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Given an imaginary quadratic extension $K$ of $\mathbb{Q}$, we give a classification of the maximal nonelementary subgroups of the Picard modular group $\operatorname{PSU}_{1,2}(\mathcal{O}_K)$ preserving a complex geodesic in the complex hyperbolic plane $\mathbb{H}^2_\mathbb{C}$. Complementing work of Holzapfel, Chinburg-Stover and M\"oller-Toledo, we show that these maximal $\mathbb{C}$-Fuchsian subgroups are arithmetic, arising from a quaternion algebra $\Big(\!\begin{array}{c} D\,,D_K\\\hline\mathbb{Q}\end{array} \!\Big)$ for some explicit $D\in\mathbb{N}-\{0\}$ and $D_K$ the discriminant of $K$. We thus prove the existence of infinitely many orbits of $K$-arithmetic chains in the hypersphere of $\mathbb{P}_2(\mathbb{C})$.

We prove that for a strongly pseudoconvex domain $$D\subset \mathbb {C}^n$$ , the infinitesimal Caratheodory metric $$g_C(z,v)$$ and the infinitesimal Kobayashi metric $$g_K(z,v)$$ coincide if z is sufficiently close to bD and if v is sufficiently close to being tangential to bD. Also, we show that every two close points of D sufficiently close to the boundary and whose difference is almost tangential to bD can be joined by a (unique up to reparameterization) complex geodesic of D which is also a holomorphic retract of D. The same continues to hold if D is a worm domain, as long as the points are sufficiently close to a strongly pseudoconvex boundary point. We also show that a strongly pseudoconvex boundary point of a worm domain can be globally exposed; this has consequences for the behavior of the squeezing function.

We give a description of complex geodesics and we study the structure of stationary discs in some non-convex domains for which complex geodesics are not unique., Comment: 9 pages

Using the Kobayashi distance, we provide sufficient conditions for the intersection of a family of holomorphic retracts in a Banach space to also be a holomorphic retract.

Let be convex domains in complex normed spaces respectively. When a mapping is holomorphic with f(0) = 0, we obtain some results like the Schwarz lemma. Furthermore, we discuss a condition whereby f is linear or injective or isometry.

The purpose of the article is two-fold. First of all, we will show that in general gap rigidity already fails in the complex topology. More precisely, we show that gap rigidity fails for $\Delta^2, \Delta \ti 0$ by constructing a sequence of ramified coverings fi : Si goes to Ti between hyperbolic compact Riemann surfaces such that, with respect to norms defined by the Poincare metrics. Since any bounded symmetric domain of rank greater than or equal to 2 contains a totally geodesic bidisk, this implies that gap rigidity fails in general on any bounded symmetric domain of rank greater than or equal to 2. Our counter examples make it all the more interesting to find sufficient conditions for pairs(OMEGA,D)for which gap rigidity holds. This will be addressed in the second part of the article, where for irreducible, we generalize the results for holomorphic curves in [Mok2002]to give a sufficient condition for gap rigidity to hold for (OMEGA,D)in the Zariski topology.

We prove that any proper holomorphic mapping from D I−1 to D I (p ≥ 3) is necessarily a totally geodesic isometric embedding with respect to their Bergman metrics and therefore is the standard linear em- bedding up to their automorphisms. The proof of the main result in this paper will be achieved by a differential-geometric study of a special class of complex geodesic curves on the bounded symmetric domains with respect to their Bergman metrics.

Let D be a balanced convex domain in a sequentially complete locally convex space E. If f : D E is a convex biholomorphic mapping with f(0) = 0 and df(0) = id, we have an upper bound of the growth of f. Also let D1, D2 be bounded balanced pseudoconvex domains in complex normed spaces E1, E2 respectively. When f : D1 D2 is a holomorphic mapping, we discuss a condition whereby f is linear or injective.

We prove that any proper holomorphic mapping between two equidimensional irreducible bounded symmetric domains with rank ≥ 2 \geq 2 is a biholomorphism. The proof of the main result in this paper will be achieved by a differential-geometric study of a special class of complex geodesic curves on the bounded symmetric domains with respect to their Bergman metrics.

