An analogue of the squeezing function for projective maps
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An analogue of the squeezing function for projective maps Nikolai Nikolov1,2 · Pascal J. Thomas3 Received: 24 September 2019 / Accepted: 23 January 2020 © Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract In the spirit of Kobayashi’s applications of methods of invariant metrics to questions of projective geometry, we introduce a projective analogue of the complex squeezing function. Using Frankel’s work, we prove that for convex domains it stays uniformly bounded from below. In the case of strongly convex domains, we show that it tends to 1 at the boundary. This is applied to get a new proof of a projective analogue of the Wong–Rosay theorem. Keywords Projective maps · Invariant distances · Squeezing function Mathematics Subject Classification 52A20 · 53A20 · 32F45
1 Introduction The projective maps are the ones that preserve lines in projective space. They are linear in the homogeneous coordinates and in affine space yield linear-fractional maps (which we will call projective too, with a slight abuse of language). There is a long tradition of applying the appropriate analogues of convex objects to complex analysis. Surprisingly, it is also sometimes useful to study geometrically convex domains and projective maps, which are rather rigid objects, with the methods developed for complex analysis in several The first named author is partially supported by the Bulgarian National Science Fund, Ministry of Education and Science of Bulgaria under contract DN 12/2. This paper was started while he was visiting the Paul Sabatier University, Toulouse, in November 2018 as a guest professor. * Pascal J. Thomas [email protected]‑toulouse.fr Nikolai Nikolov [email protected] 1
Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Acad. G. Bonchev 8, 1113 Sofia, Bulgaria
2
Faculty of Information Sciences, State University of Library Studies and Information Technologies, Shipchenski prohod 69A, 1574 Sofia, Bulgaria
3
Institut de Mathématiques de Toulouse, UMR5219, CNRS, UPS, Université de Toulouse, 31062 Toulouse Cedex 9, France
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variables. For instance, one can use projective mappings from an interval into domains to construct metrics analogous to the Carathéodory and Kobayashi metrics, which recover the classical Hilbert metric in the case of convex domains. Shoshichi Kobayashi developed this approach in [10], and László Lempert summarized the analogy and built upon it in [11]. This was pursued in papers such as [6, 20]. The complex squeezing function was defined under this name in [1], which provides a good overview of the motivations to study it. It has been the object of numerous further works in recent years. We will be using a “projective” analogue of the squeezing function and study its relationship with the properties of convex sets. Some of the results of Sidney Frankel’s pioneering paper [6] can be rephrased as the fact that the projective squeezing function of co
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