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Pseudoconvexity and the Envelope of Holomorphy for Functions of Several Complex Variables

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Pseudoconvexity and the Envelope of Holomorphy for Functions of Several Complex Variables PSEUDOCONVEXITY AND THE ENVELOPE OF HOLOMORPHY FOR FUNCTIONS OF SEVERAL COMPLEX VARIABLES by LORNE BARRY MULLETT B.Sc, University of British Columbia, 1964 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF ARTS in the Department of MATHEMATICS ^ We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA August, 1966 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. 1 further agree that permission, for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Depa-rtment or by his representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed •without my written permission. Department of MATHEMATICS The University of British Columbia Vancouver 8, Canada Date October 17th, 1966 ii ABSTRACT We first handle some generalizations from the theory of functions of a single complex variable, including results regarding analytic continuation. Several "theorems of continuity" are considered, along with the associated definitions of pseudoconvexity, and these are shown to be equivalent up to a special kind of trans• formation. By successively applying a form of analytic contin• uation to a function f , a set of pseudoconvex domains is constructed, and the union of these domains is shown to be the envelope of holomorphy of f . iii TABLE OF CONTENTS Page I. Introduction 1 II. Pseudoconvexity and the Envelope of Holomorphy 7 IIT. Bibliography 27 iv ACKNOWLEDGEMENT I wish to thank Dr. A.H. Cayford for his aid both in the preparation of this thesis and in research in the field of complex analysis. I also with to thank Dr. CA. Swanson for hi s encouragement. The financial support of the University of British Columbia and the National Research Council is gratefully acknowledged. 1. INTRODUCTION The theory of several complex variables is about seventy years old. A relatively dormant period was reached when scholars found that their generalizations from the theory of a single complex variable began to fail, notable exceptions being discovered principally by P. Hartogs about 1910. Some striking differences in the theories are listed. Except for the case n=l , for f(Bva0,...^ ) in Dc$^ , we 'have in general (i) holomorphic functions do not produce conformal maps (rather the result is a "pseudoconformal" map, in which only the projection onto $~ is necessarily conformal) . (ii) holomorphic functions do not have isolated zeros (or singularities). (iii) not every domain Bc(|Jn is a domain of holomorphy. In particular, we also have except for n=l , (iv) the boundary of a domain of holomorphy is always connected. i (v) the exterior of a bounded set cannot be a domain of holomorphy. About twenty years later, due to the work of H. Behnke, H. Cartan, and P. Thullen on domains, and the contribution by S. Bergman of the invariant metric and the "kernel function", the theory "branched off in several new directions and became recognized as being independent. However, another period of dormancy was reached] this time the restriction was the old- fashioned machinery mathematicians were using in an attempt to solve contemporary problems. The most recent breakthrough is the language of "fibre-bundles'1, which is now being applied successfully to the study of "complex spaces", which are the extensions to n complex dimensions of the familiar Riemann surfaces. z z We define f(a) =f( i> 2> ••• n) to be "holomorphic" n in Dc$Y iff at all H^eD , exists, and f satisfies a — aak the Cauchy-Riemann equations = 0 k = 1,2,...,n . Observe that = + i|= ) . Let d 3X 3y 9ik k k 0(D) ={f : f(ja) is holomorphic in D} Equivalently, fe<0>(D) <=> f can oe developed into a multiple power series a (o) kl a (o) k2 a f a ^K,,k0,?..,k =o k,v..k (v i ) (v 2 ) ---(v 12 ' n 12 n (o) that converges uniformly in a neighbourhood of _av 1 for any choice of jJ^eD . One of the early generalizations was Cauchy's integral formula, namely that if f(a) satisfies the Cauchy-Riemann equations for ae(jj , then f(a) = 75^- J' dt, where C is a 1 C ' 'contour" (simple, closed, rectifiable, oriented). The generalization yields the following:' If f(ia) is holomorphic in a neighbourhood of the product domain D-^xDgX. .xDn , then aD^^DgX. .xaDn (t1-a1)(t2-a2) .. -(tn-an) We state without proof the following fundamental theorem, the proof of wnich depends basically on the above formula. Theorem A: If f(z)=f(z^yz2,...,an) e o(P) , where 0 0 P- {|a1-a1( )|<r1,|a2-B2( )|<r2,...,|an-an(°)|<rn3 , .then f a a ))kl(a (D=kl,kp,.?.,Vo V2--V i~ i° ^ I3 23 3 n 12 n uniquely, for all aeP , where 4. Moreover, f(js) converges absolutely and uniformly in P . An important and very useful tool is the maximum principle. Theorem B: Let f(j3)'€<>(D) > not constant. Then |f(is)| does not assume its maximum in D . Proof: Assume the theorem is false. Let eS°}<= intD be a maximal point of f(js) , and let N(^°^;r_) be a (polycylindrical) neighbourhood of such that NeD . Wow the holomorphic functions ffA Ba a a (°h fCa (°) a a (°' a (°h £(*^9 a^a ) \ p°2 n 1 23 »•••» °n /*»*«y-V'2_ **"n-l,n of a single complex variable must then be constant, for they assume their maxima at the centres of the composite disks of N . But this implies that f(js) is constant in N with respect B to all of a^Bg, •' • > n > ^ Theorem A . We assert that f(_a) must have been constant, contrary to our assumption. Por if there were a point ^^^B^0'' in D , by the connectivity of D we can construct an arc from a^0^to a/1/1 and cover it with polycylindrical neighbourhoods {IL } . By the Heine-Borel A. theorem there exists a finite subcovering {.Nj} which we may choose to overlap in such a way that each neighbourhood contains the centre of the previous. Prom our earlier result, f(js) is constant in all these neighbourhoods by construction, and 5. thus f(a/°)) = f(B^) . Since our choice of a/1) was arbi• trary, the contradiction follows. Corollary: |f(a)| assumes its maximum on dD , the boundary of D , provided the function is continuous in D , the closure of D . Proof: Weierstrass' theorem on the maximum of a continuous function. This thesis is concerned with the "envelope of holomorphy" of a given (univalent) domain D , which is the smallest domain of holomorphy containing D . The generalization of the convex envelope from n real dimensions leads us to con• sider non-univalent regions. This is analogous to continuing a holomorphic function of one complex variable and getting a multiple-valued function as the result. We can make the contin• uation single-valued by considering the associated Riemann surface. We may allow complex manifolds to be envelopes, but examples show they will often be non-univalent. Hence there are no norms, and we must find a more general definition for a pseudoconvex region (equivalent to a region of holomorphy), preferably not de• pendent on the concept of "plurisubharmonic function", to be de• fined later. From the familiar unit disk of complex analysis, we can branch off in our generalization in one of two ways. We could take as our basic domain the polycylinder, which in fact b. we do in this paper for some of the results, or we could choose our domain to he the n - hall. Modern theories have extended the polycylinder to the analytic polyhedron, and the n - hall to the pseudoconvex domain. The latter parts of this thesis are concerned with the second extension. 7. PSEUDOCONVEXITY AND THE ENVELOPE OP HOLOMORPHY i We begin by recalling some definitions from complex analysis, and by stating some results and definitions which are directly relevant to the topic. The space under consideration is ^n=^ , consisting of n-tuples of complex variables a=:(Bj,a2,...,an) . The topology will be that generated by the euclidean norm; all topologies generated by arbitrary norms are equivalent. A "region" is an open set, and a "domain" is a connected region. We let Q(D) denote the convex hull, and E(D) the "envelope of holomorphy" i of the domain D . E(D) • is the largest domain to which every function'holomorphic in D may be continued. The envelope of holomorphy has become useful to theoretical physicists in the derivation of dispersion relations of quantum field theory. We will call D a "domain of holomorphy" iff D=E(D), and some equivalent definitions are clear. H is a "region of , holomorphy" iff there exists a function holomorphic in H but not in any larger region. A "Riemann domain" is a pair <X,h> , where IT is a topological space and h:l[ -» (jJn is a local homeomorphism. We will restrict ourselves to univalent (schlicht) domains, with occasional reference to non-univalent domains. A sufficient condition for the existence of E(D) is that D be a Riemann domain. o. E(D) has been determined only for certain special domains, the definitions of which follow. I. The "polycylinder" is probably the most commonly en• countered domain in dealing with functions of several complex variables. It is represented by {a_: | a1|<r1, | B2|<r2, . .., | a |<r } . This domain is the product of n disks {a1: |a1|<r1}x{a2: |a2 |<r2}x-• -x{an: |anl<rn} .
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