Isoperimetric inequalities in Riemann surfaces of infinite type.
Lengths of geodesics on Riemann surfaces with boundary.
Les groupes hyperboliques
Lipschitz continuity of solutions of the Liouville equation. (La continuité de Lipschitz des solutions de l'équation de Liouville.)
Local convexity properties of -metric balls.
Metric distorsion and triangle maps.
Nevanlinna theory, Fuchsian functions and Brownian motion windings.
Atsuji proposed some integrals along Brownian paths to study the Nevanlinna characteristic function T(f,r) when f is meromorphic in the unit disk D. We show that his criterios does not apply to the basic case when f is a modular elliptic function. The divergence of similar integrals computed along the geodesic flow is also proved. (A)
Quasihyperbolic geodesics in John domains.
Stability of the bases and frames reproducing kernels in model spaces
We study the bases and frames of reproducing kernels in the model subspaces of the Hardy class in the upper half-plane. The main problem under consideration is the stability of a basis of reproducing kernels under “small” perturbations of the points . We propose an approach to this problem based on the recently obtained estimates of derivatives in the spaces and produce estimates of admissible perturbations generalizing certain results of W.S. Cohn and E. Fricain.
Strengthening pseudoconvexity of finite-dimensional Teichmüller spaces.
Systole et invariant d'Hermite.
Systoles of arithmetic surfaces and the Markoff spectrum.
Systoles on Riemann surfaces.
The Apollonian metric: limits of the comparison and bilipschitz properties.
The Apollonian metric: uniformity and quasiconvexity.
The asymptotic behavior of Green's functions for quasi-hyperbolic metrics on degenerating Riemann surfaces.
The hyperbolic Carnot theorem in the Poincaré disc model of hyperbolic geometry.
The hyperbolic geometry of continued fractions .
The hyperbolic metric in a rectangle. II.
The hyperbolic metric of a rectangle.