Kang-Tae Kim

The geometry of complex domains

Preface -- 1 Preliminaries -- 2 Riemann Surfaces and Covering Spaces -- 3 The Bergman Kernel and Metric -- 4 Applications of Bergman Geometry -- 5 Lie Groups Realized as Automorphism Groups -- 6 The Significance of Large Isotropy Groups -- 7 Some Other Invariant Metrics -- 8 Automorphism Groups and Classification of Reinhardt Domains -- 9 The Scaling Method, I -- 10 The Scaling Method, II -- 11 Afterword -- Bibliography -- Index.

A Note on the Wong-Rosay Theorem in Complex Manifolds

The authors prove a version, in utmost generality, of the Bun Wong-Rosay theorem on a complex manifold M. The essence of the result is that a domain Ω⫅M with non-compact automorphism group and boundary orbit accumulation point that is strongly pseudoconvex must be biholomorphic to the unit ball in C n .

Characterization of the Hilbert ball by its automorphism group

Let Ω be a bounded, convex domain in a separable Hilbert space. The authors prove a version of the theorem of Bun Wong, which asserts that if such a domain admits an automorphism orbit accumulating at a strongly pseudoconvex boundary point, then it is biholomorphic to the ball. Key ingredients in the proof are a new localization argument using holomorphic peaking functions and the use of new normal families arguments in the construction of the limit biholomorphism.

ANALYSIS OF THE WU METRIC. I: THE CASE OF CONVEX THULLEN DOMAINS

We present an explicit description of the Wu metric on the convex Thullen domains which turns out to be the first natural example of a purely Hermitian, non-Kahlerian invariant metric. Also, we show that the Wu metric on these Thullen domains is in fact real analytic everywhere except along a lower dimensional subvariety, and is Cl smooth overall. Finally, we show that the holomorphic curvature of the Wu metric on these Thullen domains is strictly negative where the Wu metric is real analytic, and is strictly negative everywhere in the sense of current.

Normal families of holomorphic functions and mappings on a Banach space

The authors lay the foundations for the study of normal families of holomorphic functions and mappings on an infinite-dimensional normed linear space. Characterizations of normal families, in terms of value distribution, spherical derivatives, and other geometric properties are derived. Montel-type theorems are established. A number of different topologies on spaces of holomorphic mappings are considered. Theorems about normal families are formulated and proved in the language of these various topologies. Normal functions are also introduced. Characterizations in terms of automorphisms and also in terms of invariant derivatives are presented.

Complexn-dimensional manifolds with a realn2-dimensional automorphism group

The main theorem of this article is a characterization of non compact simply connected complete Kobayashi hyperbolic complex manifold of dimension n≽ 2 with real n2-dimensional holomorphic automorphism group. Together with the earlier work [11, 12] and [13] of Isaev and Krantz, this yields a complete classification of the simply-connected, complete Kobayashi hyperbolic manifolds with dimℝ Aut (M) ≽ (dimℂM)2.

Weak-type normal families of holomorphic mappings in Banach spaces and characterization of the Hilbert ball by its automorphism group

We present a characterization of the open unit ball in a separable infinite dimensional Hilbert space by the property of automorphism orbits among the domains that are not necessarily bounded. This generalizes the recent work of Kim and Krantz [6]. Key new features of this article include: a lower bound estimate of the Kobayashi metric and distance near a pluri-subharmonic peak boundary point of the domains in Banach spaces, an effective localization argument, and an improvement of weak-type convergence of sequences of biholomorphic mappings of domains in Banach spaces.

COMPLEX SCALING AND GEOMETRIC ANALYSIS OF SEVERAL VARIABLES

The authors study the method of scaling in the context of the study of automorphism groups of complex domains in multiple dimensions. Various types of scaling techniques are compared and contrasted. Applications are given in a number of areas of complex geometric analysis. Relations with other parts of mathematics are described.

A Kobayashi metric version of Bun Wong's theorem

We study the classical theorem of Bun Wong and Rosay about domains with non-compact automorphism group and strongly pseudoconvex orbit accumulation point. We formulate and prove a version of the result in the language of the Kobayashi metric

On an explicit construction of weakly sufficient sets for the function algebra A −∞(Ω)

This article studies the space A −∞(Ω), the function algebra of holomorphic functions on a domain Ω with 𝒞1 boundary in ℂ n with polynomial growth. In particular, we present an explicit construction of countable subset that is weakly sufficient.