Nonexistence of Proper Holomorphic Maps Between Certain Classical Bounded Symmetric Domains

The author,motivated by his results on Hermitian metric rigidity,conjectured in [4] that a proper holomorphic mapping f：Ω→Ω′from an irreducible bounded symmetric domainΩof rank≥2 into a bounded symmetric domainΩ′is necessarily totally geodesic provided that r′：=rank（Ω′）≤rank（Ω）：=r.The Conjecture was resolved in the affirmative by I.-H.Tsai [8].When the hypothesis r′≤r is removed,the structure of proper holomorphic maps f：Ω→Ω′is far from being understood,and the complexity in studying such maps depends very much on the difference r′-r,which is called the rank defect.The only known nontrivial non-equidimensional structure theorems on proper holomorphic maps are due to Z.-H.Tu [10],in which a rigidity theorem was proven for certain pairs of classical domains of type I,which implies nonexistence theorems for other pairs of such domains.For both results the rank defect is equal to 1,and a generaliza- tion of the rigidity result to cases of higher rank defects along the line of arguments of [10] has so far been inaccessible. In this article, the author produces nonexistence results for infinite series of pairs of （Ω→Ω′） of irreducible bounded symmetric domains of type I in which the rank defect is an arbitrarily prescribed positive integer. Such nonexistence results are obtained by exploiting the geometry of characteristic symmetric subspaces as introduced by N. Mok and L-H Tsai [6] and more generally invariantly geodesic subspaces as formalized in [8]. Our nonexistence results motivate the formulation of questions on proper holomorphic maps in the non-equirank case.     

Rigidity of proper holomorphic mappings between certain unbounded non-hyperbolic domains

The Fock–Bargmann–Hartogs domain _Dn,m(μ)$D_{n,m}(\mu )$ (μ>0$\mu >0$) in C^n+m${\mathbf{C}}^{n+m}$ is defined by the inequality ‖w‖²<e^{−μ‖z‖2}${\Vert w\Vert }^2<e^{-\mu {\Vert z\Vert }^2}$, where (z,w)∈Cⁿ×C^m$(z,w)\in {\mathbf{C}}^n\times {\mathbf{C}}^m$, which is an unbounded non-hyperbolic domain in C^n+m${\mathbf{C}}^{n+m}$. Recently, Yamamori gave an explicit formula for the Bergman kernel of the Fock–Bargmann–Hartogs domains in terms of the polylogarithm functions and Kim–Ninh–Yamamori determined the automorphism group of the domain _Dn,m(μ)$D_{n,m}(\mu )$. In this article, we obtain rigidity results on proper holomorphic mappings between two equidimensional Fock–Bargmann–Hartogs domains. Our rigidity result implies that any proper holomorphic self-mapping on the Fock–Bargmann–Hartogs domain _Dn,m(μ)$D_{n,m}(\mu )$ with m≥2$m\ge 2$ must be an automorphism.     

Proper Holomorphic Mappings Between Some Nonsmooth Domains

61. IntroductionLet DI, DZ be two bounded domains in C", a holomorphic mapping F: DI ~ DZ isproper if F--'(K) is compact whenever K is compact. It is Obvious that for every boUndeddomain in C", there always edests proper holomorphic self-mapping. However, given tabbounded domains DI, DZ in C", it does not seem easy to answer whether there ealsts properholomorphic mapping F: DI ~ D2. A main result obtained in recellt y6ars isTheorem 1.1.[1] Let Z (a) and Z (P) be two Generalized P…     

Proper Holomorphic Maps from Domains in C2 with Transverse Circle Action

The authors consider proper holomorphic mappings between smoothly bounded pseudoconvex regions in complex 2-space,where the domain is of finite type and admits a transverse circle action.The main result is that the closure of each irreducible component of the branch locus of such a map intersects the boundary of the domain in the union of finitely many orbits of the group action.     

