Поредното заседание на Общия семинар на секция “Анализ, геометрия и топология” ще се проведе
на 29 март 2022 г. от 13:30 часа в зала 478 на ИМИ – БАН.
Доклад на тема:

On Squeezing Function for Planar Domains

ще изнесе Ahmed Yekta Ökten, Institut de Mathématiques de Toulouse, France.

Поканват се всички интересуващи се.

Резюме. Let Ω be a domain in ℂ^𝑛 such that the set 𝐸(Ω, 𝐵^𝑛) of injective holomorphic maps from Ω into the unit ball 𝐵^𝑛 ⊂ ℂ^𝑛 is non-empty. The squeezing function of Ω, denoted by 𝑆_Ω is defined as

𝑆_Ω(𝑧) = sup{𝑟 ∈ (0, 1): 𝑟𝐵^𝑛 ⊂ 𝑓(Ω),      𝑓 ∈ 𝐸(Ω, 𝐵^𝑛),      𝑓 (𝑧) = 0}.

It follows from the definition that the squeezing function is biholomorphically invariant and roughly speaking, it measures how much a domain looks like the unit ball looking at a fixed point. As expected, the study of the squeezing function leads to nice results about the properties of the invariant metrics on complex domains. The behaviour of the squeezing function is well studied however very few non-trivial explicit formulas of squeezing functions have been found.

In this talk we will establish the explicit formulas of squeezing functions on doubly connected planar domains in an elementary way. With the same method we will also provide bounds to squeezing functions of higher connected domains. Finally, we will conclude by mentioning other results and further questions about explicit formulas of squeezing functions on planar domains.