Perspectives on Riemannian geometry 9780821838525, 0821838520

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Perspectives in Riemannia n Geometiy

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https://doi.org/10.1090/crmp/040

Volume 4 0

CR M R PROCEEDING S & M LECTUR E NOTE S r

Centre d e Recherche s Mathematique s Universite d e Montrea l

Perspectives in Riemannia n Geometry Vestislav Apostolo v Andrew Dance r Nigel Hitehi n McKenzie Wan g Editors The Centr e d e Recherche s Mathematique s (CRM ) of th e Universite d e Montrea l wa s create d i n 1 96 8 t o promot e research i n pur e an d applie d mathematic s an d relate d disciplines. Among it s activitie s ar e specia l them e years , summer schools , workshops , postdoctora l programs , an d publishing. Th e CR M is supporte d b y th e Universit e d e Montreal, th e Provinc e o f Quebec (FCAR) , and th e Natural Science s an d Engineerin g Researc h Counci l o f Canada. I t is affiliate d wit h th e Institu t de s Science s Mathematiques (ISM ) of Montreal, whos e constituen t members ar e Concordi a University , McGil l University , th e Universite d e Montreal , th e Universit e d u Quebe c a Montreal, an d th e Ecol e Pol y technique. Th e CR M may b e reached o n th e We b a t www.crm.umontreal.ca .

American Mathematical Societ y Providence, Rhode Island US A

T h e p r o d u c t i o n o f thi s volum e wa s s u p p o r t e d i n p a r t b y t h e Fond s p o u r l a F o r m a t i o n de C h e r c h e u r s e t l'Aid e a l a Recherch e (Fond s F C A R ) a n d t h e N a t u r a l Science s a n d Engineering Researc h Counci l o f C a n a d a ( N S E R C ) .

2000 Mathematics Subject

Classification. P r i m a r y 53Cxx , 53Bxx , 53C26 ; Secondary 53Dxx , 32Qxx .

Library o f Congres s C a t a l o g i n g - i n - P u b l i c a t i o n D a t a Perspectives i n Riemannia n geometr y / Vestisla v Apostolov.. . [e t al.] , editors . p. cm . — (CR M proceeding s & lectur e notes , ISS N 1 065-858 0 ; v. 40 ) Includes bibliographica l references . ISBN 0-821 8-3852- 0 (alk . paper ) 1. Geometry, Riemannian . 2 . Riemannian manifolds . 3 . Geometry, Algebraic . I . Apostolov , Vestislav, 1 971 - II . Title . III . Series . QA649 .P384 200 6 516.3'73—dc22 20060428

8

C o p y i n g an d reprinting . Materia l i n thi s boo k ma y b e reproduce d b y an y mean s fo r edu cational an d scientifi c purpose s withou t fe e o r permissio n wit h th e exceptio n o f reproductio n b y services tha t collec t fee s fo r deliver y o f document s an d provide d tha t th e customar y acknowledg ment o f th e sourc e i s given . Thi s consen t doe s no t exten d t o othe r kind s o f copyin g fo r genera l distribution, fo r advertisin g o r promotiona l purposes , o r fo r resale . Request s fo r permissio n fo r commercial us e o f materia l shoul d b e addresse d t o th e Acquisition s Department , America n Math ematical Society , 20 1 Charle s Street , Providence , Rhod e Islan d 02904-2294 , USA . Request s ca n also b e mad e b y e-mai l t o [email protected] . Excluded from thes e provision s i s materia l i n article s fo r whic h th e autho r hold s copyright . I n such cases , request s fo r permissio n t o us e o r reprin t shoul d b e addresse d directl y t o th e author(s) . (Copyright ownershi p i s indicate d i n th e notic e i n th e lowe r right-han d corne r o f th e firs t pag e o f each article. ) © 200 6 b y th e America n Mathematica l Society . Al l right s reserved . The America n Mathematica l Societ y retain s al l right s except thos e grante d t o th e Unite d State s Government . Copyright o f individua l article s ma y rever t t o th e publi c domai n 2 8 year s after publication . Contac t th e AM S fo r copyrigh t statu s o f individua l articles . Printed i n th e Unite d State s o f America . @ Th e pape r use d i n thi s boo k i s acid-fre e an d fall s withi n th e guideline s established t o ensur e permanenc e an d durability . This volum e wa s submitte d t o th e America n Mathematica l Societ y in camer a read y for m b y th e Centr e d e Recherche s Mathematiques . Visit th e AM S hom e pag e a t http://www.ams.org / 10 9 8 7 6 5 4 3 21 1 1 1 0 09 08 0 7 0 6

