arXiv subjects

Burglind Jöricke

Publications and source records attributed to Burglind Jöricke.

7 recordsLinked to original sources

arXiv2023

Braids, conformal module and entropy

In the work we discuss two invariants of conjugacy classes of braids. The first invariant is the conformal module which occurred in connection with the interest in the 13th Hilbert Problem. The second is a popular dynamical invariant, the entropy. It occurred in connection with Thurston's theory of surface homeomorphisms. We prove that these invariants are related: They are inverse proportional. This allows us on one hand to use known results on entropy for applications to the concept of conformal module, and on the other hand, to apply methods from quasi-conformal mappings to problems related to the entropy. In particular, we give a short conceptional proof of a theorem which appeared in connection with research on the Thirteen's Hilbert Problem. Mainly, we give applications of the concept of conformal module to the problem of the existence of homotopies (or isotopies, respectively) of continuous objects to the respective holomorphic objects, i.e. to Gromov's Oka Principle. We study objects involving braids and, more generally, continuous mappings from Riemann surfaces to complex manifolds, and focus on the case, when there are obstructions to Gromov's Oka Principle.

math.GT↗

arXiv2021

Gromov's Oka principle, fiber bundles and the conformal module

The conformal module of conjugacy classes of braids is an invariant that appeared earlier than the entropy of conjugacy classes of braids, and is inverse proportional to the entropy. Using the relation between the two invariants we give a short conceptional proof of an earlier result on the conformal module. Mainly, we consider situations, when the conformal module of conjugacy classes of braids serves as obstruction for the existence of homotopies (or isotopies) of smooth objects involving braids to the respective holomorphic objects, and present theorems on the restricted validity of Gromov's Oka principle in these situations.

math.CV↗

arXiv2020

Conformal invariants of $3$-Braids and Counting Functions

We consider a conformal invariant of braids, the extremal length with totally real horizontal boundary values $λ_{tr}$. The invariant descends to an invariant of elements of $\mathcal{B}_n\diagup\mathcal{Z}_n$, the braid group modulo its center. We prove that the number of elements of $\mathcal{B}_3\diagup\mathcal{Z}_3$ of positive $λ_{tr}$ grows exponentially. The estimate applies to obtain effective finiteness theorems in the spirit of the geometric Shafarevich conjecture over Riemann surfaces of second kind. As a corollary we obtain another proof of the exponential growth of the number of conjugacy classes of $\mathcal{B}_3\diagup\mathcal{Z}_3$ with positive entropy not exceeding $Y$.

math.GT↗

arXiv2017

Fundamental groups, slalom curves and extremal length

We define the extremal length of elements of the fundamental group of the twice punctured complex plane and give upper and lower bounds for this invariant. The bounds differ by a multiplicative constant. The main motivation comes from $3$-braid invariants and their application.

math.CV↗

arXiv2017

Analytic knots, satellites and the 4-ball genus

Call a smooth knot (or smooth link) in the unit sphere in $\mathbb{C}^2$ analytic (respectively, smoothly analytic) if it bounds a complex curve (respectively, a smooth complex curve) in the complex ball. Let $K$ be a smoothly analytic knot. For a small tubular neighbourhood of $K$ we give a sharp lower bound for the 4-ball genus of analytic links $L$ contained in it.

math.GT↗

arXiv2012

Braids, Conformal Module and Entropy

The conformal module of conjugacy classes of braids implicitly appeared in a paper of Lin and Gorin in connection with their interest in the 13. Hilbert Problem. This invariant is the supremum of conformal modules (in the sense of Ahlfors) of certain annuli related to the conjugacy class. This note states that the conformal module is inverse proportional to a popular dynamical braid invariant, the entropy. The entropy appeared in connection with Thurston's theory of surface homeomorphisms. An application of the concept of conformal module to algebraic geometry is given.

math.AG↗

arXiv2004

A Cantor set in the unit sphere in $\mathbb{C}^2$ with large polynomial hull

It is an old question how massive polynomial hulls of Cantor sets in $\mathbb{C}^n$ can be. In contrast to expectation e.g. Rudin, Vitushkin and Henkin showed on examples that it can be rather massive. Motivated by problems of holomorphic convexity of subsets of strictly pseudoconvex boundaries and removable singularities the question was asked for Cantor sets in the unit sphere. It was known that tame Cantor sets in the unit sphere are polynomially convex. We give an example of a wild Cantor set in the sphere whose polynomial hull contains a large ball. In some sense this can be opposed to a still open conjecture of Vitushkin on the existence of a lower bound for the diameter of the largest boundary component of a relatively closed complex curve in the ball passing through the origin.

math.CV↗