Dror Varolin

Stable manifolds of holomorphic diffeomorphisms

where ρ = ρp < 1 and C = Cp > 0. It turns out that often W s p is an immersed complex manifold. Assuming this to be the case, the following problem was posed by E. Bedford [B]. Problem: Determine the complex structure of the stable manifolds of f . In many cases it can be shown that W s p is a monotone union of balls, and this in turn implies [Br] that it is diffeomorphic to real Euclidean space. Moreover, by the contracting nature of the dynamics, one sees that the Kobayashi pseudometric of W s p vanishes identically. However, when dim(W s p ) ≥ 3, it is not possible to deduce only from these properties that W s p is biholomorphic to Euclidean space. For example, there exist monotone unions of balls which are not Stein [F]. (The question of Steinness of monotone unions of balls in complex dimension 2 is open.) When dim(W s p ) = 1, the Uniformization Theorem implies that W s p is biholomorphic to C [BLS, W]. The main results of this paper are proved in the non-uniform setting, i.e. with respect to compactly supported invariant measures. More precisely, we say that a subset A ⊂ M is invariant if fA = A, and that it has total measure if μ(A) = 1 for every compactly supported invariant probability measure μ. Our main objective in this paper is to prove the following theorem.

Analytic inversion of adjunction: L^2 extension theorems with gain@@@L’inversion analytique d’adjonction : théorèmes de prolongement avec gain

We establish new results on weighted

Holomorphic diffeomorphisms of semisimple homogeneous spaces

L^2

Interpolation and Sampling for Generalized Bergman Spaces on finite Riemann surfaces

extension of holomorphic top forms with values in a holomorphic line bundle, from a smooth hypersurface cut out by a holomorphic function. The weights we use are determined by certain functions that we call denominators. We give a collection of examples of these denominators related to the divisor defined by the submanifold.

Sampling Sequences for Bergman Spaces for p < 1

The density property for a Stein manifold X implies that the group of holomorphic diffeomorphisms of X is infinite-dimensional and, in a certain well-defined sense, as large as possible. We prove that if G is a complex semisimple Lie group of adjoint type and K is a reductive subgroup, then G/K has the density property. This theorem is a non-trivial extension of an earlier result of ours, which handles the case of complex semi-simple Lie groups. We also establish the density property for some other complex homogeneous spaces by ad hoc methods. Finally, we introduce a lifting method that extends many results on complex manifolds with the density property to covering spaces of such manifolds.

Analytic inversion of adjunction: L2 extension theorems with gain

The goal of this paper is to establish sufficient conditions for a uniformly separated set on a finite Riemann surface to be interpolating or sampling for a generalized Bergman space of holomorphic functions on that surface. Let us fix an open Riemann surface X. Much of the geometry used in the statements and proofs of our results arises from potential theory on X. If X is hyperbolic, then X admits a Green’s function, while if X is parabolic, then X admits a so-called Evans Kernel. (See Section 2 for definitions.) After a normalization of the latter, both objects are unique, and we refer to either as the extremal fundamental solution E(z, ζ). Associated to this extremal fundamental solution is a distance ρ(z, ζ) = e, and we denote by De(z) the e-disk with respect to this distance. The geometry and potential theory we use in this paper is discussed in greater detail in Section 2 To a conformal metric g = e−ψ|dz|2 on X, a smooth function φ : X → R and a discrete subset Γ ⊂ X, uniformly separated with respect to the distance ρ above, we associated the following two Hilbert spaces:

Toeplitz Operators and Carleson Measures on Generalized Bargmann–Fock Spaces

We provide a proof of the sufficiency direction of Seips characterization of sampling sequences for Bergman spaces for p < 1 based on the methods of Berndtsson and Ortega-Cerdà.