This paper deals with the characterization of a domain D in Cn by the Kobayashi infinitesimal metric in a neighborhood of a point a of D. I prove this characterization in the following cases: a domain D in C analytically isomorphic to the open unit disc, an hyperbolic domain D in C, a bounded strictly convex domain D in Cn and also a bounded convex domain D in Cn which is isomorphic to an open unit ball. The proofs use the result of L. Lempert on the equality of the Carathéodory and Kobayashi infinitesimal metric on convex domains and the notion of complex geodesic.RésuméThis paper deals with the characterization of a domain D in Cn by the Kobayashi infinitesimal metric in a neighborhood of a point a of D. I prove this characterization in the following cases: a domain D in C analytically isomorphic to the open unit disc, an hyperbolic domain D in C, a bounded strictly convex domain D in Cn and also a bounded convex domain D in Cn which is isomorphic to an open unit ball. The proofs use the result of L. Lempert on the equality of the Carathéodory and Kobayashi infinitesimal metric on convex domains and the notion of complex geodesic.

LetD be the class of domains\(D_{a,p} = \left\{ {z \in \mathbb{C}^n |\left| {z_1 } \right|^{2_{p1} } + 2a\left| {z_1 } \right|^{P1} \left| {z_2 } \right|^{_{P2} } + \left| {z_2 } \right|^{2_{P2} } + \sum\limits_{j = 3}^n {\left| {z_j } \right|^{2_{Pj} }< 1} } \right\}\) forn≥2,a≥0 and p=(p1,...,n) ∈ (ℝ+) n such thatD a,p is convex. The classD is a class of convex bounded Reinhardt domains of ℂ n which are a generalization of complex ellipsoids. In this paper we compare Caratheodory balls and norm balls of the domainsD∈D. We prove that in this case a Caratheodory ball inD∈D is a norm ball if, and only if,D is a complex ellipsoid\(\left\{ {z \in \mathbb{C}\left| {\Sigma _{j = 1}^n \left| {z_j } \right|^{2_{Pj} }< 1} \right.} \right\}\) such thatp k=1 for exactly onek∈{1,…,n},p j=1/2 for allj≠k and the centre lies on thez k-axis.

In the paper we generalize the notion of problem (P) introduced by Poletsky. We introduce the notion of (P_m) extremals. For example, geodesics are (P_1) extremals. Using obtained results we present a description of (P_m) extremals in arbitrary complex ellipsoids. It is a generalization of the result obtained by Jarnicki-Pflug-Zeinstra. We also have a proof of conjecture put forward by Pflug-Zwonek concerning the formulas for geodesics in non-convex complex ellipsoids.

We prove that Kobayashi isometries between strongly convex domains are holomorphic or anti-holomorphic. More precisely, let $n_1, n_2$ be positive integers and let $\Omega_i \subset \C^{n_i}, \ i=1,2$, be bounded $C^3$ strongly convex domains. If $\phi: (\Omega_1, d^K_{\Omega_1}) \rightarrow (\Omega_2, d^K_{\Omega_2})$ is an isometry, i.e. $ d^K_\Omega_{n_2}(f(\zeta),f(\eta)) = d^K_{n_1} (\zeta,\eta)$ for all $\zeta,\eta \in \Omega_1,$ then $\phi$ is either holomorphic or anti-holomorphic., Comment: 12 pages

LetH be the domain inC 2 defined byH={Z=(z 1,z 2):║Z║1=│z║1│+│z║2│

In this work we examine the conditions which guarantee the uniqueness of a complex geodesic whose range contains two fixed points of a holomorphic mapf of a bounded convex circular domain in itself and is contained in the fixed points set off.

The tetrablock is a domain in 3-dimensional complex space that meets 3-dimensional Euclidean space in a regular tetrahedron. It is shown to be inhomogeneous and its automorphism group is determined. A type of Schwarz lemma for the tetrablock is proved. The action of the automorphism group is described in terms of a certain natural foliation of the tetrablock by complex geodesic discs., 13 pages, 0 figures

We introduce a geometric condition of Bloch type which guarantees that a subset of a bounded convex domain in several complex variables is degenerate with respect to every iterated function system. Furthermore we discuss the relations of such a Bloch type condition with the analogous hyperbolic Lipschitz condition., Comment: 6 pages

The set G = def { ( z + w , z w ) : | z | 1 , | w | 1 } ⊂ C 2 has intriguing complex-geometric properties; it has a 3-parameter group of automorphisms, its distinguished boundary is a ruled surface homeomorphic to the Mobius band and it has a special subvariety which is the only complex geodesic of G that is invariant under all automorphisms. We exploit the geometry of G to develop an explicit and detailed structure theory for the rational maps from the unit disc to the closure Γ of G that map the boundary of the disc to the distinguished boundary of Γ.