Based on some ideas of Greene and Krantz, we study the semicontinuity of automorphism groups of domains in one and several complex variables. We show that semicontinuity fails for domains in , , with Lipschitz boundary, but it holds for domains in with Lipschitz boundary. Using the same ideas, we develop some other concepts related to mappings of Lipschitz domains. These include Bergman curvature, stability properties for the Bergman kernel, and also some ideas about equivariant embeddings. 1. Introduction A domain in is a connected open set. If is a domain, then we let denote the group (under the binary operation of composition of mappings) of biholomorphic self-maps of . When is a bounded domain, is a real (never a complex) Lie group. A notable theorem of Greene and Krantz [1] says the following. Theorem 1.1. Let be a smoothly bounded, strongly pseudoconvex domain with defining function (see [2] for the concept of defining function). There is an so that, if is a defining function for a smoothly bounded, strongly pseudoconvex domain with (some large ) then the automorphism group of is a subgroup of the automorphism group of . Furthermore, there is a diffeomorphism such that the mapping is an injective group homomorphism of into . In what follows we shall refer to this result as the “semicontinuity theorem.” It should be noted that, although this theorem was originally proved for strongly pseudoconvex domains in , the very same proof shows that the result is true in for any smoothly bounded domain . In fact the proof, while parallel to the original proof in [1], is considerably simpler in the one-dimensional context. The original proof of this result, which was rather complicated, used stability results for the Bergman kernel and metric established in [3] and also the idea of Bergman representative coordinates. An alternative approach, using normal families, was developed in [4]. The paper [5] produced a method for deriving a semicontinuity theorem when the domain boundaries are only . The more recent work [6] gives a new and more powerful approach to this matter of reduced boundary smoothness. The paper [7] gives yet another approach to the matter and proves a result for finite type domains. It is geometrically natural to wonder whether there is a semicontinuity theorem when the boundary has smoothness of degree less than 2. On the one hand, experience in geometric analysis suggests that is a natural cutoff for many positive results (see [8]). On the other hand, Lipschitz boundary is very natural from the point of view of dilation and other geometric
References
[1]
R. E. Greene and S. G. Krantz, “The automorphism groups of strongly pseudoconvex domains,” Mathematische Annalen, vol. 261, no. 4, pp. 425–446, 1982.
[2]
S. G. Krantz, Function Theory of Several Complex Variables, American Mathematical Society, Providenc, RI, USA, 2nd edition, 2001.
[3]
R. E. Greene and S. G. Krantz, “Deformation of complex structures, estimates for the -equation, and stability of the Bergman kernel,” Advances in Mathematics, vol. 43, no. 1, pp. 1–86, 1982.
[4]
Y. W. Kim, “Semicontinuity of compact group actions on compact differentiable manifolds,” Archiv der Mathematik, vol. 49, no. 5, pp. 450–455, 1987.
[5]
R. E. Greene and S. G. Krantz, “Normal families and the semicontinuity of isometry and automorphism groups,” Mathematische Zeitschrift, vol. 190, no. 4, pp. 455–467, 1985.
[6]
R. E. Greene, K.-T. Kim, S. G. Krantz, and A.-R. Seo, “Semi-continuity of automorphism groups of strongly pseudoconvex domains: the low differentiability case,” The Pacific Journal of Mathematics, In press.
[7]
S. G. Krantz, “Convergence of automorphisms and semicontinuity of automorphism groups,” Real Analysis Exchange. In press.
[8]
S. G. Krantz and H. R. Parks, The Geometry of Domains in Space, Birkh？user, Boston, Mass, USA, 1996.
[9]
G. M. Henkin and J. Leiterer, “Theory of Functions on Strictly Pseudoconvex Sets with Nonsmooth Boundary, with German and Russian summaries,” Report MATH 1981, Akademie der Wissenschaften der DDR, Institut für Mathematik, Berlin, Germany, 1981.
[10]
L. Lempert and L. Rubel, “An independence result in several complex variables,” Proceedings of the American Mathematical Society, vol. 113, pp. 1055–1065, 1991.
[11]
W. Rudin, Function Theory in the Unit Ball of ？n, Springer, New York, NY, USA, 1980.
[12]
S. G. Krantz, Cornerstones of Geometric Function Theory: Explorations in Complex Analysis, Birkh？user, Boston, Mass, USA, 2006.
[13]
B. Maskit, “The conformal group of a plane domain,” American Journal of Mathematics, vol. 90, pp. 718–722, 1968.
[14]
J.-P. Rosay, “Sur une characterization de la boule parmi les domains de par son groupe d’automorphismes,” Annales de l'institut Fourier Grenoble, vol. 29, pp. 91–97, 1979.
[15]
B. Wong, “Characterization of the unit ball in ？n by its automorphism group,” Inventiones Mathematicae, vol. 41, no. 3, pp. 253–257, 1977.
[16]
S. G. Krantz, “Characterizations of smooth domains in by their biholomorphic self maps,” American Mathematical Monthly, vol. 90, pp. 555–557, 1983.
[17]
G. Julia, “Le？ons sur la représentation conforme des aires multiplement connexes,” Paris, France, 1934.
[18]
M. Heins, “On the number of 1-1 directly conformal maps which a multiply-connected plane regions of finite connectivity admits onto itself,” Bulletin of the AMS, vol. 52, pp. 454–457, 1946.
[19]
L. Apfel, Localization properties and boundary behavior of the Bergman kernel [thesis], Washington University in St. Louis, 2003.
