We give a counterexample to the following theorem of Bremermann on Shilov boundaries ([Bre 1959])... more We give a counterexample to the following theorem of Bremermann on Shilov boundaries ([Bre 1959]): if D is a bounded domain in C n having a univalent envelope of holomorphy, say D, then the Shilov boundary of D with respect to the algebra A(D), call it ∂ S D, coincides with the corresponding one for D, called ∂ S D.
Kobayashi initiated the study of the pseudodistance k, which now is called the Kobayashi pseudodi... more Kobayashi initiated the study of the pseudodistance k, which now is called the Kobayashi pseudodistance; cf. [311, 317] (see § § 3.1, 3.3). An extended study of taut domains and their relation to the topics discussed here can be found in § 3.2. H. L. Royden published the infinitesimal form~in 1971 [459] (see § 3.5). Most of the material collected in this chapter is due to these two authors and to T. J. Barth (cf. [33, 36]). A fairly complete survey about the results up to 1976 can be found in the report of S. Kobayashi (see [314]). In 1990 S. Kobayashi introduced the pseudometric ỹ, which is here called the Kobayashi-Buseman pseudometric (see [315]) (see § 3.6). The chapter concludes with a discussion of higher-order Lempert functions and Kobayashi-Royden pseudometrics (see § 3.8). Introduction. In the previous chapter, we discussed the Carathéodory pseudodistance and we observed that if d G W G G ! R C is a function with p.f .z 0 /; f .z 00 // Ä d G .z 0 ; z 00 /, z 0 ; z 00 2 G; f 2 O.G; D/, then c G Ä d G (cf. Remark 2.1.2). In this chapter, we will study the opposite case, namely, a family of pseudodistances .k G / G such that if d G is a pseudodistance on G satisfying d G .f. 0 /; f. 00 // Ä p. 0 ; 00 /, 0 ; 00 2 D; f 2 O.D; G/, then d G Ä k G. Similar investigations will be presented on the level of pseudometrics. The main tool will be the space O.D; G/ of "analytic discs" in G "antipodal" to the space O.G; D/ that was the basis for our studies in Chapter 2.
Continuous Nowhere-Differentiable Functions--an Application of Contraction Mappings
The American Mathematical Monthly, 1991
then f is also a continuous nowhere-differentiable function. (See [3, p. 115].) The above example... more then f is also a continuous nowhere-differentiable function. (See [3, p. 115].) The above examples have concise definitions and establish the existence of continuous nowhere-differentiable functions. However, it is not easy to visualize or guess what their graphs look like, let alone to see intuitively why they work. Our continuous nowhere-differentiable function f: [0, 11 -] R is the uniform limit of a sequence of piecewise linear continuous functions fn: [0, 1] -+ R with steep slopes. In constructing the sequence (f,,>, we will be using a contraction mapping w from the family of all closed subsets of X = [0, 1] x [0, 1] into itself with respect to the Hausdorff metric induced by the Euclidean metric. (A contraction mapping on a metric space (Y, d) is a function g: Y -+ Y for which there is a positive constant k of sets converges to the graph of f in the Hausdorff metric. If A is the diagonal of slope 1 in the square X, then wn(A) is the graph of fn, and hence, a reader can obtain an intuitive idea of the graph of f. This idea first occurred to me when I attended Mr. Gary Church's master's thesis defense at San Jose State University at which he talked about attractors of contraction mappings (see [4]).
We give a characterization of L 2 h-domains of holomorphy with the help of the boundary behavior ... more We give a characterization of L 2 h-domains of holomorphy with the help of the boundary behavior of the Bergman kernel and geometric properties of the boundary, respectively.
About the Caratheodory Completeness of all Reinhardt Domains
Elsevier eBooks, 1984
Publisher Summary In the theory of complex analysis, there are different notions of distances on ... more Publisher Summary In the theory of complex analysis, there are different notions of distances on a bounded domain, for example, the Caratheodory-distance dealing with bounded holomorphic functions, the Bergmann-metric measuring how many L 2 -holomorphic functions exist, or the Kobayashi-distance, describing the sizes of analytic discs in G . The main problem working with these distances is to decide the domain G that is complete with respect to one of the distances. Any pseudo-convex domain with C 1 -boundary is complete with respect to the Bergmann-metric. The Caratheodory-distance can be compared with the other two, in fact, it is the smallest one, but there is no relation between the Bergmann-metric and the Kobayashi-metric. Any bounded complete Reinhardt domain G that is pseudo-convex is complete in the sense of the Caratheodory-distance; in fact, it is proved that any Caratheodory ball is a relatively compact subset of G.
