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In this paper we study the following "slice rigidity property": given two Kobayashi complete hyperbolic manifolds $M, N$ and a collection of complex geodesics $\mathcal F$ of $M$, when is it true that every holomorphic map $F:M\to N$ which maps isometrically every complex geodesic of $\mathcal F$ onto a complex geodesic of $N$ is a biholomorphism? Among other things, we prove that this is the case if $M, N$ are smooth bounded strictly (linearly) convex domains, every element of $\mathcal F$ contains a given point of $\overline{M}$ and $\mathcal F$ spans all of $M$. More general results are provided in dimension $2$ and for the unit ball., Comment: 19 pages - final version, to appear in J. Geom. Anal

The symmetrized bidisc $$\begin{aligned} G {\mathop {=}\limits ^\mathrm{{def}}}\{(z+w,zw):|z| G = def { ( z + w , z w ) : | z | < 1 , | w | < 1 } , under the Carathéodory metric, is a complex Finsler space of cohomogeneity 1 in which the geodesics, both real and complex, enjoy a rich geometry. As a Finsler manifold, G does not admit a natural notion of angle, but we nevertheless show that there is a notion of orthogonality. The complex tangent bundle TG splits naturally into the direct sum of two line bundles, which we call the sharp and flat bundles, and which are geometrically defined and therefore covariant under automorphisms of G. Through every point of G, there is a unique complex geodesic of G in the flat direction, having the form $$\begin{aligned} F^\beta {\mathop {=}\limits ^\mathrm{{def}}}\{(\beta +{\bar{\beta }} z,z)\ : z\in \mathbb {D}\} \end{aligned}$$ F β = def { ( β + β ¯ z , z ) : z ∈ D } for some $$\beta \in \mathbb {D}$$ β ∈ D , and called a flat geodesic. We say that a complex geodesic Dis orthogonal to a flat geodesic F if D meets F at a point $$\lambda $$ λ and the complex tangent space $$T_\lambda D$$ T λ D at $$\lambda $$ λ is in the sharp direction at $$\lambda $$ λ . We prove that a geodesic D has the closest point property with respect to a flat geodesic F if and only if D is orthogonal to F in the above sense. Moreover, G is foliated by the geodesics in G that are orthogonal to a fixed flat geodesic F.

Following the work of Burger, Iozzi and Wienhard for representations, in this paper we introduce the notion of maximal measurable cocycles of a surface group. More precisely, let $\mathbf{G}$ be a semisimple algebraic $\mathbb{R}$-group such that $G=\mathbf{G}(\mathbb{R})^\circ$ is of Hermitian type. If $\Gamma \leq L$ is a torsion-free lattice of a finite connected covering of $\text{PU}(1,1)$, given a standard Borel probability $\Gamma$-space $(\Omega,\mu_\Omega)$, we introduce the notion of Toledo invariant for a measurable cocycle $\sigma:\Gamma \times \Omega \rightarrow G$. The Toledo remains unchanged along $G$-cohomology classes and its absolute value is bounded by the rank of $G$. This allows to define maximal measurable cocycles. We show that the algebraic hull $\mathbf{H}$ of a maximal cocycle $\sigma$ is reductive and the centralizer of $H=\mathbf{H}(\mathbb{R})^\circ$ is compact. If additionally $\sigma$ admits a boundary map, then $H$ is of tube type and $\sigma$ is cohomologous to a cocycle stabilizing a unique maximal tube-type subdomain. This result is analogous to the one obtained for representations. In the particular case $G=\text{PU}(n,1)$ maximality is sufficient to prove that $\sigma$ is cohomologous to a cocycle preserving a complex geodesic. We conclude with some remarks about boundary maps of maximal Zariski dense cocycles., Comment: 29 pages, more general definition of pullback added, explicit example of $G=\text{PU}(n,1)$. To appear on Geometriae Dedicata

