Complex geodesics in tube domains and their role in the study of harmonic mappings in the disc

Complex geodesics in convex tube domains

We describe all complex geodesics in convex tube domains. In the case when the base of a convex tube domain does not contain any real line, the obtained description involves the notion of boundary measure of a holomorphic map and it is expressed in the language of real Borel measures on the unit circle. Applying our result, we calculate all complex geodesics in convex tube domains with unbounded base covering special class of Reinhardt domains.

Complex geodesics in convex tube domains II

We give a description (direct formulas) of all complex geodesics in a convex tube domain in $\CC^n$ containing no complex affine lines, expressed in terms of geometric properties of the domain. We next apply that result to give formulas (a necessary condition) for extremal mappings with respect to the Lempert function and the Kobayashi-Royden metric in a big class of bounded, pseudoconvex, complete Reinhardt domains: for all of them in $\CC^2$ and for those of them in $\CC^n$ which logarithmic image is strictly convex in geometric sense.

Regularity of complex geodesics and (non)-Gromov hyperbolicity of convex tube domains

We deliver examples of non-Gromov hyperbolic tube domains with convex bases (equipped with the Kobayashi distance). This is shown by providing a criterion on non-Gromov hyperbolicity of (non-smooth) domains.The results show the similarity of geometry of the bases of non-Gromov hyperbolic tube domains with the geometry of non-Gromov hyperbolic convex domains. A connection between the Hilbert metric of a convex domain $Ω$ in $\mathbb R^n$ with the Kobayashi distance of the tube domain over the domain $Ω$ is also shown. Moreover, continuity properties up to the boundary of complex geodesics in tu…

复双曲流形中的管域

Tubes in Complex Hyperbolic Manifolds

本研究证明复双曲2-流形中嵌入复测地曲面的管状邻域定理，给出宽度显式估计，推导流形体积下界与拉普拉斯第一特征值上界，并通过PU(2,1)富克斯子群的几何组合定理得到管域最优宽度的渐近上下界。

A brief proof of Bochner's tube theorem and a generalized tube

The aim of this note is firstly to give a new brief proof of classical Bochner's Tube Theorem (1938) by making use of K. Oka's Boundary Distance Theorem (1942), showing directly that two points of the envelope of holomorphy of a tube can be connected by a line segment. We then apply the same idea to show that if an unramified domain $\mathfrak{D}:=A_1+iA_2 \to \mathbf{C}^n$ with unramified real domains $A_j \to \mathbf{R}^n$ is pseudoconvex, then the both $A_j$ are univalent and convex (a generalization of Kajiwara's theorem). From the viewpoint of this result we discuss a generalization by M.…

Theta functions on tube domains

We discuss generalizations of classical theta series, requiring only some basic properties of the classical setting. As it turns out, the existence of a generalized theta transformation formula implies that the series is defined over a quasi-symmetric Siegel domain. In particular the exceptional symmetric tube domain does not admit a theta function.