Related papers: Inverse problem for Moebius geometry on the circle

We show that any measurable solution of the cohomological equation for a H\"older linear cocycle over a hyperbolic system coincides almost everywhere with a H\"older solution. More generally, we show that every measurable invariant…

This paper continues a geometric study of Harvey's Complex of Curves, whose ultimate goal is to apply the theory of hyperbolic spaces and groups to algorithmic questions for the Mapping Class Group and geometric properties of Kleinian…

We consider an inverse problem associated with some 2-dimensional non-compact surfaces with conical singularities, cusps and regular ends. Our motivating example is a Riemann surface $\mathcal M = \Gamma\backslash{\bf H}^2$ associated with…

We introduce the notion of topological hyperbolicity to characterize the largeness of the topological fundamental group of a complex variety. Inspired by the Shafarevich conjecture, we propose to study the topological hyperbolicity of…

We study lattice points on hyperbolic circles centred at Heegner points of class number one. Our main result is that, on a density one subset of radii tending to infinity, the angles of such points equidistribute on the unit circle. To…

We show how a type of multi-Frobenius nonclassicality of a curve defined over a finite field $\mathbb{F}_q$ of characteristic $p$ reflects on the geometry of its strict dual curve. In particular, in such cases we may describe all the…

We extend the old definition of the Apollonius circle in such a way that it results in the same curve in Euclidean geometry but will be more convenient in hyperbolic and spherical geometries. We show that there exists an Apollonius circle…

Here we study the behaviour of the horocyclic orbit of a vector on the unit tangent bundle of a geometrically infinite surface with variable negative curvature, when the corresponding geodesic ray is almost minimizing and the injectivity…

Many geometric structures associated to surface groups can be encoded in terms of invariant cross ratios on their circle at infinity; examples include points of Teichm\"uller space, Hitchin representations and geodesic currents. We add to…

In this paper we solve the inverse problem for the cubic mean-field Ising model. Starting from configuration data generated according to the distribution of the model we reconstruct the free parameters of the system. We test the robustness…

In this paper, we systematically investigate the geometry and topology of manifolds with integral radial curvature bounds, and obtain many interesting and important conclusions.

We introduce a structure of an infinite-dimensional Frobenius manifold on a subspace in the space of pairs of functions analytic inside/outside the unit circle with simple poles at 0/infinity respectively. The dispersionless 2D Toda…

We study deformations of complex hyperbolic surfaces which furnish the simplest examples of: (i) negatively curved K\"ahler manifolds and (ii) negatively curved Riemannian manifolds not having {\it constant} curvature. Although such complex…

We show that every plumbing of disc bundles over surfaces whose genera satisfy a simple inequality may be embedded as a convex submanifold in some closed hyperbolic four-manifold. In particular its interior has a geometrically finite…

We consider principal fibre bundles with a given connection and construct almost complex structures on the total space if the adjoint bundle is isomorphic to the tangent bundle of the base. We derive the integrability condition. If the…

In this paper, we extend the work of Ge-Hua-Zhou \cite{GHZ} on combinatorial Ricci flows for ideal circle patterns to combinatorial Calabi flows in both hyperbolic and Euclidean background geometry. We prove the solution to the…

A generic geodesic on a finite area, hyperbolic 2-orbifold exhibits an infinite sequence of penetrations into a neighborhood of a cone singularity, so that the sequence of depths of maximal penetration has a limiting distribution. The…

To construct a curve with a monotonic curvature (spiral), and given tangents and curvatures at the ends, the author proposed the following method. From given boundary conditions, the values of two inverse invariants are determined. Then, on…

Based on properties of n-subharmonic functions we show that a complete, noncompact, properly embedded hypersurface with nonnegative Ricci curvature in hyperbolic space has an asymptotic boundary at infinity of at most two points. Moreover,…

We study the intrinsic geometry of area minimizing (and also of almost minimizing) hypersurfaces from a new point of view by relating this subject to quasiconformal geometry. For any such hypersurface we define and construct a so-called…