions ensuring the existence of a Monge-Ampère exhaustion as in the case of strongly convex domains in C and to show that the metric is (a suitable multiple of) the Kobayashi metric of M . We like to thank J. Bland for some very useful remarks and his interest in our work.

UNIQUENESS OF COMPLEX GEODESICS AND CHARACTERIZATION OF CIRCULAR DOMAINS

We study complex geodesics for complex Finsler metrics and prove a uniqueness theorem for them. The results obtained are applied to the case of the Kobayashi metric for which, under suitable hypotheses, we describe the exponential map and the relationship between the indicatrix and small geodesic balls. Finally, exploiting the connection between intrinsic metrics and the complex Monge-Ampère equation, we give characterizations for circular domains in ℂn.

Finite type Monge–Ampère foliations

In this paper we extend our previous work on singularities of Monge-Amp\`ere foliations to the case of pseudoconvex finite type domains. We are able to answer the questin of Burns on homogeneous polynomials whose logarithm satisfies the complex Monge-Amp\`ere equation completely in dimension 2 . We are also able to generalize the work of P.M. Wong in dimension 2 on the classification of complete weighted circular domains to include finite type domains.

Stationary disks and Green functions in almost complex domains

Using generalized Riemann maps, normal forms for almost complex domains (D, J) with singular foliations by stationary disks are defined. Such normal forms are used to construct counterexamples and to determine intrinsic conditions, under which the stationary disks are extremal disks for the Kobayashi metric or determine solutions to almost complex Monge-Ampere equation.

Complex geodesics and Finsler metrics

0. Introduction. In the study of intrinsic metrics and distances on complex manifolds, a crucial role is played by the notion of complex geodesic introduced by Vesentini [V]. Roughly speaking a complex geodesic is a holomorphic embedding of the unit disk with the hyperbolic metric which is an isometry with respect to the intrinsic metric or distance (or both) which is defined on the manifold under consideration. As it is well known, the problem of existence of complex geodesics is satisfactory solved only for convex domains by the work of Lempert [L] who also proves uniqueness for strictly convex domains (see also [A, chapter 2.6]). In order to find a different approach to this problem, which may eventually lead to an understanding of it on a larger class of complex manifolds, in [AP1] and [AP2] it was studied the same problem from a differential geometric point view looking for minimal conditions on an abstract complex Finsler metric which imply the existence and uniqueness of complex geodesics. A complete solution to the problem was achieved in terms of the holomorphic curvature of the metric, which must be a negative constant, and the vanishing of suitable torsion tensors. We give a brief account of these results at the beginning of section 2. In this general framework it is very natural to ask whether it is possible to solve the same kind of problems for isometric holomorphic embeddings of C with the euclidean metric and of P1 with the Fubini-Study metric. In this paper we show that the methods of [AP1] and [AP2] work also in this case and that it is

Regularity of Kobayashi Metric

We review some recent results on existence and regularity of Monge-Amp\`ere exhaustions on the smoothly bounded strongly pseudoconvex domains, which admit at least one such exhaustion of sufficiently high regularity. A main consequence of our results is the fact that the Kobayashi pseudo-metric k on each of the above domains is actually a smooth Finsler metric. The class of domains to which our result apply is very large. It includes for instance all smoothly bounded strongly pseudoconvex complete circular domains and all their sufficiently small deformations.