Contents Preface vi

i

List o f Participant s i

x

Topics i n Conformall y Compac t Einstei n Metric s Michael T. Anderson Cauchy-Riemann 3-Manifold s an d Einstei n Filling s Olivier Biquard 2

7

Sasakian Geometr y an d Einstei n Metric s o n Sphere s Charles P. Boyer an d Krzysztof Galicki 4

7

Second Orde r Familie s o f Specia l Lagrangia n 3-Fold s Robert L. Bryant 6

3

Einstein Equations , Superpotential s an d Conve x Polytope s Andrew Dancer an d McKenzie Y. Wang 9

9

The Bochner-Fla t Geometr y o f Weighte d Projectiv e Space s 1 Liana David an d Paul Gauduchon 0

9

Aspects o f Compariso n Geometr y Karsten Grove 5

7

Low-Dimensional Geometry— A Variationa l Approac h Nigel Hitchin 8

3

Twistors, Holomorphi c Disks , an d Rieman n Surface s wit h Boundar y Claude LeBrun 20

9

Combinatorics o f th e Space s o f Riemannia n Structure s an d Logi c Phenomena o f Euclidea n Quantu m Gravit y Alexander Nabutovsky 22

3

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Preface The paper s i n thi s volum e ar e writte n b y som e o f the participant s o f the Shor t Program o n Riemannia n Geometr y hel d i n June-Jul y 200 4 i n Montreal . Thi s event wa s a par t o f th e genera l scientifi c progra m o f th e Centr e d e Recherche s Mathematiques i n Montrea l durin g th e summe r o f 2004 . Th e three-wee k progra m opened wit h a week of introductory shor t courses , followed b y a two-week workshop , and thu s gathere d clos e t o on e hundre d participants . Here is a brief descriptio n o f the volum e contents . Highlightin g th e mai n topic s of th e Proceeding s ar e th e thre e paper s whic h ar e base d o n th e lecture s give n during th e introductor y shor t course s b y Professor s M . Anderson , K . Grov e an d N. J . Hitchin . The firs t i s Michae l Anderson' s comprehensiv e surve y o f recen t result s o n th e existence o f Einstei n metric s o n ope n manifold s wit h a certai n structur e a t infin ity. Asymptotically , th e simples t structure s ar e thos e o f constan t curvature , an d this pape r cover s i n grea t detai l globa l aspect s o f th e hyperboli c (o r conformally compact) case , includin g application s o f thi s theor y t o th e AdS/CF T correspon dence i n physics . Anothe r pape r complementin g thi s topi c i s Biquard' s surve y of Einstei n metric s wit h asymptoti c structur e a t infinit y modele d o n th e comple x hyperbolic space , an d thei r applicatio n t o C R geometry . Karsten Grove' s lectur e note s surve y compariso n geometry . Th e reade r wil l find i n i t a n exhaustiv e accoun t o f th e subject , mainl y i n th e framewor k o f lowe r bounds o f sectiona l curvatures , startin g fro m th e basi c compariso n theorem s o f Myers, Synge , Rauch , Alexandrov , Toponogo v an d Bishop , the n discussin g impor tant development s i n conjunctio n wit h Mors e theor y an d convexity , th e Gromov Hausdorff topolog y o n space s o f Riemannia n manifolds , th e geometr y o f singula r spaces, an d som e recent result s i n th e presenc e o f symmetries. Riemannia n metric s with bounde d geometr y pla y a central rol e in A . Nabutovsky's pape r whic h survey s the relationshi p betwee n th e spac e o f Riemannia n structure s o n a close d manifold , computability theor y an d algorithmi c informatio n theory . What i s specia l abou t 6 , 7 an d 8 dimensions ? Wh y d o w e stud y Calabi-Ya u threefolds, G2 an d Spin(7 ) manifolds ? Thes e ar e th e question s addresse d i n Nige l HitchinTs lectur e note s (writte n wit h th e hel p o f Marc o Gualtieri ) o n thes e specia l geometries o f grea t interes t i n strin g theory . Base d o n th e fundamenta l principl e of lookin g a t th e geometr y o f ope n orbit s o f Li e groups , the y presen t a n origina l entrance int o the exciting world of generalized geometry . Anothe r tw o papers i n th e volume ar e concerned wit h specia l geometric structure s o n differentiabl e manifolds . R. Bryant' s pape r provide s a systematic stud y o f a class o f special Lagrangia n sub manifolds i n C n . I n their paper , A . Dancer an d M . Wang explain thei r Hamiltonia n

