L ectures onCounterexamples in Several

Complex Variables

J ohn Erik Fornass Berit Stensones

AMS CHELSEA PUBLISHING American Mathematical Society * Providence, Rhode Island

Lectures onCounterexamples in Several

Complex Variables

John Erik Forn/ESS Berit Stensones

AMS CHELSEA PUBLISHINGAmerican 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 ataFornaess, 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 2007515'. 94—dc22 2007026106

C opying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given.

Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294, USA. Requests can also be made by e-mail to reprint-permission@ams.org.

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

Printed in the United States of America.

@ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability.

Visit the AMS home page at http://www.ams.org/

10 9 8 7 6 5 4 3 2 1 12 11 10 09 08 07

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 icF 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 icF 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 icF 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 go to E. Low, A. N o e ll , P. Sm ith and B. S ten s^ n es who gave

some o f th e l e c t u r e s , w h ile th e n o te s were w r i t t e n by B. S te n s^ n e s .

My th a n k s a ls o go to Jay B elanger who p ro o fre a d th e m a n u sc r ip t.

The l e c tu r e s a re in th e same o rd e r a s th e y were g iven and hence th e r e

a re jumps back and f o r th between v a r io u s to p ic s b u t th e v a r io u s exam ples can

be re a d in d e p e n d e n tly .

John E rik F o rn æ ss

2k2

BIBLIOGRAPHY

[ALE 1] A leksand rov , A. B .; E x is te n c e o f in n e r fu n c tio n s in th e h a l l , Mat. Sh. 117 (1 9 8 2 ), 1U7-163 (R u ss ia n ) .

[ALE 2] A leksand rov , A.' B .; P r iv a te com m unication to E. L</w, O cto b er, 1982.

[ARN l ] A rn o ld , V . I . ; O rd inary d i f f e r e n t i a l e q u a tio n s . Cam bridge, MA, London: MIT, 1973.

[BER 1] B e rn d tsso n , B . ; I n te g r a l Form ulas and ze ro s o f bounded holom orphic fu n c tio n s in th e u n i t b a l l , Math. Ann. 2k9( 1980) , 163- 176 .

[B-F 1] B ed fo rd , E. F o rn æ ss , J . E .; Counterexam ples to r e g u la r i t y f o r th e complex Monge-Ampère e q u a tio n , In v . Math. 50 (1 9 7 9 ), 129-13*+.

[B-T 1] B edford , E . , T a y lo r , A .; The D ir ic h l e t problem f o r a complex Monge-Ampère e q u a tio n , In v . Math. 37 (1 9 7 6 ), 1-*+*+.

[BLO l ] Bloom, T . , Peak fu n c tio n s f o r pseudoconvex domains o f s t r i c t ty p e , Luke M ath. J . ¿+5 (1 9 7 8 ), 133-l*+7.

[BOU 1] B o u te t de M onvel, L . ; I n te g r a t io n des e q u a tio n s de Cauchy- Riemann in d u i t e s fo rm e lle s , S ém inaire G ou laou ic-L ions-S chw artz 197^-19755 Expose No. 9 .

[B-S 1] Behnke, H . , S te in , K . , ; K onvergente F olgen von R e g u la r i t a t s - b e re ic h e n und d ie M erom orph iekonvex ita t, Math. Ann. I l 6 (1938-1939), 201+-216.

[CAT 1] C a t l i n , D .; Boundary b eh a v io r o f holom orphic fu n c tio n s on pseudoconvex, J . D i f f . Geometry 15 (1 9 8 0 ), 605-625.

[CAT 2] C a t l i n , D .; N ecessa ry c o n d it io n s o f s u b e l l i p t i c i t y and H y p o e l l ip t ic i ty f o r th e T-Neumann problem on pseudoconvex dom ains, in : Recent developm ents in s e v e ra l complex v a r ia b le s , Ann. o f Math. S tu d ie s , P r in c e to n U n iv e r s i ty P re ss 100 (1 9 8 1 ), 93-100,

[C-C 1] Chaumat, J . , C h o l le t , A. M .; C h a ra c te r is a t io n e t p r o p r ié té s

des ensem bles lo ca lem en t p ic de A (D) , Duke Math. J . Vf (1 9 8 0 ), 763-787.