Proper Holomorphic Maps from Domains in ?2 with Transverse Circle Action

The authors consider proper holomorphic mappings between smoothly bounded pseudoconvex regions in complex 2-space, where the domain is of finite type and admits a transverse circle action. The main result is that the closure of each irreducible component of the branch locus of such a map intersects the boundary of the domain in the union of finitely many orbits of the group action.     

Holomorphic Maps of Compact Generalized Hopf Manifolds

We investigate holomorphic maps between compact generalized Hopf manifolds (i.e., locally conformal Kähler manifolds with parallel Lee form). We show that they preserve the canonical foliations. Moreover, we study compact complex submanifolds of g.H. manifolds and holomorphic submersions from compact g.H. manifolds.     

A classification of proper holomorphic mappings between generalized pseudoellipsoids of different dimensions

ABSTRACT

We classify proper holomorphic mappings between generalized pseudoellipsoids of different dimensions. Those domains are parametrized by the exponents. The relations among them are also obtained. Main tool is the orthogonal decomposition of a CR bundle. Such a decomposition derives the ‘variable-splitting’ of the mapping.     

一类全纯映射的增长性

§ 1.　NotationsandFormulas　　Let(N ,dS2 N)beaHermitianmanifoldofdimensionn ,anddS2 NitsHermitianmertric.Let{e1～ ,Λen～}oftype (1 ,0 )tangentvectorsconstitutealocalunitaryframefieldinanopensetofM .Let{ω1 ,Λωn}betheunitarycoframewhichconsistsofncomplex valuedlineardifferen tialformsoftype (1 ,0 )suchthatdS2 N =ωαω-α. (1 )Hereandbelowthesummationconventionisused ,i.e .,therepeatedindexmeanssummation .LetBbethebundleofallunitaryframesofN .Thentheforms{ωα}areformsinB .Itiswellkn…     

A local version of Wong-Rosay's theorem for proper holomorphic mappings

In the present paper, we generalize Wong-Rosay's theorem for proper holomorphic mappings with bounded multiplicity. As an application, we prove the non-existence of a proper holomorphic mapping from a bounded, homogenous domain in onto a domain in whose boundary contains strongly pseudoconvex points.

     

The Classification of Proper Holomorphic Mappings Between Special Hartogs Triangles of Different Dimensions

The authors discuss the proper holomorphic mappings between special Hartogs triangles of different dimensions and obtain a corresponding classification theorem.     

C~2中某类Hartogs域的逆紧全纯自映射

对C2中某类Hartogs域的逆紧全纯自映射证明了刚性定理,即逆紧全纯自映射必定为全纯自同构．此类域是光滑有界拟凸完全的Hartogs域,且它的边界上具有无限型点．     

Minkowski空间中保高斯映射的共形形变

设Ｍｎ为Ｒｉｅｍａｎｎ流形，给定类空浸入：Ｍｎ→Ｒｎ，ｐ，如果存在另一个类空浸入：Ｍｎ→Ｒｎ，ｐ，使与在共形对应之下且对应点的地空间平行，则称类空子流形是可保高斯映射共形形变的．本文给出可保高斯映射共形形变的充要条件．对ｎ＝２，ｐ＝１的情形，如果上述形变是同向的，我们分类了曲面；如果是反向的，我们用主曲率满足的方程来描述．     

关于逆紧全纯映射刚性的一个注记

本文对某类广义Hartogs三角形上的逆紧全纯自映射证明了刚性定理,即逆紧全纯自映射必定为全纯自同构．同时完全刻画了其全纯自同构群,并且给出了关于其全纯自同构以及两个这类域之间逆紧全纯映射的分类。     

Immersed surfaces of prescribed Gauss curvature into Minkowski space

Given a positive real valued function on the disc, we will immerse the disc into three dimensional Minkowski space in such a way that Gauss curvature at the image point of is . Our approach lies on the construction of Gauss map of surfaces.

