2020/07/20 by Tuen Wai Ng, Ng, Tuen Wai, Chiu Chak Tang +3
Mathematics · #30C35 #30C75 #32F45 and 32H02 #Analytic and geometric function theory #Classical Analysis and ODEs (math.CA) #Complex Variables (math.CV) #Differential Geometry (math.DG) #FOS: Mathematics #Holomorphic and Operator Theory #Meromorphic and Entire Functions
paper · pdf · doi:10.48550/arxiv.2007.10010
openalex publication_date 2020/07/20 · openalex created_date 2022/07/26 · openalex updated_date 2026/07/28
For any bounded domains \Ω in \ℂn, Deng, Guan and Zhang\nintroduced the squeezing function S_\Ω (z) which is a biholomorphic\ninvariant of bounded domains. We show that for n=1, the squeezing function on\nan annulus Ar = lbrace z \∈ \ℂ : r <|z| <1 rbrace is given by\nSAr(z)= \max \lbrace |z| ,\(r)/(|z|) \rbrace for all\n0<r<1. This disproves the conjectured formula for the squeezing function\nproposed by Deng, Guan and Zhang and establishes (up to biholomorphisms) the\nsqueezing function for all doubly-connected domains in \ℂ other than\nthe punctured plane. It provides the first non-trivial formula for the\nsqueezing function for a wide class of plane domains and answers a question of\nWold. Our main tools used to prove this result are the Schottky-Klein prime\nfunction (following the work of Crowdy) and a version of the Loewner\ndifferential equation on annuli due to Komatu. We also show that these results\ncan be used to obtain lower bounds on the squeezing function for certain\nproduct domains in \ℂn.\n