[C-C 2] Chaumat, J . , C h o l le t , A. M .; Ensem bles p ic pour A (D) , Ann. I n s t . F o u r ie r (G renoble) 29 (1 9 7 9 ), 171-200.

[C-C 3] Chaumat, J . , C h o l le t , A. M. ;• Ensem bles p ic s pour A (D) non g lobalem ent in c lu s dans une v a r i é t é i n t e g r a l , M ath. Ann. 258 (1 9 8 2 ), 213-252.

21*3

[C-K-N-S

[C-T 1]

[C-Y 1]

[DM 1]

[DM 2]

[D-F 1]

[D-F 2]

[D-F 3]

[D-P 1]

[D-S 1]

[FED l ]

[F-N l ]

[FOR l ]

[FOR 2]

[FOR 3]

l ] C a f f a r e l l i , L . , Kohn, J . J . , N ire n b e rg , L. , S p ruck , J . ; The D i r i c h l e t problem fo r n o n lin e a r second o rd e r e l l i p t i c e q u a tio n s , I I . Complex Monge-Ampere, and u n ifo rm ly e l l i p t i c , e q u a tio n s .

C a rta n , H ., T h u lle n , P. ; R e g u la r i ta ts - u n d K o n v erg en z -b e re ic h e , Math. Ann. 106 (1 9 3 2 ), 617-6^7.

Cheng, S. Y. , Yau, S. T . ; On th e e x is te n c e o f a com plete K ahler m e tr ic on noncompact complex m a n ifo ld s and th e r e g u la r i t y o f F e ffe rm an ’s e q u a tio n , Comm. P ure A ppl. M ath. 33 (1 9 8 0 ),507- 5WL

D’A ngelo , J . P . ; S u b e l l i p t i c e s t im a te s and f a i l u r e o f sem i­c o n t in u i ty fo r o rd e rs o f c o n ta c t , Duke M ath. J . h i (1 9 8 0 ), 955-957.

D’A ngelo, J . P . ; R eal h y p e r s u r fa c e s , o rd e rs o f c o n ta c t , and a p p l i c a t io n s , Ann. o f Math. 115 (1 9 8 2 ), 615-637-

D ie d e r ic h , K ., Fornaess , J . E. ; A smooth pseudoconvex domain w ith o u t pseudoconvex e x h a u s tio n , m a n u sc r ip ta M ath. 39 (1 9 8 2 ), 119-123.

D ie d e r ic h , K . , F o rn a e s s , J . E .; An example w ith n o n t r iv i a l n e b e n h u e l le , Math. Ann. 225 (1 9 7 7 ), 275-292.

D ie d e r ic h , K . , Fornaess , J . E . ; Pseudoconvex dom ains:Bounded s t r i c t l y p lu risu b h a rm o n ic e x h a u s tio n f u n c t io n s , In v . In v . Math. 39 (1 9 7 7 ), 129-1*0..

D ie d e r ic h , K ., P f lu g , R. P . ; N ecessary c o n d it io n s f o r hypo- e l l i p t i c i t y o f th e TT-problem, in : R ecent developm ents ins e v e r a l complex v a r i a b l e s , Ann. o f Math. S tu d ie s , P rin c e to n U n iv e rs i ty P re ss 100 (1 9 8 1 ), 93-100.

Duchamp, T . , S to u t , E. L. ; Maximum modulus s e t s , Ann. I n s t . F o u r ie r (G renoble) 31 ( l 9 8 l ) , 37-89-

F e d e re r , H .; G eom etric m easure th e o ry , B e r l in , H e id e lb e rg ,New Y ork; S p r in g e r , 1969-

Fornaess , J . E . , N arasim han, R . ; The Levi problem on complex spaces w ith s i n g u l a r i t i e s , Math. Ann. 2U8 (1980) , ^7-72 .