     

Bifurcations of Periodic Points of Holomorphic Maps From C2 into C2

Let F:Cⁿ Cⁿ be a holomorphic map, F^k be the kth iterate ofF, and p Cⁿ be a periodic point of F of period k. That is,F^k(p) = p, but for any positive integer j with j < k, F^j(p) p. If p is hyperbolic, namely if DF^k(p) has no eigenvalue ofmodulus 1, then it is well known that the dynamical behaviourof F is stable near the periodic orbit = {p, F(p),..., F^k–1(p)}.But if is not hyperbolic, the dynamical behaviour of F near may be very complicated and unstable. In this case, a veryinteresting bifurcational phenomenon may occur even though may be the only periodic orbit in some neighbourhood of : forgiven M N\{1}, there may exist a C^r-arc {F_t: t [0,1]} (wherer N or r = ) in the space H(Cⁿ) of holomorphic maps from Cⁿinto Cⁿ, such that F₀ = F and, for t (0,1], F_t has an Mk-periodicorbit _t with as t 0. Theperiod thus increases by a factor M under a C^r-small perturbation!If such an F_t does exist, then , as well as p, is said to beM-tupling bifurcational. This definition is independent of r. For the above F, there may exist a C^r-arc in H(Cⁿ), with t [0,1], such that and, for t (0,1], has two distinct k-periodic orbits _t,1 and _t,2 with d(_t,i, ) 0 as t 0 for i = 1,2. If such an does exist, then , as well as p, is said to be 1-tupling bifurcational. In recent decades, there have been many papers and remarkableresults which deal with period doubling bifurcations of periodicorbits of parametrized maps. L. Block and D. Hart pointed outthat period M-tupling bifurcations cannot occur for M >2 in the 1-dimensional case. There are examples showing thatfor any M N, period M-tupling bifurcations can occur in higher-dimensionalcases. An M-tupling bifurcational periodic orbit as defined here actsas a critical orbit which leads to period M-tupling bifurcationsin some parametrized maps. The main result of this paper isthe following. Theorem. Let k N and M N, and let F: C² C² be a holomorphicmap with k-periodic point p. Then p is M-tupling bifurcationalif and only if DF^k(p) has a non-zero periodic point of periodM. 1991 Mathematics Subject Classification: 32H50, 58F14.     

Extension of Fourier algebra homomorphisms to duals of algebras of uniformly continuous functionals

For a locally compact group G, let XG be one of the following introverted subspaces of VN(G): , the C^∗-algebra of uniformly continuous functionals on A(G); , the space of weakly almost periodic functionals on A(G); or , the C^∗-algebra generated by the left regular representation on the measure algebra of G. We discuss the extension of homomorphisms of (reduced) Fourier-Stieltjes algebras on G and H to cb-norm preserving, weak^∗-weak^∗-continuous homomorphisms of into , where (XG,XH) is one of the pairs , , or . When G is amenable, these extensions are characterized in terms of piecewise affine maps.     

Weak lower semicontinuity of integral functionals

A lower semicontinuity theorem for integral functionals is proved underL ₁-strong convergence of the trajectories andL ₁-weak convergence of the control functions. An alternative statement is also proved under pointwise convergence of the trajectories.     

Random iterations of holomorphic maps in complex Banach spaces

Conditions guaranteeing the uniform convergence to constant maps of random iterations of holomorphic contractions on unbounded domains in complex Banach spaces are established.

     

关于C.Fefferman定理与全纯映射的边界扩充

C．Fefferman定理证明了光滑有界强拟凸域之间的双全纯映射可以光滑延拓到边界，这个结果已经被推广到各种情形．其中Bell和Catlin以及Diederich和Fornaess独立地将其推广到拟凸域的逆紧全纯映射．本文较全面地综述了C．Fefferman定理的推广情况以及Bergman投射的边界正则性问题，同时对如何去掉Bell和Catlin以及Diederich和Fornaess定理条件中的拟凸性给出一个新观察，提出一个解决方向并且说明在具体情况下这个新观察确实是可以提供答案的．