Conical plurisubharmonic measure and new cross theorems

We study the boundary behavior of the conical plurisubharmonic measure. As an application, we establish some new extension theorems for separately holomorphic mappings defined on cross sets.     

Regularity and decay of solutions of nonlinear harmonic oscillators

We prove sharp analytic regularity and decay at infinity of solutions of variable coefficients nonlinear harmonic oscillators. Namely, we show holomorphic extension to a sector in the complex domain, with a corresponding Gaussian decay, according to the basic properties of the Hermite functions in ${\mathbb{R}}^d$. Our results apply, in particular, to nonlinear eigenvalue problems for the harmonic oscillator associated to a real-analytic scattering, or asymptotically conic, metric in ${\mathbb{R}}^d$, as well as to certain perturbations of the classical harmonic oscillator.     

Functions Holomorphic along Holomorphic Vector Fields

The main result of the paper is the following generalization of Forelli’s theorem (Math. Scand. 41:358–364, 1977): Suppose F is a holomorphic vector field with singular point at p, such that F is linearizable at p and the matrix is diagonalizable with eigenvalues whose ratios are positive reals. Then any function φ that has an asymptotic Taylor expansion at p and is holomorphic along the complex integral curves of F is holomorphic in a neighborhood of p. We also present an example to show that the requirement for ratios of the eigenvalues to be positive reals is necessary. K.T. Kim and G. Schmalz were supported by the Scientific visits to Korea program of the AAS and KOSEF. E. Poletsky was supported by NSF Grant DMS-0500880. G. Schmalz gratefully acknowledges support and hospitality of the Max-Planck-Institut für Mathematik Bonn.     

On Univalence of the Power Deformation z（f（z）/z）c

The authors mainly concern the set U _f of c ?? ? such that the power deformation $ z(frac{{f(z)}} {z})^c $ is univalent in the unit disk |z| < 1 for a given analytic univalent function f(z) = z + a ₂ z ² + ?? in the unit disk. It is shown that U _f is a compact, polynomially convex subset of the complex plane ? unless f is the identity function. In particular, the interior of U _f is simply connected. This fact enables us to apply various versions of the ??-lemma for the holomorphic family $ z(frac{{f(z)}} {z})^c $ of injections parametrized over the interior of U _f . The necessary or sufficient conditions for U _f to contain 0 or 1 as an interior point are also given.     

A general approach to index theorems for holomorphic maps and foliations

Let M be a smooth complex manifold, and S(⊂ M) be a compact irreducible subvariety with dim _C S > 0. Let be given either a holomorphic map f : M → M with f _|S  = id _S , f ≠ id _M , or a holomorphic foliation on M: we describe an approach that can be applied to both map and foliation in order to obtain index theorems. Partially supported by GNSAGA, Centro de Giorgi, M.U.R.S.T.     

The equality case in Wu–Yau inequality

In recent papers Wu-Yau, Tosatti-Yang and Diverio-Trapani, used some natural differential inequalities for compact Kähler manifolds with quasi negative holomorphic sectional curvature to derive positivity of the canonical bundle. In this note we study the equality case of these inequalities.     

Inverses depending holomorphically on a parameter in a Banach space

We show that the existence of a right inverse at each point for a holomorphic mapping from a pseudo-convex domain in a Banach space with an unconditional basis into a unital Banach algebra implies the existence of a holomorphic right inverse. Variations of this result are given.     

Reflection about k-jets and holomorphic extension of CR mappings

Let be a CR mapping between real analytic generic submanifolds M, M₁ of and , respectively. According to Webster's theory (Proc. Amer. Math. Soc. 86 (1982) 236-240) and its further developments, f has holomorphic extension to a full neighborhood of M in when the following requirements are fulfilled: f extends to a wedge W continuous up to M; f is of class C^k; (where denotes the complex tangent bundle); M₁ is “k-nondegenerate.” We deal here with the case where is strictly smaller than but is still real analytic in suitable sense. We show that a suitably refined condition of k-nondegeneracy still entails holomorphic extension of f.     

一类广义Hartogs三角形之间逆紧映射的分类

本文主要证明一类广义Hartogs三角形之间的逆紧全纯映射在相差全纯自同构的意义下是唯一的。     

Embedding Certain Infinitely Connected Subsets of Bordered Riemann Surfaces Properly into ℂ2

We prove that certain infinitely connected domains D in a bordered Riemann surface, which admits a holomorphic embedding into C ², admit a proper holomorphic embedding into C ². We also prove that certain infinitely connected open subsets D⊂ℂ admit a proper holomorphic embedding into ℂ².      

q extension of Wielandt's Theorem

It is well known that the q-Gamma function cannot be characterised by its functional equation o q ( z + 1) =/((1 m q z )(1 m q )) o q ( z ) and the condition o q (1) = 1. In 1980 Richard Askey showed in [1] that the additional assumption of logarithmic convexity yields the uniqueness of o q ( x ) for real x > 0. The goal of this note is to establish a q-extension of Wielandt's theorem which gives a characterization of o q ( z ) for all $ zin {zin{shadC}:{rm Re}(z)gt 0} $ .     

Complete polynomial vector fields on the complex plane

We give a full classification, up to polynomial automorphisms, of complete polynomial vector fields in two complex variables.