Fornaess, J . E . ; Peak p o in ts on w eakly pseudoconvex dom ains. Math. Ann. 227 (1 9 8 8 ), 173-175.

Forneess , J . E . ; Embedding s t r i c t l y pseudoconvex domains in convex dom ains, Am. J . M ath. 98 (1 9 7 8 ), 529-569«

Fornsess , J . E . ; P lu r i subharm onic d e t i r in g f u n c t io n s , P a c i f i c J . Math. 80 (1 9 7 9 ), 381-388.

2hh

[FO-S 1] Fornæ ss , J . E . , S to u t , E. L .; S p read ing p o ly d is c s on complex m a n ifo ld s , Amer. J . Math. 99 (1977)» 933-960.

[F-S 1 ] F ornæ ss , J . E . , S ten s^n es (H e n rik se n ) , B . ; C h a ra c te r iz a t io n

o f g lo b a l peak s e ts f o r A°°(D) , Math. Ann. 259 (19Ö2), 125- 130 .

[F-Z 1 ] F o rnæ ss , J . E . , Zame, W. ( to a p p e a r) .

[G-L 1] G ra u e r t, H ., L ie b , I . ; Das R am irezsche I n te g r a l und d ie Lösung TTf = a in B e re ic h d e r b esch rän k te n Formen, R ice U n iv e rs i ty S tu d ie s 56 (1 970 ), 26-50.

[G-R 1] G ra u e r t, H. , Reramert, R. *, K o n v ex itä t in d e r kompleen A n a ly s is N ich t-ho lom orph-konvexe H olom orph iegeb ie te und Anwendungen a u f d ie A b b ild u n g s th e o r ie . Comm. Math. H e lv e t . 31 (1 9 5 6 ), 152-183.

[HEN l ] H enkin, G .; An a n a ly t i c po lyhed ron i s n o t h o lo m o rp h ica lly e q u iv a le n t to a s t r i c t l y pseudoconvex dom ain, D okl. A k ad ., Nauk. SSSR 210 (1 9 7 3 ), 1026-1029 - S o v ie t M ath. D okl. I k (1 9 7 3 ), 858- 862.

[HEN 2] H enkin, G. M .; I n te g r a l r e p r e s e n ta t io n s o f f u n c tio n s h o lo - m orphic in s t r i c t l y pseudoconvex domains and a p p l ic a t io n s to th e IT-problem, Math. Sb. 82 (1 9 7 0 ), 300-308, Mat USSR S b ll (1 9 7 0 ), 273-281.

[HEN 3] H enkin, G. M .; C o n s tru c tio n o f f u n c tio n s o f th e N evanlinna c la s s w ith p r e s c r ib e d ze ro s in s t r i c t l y pseudoconvex dom ains, D okl. Akad. Nauk. SSSR 22k (1 9 7 5 ), 3-13 .

[HO-W 1] Hörm ander, L . , Wermer, J . ; U niform ap p ro x im atio n on compact

s e t s in (Cn , Math. Scand. 23 (1 9 6 8 ), 5-21.

[H-R 1] H enkin, G. M. , Romanov, A .; Exact H older e s t im a te s o f s o lu t io n s o f th e 9* e q u a t io n s , l z v e s t i j a Akad. SSR, S e r .Mat. 35 (1 9 7 1 ), 1171-1183, Math. USSR Sb S (1 9 7 1 ), 1180-1192.

[H-S 1] Hakim, M ., S ibony , N . ; Q uelques c o n d itio n s pour £ r e x is te n c e de fo n c tio n s p ic s dans des dom aines pseudoc o n v e x es , Duke Math. J . kh (1977)» 399-^06.

