The Teichmüller distance on the space of spherical CR structures

We define the Teichmüller pseudodistance on the space of spherical CR structures on a fixed compact manifold by using quasiconformal mappings between spherical CR manifolds. The pseudodistance is shown to be a complete distance. Dedicated to Professor Sheng GONG on the occasion of his 75th birthday     

Analytic sets and the boundary regularity of CR mappings

It is shown that if a continuous CR mapping between smooth real analytic hypersurfaces of finite type in extends as an analytic set, then it extends as a holomorphic mapping.

     

A generalization of the little Picard theorem

We show the little Picard theorem for a map of a quasicompact manifold to a relatively compact domain of Pn whose degeneracy locus of the Kobayashi pseudodistance is contained in some algebraic hypersurface.     

The maximum principle for the Bergman space and the Möbius pseudodistance for the annulus

It is shown that the formula for the Möbius pseudodistance for the annulus yields better estimates than previously known for the constant in the Bergman space maximum principle.

     

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.     

The Harnack Estimate for the Yamabe Flow on CR Manifolds of Dimension 3

We deform the contact form by the amount of the Tanaka–Webstercurvature on a closed spherical CR three-manifold.We show that if a contact form evolves with free torsion from initial datawith positive Tanaka–Webster curvature, then a certainHarnack inequality for the Tanaka–Webster curvature holds.     

Compactness of pseudohermitian structures with integral bounds on curvature

In this paper, we show a compactness criterion for contact forms in a fixed CR structure(i.e., conformal pseudohermitian structures), assuming a volume bound and L^p bounds, p>2, on the Tanaka-Webster curvature, the pseudohermitian torsion and its covariant derivative. We also need the L² bound on the derivative of the Tanaka-Webster curvature along the characteristic vector field. As an application, we can show that the CR automorphism group is compact if M is not CR spherical or the CR Yamabe constant is negative.     

A Brody theorem for orbifolds

We study the Kobayashi pseudodistance for orbifolds, proving an orbifold version of Brody’s theorem and classifying which one-dimensional orbifolds are hyperbolic.     

On the Schwarz symmetry principle in a model case

In this article, we prove that smooth CR diffeomorphisms between two real analytic holomorphically nondegenerate hypersurfaces, one of which is rigid and polynomial, extend to be locally biholomorphic. It turns out that the result can be generalized to not totally degenerate mappings, in the sense of Baouendi and Rothschild.

     

Flatness of CR submanifolds in a sphere

In Euclidean geometry, for a real submanifold M in E n+a , M is a piece of E n if and only if its second fundamental form is identically zero. In projective geometry, for a complex submanifold M in CP n+a , M is a piece of CP n if and only if its projective second fundamental form is identically zero. In CR geometry, we prove the CR analogue of this fact in this paper.     

Discrete groups and holomorphic functions

We determine conditions under which the quotient of a Kähler-Hadamard manifold by a torsion-free, discrete group of isometries admits a big set of nontrivial holomorphic functions. We also generalize a theorem of G. A. Margulis on centerless of cocompact discrete groups of automorphisms of a Carathéodory hyperbolic manifold by a different approach based on the Carathéodory pseudodistance.     

Geometric structures associated to triangulations as fixed point sets of involutions

We establish that hyperbolic structures and spherical CR structures on a three-dimensional manifold are contained in fixed point sets of a larger class of structures associated to a triangulation of the manifold. We generalize the 5 term relation to this setting.     

Reproducing Kernel for D2(Ω, ρ) and Metric Induced by Reproducing Kernel

An important property of the reproducing kernel of D2(Ω, ρ) is obtained and the reproducing kernels for D2(Ω, ρ) are calculated when Ω = Bn × Bn and ρ are some special functions. A reproducing kernel is used to construct a semi-positive definite matrix and a distance function defined on Ω×Ω. An inequality is obtained about the distance function and the pseudodistance induced by the matrix.     

On an example concerning the Kobayashi pseudodistance

In this short note we calculate the Kobayashi pseudodistance for elementary Reinhardt domains in . They deliver us a number of examples giving a negative answer to a problem posed by S. Kobayashi.

     

Heights on a subvariety of an abelian variety

Extending Ullmo-Zhang's result on the Bogomolov conjecture, we give conditions that a closed subvariety of an abelian variety A defined over a number field is isomorphic to an abelian variety in terms of the value distribution of a Neron-Tate height function on the subvariety. As a corollary of the result, we prove the Bogomolov conjecture which claims that if an irreducible curve X in A is not isomorphic to an elliptic curve, then for the pseudodistance defined by the Neron-Tate height, the distribution of algebraic points on X is uniformly discrete. These results can be extended in the case where base fields are finitely generated over via Moriwaki's height theory.     

Hartogs域上延拓算子的性质

主要研究Roper-Suffridge延拓算子在推广的Hartogs域上的性质.借助双全纯映照的偏差定理,得到延拓算子在Ω_N上保持强α次殆β型螺形映照、α次殆β型螺形映照和α次β型螺形映照的性质,进而得到B~n上相应的结论.所得结论包含已有的结果并为研究C~n中的双全纯映照提供了新的途径.     

B~n上推广的Roper-Suffridge算子的性质

讨论改进后的Roper-Suffridge延拓算子保持双全纯映照子族的性质.借助双全纯映照子族的解析特征及其偏差结论,得到改进后的Roper-Suffridge延拓算子在一定条件下保持α次殆β型螺形映照、α次β型螺形映照、强β型螺形映照的性质,从而得到改进后的算子在一定条件下保持α次殆星形性、α次星形性和强星形性.所得结论为在多复变数空间中构造这些双全纯映照提供了一种新的途径.     

一类凸模糊映射的注记

本文讨论了一些凸模糊映射与拟凸模糊映射的性质及其在凸模糊优化的应用 .     

A Simple Proof of the Riemann Mapping Theorem for Domains with Spherical Boundary

We give a simple proof of a Riemann mapping theorem for domains in a Stein manifold with a spherical boundary. Those boundaries are real hypersurfaces which are locally CR equivalent to the sphere inside complex space. The main observation is that for higher dimensional Stein manifolds, the fundamental group of its boundary coincides with that of its interior.     

Cn中拟凸映射和星形映射的系数问题

§ 1.　Introduction　　Theconvexfunctionshavebeenextensivelystudiedinonecomplexvariable .Thesefunc tionsareeasilycharacterizedbysimpleanalyticorgeometricconditionsandtherearemanywellknownresultswhichcanhelpusunderstandtheirnature.However,thedifficultiesariseinsev eralcomplexvariables.Imposingtheconditionthatamappingbeconvexmappingturnsouttobeveryrestrictiveandsothefamilyofquasi convexmappingswasintroducedbyROPERKAandSUFFRIDGETJin [1 ].In [2 ]wehaveshown :thefamilyofquasi convexmappings…