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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

One of the long-standing problems in the quasiconformal theory is finding sharp distortion bounds for k-quasiconformal maps for arbitrary $$k

Complex geodesics are fundamental constructs for complex analysis and as such constitute one of the most vital research objects within this discipline. In this paper, we formulate a rigorous description, expressed in terms of geometric properties of a domain, of all complex geodesics in a convex tube domain in \({\mathbb {C}}^n\) containing no complex affine lines. Next, we illustrate the obtained result by establishing a set of formulas stipulating a necessary condition for extremal mappings with respect to the Lempert function and the Kobayashi–Royden metric in a large class of bounded, pseudoconvex, complete Reinhardt domains: for all of them in \({\mathbb {C}}^2\) and for those in \({\mathbb {C}}^n\) whose logarithmic image is strictly convex in the geometric sense.

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.

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.

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.

We solve the Caratheodory and Kobayashi extremal problems for the open symmetrized bidisc % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfKttLearuqr1ngBPrgarmWu51MyVXgatC% vAUfeBSjuyZL2yd9gzLbvyNv2CaeHbd9wDYLwzYbItLDharyavP1wz% ZbItLDhis9wBH5garqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbb% L8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpe% pae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaaiaabeqaam% aaeaqbaaGcbaGaem4raCucKbai-pacacidaqZ-meaabGaGaYaa08pc% eaGaiaiGmaan--hzaiacacidaqZ--vgacGaGaYaa08-FMbaaaeHbbj% xAHXgaiyaacaGFGaGaa4hiaKaaGiadaALG7bWEcWaGwjikaGIamaOv% dQha6PWaiaOvBaaaleacaALamaOvigdaXaqajaOvaOGamaOvgUcaRK% aaGiadaA1G6bGEkmacaA1gaaWcbGaGwjacaA1FYaaabKaGwbGccWaG% wjilaWIaiaOv+bcajaaicWaGwnOEaONcdGaGwTbaaSqaiaOvcWaGwH% ymaedabKaGwbqcaaIamaOvdQha6PWaiaOvBaaaleacaALaiaOv-jda% aeqcaAfajaaicWaGwjykaKIamaOvcQda6iadgALG8baFcWaGwnOEaO% NcdGaGwTbaaSqaiaOvcWaGwHymaedabKaGwbqcaaIamaOvcYha8jad% aALH8aapcWaGwHymaeJamaOvcYcaSiadgALG8baFcWaGwnOEaONcdG% aGwTbaaSqaiaOvcGaGw9NmaaqajaOvaKaaGiadaALG8baFcWaGwzip% aWJamaOvigdaXiadaALG9bqFcWaGwzOGIW8efv3ySLgznfgDOjdarC% qr1ngBPrginfgDObcv39gaiCqacWaGw1NaHmKcdGaGwXbaaSqajaOv% bGaGwjacaA1FYaaaaOGamaOvc6caUaaa!AF58! $$G\underline{\underline {def}} \{ (z_1 + z_2 , z_1 z_2 ):|z_1 |< 1,|z_2 |< 1\} \subset \mathbb{C}^2 .$$ We prove the equality of the Caratheodory and Kobayashi distances on G and describe the extremal functions for the two problems; they are rational of degree 1or 2.G is the first example of a non convexifiable domain for which the two distances coincide.

In his famous 1981 paper, Lempert proved that given a point in a strongly convex domain the complex geodesics (i.e., the extremal disks) for the Kobayashi metric passing through that point provide a very useful fibration of the domain. In this paper we address the question whether, given a smooth complex Finsler metric on a complex manifoldM, it is possible to find purely differential geometric properties of the metric ensuring the existence of such a fibration in complex geodesies ofM. We first discuss at some length the notion of holomorphic sectional curvature for a complex Finsler metric; then, using the differential equation of complex geodesies we obtained in [AP], we show that for every pair (p;v) ∈T M, withv ≠ 0, there is a (only a segment if the metric is not complete) complex geodesic passing throughp tangent tov iff the Finsler metric is Kahler, has constant holomorphic sectional curvature −4, and its curvature tensor satisfies a specific simmetry condition—which are the differential geometric conditions we were after. Finally, we show that a complex Finsler metric of constant holomorphic sectional curvature −4 satisfying the given simmetry condition on the curvature is necessarily the Kobayashi metric.

Vesentini studied complex geodesics of a bounded domain D in a complex Banach space and applied his results to the set of fixed points of an holomorphic map from D to D. In a previous paper, I studied the fixed points of an holomorphic map f from a bounded convex domain D in Cn into itself, and I proved that, given two fixed points of f, there exists a complex geodesic containing these two points and contained in the set of fixed points of f. In this paper, I prove the same result for a finite product of open unit balls in complex Hilbert spaces. The proof is more difficult, is based on a careful study of the Caratheodory distance, and uses results of Goebel, Sekowski and Stachura.