[H-S 2] Hakim, M ., S ibony , N .; Ensem bles p ic s dans des domains s t r ic te m e n t p seudoconves, Duke Math. J . 1+5 (1 9 7 8 ), 60I - 617 .

[H-S 3] Hakim, M ., S ibony , N .; F o n c tio n s holom orphes bornée su r l a

b o u le u n i té de Œn , Xnv. Math. 67 (1 9 8 2 ), 213-222.

[H-S it] Hakim, M ., S ibony , N .; Ensem bles des zé ro s d fune fo n c tio n holomorphe bo rnée dans l a bou le u n i t é , Math. Ann. 260 (1 9 8 2 ),U69-I+7 I+.

2k5

[H-S 5]

[H-S 6]

[H-W 1]

[J-T 1]

[KER 1]

[K-N l j

[KOH 1]

[KOH 2]

[KOH 3]

[KOR 1]

[KRA 1]

[KUR 1]

[LEL 1]

[LEV 1]

Hakim, M ., S ib o n y , N .; V aleu rs an bord des m odules de fo n c tio n s holom orphes.P r e p u b l ic a t io n s , U n iv e r s i té de P a r is -S u d (1 9 8 3 ).

Hakim, M ., S ibony , N .; F o n c tio n s holom orphes bornée e t l im i te s t a n g e n t i e l l e s .P re p u b l ic a t io n s , U n iv e r s i té de P a r is -S u d (1982 ).

H arvey, F . R . , W e lls , R. 0 . ; Zero s e t s o f non n e g a tiv e s t r i c t l y p l u r i subharm onic f u n c t io n s , Math. Ann. 201 (1 9 7 3 ), 165- 170 .

Ja co b o w itz , H ., T rê v es , F . ; N o n re a liz a b le CR s t r u c t u r e s , In v e n tio n e s Math. 66 (1 9 8 2 ), 231-2^9.

Kerzman, N .; H older and e s t im a te s f o r s o lu t io n s o f"5u = f on s tro n g ly pseudoconvex dom ains, Commun. P ure A ppl. Math. 2h (1 9 7 1 ), 301-380.

Kohn, J . J . ; N Iren b e rg , L . ; A pseudoconvex domain n o t a d m ittin g a holom orphic su p p o rt f u n c t io n , Math. Ann. 201 (1 9 7 3 ), 265-268.

Kohn, J . J . ; G lobal r e g u la r i t y f o r 3 on w eakly pseudo­convex m a n ifo ld s , T ran s. Amer. Math. Soc. l 8 l (1 9 7 3 ), 273-292.

Kohn, J . J . ; S u b e l l i p t i c i t y o f th e 3-Neumann problem on pseudoconvex dom ains: S u f f ic i e n t c o n d i t io n s , A cta Math.1^2 (1 9 7 9 )» 79-122.

Kohn, J . J . ; Boundary b e h a v io r o f 3 on w eakly pseudoconvex m a n ifo ld s o f d im ension tw o , J . D i f f . Geom. 6 (1 9 7 2 ), 523-5^2.

K o ran y i, A .; Harmonic fu n c tio n s on H erm itian h y p e rb o lic sp a ce , T ran s. Amer. Math. Soc. 135 (1 9 6 9 ), 507-516.

K ra n tz , S. G .; F u n ctio n th e o ry o f s e v e ra l complex v a r i a b l e s , John W iley & Sons (1982 ).

K u ra n ish i, M .; S tro n g ly pseudoconvex CR s t r u c tu r e s over s m a l l - b a l l s , I . An a p r i o r i e s t im a te , Ann. o f Math. 115 (1 9 8 2 ), U51-500, I I . A r e g u la r i t y th eo rem , Ann. o f Math.I l 6 (1 9 8 2 ), 1-6U, I I I . An embedding theo rem , Ann. o f Math.116 (1 9 8 2 ), 2U9-330.

