We study whether the basin of attraction of a sequence of automorphisms of C is biholomorphic to C. In particular we show that given any sequence of automorphisms with the same attracting fixed point, the basin is biholomorphic to C if the maps are repeated often enough. We also construct Fatou-Bieberbach domains whose boundaries are 4-dimensional.

Throughout this paper Σ is a smooth, oriented, closed Riemann surface of genus g, with n punctures and 3g−3+n > 1. Teichmüller space Tg,n is the space of conformal structures on Σ, where two conformal structures σ and ρ are equivalent if there is a biholomorphic map between (Σ, σ) and (Σ, ρ) in the homotopy class of the identity map. The moduli space Mg of Riemann surfaces can be obtained as th...

We study whether the basin of attraction of a sequence of automorphisms of Ck is biholomorphic to Ck. In particular we show that given any sequence of automorphisms with the same attracting fixed point, the basin is biholomorphic to Ck if every map is iterated sufficiently many times. We also construct Fatou-Bieberbach domains in C2 whose boundaries are 4-

We show that the Bergman, Szegő, and Poisson kernels associated to an n-connected domain in the plane are not genuine functions of two complex variables. Rather, they are all given by elementary rational combinations of n+ 1 holomorphic functions of one complex variable and their conjugates. Moreover, all three kernel functions are composed of the same basic n+ 1 functions. Our results can be i...

Let Ω1 and Ω2 be strongly pseudoconvex domains in C and f : Ω1 → Ω2 an isometry for the Kobayashi or Carathéodory metrics. Suppose that f extends as a C map to Ω̄1. We then prove that f |∂Ω1 : ∂Ω1 → ∂Ω2 is a CR or anti-CR diffeomorphism. It follows that Ω1 and Ω2 must be biholomorphic or anti-biholomorphic. The main tool is a metric version of the Pinchuk rescaling technique.

There exist polynomial identities asociated to normal form, which yield an existence and uniqueness theorem. The space of normalized real hypersurfaces has a natural group action. Umbilic point is defined via normal form. A nondegenerate analytic real hypersurface is locally biholomorphic to a real hyperquadric if and only if every point of the real hypersurface is umbilic. 0. Introduction An a...

We prove that a complete noncompact Kähler manifold Mof positive bisectional curvature satisfying suitable growth conditions is biholomorphic to a pseudoconvex domain of C and we show that the manifold is topologically R2n. In particular, when M is a Kähler surface of positive bisectional curvature satisfying certain natural geometric growth conditions, it is biholomorphic to C2.

Let 1 and 2 be strongly pseudoconvex domains in Cn and f : 1 → 2 an isometry for the Kobayashi or Carathéodory metrics. Suppose that f extends as a C1 map to
̄1. We then prove that f |∂ 1 : ∂ 1 → ∂ 2 is a CR or anti-CR diffeomorphism. It follows that 1 and 2 must be biholomorphic or anti-biholomorphic. Mathematics Subject Classification (2000): 32F45 (primary); 32Q45 (secondary).