Related papers: Teichm\"uller's problem for Gromov hyperbolic doma…

We prove a special case of Erd\H{o}s' unit distance problem using a corollary of the subspace theorem bounding the number of solutions of linear equations from a multiplicative group. We restrict our attention to unit distances coming from…

We prove that the every quasi-isometry of Teichm\"uller space equipped with the Teichm\"uller metric is a bounded distance from an isometry of Teichm\"uller space. That is, Teichm\"uller space is quasi-isometrically rigid.

We construct new coordinates for the Teichm\"uller space Teich of a punctured torus into $\bold{R} \times\bold{R}^+$. The coordinates depend on the representation of Teich as a space of marked Kleinian groups $G_\mu$ that depend…

In this paper we determine the automorphism group of the Fock-Bargmann-Hartogs domain $D_{n,m}$ in $\mathbb{C}^n\times\mathbb{C}^m$ which is defined by the inequality ${\|\zeta\|}^2<e^{-\mu{\|z\|}^2}$.

We derive new estimates for distances between optimal matchings of eigenvalues of non-normal matrices in terms of the norm of their difference. We introduce and estimate a hyperbolic metric analogue of the classical spectral-variation…

We investigate isoperimetric upper bounds for sums of consecutive Steklov eigenvalues of planar domains. The normalization involves the perimeter and scale-invariant geometric factors which measure deviation of the domain from roundness. We…

We determine the asymptotic behaviour of extremal length along arbitrary Teichm\"uller rays. This allows us to calculate the endpoint in the Gardiner-Masur boundary of any Teichm\"uller ray. We give a proof that this compactification is the…

In this article, we study acylindrical graphs of groups, local quasiconvexity, and Cannon-Thurston maps in the setting of totally disconnected locally compact (TDLC) hyperbolic groups, extending several fundamental notions and results from…

We prove a criterion of convergence in the augmented Teichmueller space that can be phrased in terms of convergence of the hyperbolic metrics or of quasiconformal convergence away from the nodes.

Let $\phi$ be an atoroidal outer automorphism of the free group $F_n$. We study the Gromov boundary of the hyperbolic group $G_{\phi} = F_n \rtimes_{\phi} \mathbb{Z}$. We explicitly describe a family of embeddings of the complete bipartite…

Consider the geometric inverse problem: There is a set of delta-sources in spacetime that emit waves travelling at unit speed. If we know all the arrival times at the boundary cylinder of the spacetime, can we reconstruct the space, a…

We study planar domains $G$ equipped with a hyperbolic type metric and approximate geodesics that join two points $x,y \in G$ and their lengths. We present an algorithm that enables one to approximate the shortest distance in polygonal…

In this article we study a controllability problem for a parabolic and a hyperbolic partial differential equations in which the control is the shape of the domain where the equation holds. The quantity to be controlled is the trace of the…

We study the homeomorphic extension of biholomorphisms between convex domains in $\mathbb C^d$ without boundary regularity and boundedness assumptions. Our approach relies on methods from coarse geometry, namely the correspondence between…

We propose and study several inverse boundary problems associated with a quasilinear hyperbolic equation of the form ${c(x)^{-2}}\partial_t^2u=\Delta_g(u+F(x, u))+G(x, u)$ on a compact Riemannian manifold $(M, g)$ with boundary. We show…

We prove, for any n, that there is a closed connected orientable surface S so that the hyperbolic space H^n almost-isometrically embeds into the Teichm\"uller space of S, with quasi-convex image lying in the thick part. As a consequence,…

We prove a semisimplicity result for the boundary, in the corresponding Deligne-Mumford compactification, of a totally geodesic subvariety of a moduli space of Riemann surfaces. At the level of Teichm\"uller space, this semisimplicity…

This paper considers how the eigenvalues of the Neumann problem for an elliptic operator depend on the domain. The proximity of two domains is measured in terms of the norm of the difference between the two resolvents corresponding to the…

The Fock-Bargmann-Hartogs domain $D_{n,m}(\mu)$ ($\mu>0$) in $\mathbb{C}^{n+m}$ is defined by the inequality $\|w\|^2<e^{-\mu\|z\|^2},$ where $(z,w)\in \mathbb{C}^n\times \mathbb{C}^m$, which is an unbounded non-hyperbolic domain in…

We study spectral properties of divergence form elliptic operators $-\textrm{div} [A(z) \nabla f(z)]$ with the Neumann boundary condition in planar domains (including some fractal type domains), that satisfy to the quasihyperbolic boundary…