L elong , P . ; F o n c t io n e l le s a n a ly tiq u e s e t fo n c tio n s e n t i è r e s (n v a r i a b l e s ) , M ontréal Les P re s s e s de L ’U n iv e r s i té de M o n tréa l, 1968 (S ém ina ire de M athém atiques s u p é r ie u r e s , é té 1967, n° 28. ) .

L ev i, E. E . ; S tu d i i su i p u n ti s in g o la r i e s s e n z i a l i d e l l e fu n z io n i a n a l i t i c h e d i due o p iu v a r i a b i l i com plesse ,Ann. Math. P ura Appl. 17 (1 9 1 0 ), 61-87•

21*6[LEW 1]

[LOW l ]

[LOW 2]

[MA 1]

[MAR l ]

[MER 1]

[WIR l ]

[WOE 1]

[OKA 1]

[PFL 1]

[RAW 1]

[RAW 2]

[RIC 1]

[ROS 1]

[RUD 1]

[SIB 1]

[SIB 2]

Levy, H .; An example o f a smooth l i n e a r p a r t i a l d i f f e r e n t i a l e q u a tio n w ith o u t s o lu t io n , Ann. o f Math. 66 (1 9 5 7 ), 155-158.

Ltf$w, E . ; T o p p in te rp o las jo n s-m en g d er i ran d a t i l s t r e n g t pseudokonvekse om rader (Cand. r e a l t h e s i s . , U niv. o f O slo , 1979).

Lj6w, E. ; A c o n s tru c t io n o f in n e r f u n c tio n s on th e u n i t h a l l

in ffin , In v . Math. 67 (1 9 8 2 ), 223-229.

M arkoe, A .; Runge f a m i l ie s and in d u c tiv e l im i t s o f S te in sp a c e s , Ann. I n s t . F o u r ie r (G ren o b le ) , 27 (1 9 7 7 ), 117-127*

M a rg u lis , G. A .; A ll-U n ion C onference on th e Theory o f F u n c tio n s , H ar 'k o v , 1971» p . 137.

M ergelyan, S. N .; Uniform ap p ro x im atio n to fu n c tio n s o f a complex v a r ia b le , Amer. Math. S oc. T ra n s i . 101 (195*0.

N iren b e rg , L . ; On a problem o f Hans Lewy, U speki Math. Nauk. 292 (197*0 » 2*+l-251.

N o e ll , A .; P ro p e r t ie s o f peak s e t s in w eakly pseudoconvex

b o u n d arie s in Π, Ph.D . T h e s is , P r in c e to n U n iv e r s i ty , 1983.

Oka, K .; Domaines f i n i s sans p o in t c r i t i q u e i n t é r i e u r , Jap an ese J . Math. 23 (1 9 5 3 ), 97-155.

P f lu g , R. P . ; Q u a d ra t in te g ra b le holomorphe F unk tionen und d ie Serre-V erm utung , Math. Ann. 216 (1 9 7 6 ), 285-288.

Range, R. M .; Holomorphic ap p ro x im atio n n e a r s t r i c t l y pseudo­convex boundary p o in ts , Math. Ann. 201 (1 9 7 3 ), 9 -17 .

— 2R an g e,,M .; H older e s t im a te s fo r 3 on convex domains in Œ w ith r e a l a n a ly t i c boundary , P ro c . Symp. Pure A ppl. Math.39 (1 9 7 7 ), 31-33.

R ich b e rg , R . ; S te t ig e s tre n g pseudokonvexe F u n k tio n en ,Math. Ann. 175 (1 9 6 8 ), 251-286.

R o s s i, H .; The lo c a l maximum modulus p r i n c i p l e , Ann. Math.72 (I960), 1-11.R udin , W .; F u n c tio n th e o ry in th e u n i t b a l l o f ffin ,New York: S p r in g e r , 1980.

S ibony , N .; P rolongem ent des fo n c tio n s holom orphes b o rnées e t m é triq u e de C a ra th e o d o ry , In v . Math. 29 (1 9 7 5 ), 205-230.