We deliver examples of non-Gromov hyperbolic tube domains with convex bases (equipped with the Ko... more We deliver examples of non-Gromov hyperbolic tube domains with convex bases (equipped with the Kobayashi distance). This is shown by providing a criterion on non-Gromov hyperbolicity of (non-smooth) domains.The results show the similarity of geometry of the bases of non-Gromov hyperbolic tube domains with the geometry of non-Gromov hyperbolic convex domains. A connection between the Hilbert metric of a convex domain Ω in R n with the Kobayashi distance of the tube domain over the domain Ω is also shown. Moreover, continuity properties up to the boundary of complex geodesics in tube domains with a smooth convex bounded base are also studied in detail.
We discuss some basic properties of the Sibony functions and pseudometrics. 2010 Mathematics Subj... more We discuss some basic properties of the Sibony functions and pseudometrics. 2010 Mathematics Subject Classification. 32F45.
Let X, Y be two complex manifolds of dimension 1 which are countable at infinity, let D ⊂ X, G ⊂ ... more Let X, Y be two complex manifolds of dimension 1 which are countable at infinity, let D ⊂ X, G ⊂ Y be two open sets, let A (resp. B) be a subset of ∂D (resp. ∂G), and let W be the 2-fold cross ((D ∪ A) × B) ∪ (A × (B ∪ G)). Suppose in addition that D (resp. G) is Jordan-curve-like on A (resp. B) and that A and B are of positive length. We determine the "envelope of holomorphy" W of W in the sense that any function locally bounded on W, measurable on A × B, and separately holomorphic on (A × G) ∪ (D × B) "extends" to a function holomorphic on the interior of W .
We give a counterexample to the following theorem of Bremermann on Shilov boundaries ([Bre 1959])... more We give a counterexample to the following theorem of Bremermann on Shilov boundaries ([Bre 1959]): if D is a bounded domain in C n having a univalent envelope of holomorphy, say D, then the Shilov boundary of D with respect to the algebra A(D), call it ∂ S D, coincides with the corresponding one for D, called ∂ S D.
Kobayashi initiated the study of the pseudodistance k, which now is called the Kobayashi pseudodi... more Kobayashi initiated the study of the pseudodistance k, which now is called the Kobayashi pseudodistance; cf. [311, 317] (see § § 3.1, 3.3). An extended study of taut domains and their relation to the topics discussed here can be found in § 3.2. H. L. Royden published the infinitesimal form~in 1971 [459] (see § 3.5). Most of the material collected in this chapter is due to these two authors and to T. J. Barth (cf. [33, 36]). A fairly complete survey about the results up to 1976 can be found in the report of S. Kobayashi (see [314]). In 1990 S. Kobayashi introduced the pseudometric ỹ, which is here called the Kobayashi-Buseman pseudometric (see [315]) (see § 3.6). The chapter concludes with a discussion of higher-order Lempert functions and Kobayashi-Royden pseudometrics (see § 3.8). Introduction. In the previous chapter, we discussed the Carathéodory pseudodistance and we observed that if d G W G G ! R C is a function with p.f .z 0 /; f .z 00 // Ä d G .z 0 ; z 00 /, z 0 ; z 00 2 G; f 2 O.G; D/, then c G Ä d G (cf. Remark 2.1.2). In this chapter, we will study the opposite case, namely, a family of pseudodistances .k G / G such that if d G is a pseudodistance on G satisfying d G .f. 0 /; f. 00 // Ä p. 0 ; 00 /, 0 ; 00 2 D; f 2 O.D; G/, then d G Ä k G. Similar investigations will be presented on the level of pseudometrics. The main tool will be the space O.D; G/ of "analytic discs" in G "antipodal" to the space O.G; D/ that was the basis for our studies in Chapter 2.