Vll

viii P R E F A C

E

approach t o cohomogeneity-on e Einstei n metrics , whic h lead s to ne w example s an d in specia l case s link s t o integrabl e systems . Three o f th e paper s ar e devote d t o th e interactio n betwee n Riemannia n an d complex geometry. Th e paper b y C. Boyer an d K. Galick i is a comprehensive surve y of new Sasaki-Einstein metric s build ou t o f Kahler-Einstein orbifolds . L . David an d P. Gauduehon' s pape r provide s a thoroug h stud y o f Boclmer-na t Kahle r orbifold s from th e vie w point o f the C R geometr y o f the standar d sphere , building o n a n ide a of S. Webster. C . LeBrun's paper studie s stability o f complex curves-with-boundar y in th e twisto r space s appearin g i n th e spectacula r ne w approac h t o Zol l manifold s and split-signatur e pseudo-Riemannia n manifold s tha t h e an d L . J . Maso n hav e recently found . We hop e tha t th e contribution s presente d i n thi s volum e reflec t t o a certai n extent th e variet y o f topic s presente d durin g th e workshop . W e ar e gratefu l t o th e Centre d e Recherche s Mathematique s fo r thei r kin d an d efficien t assistanc e durin g the worksho p an d i n th e preparatio n o f thi s volum e an d t o Arthu r Greenspoo n fo r a carefu l readin g o f th e manuscripts . Vestislav Apostolo v Andre w Dance r Nige l Hitchi n McKenzie Wan g Montreal, 200 6

List o f Participant s Alexandrov, Bogda n Universitat Greifswal d [email protected]

Bulawa, Andre w SUNY—Stony Broo k abulawa@math. sunysb. edu

Anderson, Michae l T . SUNY—Stony Broo k andersonOmath.sunysb.edu

Butscher, Adria n University o f Toront o butscherOutsc.utoronto.ca Calderbank, Davi d J . M . University o f Edinburg h davidm j c@maths . ed.ac.uk

Apostolov, Vestisla v Univeisite d u Quebe c a Montrea l [email protected]

Chan, Ya t Min g Oxford Universit y

Armstrong, Stuar t Oxford Universit y stuart.armstrongOstx.ox.ac.uk

[email protected]

Chen, Xiuxion g University o f Wisconsin-Madiso n

Bergin, Ciar a National Universit y o f Irelan d Maynooth

[email protected]

Clarke, Andre w SUNY—Stony Broo k [email protected]

[email protected]

Bielawski, Roge r University o f Glasgo w [email protected]

Craig, Gordo n McGill Universit y [email protected]

Biquard, Olivie r Universite Loui s Pasteu r e t CNR S [email protected]

Dancer, Andre w Oxford Universit y [email protected]

Bohm, Christop h University o f Mlinste r [email protected]

Davaux, Helen e U M P A — E N S Lyo n [email protected]

Boyer, Charle s P . University o f Ne w Mexic o [email protected]

Davidov, J o h a n n Bulagarian Academ y o f Science s [email protected]

Bryant, Rober t Duke Universit y [email protected]

De l a Cru z d e Oha , Artori x Concordia Universit y [email protected] IX

PARTICIPANTS

Draghici, Ted i Florida Internationa l Universit y [email protected]

Duchemin, Davi d Universite d u Quebe c Montrea l

Gyrya, Pave l Cornell Universit y pgyryaOmath.Cornell.edu

[email protected]

Herzlich, Mar c Universite Montpellie r I I [email protected]

Dunajski, Macie j University o f Cambridg e m.dunaj skiOdamtp.cam.ac.u k

Hitchin, Nige l J . Oxford Universit y hitchinOmaths. ox. ac. uk

D u t t a , Satyak i State Universit y o f Ne w York , StonyBrook

Hurtubise, Jacque s McGill Universit y [email protected]

sunnyOmath.sunysb.edu

Fino, Anna-Mari a Universita' d i Torin o [email protected] Flores, Jos e L . State Universit y o f Ne w York , Ston y Brook [email protected]

Fraser, Ailan a University o f Britis h Columbi a [email protected] Gauduchon, Pau l Ecole Polytechniqu e Paul.GauduchonOmath.polytechnique.fr

Gegenberg, Jac k D . University o f Ne w Brunswic k [email protected] Grantcharov, Gue o Florida Internationa l Universit y [email protected]