S ibony , N .; Un example de domain pseudoconvexe r é g u l i e r ou 1 ' e q u a tio n 3u = f n 'adm et pas de s o lu t io n bornée pour f b o rn é e , In v . Math. 62 (1 9 8 0 ), 235-2^2.

2hj

[SIB 3] S ibony , N .; V aleu rs au bord de fo n c tio n s holom orphes e t ensem bles polynom ialem ent convexes, L ec tu re N otes in Math. 5T8î 300-313- B e rlin -H eide lberg -N ew York: S p r in g e r , 1977*

[SIL 1] S i lv a , A .; Rungescher S a tz and a c o n d it io n fo r S te in n e s fo r th e l im i t o f an in c re a s in g sequence o f S te in sp a c e s , Ann. I n s t . F o u r ie r (G renoble) 28 (1978) , 187- 200.

[SKO 1] Skoda, H .; V aleu rs au bond pour l ’o p e ra tu e r d e t zé ro s des fo n c tio n s de l a c l a s s e de N evan linna . B u ll . Cos. Math. F rance lOh (1 9 7 6 ), 225-299-

[ST 1] S te in , E. M .; Boundary b eh a v io r o f holom orphic fu n c tio n s o f s e v e r a l complex v a r i a b l e s , P r in c e to n U n iv e rs i ty P r e s s , P r in c e to n , NJ, 1972.

[ST 2] S te in , E. M. -, S in g u la r i n te g r a l s and d i f f e r e n t i a b i l i t y p r o p e r t ie s o f f u n c t io n s , P r in c e to n U n iv e r s i ty P re s s , P r in c e to n , 1970.

[STE 1] S ten s^n es (H e n rik se n ) , B.*, A peak s e t o f Haus dor f f-d im e n s io n 2 n - l f o r th e a lg e b ra A(D) in th e boundary o f a domain D

w ith C -boundary in Œn , Math. Ann. 259 (1 9 8 2 ), 271-277*

[TUM 1] Tumanov, A. E . ; A peak s e t f o r th e d is c a lg e b ra o f m e tr ic dim ension 2 .5 in th e th re e -d im e n s io n a l u n i t sp h e re , Math. USSR-Izv. 11 (1 9 7 7 ), 353-369.

[VOR 1] Vormoor, N .; T opo log ische F o rts e tz u n g b iho lom orpher F unk tionen a u f dem Rande b e i b esch rän k te n s tr e n g pseudokonvexen G eb ie ten

in ffin m it C“ -R and, Math. Ann. 20U (1 9 7 3 ), 231-261.

[WER l ] Wermer, J . ; Banach a lg e b ra s and s e v e r a l complex v a r ia b le s , Markham, 1971-

[WER 2] Wermer, J . ; An example co n cern in g po lynom ial c o n v e x ity , Math. Ann. 139 (1 9 5 9 ), 1^7-150; Addendum: Math. Ann. ih o (1 9 5 9 ), 322-323.

[WER 3] Wermer, J . ; On a domain e q u iv a le n t t o th e b id i s c , Math. Ann.2U8 (1980 ), 193-19^-

ISBN 978-0-8218-4422-9

7 8 0 8 2 1 8 4 4 2 2 9CHEL/363.H

9

9780821844229

About this book

C ounterexam ples are rem arkably effective for u n d erstand ing th e m eaning, and th e lim itations, o f m athem atical results. Fornæss and Stensones look a t som e o f th e m ajor ideas o f several com plex variables by considering counterexam ples to w hat m ig h t seem like reasonable variations o r g en ­eralizations. T he first p a r t o f th e b ook reviews som e o f th e basics o f the theory , in a self-contained in tro d u c tio n to several com plex variables. T he counterexam ples cover a variety o f im p o rtan t topics: th e Levi p rob lem , p lurisubharm onic functions, M onge-A m père equations, C R geom etry, function theory , and th e d equation .

T he book w ou ld be an excellent supp lem ent to a g raduate course on sev­eral com plex variables.

9780821844229