The squeezing function on doubly-connected domains via the Loewner differential equation
- PDF / 830,862 Bytes
- 26 Pages / 439.37 x 666.142 pts Page_size
- 43 Downloads / 191 Views
Mathematische Annalen
The squeezing function on doubly-connected domains via the Loewner differential equation Tuen Wai Ng1
· Chiu Chak Tang1 · Jonathan Tsai1
Received: 21 February 2019 / Revised: 13 January 2020 © The Author(s) 2020
Abstract For any bounded domains Ω in Cn , Deng, Guan and Zhang introduced the squeezing function SΩ (z) which is a biholomorphic invariant of bounded domains. We show that for n = 1, the squeezing function on an annulus Ar = {z ∈ C : r < |z| < 1} is given r for all 0 < r < 1. This disproves the conjectured formula by S Ar (z) = max |z|, |z| for the squeezing function proposed by Deng, Guan and Zhang and establishes (up to biholomorphisms) the squeezing function for all doubly-connected domains in C other than the punctured plane. It provides the first non-trivial formula for the squeezing function for a wide class of plane domains and answers a question of Wold. Our main tools used to prove this result are the Schottky–Klein prime function (following the work of Crowdy) and a version of the Loewner differential equation on annuli due to Komatu. We also show that these results can be used to obtain lower bounds on the squeezing function for certain product domains in Cn . Mathematics Subject Classification 30C35 · 30C75 · 32F45 · 32H02
1 Introduction In 2012, Deng, Guan and Zhang [8] introduced the squeezing function of a bounded domain Ω in Cn as follows. For any z ∈ Ω, let FΩ (z) be the collection of all embed-
Communicated by Ngaiming Mok. Dedicated to Professor Alan Frank Beardon on the occasion of his 80th birthday.
B
Tuen Wai Ng [email protected] Chiu Chak Tang [email protected] Jonathan Tsai [email protected]
1
The University of Hong Kong, Pokfulam, Hong Kong
123
T. W. Ng et al.
dings f from Ω to Cn such that f (z) = 0. Let B(0; r ) = {z ∈ Cn : z < r } denote the n-dimensional open ball centered at the origin 0 with radius r > 0. Then the squeezing function SΩ (z) of Ω at z is defined to be a SΩ (z) = sup : B(0; a) ⊂ f (Ω) ⊂ B(0; b) . f ∈FΩ (z) b Remark 1. For the supremum in the definition of the squeezing function, we can restrict the family FΩ (z) to the subfamily of functions f such that f (Ω) is bounded. 2. For any λ = 0, we have f ∈ FΩ (z) if and only if λ f ∈ FΩ (z). As a consequence, we may assume that b = 1. It is clear from the definition that the squeezing function on Ω is positive and bounded above by 1. Also, it is invariant under biholomorphisms, that is, Sg(Ω) (g(z)) = SΩ (z) for any biholomorphism g of Ω. If the squeezing function of a domain Ω is bounded below by a positive constant, i.e., if there exists a positive constant c such that SΩ (z) ≥ c > 0 for all z ∈ Ω, then the domain Ω is said to be holomorphic homogeneous regular by Liu, Sun and Yau [23] or with uniform squeezing property by Yeung [27]. The consideration of such domains appears naturally when one applies the Bers embedding theorem to the Teichmüller space of genus g hyperbolic Riemann surfaces. The squeezing function is interesting because it provides some
Data Loading...
