Download Complex Analysis-1986 by S.G. Krantz PDF

By S.G. Krantz

This convention accumulated jointly a small staff of individuals with related pursuits within the geometric functionality concept of numerous advanced variables. whereas the speeches have been of a really expert nature, the papers within the court cases are principally of a survey and speculative nature. the amount is meant to serve either scholars and researchers as a call for participation to energetic new parts of analysis. the extent of the writing has been deliberately set in this sort of means that the papers might be obtainable to a wide viewers.

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It should be remarked that this fact could have been deduced from the finite type condition on ~2 because a domain of finite type cannot contain a non-trivial holomorphic curve in its boundary [14]). Similarly, it can be shown that there is a point Pl E b£21 such that Fi(w) tends to p~ uniformly on compact subsets of £22. Let h(z) = 1. Arguing as in the proof of Theorem I, part A, we assert that given a positive integer s, there exist finitely many points {wj} in Oz with the following property.

The exponential map centred at p is a diffeomorphism). Suppose that the norm of the sectional curvature is _< A/(1 + r2) l+e for some constants e, and A depending on e. Then M -- C n. If, in addition, M has nonpositive sectional curvature, then M is isometrically biholomorphic to C n. Notice that this theorem not only gives intrinsic geometric characterizations of open complex manifolds which are biholomorphically equivalent to C n, but it also establishes an important gap theorem for complete K~hler manifolds with nonpositive sectional curvature: if the curvature dies too quickly at co, then the metric is identically flat.

L e t k b e a p o s i t i v e i n t e g e r a n d let ~ d e n o t e t h e s e l f - m a p of t h e c o m p l e x p l a n e g i v e n b y ~(z)=z/z[ 2k. (~ is a h o m e o m o r p h i s m b u t n e i t h e r a b i h o l o m o r p h i s m n o r a d i f f e o m o r p h i s m . ) L e t f2 be a b o u n d e d 26 planar d o m a i n with s m o o t h boundary. Then the d o m a i n ~-If2 also has s m o o t h boundary. F u r t h e r m o r e if f2 has order of contact s %vith its tangent line at 0 then the corresponding order of contact for ~-ibf2 is s+2k(s-l).

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