SEMINARIO DE ANÁLISIS COMPLEJO (COMPLEX ANALYSIS SEMINAR)

Hyperbolic convexity of holomorphic level sets

Iason EFRAIMIDIS

UAM
Wednesday, November 13, 2024 at 15:30 hr.

Room 520, Module 17, Department of Mathematics,
Universidad Autónoma de Madrid (Autonomous University of Madrid)

Abstract:
Given a holomorphic self-map f of the unit disk we consider
(i) the quotient of the density of the hyperbolic metric at a preimage z over the
density at the image f (z), and
(ii) the difference between the hyperbolic distance from some fixed z0 to z and the
distance from some fixed w0 to f (z).
We prove that the sublevel sets of these functions are convex with respect to the
hyperbolic geometry of the disk. The result concerning (i) answers a question of
Arango, Mej´ıa and Pommerenke (2019), while (ii) seems more natural in more general
contexts.
This is joint work with Pavel Gumenyuk.