Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userA new gap phenomenon for proper holomorphic mappings from B^n into B^N
66
0
0.0
(
0
)
Authors
Xiaojun Huang
Ask ChatGPT about the research
No Arabic abstract
In this paper (Math. Res. Lett. 13 (2006). No 4, 509-523), the authors established a pseudo-normal form for proper holomoprhic mappings between balls in complex spaces with degenerate rank. This then was used to give a complete characterization for all proper holomorphic maps with geometric rank one, which, in particular, includes the following as an immediate application: Theorem: Any rational holomorphic map from B^n into B^N with $4le nle Nle 3n-4$ is equivalent to the DAngelo map $$F_{theta}(z,w)=(z,(costheta)w,(sintheta)z_1w, ..., (sintheta)z_{n-1}w, (sintheta)w^2, 0), 0le thetaleq pi/2.$$ It is a well-known (but also quite trivial) fact that any non-constant rational CR map from a piece of the sphere $partial {B^n}$ into the sphere $partial {B^N}$ can be extended as a proper rational holomoprhic map from $B^n$ into $B^N$ ($Nge nge 2$). By using the rationality theorem that the authors established in [HJX05], one sees that the the above theorem (and also the main theorem of the paper) holds in the same way for any non-constant $C^3$-smooth CR map from a piece of $partial {B^n}$ into $partial{B^N}$. The paper [Math. Res. Lett. 13 (2006). No 4, 509-523] was first electronically published by Mathematical Research Letters several months ago at its home website: http://www.mrlonline.org/mrl/0000-000-00/Huang-Ji-Xu2.pdf. (The pdf file of the printed journal version can also be downloaded at http://www.math.uh.edu/~shanyuji/rank1.pdf).
rate research
Read More
We introduce several homotopy equivalence relations for proper holomorphic mappings between balls. We provide examples showing that the degree of a rational proper mapping between balls (in positive codimension) is not a homotopy invariant. In domain dimension at least 2, we prove that the set of homotopy classes of rational proper mappings from a ball to a higher dimensional ball is finite. By contrast, when the target dimension is at least twice the domain dimension, it is well known that there are uncountably many spherical equivalence classes. We generalize this result by proving that an arbitrary homotopy of rational maps whose endpoints are spherically inequivalent must contain uncountably many spherically inequivalent maps. We introduce Whitney sequences, a precise analogue (in higher dimensions) of the notion of finite Blaschke product (in one dimension). We show that terms in a Whitney sequence are homotopic to monomial mappings, and we establish an additional result about the target dimensions of such homotopies.
We construct a complete proper holomorphic embedding from any strictly pseudoconvex domain with $mathcal{C}^2$-boundary in $mathbb{C}^n$ into the unit ball of $mathbb{C}^N$, for $N$ large enough, thereby answering a question of Alarcon and Forstneric.
We make several new contributions to the study of proper holomorphic mappings between balls. Our results include a degree estimate for rational proper maps, a new gap phenomenon for convex families of arbitrary proper maps, and an interesting result about inverse images.
Cartan-Thullen theorem is a basic one in the theory of analytic functions of several complex variables. It states that for any open set $U$ of ${mathbb C}^k$, the following conditions are equivalent: (a) $U$ is a domain of existence, (b) $U$ is a domain of holomorphy and (c) $U$ is holomorphically convex. On the other hand, when $f , (, =(f_1,f_2,cdots,f_n), )$ is a $mathbb C^n$-valued function on an open set $U$ of $mathbb C^{k_1}timesmathbb C^{k_2}timescdotstimesmathbb C^{k_n}$, $f$ is said to be $mathbb C^n$-analytic, if $f$ is complex analytic and for any $i$ and $j$, $i ot=j$ implies $frac{partial f_i}{partial z_j}=0$. Here, $(z_1,z_2,cdots,z_n) in mathbb C^{k_1}timesmathbb C^{k_2}timescdotstimesmathbb C^{k_n}$ holds. We note that a $mathbb C^n$-analytic mapping and a $mathbb C^n$-analytic manifold can be easily defined. In this paper, we show an analogue of Cartan-Thullen theorem for a $mathbb C^n$-analytic function. For $n=1$, it gives Cartan-Thullen theorem itself. Our proof is almost the same as Cartan-Thullen theorem. Thus, our generalization seems to be natural. On the other hand, our result is partial, because we do not answer the following question. That is, does a connected open $mathbb C^n$-holomorphically convex set $U$ exist such that $U$ is not the direct product of any holomorphically convex sets $U_1, U_2, cdots, U_{n-1}$ and $U_n$ ? As a corollary of our generalization, we give the following partial result. If $U$ is convex, then $U$ is the direct product of some holomorphically convex sets. Also, $f$ is said to be $mathbb C^n$-triangular, if $f$ is complex analytic and for any $i$ and $j$, $i<j$ implies $frac{partial f_i}{partial z_j}=0$. Kasuya suggested that a $mathbb C^n$-analytic manifold and a $mathbb C^n$-triangular manifold might, for example, be related to a holomorphic web and a holomorphic foliation.
In this paper, we generalize a recent work of Liu et al. from the open unit ball $mathbb B^n$ to more general bounded strongly pseudoconvex domains with $C^2$ boundary. It turns out that part of the main result in this paper is in some certain sense just a part of results in a work of Bracci and Zaitsev. However, the proofs are significantly different: the argument in this paper involves a simple growth estimate for the Caratheodory metric near the boundary of $C^2$ domains and the well-known Grahams estimate on the boundary behavior of the Caratheodory metric on strongly pseudoconvex domains, while Bracci and Zaitsev use other arguments.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
