Search

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.

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