Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userOn the complexity of proper holomorphic mappings between balls
131
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
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.
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.
Let $f:{mathbb B}^n to {mathbb B}^N$ be a holomorphic map. We study subgroups $Gamma_f subseteq {rm Aut}({mathbb B}^n)$ and $T_f subseteq {rm Aut}({mathbb B}^N)$. When $f$ is proper, we show both these groups are Lie subgroups. When $Gamma_f$ contains the center of ${bf U}(n)$, we show that $f$ is spherically equivalent to a polynomial. When $f$ is minimal we show that there is a homomorphism $Phi:Gamma_f to T_f$ such that $f$ is equivariant with respect to $Phi$. To do so, we characterize minimality via the triviality of a third group $H_f$. We relate properties of ${rm Ker}(Phi)$ to older results on invariant proper maps between balls. When $f$ is proper but completely non-rational, we show that either both $Gamma_f$ and $T_f$ are finite or both are noncompact.
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).
In this paper we give some quantative characteristics of boundary asymptotic behavior of semigroups of holomorphic self-mappings of the unit disk including the limit curvature of their trajectories at the boundary Denjoy--Wolff point. This enable us to establish an asymptotic rigidity property for semigroups of parabolic type.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
