Vol. 296, No. 1, 2018

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Two applications of the Schwarz lemma

Bingyuan Liu

Vol. 296 (2018), No. 1, 141–153

We exhibit two applications of Schwarz lemmas in several complex variables. The first application extends Fornæss and Stout’s theorem on monotone unions of balls to monotone unions of ellipsoids. The second application extends Yang’s theorem on bidiscs to the generalized bidisc defined by the author in his previous work. These applications reveal a connection between the geometry of domains and their curvatures. The proof contains a careful study of biholomorphisms, a detailed analysis on convergences, and a modified argument of Yang.

the Schwarz lemma, applications, the generalized bidisc, ellipsoids
Mathematical Subject Classification 2010
Primary: 32M17, 32Q05, 32Q15
Received: 15 February 2017
Revised: 4 June 2017
Accepted: 22 December 2017
Published: 1 May 2018
Bingyuan Liu
Department of Mathematics
University of California, Riverside
Riverside, CA
United States