The set \[ \overline{\mathbb{E}}= \{ x \in {\mathbb{C}}^3: \quad 1-x_1 z - x_2 w + x_3 zw \neq 0 \mbox{ whenever } |z| < 1, |w| < 1 \} \] is called the tetrablock and has intriguing complex-geometric properties. It is polynomially convex, nonconvex and starlike about $0$. It has a group of automorphisms parametrised by ${\mathrm{Aut}~} {\mathbb{D}} \times {\mathrm{Aut}~} {\mathbb{D}} \times {\mathbb{Z}}_2$ and its distinguished boundary $b\overline{\mathbb{E}}$ is homeomorphic to the solid torus $\overline{\mathbb{D}} \times {\mathbb{T}}$. It has a special subvariety \[\mathcal{R}_{\mathbb{\overline{E}}} = \big\{ (x_{1}, x_{2}, x_{3}) \in \overline{\mathbb{E}} : x_{1}x_{2}=x_{3} \big\}, \] called the royal variety of $\overline{\mathbb{E}}$, which is a complex geodesic of ${\mathbb{E}}$ that is invariant under all automorphisms of ${\mathbb{E}}$. We exploit this geometry to develop an explicit and detailed structure theory for the rational maps from the unit disc ${\mathbb{D}}$ to $\overline{\mathbb{E}}$ that map the unit circle ${\mathbb{T}}$ to the distinguished boundary $b\overline{\mathbb{E}}$ of $\overline{\mathbb{E}}$. Such maps are called rational $\mathbb{ \overline{ E}}$-inner functions. We show that, for each nonconstant rational $\mathbb{ \overline{ E}}$-inner function $x$, either $x(\overline{\mathbb{D}}) \subseteq \mathcal{R}_{\mathbb{\overline{E}}} \cap \overline{\mathbb{E}}$ or $x(\overline{\mathbb{D}})$ meets $\mathcal{R}_{\mathbb{\overline{E}}}$ exactly $deg(x)$ times. We study convex subsets of the set $\mathcal{J}$ of all rational $\mathbb{ \overline{ E}}$-inner functions and extreme points of $\mathcal{J}$., 47 pages. This version includes minor revisions. It has been accepted for publication by the Journal of Mathematical Analysis and Applications

We show that any totally geodesic submanifold of Teichmuller space of dimension greater than one covers a totally geodesic subvariety, and only finitely many totally geodesic subvarieties of dimension greater than one exist in each moduli space., Comment: Final copy; very minor revisions

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 $\mathbb{P}_2(\mathbb{C})$.

International audience; The present volume of the Handbook of Teichmüller theory is divided into three parts.The first part contains surveys on various topics in Teichmüller theory, including the complex structure of Teichmüller space, the Deligne–Mumford compactification of the moduli space, holomorphic quadratic differentials, Kleinian groups, hyperbolic 3-manifolds and the ending lamination theorem, the universal Teichmüller space, barycentric extensions of maps of the circle, and the theory of Higgs bundles.The second part consists of three historico-geometrical articles on Tissot (a precursor of the theory of quasiconfomal mappings), Grötzsch and Lavrentieff, the two main founders of the modern theory of quasiconformal mappings.The third part comprises English translations of five papers by Grötzsch, a paper by Lavrentieff, and three papers by Teichmüller. These nine papers are foundational essays on the theories of conformal invariants and quasiconformal mappings, with applications to conformal geometry, to the type problem and to Nevanlinna's theory. The papers are followed by commentaries that highlight the relations between them and between later works on the subject. These papers are not only historical documents; they constitute an invaluable source of ideas for current research in Teichmüller theory.

The symmetrized bidisc \[ G \stackrel{\rm{def}}{=}\{(z+w,zw):|z, 45 pages, 1 figure, with index. To appear in J. Math. Anal. Applic

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.

In this paper we survey projective curvature invariants of Douglas type and use these to give some generalizations for the notion of a complex Berwald space. Various descriptions of complex Douglas spaces are given in relation to other special classes of complex Finsler spaces. This study was performed from the viewpoint of the equations of a complex geodesic curve. Complex Randers spaces offer examples of complex Douglas spaces.

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.

In this paper, we first provide an updated survey of the geometry of complex Cartan spaces. New characterizations for some particular classes of complex Cartan spaces are pointed out, e.g. Landsberg-Cartan, strongly Berwald-Cartan and others. We introduce the Cartan-Randers spaces which offer examples of Berwald-Cartan and strongly Berwald-Cartan spaces. Then, we investigate the complex geodesic curves of a complex Cartan space, using the image by Legendre transformation ($\mathcal{L}-$ duality) of complex geodesic curves of a complex Finsler space. Assuming the weakly K\"{a}hler condition for a complex Cartan space, we establish that its complex geodesic curves derive from Hamilton-Jacobi equations. Also, by $\mathcal{L}-$ duality, we introduce the corespondent notion of the projectively related complex Finsler metrics, on the complex Cartan spaces. Various descriptions of the projectively related complex Cartan metrics are given. As applications, the projectiveness of a complex Cartan-Randers metric and the locally projectively flat complex Cartan metrics are analyzed.

