Related papers: Blaschke product for bordered surface

Let $\mathbb{D}$ be the unit disk in the complex plane. Among other results, we prove the following curious result for a finite Blaschke product: $$B(z)=e ^{is}\prod_{k=1}^d \frac{z-a_k}{1-z \overline{a_k}}.$$ The Lebesgue measure of the…

In this paper we will deal with problems in approximation theory of bounded analytic functions on the unit disc and their boundary behavior on the unit circle. We will attempt to unify two known such theorems to create a stronger theorem.…

In this paper we study the smooth moduli space of closed Riemann surfaces. This smooth moduli is an infinite cover of the usual moduli space $\mathscr{M}_g$ of closed Riemann surfaces, and is identified with the Schottky space of rank $g.$…

A proof of the uniformization theorem of Riemann surface is given with only elementary properties of holomorphic functions and not using the paracompacity of the surface. This proof leans on an holomorphic version of the topological…

Interpreting the number of ramified covering of a Riemann surface by Riemann surfaces as the relative Gromov-Witten invariants and applying a gluing formula, we derive a recursive formula for the number of ramified covering of a Riemann…

We introduce Nevanlinna classes of holomorphic functions associated to a closed set on the boundary of the unit disc in the complex plane and we get Blaschke type theorems relative to these classes by use of several complex variables…

The Blaschke rolling disk theorem is a classical inclusion principle in differential geometry. This states that a planar convex domain whose boundary is a curve of class $C^2$ with (signed) curvature not exceeding a positive constant…

Let $\D$ be the unit disk. Kutzschebauch and Studer \cite{KS} recently proved that, for each continuous map $A:\overline D\to \mathrm{SL}(2,\C)$, which is holomorphic in $\D$, there exist continuous maps $E,F:\overline \D\to…

This is the first in a series of papers on the construction of explicit solutions to the stationary axisymmetric Einstein equations which describe counter-rotating disks of dust. These disks can serve as models for certain galaxies and…

We prove a version of the Schwarz lemma for holomorphic mappings from the unit disk into the symmetric product of a Riemann surface. Our proof is function-theoretic and self-contained. The main novelty in our proof is the use of the…

For a given embedded Lagrangian in the complement of a complex hypersurface we show existence of a holomorphic disc in the complement having boundary on that Lagrangian.

We introduce a ramified covering of small categories, and we show three properties of the notion: the Riemann-Hurwitz formula holds for a ramified covering of finite categories, the zeta function of $C$ divides that of $\widetilde{C}$ for a…

We consider closed orientable hypersurfaces in a wide class of warped product manifolds, which include space forms, deSitter-Schwarzschild and Reissner-Nordstr\"{o}m manifolds. By using a new integral formula or Brendle's Heintze-Karcher…

We study the moduli spaces of flat surfaces with prescribed conical singularities. Veech showed that these spaces are diffeomorphic to the moduli spaces of marked Riemann surfaces, and endowed with a natural volume form depending on the…

Given a generic totally real torus unknotted in the unit sphere of the complex plane, we prove the following alternative : either there exists a filling of the torus by holomorphic discs and the torus is rationally convex, or its rational…

A finite Blaschke product, restricted to the unit circle, is a smooth covering map. The maximum and minimum values of the derivative of this map reflect the geometry of the Blaschke product. We identify two classes of extremal Blaschke…

We show that every cubic Blaschke product has a unique hyperbolic inflection point in the unit disk and, moreover, this point lies at the hyperbolic midpoint of the two critical points. Using this structure result for cubic Blaschke…

This is a course of lectures given for students of the Regional Mathematical Center of the Novosibirsk State University from October 20 to November 3, 2017. The course is devoted to some geometric problems of ramified coverings of the…

This paper extends the decorated Teichm\"uller theory developed before for punctured surfaces to the setting of ``bordered'' surfaces, i.e., surfaces with boundary, and there is non-trivial new structure discovered. The main new result…

Let $\mathfrak g$ be an infinite-dimensional Lie algebra, and $G$ be the algebraic completion of a $\mathfrak g$-module. Using the geometric model of Schottky uniformization of Riemann sphere to obtain a higher genus Riemann surface, we…