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Extreme points, support points and the Loewner variation in several complex variables

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Abstract

In this paper we consider extreme points and support points for compact subclasses of normalized biholomorphic mappings of the Euclidean unit ball B n in ℂn. We consider the class S 0(B n) of biholomorphic mappings on B n which have parametric representation, i.e., they are the initial elements f(·, 0) of a Loewner chain f(z, t) = et z + … such that {e−t f(·, t)} t⩾0 is a normal family on B n. We show that if f(·, 0) is an extreme point (respectively a support point) of S 0(B n), then e−t f(·, t) is an extreme point of S 0(B n) for t ⩽ 0 (respectively a support point of S 0(B n) for t ∈ [0,t 0] and some t 0 > 0). This is a generalization to the n-dimensional case of work due to Pell. Also, we prove analogous results for mappings which belong to S 0(B n) and which are bounded in the norm by a fixed constant. We relate the study of this class to reachable sets in control theory generalizing work of Roth. Finally we consider extreme points and support points for biholomorphic mappings of B n generated by using extension operators that preserve Loewner chains.

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Authors and Affiliations

  1. Department of Mathematics, University of Toronto, Toronto, Ontario, M5S 2E4, Canada

    Ian Graham

  2. Faculty of Engineering, Kyushu Sangyo University, 3-1 Matsukadai 2-Chome, Higashi-ku Fukuoka, 8138503, Japan

    Hidetaka Hamada

  3. Faculty of Mathematics and Computer Science, Babeş-Bolyai University, 1 M. Kogălniceanu Str., Cluj-Napoca, 400084, Romania

    Gabriela Kohr & Mirela Kohr

Authors
  1. Ian Graham
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  2. Hidetaka Hamada
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  3. Gabriela Kohr
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  4. Mirela Kohr
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Correspondence to Ian Graham.

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Graham, I., Hamada, H., Kohr, G. et al. Extreme points, support points and the Loewner variation in several complex variables. Sci. China Math. 55, 1353–1366 (2012). https://doi.org/10.1007/s11425-012-4376-0

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