International audience; We show that the Teichmüller space of the triangle groups of type (p,q,∞) in the automorphism group of the two-dimensional complex hyperbolic space contains open sets of 0, 1 and two real dimensions. In particular, we identify the Teichmüller space near embeddings of the modular group preserving a complex geodesic.

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.

Let B be the open unit ball of a complex Hilbert space E; and let f : B → B be a holomorphic map with f (0) = 0. In this paper, we consider a condition so that f is a linear isometry on E.

Let M be a connected taut complex manifold of dimension n and let D be a bounded balanced pseudoconvex domain in with continuous Minkowski function. Assume that there exist a finite number of complex hyperplanes H j through the origin such that every point of is an extreme point for . Let f:→D be a holomorphic map. Let p be a point of M. Assume that f(p) = 0 and that df p is an isometry for the infinitesimal Kobayashi metric. In this case, we will show that f is a biholomorphic map.

In the paper we utilize the formula describing the behaviour of the pluricomplex Green function with many poles under proper holomorphic mappings to find the effective for-mulas for the function in the unit ball in with two poles.

Let M be a connected taut complex manifold of dimension n such that the set H(M) of holomorphic functions on M separates points of M and let D be a bounded complete Reinhardt domain of holomorphy in Let f:M→D be a holomorphic mapping. Let p be a point of M. Assume that f(p)=0 and that df p is an isometry for the infinitesimal Kobayashi metric, in this case, we will show that f is a biholomorphic mapping

In the paper we discuss the problem of uniqueness of left inverses (solutions of two point Nevanlinna-Pick problem) in bounded convex domains, strongly linearly convex domains, the symmetrized bidisc and the tetrablock., Comment: 24 pages, after revision with more details and the title changed, accepted for publication in Journal of Geometric Analysis

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.

0. Introduction. In the study of intrinsic metrics and distances on complex manifolds, a crucial role is played by the notion of complex geodesic introduced by Vesentini [V]. Roughly speaking a complex geodesic is a holomorphic embedding of the unit disk with the hyperbolic metric which is an isometry with respect to the intrinsic metric or distance (or both) which is defined on the manifold under consideration. As it is well known, the problem of existence of complex geodesics is satisfactory solved only for convex domains by the work of Lempert [L] who also proves uniqueness for strictly convex domains (see also [A, chapter 2.6]). In order to find a different approach to this problem, which may eventually lead to an understanding of it on a larger class of complex manifolds, in [AP1] and [AP2] it was studied the same problem from a differential geometric point view looking for minimal conditions on an abstract complex Finsler metric which imply the existence and uniqueness of complex geodesics. A complete solution to the problem was achieved in terms of the holomorphic curvature of the metric, which must be a negative constant, and the vanishing of suitable torsion tensors. We give a brief account of these results at the beginning of section 2. In this general framework it is very natural to ask whether it is possible to solve the same kind of problems for isometric holomorphic embeddings of C with the euclidean metric and of P1 with the Fubini-Study metric. In this paper we show that the methods of [AP1] and [AP2] work also in this case and that it is

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│

Before turning to our main topics we first discuss partially hyperbolic diffeomorphisms and holomorphic Anosov flows as introduced by Etienne Ghys [1995]. In Section 13.2 we discuss the geometry of the projectivized holomorphic tangent and cotangent bundles. The study of the projectivized holomorphic tangent bundle naturally raises the question of a complex geodesic flow, which we discuss in Section 13.3. In Section 13.4 we return to the projectivized holomorphic tangent bundle and develop its complex almost contact metric structure. In Section 13.5 we first discuss special directions on complex contact manifolds analogous to our treatment in the real case in Chapter 11 and then discuss complex contact structures on the Lie group SL(2, \( C\!\!\!\!C \)) in detail.

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