Continuous Nowhere-Differentiable Functions--an Application of Contraction Mappings
The American Mathematical Monthly, 1991
then f is also a continuous nowhere-differentiable function. (See [3, p. 115].) The above example... more then f is also a continuous nowhere-differentiable function. (See [3, p. 115].) The above examples have concise definitions and establish the existence of continuous nowhere-differentiable functions. However, it is not easy to visualize or guess what their graphs look like, let alone to see intuitively why they work. Our continuous nowhere-differentiable function f: [0, 11 -] R is the uniform limit of a sequence of piecewise linear continuous functions fn: [0, 1] -+ R with steep slopes. In constructing the sequence (f,,>, we will be using a contraction mapping w from the family of all closed subsets of X = [0, 1] x [0, 1] into itself with respect to the Hausdorff metric induced by the Euclidean metric. (A contraction mapping on a metric space (Y, d) is a function g: Y -+ Y for which there is a positive constant k of sets converges to the graph of f in the Hausdorff metric. If A is the diagonal of slope 1 in the square X, then wn(A) is the graph of fn, and hence, a reader can obtain an intuitive idea of the graph of f. This idea first occurred to me when I attended Mr. Gary Church's master's thesis defense at San Jose State University at which he talked about attractors of contraction mappings (see [4]).
We give a characterization of L 2 h-domains of holomorphy with the help of the boundary behavior ... more We give a characterization of L 2 h-domains of holomorphy with the help of the boundary behavior of the Bergman kernel and geometric properties of the boundary, respectively.
About the Caratheodory Completeness of all Reinhardt Domains
Elsevier eBooks, 1984
Publisher Summary In the theory of complex analysis, there are different notions of distances on ... more Publisher Summary In the theory of complex analysis, there are different notions of distances on a bounded domain, for example, the Caratheodory-distance dealing with bounded holomorphic functions, the Bergmann-metric measuring how many L 2 -holomorphic functions exist, or the Kobayashi-distance, describing the sizes of analytic discs in G . The main problem working with these distances is to decide the domain G that is complete with respect to one of the distances. Any pseudo-convex domain with C 1 -boundary is complete with respect to the Bergmann-metric. The Caratheodory-distance can be compared with the other two, in fact, it is the smallest one, but there is no relation between the Bergmann-metric and the Kobayashi-metric. Any bounded complete Reinhardt domain G that is pseudo-convex is complete in the sense of the Caratheodory-distance; in fact, it is proved that any Caratheodory ball is a relatively compact subset of G.
We deliver examples of non-Gromov hyperbolic tube domains with convex bases (equipped with the Ko... more We deliver examples of non-Gromov hyperbolic tube domains with convex bases (equipped with the Kobayashi distance). This is shown by providing a criterion on non-Gromov hyperbolicity of (non-smooth) domains.The results show the similarity of geometry of the bases of non-Gromov hyperbolic tube domains with the geometry of non-Gromov hyperbolic convex domains. A connection between the Hilbert metric of a convex domain Ω in R n with the Kobayashi distance of the tube domain over the domain Ω is also shown. Moreover, continuity properties up to the boundary of complex geodesics in tube domains with a smooth convex bounded base are also studied in detail.
We discuss some basic properties of the Sibony functions and pseudometrics. 2010 Mathematics Subj... more We discuss some basic properties of the Sibony functions and pseudometrics. 2010 Mathematics Subject Classification. 32F45.
Let X, Y be two complex manifolds of dimension 1 which are countable at infinity, let D ⊂ X, G ⊂ ... more Let X, Y be two complex manifolds of dimension 1 which are countable at infinity, let D ⊂ X, G ⊂ Y be two open sets, let A (resp. B) be a subset of ∂D (resp. ∂G), and let W be the 2-fold cross ((D ∪ A) × B) ∪ (A × (B ∪ G)). Suppose in addition that D (resp. G) is Jordan-curve-like on A (resp. B) and that A and B are of positive length. We determine the "envelope of holomorphy" W of W in the sense that any function locally bounded on W, measurable on A × B, and separately holomorphic on (A × G) ∪ (D × B) "extends" to a function holomorphic on the interior of W .
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