Bounded holomorphic functions on bounded symmetric domains

Cohen, Joel M.; Colonna, Flavia

doi:10.1090/S0002-9947-1994-1176085-6

Bounded holomorphic functions on bounded symmetric domains
HTML articles powered by AMS MathViewer

by Joel M. Cohen and Flavia Colonna PDF

Trans. Amer. Math. Soc. 343 (1994), 135-156 Request permission

Abstract:

Let D be a bounded homogeneous domain in ${\mathbb {C}^n}$, and let $\Delta$ denote the open unit disk. If $z \in D$ and $f:D \to \Delta$ is holomorphic, then ${\beta _f}(z)$ is defined as the maximum ratio $|{\nabla _z}(f)x|/{H_z}{(x,\bar x)^{1/2}}$, where x is a nonzero vector in ${\mathbb {C}^n}$ and ${H_z}$ is the Bergman metric on D. The number ${\beta _f}(z)$ represents the maximum dilation of f at z. The set consisting of all ${\beta _f}(z)$ for $z \in D$ and $f:D \to \Delta$ holomorphic, is known to be bounded. We let ${c_D}$, be its least upper bound. In this work we calculate ${c_D}$ for all bounded symmetric domains having no exceptional factors and give indication on how to handle the general case. In addition we describe the extremal functions (that is, the holomorphic functions f for which ${\beta _f} = {c_D}$) when D contains $\Delta$ as a factor, and show that the class of extremal functions is very large when $\Delta$ is not a factor of D.

References

Sur les domains bournés de l’espace de n variable complexes

11

Flavia Colonna, The Bloch constant of bounded analytic functions, J. London Math. Soc. (2) 36 (1987), no. 1, 95–101. MR 897677, DOI 10.1112/jlms/s2-36.1.95
Flavia Colonna, The Bloch constant of bounded harmonic mappings, Indiana Univ. Math. J. 38 (1989), no. 4, 829–840. MR 1029679, DOI 10.1512/iumj.1989.38.38039
Daniel Drucker, Exceptional Lie algebras and the structure of Hermitian symmetric spaces, Mem. Amer. Math. Soc. 16 (1978), no. 208, iv+207. MR 499340, DOI 10.1090/memo/0208
Kyong T. Hahn, Holomorphic mappings of the hyperbolic space into the complex Euclidean space and the Bloch theorem, Canadian J. Math. 27 (1975), 446–458. MR 466641, DOI 10.4153/CJM-1975-053-0
Maurice Heins, Selected topics in the classical theory of functions of a complex variable, Athena Series: Selected Topics in Mathematics, Holt, Rinehart and Winston, New York, 1962. MR 0162913
Sigurđur Helgason, Differential geometry and symmetric spaces, Pure and Applied Mathematics, Vol. XII, Academic Press, New York-London, 1962. MR 0145455
Mikio Ise, Bounded symmetric domains of exceptional type, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 23 (1976), no. 1, 75–105. MR 419860
Shoshichi Kobayashi, Hyperbolic manifolds and holomorphic mappings, Pure and Applied Mathematics, vol. 2, Marcel Dekker, Inc., New York, 1970. MR 0277770
Max Koecher, An elementary approach to bounded symmetric domains, Rice University, Houston, Tex., 1969. MR 0261032
I. I. Pyateskii-Shapiro, Automorphic functions and the geometry of classical domains, Mathematics and its Applications, Vol. 8, Gordon and Breach Science Publishers, New York-London-Paris, 1969. Translated from the Russian. MR 0252690
Walter Rudin, Function theory in the unit ball of $\textbf {C}^{n}$, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 241, Springer-Verlag, New York-Berlin, 1980. MR 601594
Carl L. Siegel, Analytic functions of several complex variables, Kendrick Press, Heber City, UT, 2008. Lectures delivered at the Institute for Advanced Study, 1948–1949; With notes by P. T. Bateman; Reprint of the 1950 edition. MR 2357088
Richard M. Timoney, Bloch functions in several complex variables. I, Bull. London Math. Soc. 12 (1980), no. 4, 241–267. MR 576974, DOI 10.1112/blms/12.4.241
Richard M. Timoney, Bloch functions in several complex variables. II, J. Reine Angew. Math. 319 (1980), 1–22. MR 586111, DOI 10.1515/crll.1980.319.1