Proper Holomorphic Mappings Research Articles

On the third gap for proper holomorphic maps between balls

In this paper, we study the gap rigidity phenomenon for proper holomorphic maps between balls of different dimension. We show that any $F\in prop_3({\mathbb{B}}^n, {\mathbb{B}} ^N)$, with $3n<N\leq 4n-7$ and $n\geq 7$, is equivalent to a map of the form $(G,0)$ with $G\in Rat ({\mathbb{B}}^n,{\mathbb{B}}^{3n})$. The main ingredients for the proof of our main theorem are the normal form obtained by Huang-Ji-Xu and a lemma of the first author.

Proper holomorphic mappings vs. peak points and Shilov boundary

We present a result on the existence of some kind of peak functions for $\mathbb C$-convex domains and for the symmetrized polydisc. Then we apply the latter result to show the equivariance of the set of peak points for $A(D)$ under proper holomorphic map

Strong Stein neighbourhood bases

Let Ω be a smooth bounded pseudoconvex domain in ℂ n . We give several characterizations for the closure of Ω to have a Stein neighbourhood basis in the sense that Ω has a defining function ρ such that {z ∈ ℂ n : ρ(z) < ϵ} is pseudoconvex for sufficiently small ϵ > 0. We also show that this condition is invariant under proper holomorphic maps that extend smoothly up to the boundary.

Functions preserving plurisubharmonicity under holomorphic mappings

We study functions having the property that plurisubharmonicity is preserved under holomorphic mappings between domains in ℂ n . Both pull backs and push forwards are considered. It is shown that a function introduced by Väisälä satisfies the push forward property. As an application we prove a capacity inequality for proper holomorphic mappings.

Some regularity theorems for CR mappings

The purpose of this article is to study Lipschitz CR mappings from an h-extendible (or semi-regular) hypersurface in $${\mathbb{C}^n}$$ . Under various assumptions on the target hypersurface, it is shown that such mappings must be smooth. A rigidity result for proper holomorphic mappings from strongly pseudoconvex domains is also proved.

Proper holomorphic maps between weakly spherical domains

Proper holomorphic maps between weakly spherical domains

HERMITIAN COMPLEXITY OF REAL POLYNOMIAL IDEALS

We define the Hermitian complexity of a real polynomial ideal and of a real algebraic subset of Cn. This concept is aimed at determining precise necessary conditions for a Hermitian symmetric polynomial to agree with a Hermitian squared norm on an algebraic set. The latter topic has been a central theme in modern polynomial optimization and in complex geometry, specifically related to the holomorphic embedding of pseudoconvex domain into balls, or the classification of proper holomorphic maps between balls.

Rigidity for local holomorphic isometric embeddings from $\mathbb{B}^n$ into $\mathbb{B}^{N_1} × • • • × \mathbb{B}^{N_m}$ up to conformal factors

In this article, we study local holomorphic isometric embeddings from $\mathbb{B}^n$ into $\mathbb{B}^{N_1} × • • • × \mathbb{B}^{N_m}$ with respect to the normalized Bergman metrics up to conformal factors. Assume that each conformal factor is smooth Nash algebraic. Then each component of the map is a multi-valued holomorphic map between complex Euclidean spaces by the algebraic extension theorem derived along the lines of Mok, and Mok and Ng. Applying holomorphic continuation and analyzing real analytic subvarieties carefully, we show that each component is either a constant map or a proper holomorphic map between balls. Applying a linearity criterion of Huang, we conclude the total geodesy of non-constant components.

Proper holomorphic maps between weakly spherical domains

Proper holomorphic maps between weakly spherical domains

Several complex variables and CR geometry

This paper discusses developments in complex analysis and CR geometry in the last forty years related to the Cauchy–Riemann equations, proper holomorphic mappings between balls, and positivity conditions in complex analysis. The paper includes anecdotes about some of the contributors to these developments.

Proper holomorphic mappings between symmetrized ellipsoids

We characterize the existence of proper holomorphic mappings in the special class of bounded (1, 2, . . . , n)-balanced domains in $${\mathbb C^n,}$$ called the symmetrized ellipsoids. Using this result we conclude that there are no non-trivial proper holomorphic self-mappings in the class of symmetrized ellipsoids. We also describe the automorphism groups of these domains.

Quasi-homogénéité des applications holomorphes propres d’un domaine quasi-disqué sur un domaine disqué

A result of F. Berteloot and G.Patrizio [1] states that if f is a proper holomorphic map between two bounded complete circular domains Ω1 and Ω2 in C n+1 (n≥1), such that f -1 {0}={0} and such that the principal part fp of the Taylor expansions of f at the origin is nondegenerated i.e fp -1 {0}={0}, then f≡fp . Here we propose to generalize their result in the case where Ω1 is a complete quasi-circular domain and Ω2 is a complete circular domain. Moreover this proof does not use the tools of projective dynamics of J. E. Fornaess and N. Sibony [3].

The Szegö Kernel and Proper Holomorphic Mappings to a Half Plane

We prove that all proper holomorphic mappings from a finitely connected domain in the plane to the right half plane can be expressed simply in terms of the Szego kernel associated to the domain. Our decomposition also reveals the linear structure of the semi-group of all such maps and it offers a method to construct proper holomorphic maps of arbitrary mapping degree.

A Local extension of proper holomorphic maps between some unbounded domains in ${\mathbb C}^n$

In this paper we study analytic sets extending the graph of proper holomorphic mappings. As applications, we present some results on the local holomorphic extension of proper holomorphic mappings between certain algebraic domains in ${\mathbb C}^n$, not necessarily bounded. Further supplementary results are obtained.

