Related papers: Hyperbolic geodesics, Krzyz's conjecture and beyon…
Using the Bers isomorphism theorem for Teichmuller spaces of punctured Riemann surfaces and some of their other complex geometric features, we prove a general theorem on maximization of homogeneous polynomial (in fact, more general…
A fuzzy Boolean function is a map $f:\cube^n\to [0,1]$, where $n\in\mathbb N$. We introduce and compare three ways of saying that such a function has bounded complexity. The first is a sampling property: the value $f(x)$ can be recovered,…
On a closed Riemannian surface of negative curvature, we prove a characterization for configurations of closed geodesics arising from one parameter Allen-Cahn min-max constructions. One of the facts we conclude is that every geodesic occurs…
In this paper, we prove a version of the universality theorem for the Hurwitz zeta-function in the case where the parameter is algebraic and irrational. Then we apply the result to show that many of such Hurwitz zeta-functions have…
We prove a theorem describing the limiting fine-scale statistics of orbits of a point in hyperbolic space under the action of a discrete subgroup. Similar results have been proved only in the lattice case, with two recent infinite-volume…
In this paper we prove a generalisation of Schlenk's theorem about the existence of contractible periodic Reeb orbits on stable, displaceable hypersurfaces in symplectically aspherical, geometrically bounded, symplectic manifolds, to a…
In this paper we present an unconditional proof of Wojtkowski's Ergodicity Conjecture for almost every system of 1D perfectly elastic balls falling down in a half line under constant gravitational acceleration. Namely, by introducing a new…
In 1999 Brady constructed the first example of a non-hyperbolic finitely presented subgroup of a hyperbolic group by fibring a non-positively curved cube complex over the circle. We show that his example has Dehn function bounded above by…
In most classical holomorphic function spaces on the unit disk in which the polynomials are dense, a function $f$ can be approximated in norm by its dilates $f_r(z):=f(rz)~(r<1)$, in other words, $\lim_{r\to1^-}\|f_r-f\|=0$. We construct a…
For the product $S_1\times S_2$ of any two connected compact hyperbolic surfaces $S_1$ and $S_2$, we give a finite bound $\mathcal{B}$ such that for any self-homeomorphism $f$ of $S_1\times S_2$ and any fixed point class $F$ of $f$, the…
The goal of the article is to provide different explicit quantifications of the non density of simple closed geodesics on hyperbolic surfaces. In particular, we show that within any embedded metric disk on a surface, lies a disk of radius…
For any given natural number $k$, this paper gives upper bounds on the radius of a packing of a complete hyperbolic surface of finite area by $k$ equal-radius disks in terms of the surface's topology. We show that the bounds given here are…
Motivated by various equivalent versions of the SHGH conjecture for $\mathbb{P}^2$ blown up at very general points, we propose a similar conjecture for Hirzebruch surfaces. We prove that this conjecture is true for the Hirzebruch surface…
The classical problem of steady rarefied gas flow past an infinitely thin circular disk is revisited, with particular emphasis on the gas behavior near the disk edge. The uniform flow is assumed to be perpendicular to the disk surface. An…
The purpose of this paper is to study properties of continua (closed connected sets) of nontrivial solutions of non-cooperative elliptic systems considered on geodesic balls in $S^n$. In particular, we have shown that if the geodesic ball…
Recently, the author and Melentijevi\'c resolved the longstanding Gaussian curvature problem by proving the sharp inequality \[ |\mathcal{K}| < c_0 = \frac{\pi^2}{2} \] for minimal graphs over the unit disk, evaluated at the point of the…
This paper aims to pursue some classes of normalized analytic functions $f$ with fixed second coefficient defined on open unit disk, such that ${(1+z)^2f(z)}/{z}$ and ${(1+z)f(z)}/{z}$ are functions having positive real part. The radius of…
A starlike univalent function $f$ is characterized by the function $zf'(z)/f(z)$; several subclasses of these functions were studied in the past by restricting the function $zf'(z)/f(z)$ to take values in a region $\Omega$ on the right-half…
We prove that the minimal length of a closed geodesic with self-intersection number $k$ on any finite-type hyperbolic surface is $2\cosh^{-1}(1+2k)$ for $k>1750$. This improves the previously known threshold $k > 10^{13350}$. Our proof is…
A sequence of points $z_k$ in the unit disk is said to be thin for a given decrease function $\rho$, if there is a nontrivial bounded holomorphic function such that the infinite series $\sum_k \rho(1-|z_k|)|f(z_k)|$ converges. All sequences…