The next meeting of the Joint Seminar of Analysis, Geometry and Topology Department will be held
on March 29, 2022, at 1:30 pm (UTC+2) in Room 478 of IMI-BAS..
A talk on:

On Squeezing Function for Planar Domains

will be delivered by Ahmed Yekta Ökten, Institut de Mathématiques de Toulouse, France.

Everybody is invited.

Abstract. 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.