Lectures on Counterexamples in Several Complex Variables J E F 2019. 2. 12.¢  Lectures on counterexamples

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  • L ectures on Counterexamples in Several

    Complex Variables

    J ohn Erik Fornass Berit Stensones

    AMS CHELSEA PUBLISHING American Mathematical Society * Providence, Rhode Island

  • Lectures on Counterexamples in Several

    Complex Variables

    John Erik Forn/ESS Berit Stensones

    AMS CHELSEA PUBLISHING American Mathematical Society • Providence, Rhode Island

    http://dx.doi.org/10.1090/chel/363.H

  • 2000 Mathematics Subject Classification. Primary 32-01; Secondary 32D05, 32E05, 32A38, 32U05, 32V40, 32F45.

    Library o f C ongress C ata log ing-in -P ub lication D ata Fornaess, John Erik.

    Lectures on counterexamples in several complex variables / John Erik Fornaess and Berit Stenspnes.

    p. cm. — (AMS chelsea publishing) Originally published: Princeton, N.J. : Princeton University Press ; [Tokyo] : University of

    Tokyo Press, 1987. Includes bibliographical references. ISBN 978-0-8218-4422-9 (alk. paper) 1. Functions of several complex variables. I. Stens0nes, Berit, 1956- II. Title.

    QA331.F67 2007 515'. 94—dc22 2007026106

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    © 1987 held by the American Mathematical Society. All rights reserved. Reprinted by the American Mathematical Society, 2007

    Printed in the United States of America.

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  • TABLE OF CONTENTS

    Introduction

    L ec tu re 1: Some N o ta tio n s and D e f in i t io n s ............................................................ 2

    2: Holomorphic F u n c tio n s ................................................................................... 6

    3: Holomorphic C onvexity and Domains o f H o lo m o rp h y ....................... 11

    U: S te in M a n i f o l d s .................................................................................................17

    5: Subharmonic F u n c tio n s ................................................................................... 21

    6: Subharmonic F u n c tio n s ( c o n t . ) ................................................................. 26

    7: Subharmonic F u n c tio n s ( c o n t . ) ................................................................. 31

    P lu risu b h arm o n ic F u n c tio n s ..................................................................... 3U

    8; P lu risu b h arm o n ic F u n c tio n s ( c o n t . ) ................................................... 37

    9: Pseudoconvex Domains . . ........................................................................... k2

    10: Pseudoconvex Domains ( c o n t . ) ................................................................. k6

    11: Pseudoconvex Domains ( c o n t . ) ................................................................ 50

    12; I n v a r ia n t M e trics ............................................................................................ 55

    13: B iholom orphic M a p s ........................................................................................60

    Ik : C ounterexam ples to Sm oothing o f P lu risu b h arm o n ic F u n c t i o n s ...............................................................................................................66

    15: C ounterexam ples to Smoothing o f P lu risu b h arm o n ic F u n c tio n s ( c o n t . ) ............................................................................................ 69

    16 : Counterexam ples to Sm oothing o f P lu risu b h arm o n ic F u n c tio n s ( c o n t . ) ............................................................................................ 73

    17: Counterexam ples to Smoothing o f P lu risu b h arm o n ic

    18 : Complex Monge Ampère E quation ................................................................. 83

    19:

    20:

    2 1 :

    22: CR-M anifolds ( c o n t . ) ................................................................................... 1°0

  • 2k: S te in Neighborhood B a s is ........................................................................... 105

    25: S te in N eighborhood B a s is ( c o n t . ) . . . . . .................................. 109

    26 ; S te in N eighborhood B a s is ( c o n t . ) ......................................................... 113

    27** Riemann Domains over £Cn ................................................................................ 115

    28: The K ohn-N irenberg Example .................................................................... 119

    29-* Peak P o i n t s ..................................................................................................... 123

    30: Bloom’s E x a m p l e .................................................................................................. 126

    31; D’A n g e lo 's E x a m p le ................................................... 129

    32: I n t e g r a l M a n i f o l d s .............................................................................................. 133

    33: Peak S e ts f o r A ( D ) ..............................................................................................138

    3k: Peak S e ts . S tep 1 ................................................................................................ lU l

    35: Peak S e ts . S tep 2 ................................................................................................ 1^5

    36: Peak S e ts . S tep 3 ................................................................................................ 1^8

    37: Peak S e ts . S tep h. .............................................................................. . 159

    38: Sup-Norm E s tim a te s f o r th e 3 - E q u a t i o n ................................................... 165

    39: S ib o n y ’s 3-Example ......................................................................................... l68

    1+0: H y p o e l l ip t ic i ty f o r 3 ............................................... 17^

    1+1; In n e r F u n c t i o n s ................................................... 178

    k2: In n e r F un ctio n s ( c o n t » ) - ........................ l81+

    U3: Large Maximum Modulus S e t s ............................................... 189

    kk: Zero S e ts ................................................................. 19^

    1+5; N o n ta n g e n tia l Boundary L im its o f F u n c tio n s in H (Bn ) . . . 202

    1+6: W erm er's E x a m p le .......................................................................... 212

    1+7; The Union P r o b l e m .............................................................................................. 2ll+

    1+8: Riemann D o m a i n s ................................................................................................... 218

    1+9; Runge E x h a u s t io n ...................................................................................................222

    Lecture 23; Pseudoconvex Domains without Pseudoconvex Exhaustion . . . 102

  • 51; Peak S e ts in Weakly Pseudoconvex B oundaries . , ............229

    52; Peak S e ts in Weakly Pseudoconvex B oundaries ( c o n t . ) . . . 23h

    53; The K obayashi M e t r i c .......................... . . . . . . . . . . . . 236

    B i b l i o g r a p h y .................................................................................................................................2̂ +2

    Lecture 50; Runge Exhaustion (cont.) . , . ............................... 227

  • INTRODUCTION

    These n o te s a r e from a g ra d u a te co u rse in P r in c e to n d u rin g 82 /83 and th e

    f a l l o f 83.

    The pu rpose was to c o l l e c t some o f th e coun terexam ples in th e s e v e r a l

    complex v a r ia b le th e o ry w hich w ere s c a t t e r e d th ro u g h o u t th e l i t e r a t u r e . T h is

    c o l l e c t io n i s by no means com ple te .

    D uring th e f i r s t few w eeks, th e co u rse c o n s is te d o f an in t r o d u c t io n to

    some o f th e b a s ic co n c ep ts o f th e th e o ry , in p a r t i c u l a r th o se needed l a t e r

    f o r th e exam ples.

    T h is m inim ized th e r e s u l t s needed to be ta k e n f o r g ra n te d and made i t

    p o s s ib le to s t a t e th e s e sim ply w ith a p p ro p r ia te r e f e r e n c e s .

    My th a n k s