We present a new proof of Chern-Ji's mapping theorem on a strongly pseudoconvex domain with differentiable spherical boundary. We show that a proper holomorphic self mapping of a strongly pseudoconvex domain with the real analytic boundary is biholomorphic. We shall show that a bounded domain D is biholomorphic to an open ball B n+1 whenever the boundary bD is locally biholomorphic to the bound...

let $f$ be a locally univalent function on the unit disk $u$. we consider the normalized extensions of $f$ to the euclidean unit ball $b^nsubseteqmathbb{c}^n$ given by$$phi_{n,gamma}(f)(z)=left(f(z_1),(f'(z_1))^gammahat{z}right),$$ where $gammain[0,1/2]$, $z=(z_1,hat{z})in b^n$ and$$psi_{n,beta}(f)(z)=left(f(z_1),(frac{f(z_1)}{z_1})^betahat{z}right),$$in which $betain[0,1]$, $f(z_1)neq 0$ and $...

0. Introduction and Preliminary 2 0.1. Existence and uniqueness theorem 2 0.2. Equation of a chain 7 1. Nonsingular matrices 9 1.1. A family of nonsingular matrices 9 1.2. Sufficient condition for Nonsingularity 12 1.3. Estimates 16 2. Local automorphism group of a real hypersurface 23 2.1. Polynomial Identities 23 2.2. Injectivity of a Linear Mapping 33 2.3. Beloshapka-Loboda Theorem 42 3. Com...

It is a classical fact that there is no Riemann mapping theorem in the function theory of several complex variables. Indeed, H. Poincaré proved in 1906 that the unit ball B = {z = (z1, z2) ∈ C 2 : |z| ≡ |z1| 2 + |z2| 2 < 1} and the unit bidisc D = {z = (z1, z2) ∈ C 2 : |z1| < 1, |z2| < 1} are biholomorphically inequivalent. More recently, Burns/Shnider/Wells [BSW] and Greene/Krantz [GRK1] have ...

For D, D analytic polyhedra in Cn, it is proven that a biholomorphic mapping f : D→ D extends holomorphically to a dense boundary subset under certain condition of general position. This result is also extended to a more general class of domains with no smoothness condition on the boundary.

We will develop some of the basic concepts of complex function theory and prove a number of useful results concerning holomorphic functions. We will focus on derivatives, zeros, and sequences of holomorphic functions. This will lead to a brief discussion of the significance of biholomorphic mappings and allow us to prove the Riemann mapping theorem.

In this paper we study biholomorphic maps between Teichmüller spaces and the induced linear isometries between the corresponding tangent spaces. The first main result in this paper is the following classification theorem. If M and N are two Riemann surfaces that are not of exceptional type, and if there exists a biholomorphic map between the corresponding Teichmüller spaces Teich(M) and Teich(N...

We define the domain of a linear fractional transformation in a space of operators and show that both the affine automorphisms and the compositions of symmetries act transitively on these domains. Further, we show that Liouville’s theorem holds for domains of linear fractional transformations and, with an additional trace class condition, so does the Riemann removable singularities theorem. We ...

We present a proof of the existence and uniqueness theorem of a normalizing biholomorphic mapping to Chern-Moser normal form. The explicit form of the equation of a chain on a real hyperquadric is obtained. There exists a family of normal forms of real hypersurfaces including Chern-Moser normal form. 0. Introduction Let M be an analytic real hypersurface with nondegenerate Levi form in a comple...