Geometry of quasi-circular domains and applications to tetrablock

We prove that the Shilov boundary is invariant under proper holomorphic mappings between some classes of domains (containing among others quasi-balanced domains with continuous Minkowski functionals). Moreover, we obtain an extension theorem for proper holomorphic mappings between quasi-circular domains. Using these results we show that there are no non-trivial proper holomorphic self-mappings in the tetrablock. Another important result of our work is a description of Shilov boundaries of a large class of domains (containing among other the symmetrized polydisc and the tetrablock). It is also shown that the tetrablock is not -convex.

Strongly pseudoconvex domains as subvarieties of complex manifolds

In this paper we obtain existence and approximation results for closed complex subvarieties that are normalized by strongly pseudoconvex Stein domains. Our sufficient condition for the existence of such subvarieties in a complex manifold $X$ is expressed in terms of the Morse indices and the number of positive Levi eigenvalues of an exhaustion function on $X$. Examples show that our conditions cannot be weakened in general. We obtain optimal results for subvarieties of this type in complements of compact complex submanifolds with Griffiths positive normal bundle; in the projective case these generalize classical theorems of Remmert, Bishop and Narasimhan concerning proper holomorphic maps and embeddings to ${\Bbb C}^n ={\Bbb P}^n \backslash {\Bbb P}^{n-1}$..

Factorization of proper holomorphic maps on irreducible bounded symmetric domains of rank ⩾ 2

We obtain rigidity results on arbitrary proper holomorphic maps F from an irreducible bounded symmetric domain Ω of rank ⩾ 2 into any complex space Z. After lifting to the normalization of the subvariety F(Ω) ⊂ Z, we prove that F must be the canonical projection map to the quotient space of Ω by a finite group of automorphisms. The approach is along the line of the works of Mok and Tsai by considering radial limits of bounded holomorphic functions derived from F and proving that proper holomorphic maps between bounded symmetric domains preserve certain totally geodesic subdomains. In contrast to the previous works, in general we have to deal with multivalent holomorphic maps for which Fatou’s theorem cannot be applied directly. We bypass the difficulty by devising a limiting process for taking radial limits of correspondences arising from proper holomorphic maps and by elementary estimates allowing us to define distinct univalent branches of the underlying multivalent map on certain subsets. As a consequence of our rigidity result, with the exception of Type-IV domains, any proper holomorphic map f: Ω → D of Ω onto a bounded convex domain D is necessarily a biholomorphism. In the exceptional case where Ω is a Type-IV domain, either f is a biholomorphism or it is a double cover branched over a totally geodesic submanifold which can be explicitly described.

Proper holomorphic mappings between Reinhardt domains in ℂ2

Proper holomorphic mappings between Reinhardt domains in ℂ2

RIGIDITY OF PROPER HOLOMORPHIC MAPS FROM Bn+1TO B3n-1

Let <TEX>$B^{n+1}$</TEX> be the unit ball in the complex vector space <TEX>$\mathbb{C}^{n+1}$</TEX> with the standard Hermitian metric. Let <TEX>${\Sigma}^n={\partial}B^{n+1}=S^{2n+1}$</TEX> be the boundary sphere with the induced CR structure. Let f : <TEX>${\Sigma}^n{\hookrightarrow}{\Sigma}^N$</TEX> be a local CR immersion. If N < 3n - 1, the asymptotic vectors of the CR second fundamental form of f at each point form a subspace of the CR(horizontal) tangent space of <TEX>${\Sigma}^n$</TEX> of codimension at most 1. We study the higher order derivatives of this relation, and we show that a linearly full local CR immersion f : <TEX>${\Sigma}^n{\hookrightarrow}{\Sigma}^N$</TEX>, N <TEX>$\leq$</TEX> 3n-2, can only occur when N = n, 2n, or 2n + 1. As a consequence, it gives an extension of the classification of the rational proper holomorphic maps from <TEX>$B^{n+1}$</TEX> to <TEX>$B^{2n+2}$</TEX> by Hamada to the classification of the rational proper holomorphic maps from <TEX>$B^{n+1}$</TEX> to <TEX>$B^{3n+1}$</TEX>.

Classification of rational holomorphic maps from % MathType!MTEF!2!1!+- % feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB % PrgifHhDYfgasaacH8srps0lbbf9q8WrFfeuY-Hhbbf9v8qqaqFr0x % c9pk0xbba9q8WqFfea0-yr0RYxir-Jbba9q8aq0-yq-He9q8qqQ8fr % Fve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaatuuDJX % wAK1uy0HMmaeHbfv3ySLgzG0uy0HgiuD3BaGqbaiab-fi8cjab-bW9 % YmaaCaaaleqabaacbaGaa4Nmaaaaaaa!4613! $$ \mathbb{B}^2 $$ into % MathType!MTEF!2!1!+- % feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB % PrgifHhDYfgasaacH8srps0lbbf9q8WrFfeuY-Hhbbf9v8qqaqFr0x % c9pk0xbba9q8WqFfea0-yr0RYxir-Jbba9q8aq0-yq-He9q8qqQ8fr % Fve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaatuuDJX % wAK1uy0HMmaeHbfv3ySLgzG0uy0HgiuD3BaGqbaiab-fi8cnaaCaaa % leqabaGae8xfH4eaaaaa!44B3! $$ \mathbb{B}^\mathbb{N} $$ with degree 2

Rational proper holomorphic maps from the unit ball in ℂ2 into the unit ball ℂN with degree 2 are classified, up to automorphisms of balls.