Ionel, Mariant y McMaster Universit y [email protected] Ivanov, Stefa n University o f Sofi a "St . Klimen t Ohridskr [email protected]

Joergensen, Helg e Rii s University o f Souther n Denmar k helgej Oimada.sdu.d k K a m r a n , Nik y McGill Universit y nkamranOmath. m c g i l l. c a Karigiannis, Spir o McMaster Universit y [email protected] Kim, Chang-Wa n Nankai Universit y s.cwkimOhanmail. ne t

Grove, Karste n University o f Marylan d [email protected]

Kraus, Margarit a Universitt Regensbur g [email protected] -regensburg.de

Gualtieri, Marc o T h e Field s Institut e mgualtieOfields.utoronto.ca

LeBrun, Claud e SUNY—Stony Broo k claudeOmath.sunysb.edu

Guzhvina, Galin a Universitaet Muenste r guzhvinaOmath.uni-muenster.de

Lohkamp, Joachi m Universitat Augsbur g j oachim.lohkampOmath.uni-augsburg.de

PARTICIPANTS

Maschler, Gideo n University of 1 Toronto [email protected]

Reiris, Marti n SUNY—Stony Broo k [email protected]

Matessi, Dieg o Imperial Colleg e

Rollin, Yan n MIT

[email protected]

rollinOmath.mit.edu

McKeown, Jess e McGill Universit y [email protected] Min-Oo, Maun g McMaster Universit y minooOmcmaster.ca Moraru, Da n SUNY—Stony Broo k moraruOmath.sunysb.edu Moraru, R u x a n d r a Fields Institute/Universit y o f Toront o moraruOmath.toronto.edu Mushkarov, Ole g Bulgarian Academ y o f Science s [email protected] Nabutovsky, Alexande r The Pennsylvani a Stat e Universit y nabutovOmath.psu.edu

Page, Do n N . University o f Albert a [email protected] Papadopoulos, Georg e King's Colleg e Londo n gpapasOmth.kcl.ac.uk Parton, Maurizi o Universit d i Chieti-Pescar a [email protected] Pedersen, Henri k Syddansk Universite t [email protected]

Rot man, Regin a Pennsylvania Stat e Universit y [email protected] Ruan, Wei-Don g University o f Illinois , Chicag o ruanOmath.uic.edu Schuller, Frederi c P Perimeter Institut e fo r Theoretica l Physics [email protected] Shankar, Krislma n University o f Oklahom a shankar@math. ou. edu Suzuki, Cristin a City Universit y o f Ne w Yor k [email protected]

The, Denni s McGill Universit y dtheOmath.mcgill.ca Tonnesen-Friedman, Christin a W . Union Colleg e tonnesecOunion.edu Van Coevering , Crai g SUNY—Stony Broo k craigOmath.sunysb.edu Vassilev, Dimite r University o f Arkansa s [email protected]

Petursson, Ing i Or u Universite d e Grenobl e ingiornpeturssonOhotmail.com

Vetter, Phili p Duke Universit y [email protected]

Rasdeaconu, Rare s SUNY—Stony Broo k raresOmath. sunysb. edu

Villumsen, Marti n University o f Souther n Denmar k [email protected]

xii P A R T I C I P A N T

Wang, McKenzi e Y . McMaster Universit y wangOmcmaster.ca Weber, Bria n University o f W'isconsin-Madiso n [email protected]

West, Simo n University o f Cambridg e [email protected] Wilking, Burkhar d Mathematisches Institu t de r Un i Miinster wilkingOmath.uni-muenster.de

Wong, Jerem y University o f Illinoi s j awonglOmath.uiuc.edu Xu, Min g SUNY—Stony Broo k mxuOmath.sunysb.edu

Ziller, Wolfgan g University o f Pennsylvani a [email protected]

S

https://doi.org/10.1090/crmp/040/01 Centre d e Recherche s Mathematique s CRM Proceeding s an d Lectur e Note s Volume 40 : 200 6

Topics i n Conformall y Compac t Einstei n Metric s Michael T. Anderso n

1. I n t r o d u c t i o n Conformal compactincations of Einstein metrics were introduced by Penrose [38], as a mean s t o stud y th e behaviou r o f gravitational field s a t infinity , i.e . th e asymp totic behaviou r o f solution s t o th e vacuu m Einstei n equation s a t nul l infinity . Thi s has remaine d a ver y activ e are a o f research , cf . [1 9,27 ] fo r recen t surveys . I n th e context o f Riemannian metrics , th e moder n stud y o f conformally compac t Einstei n metrics bega n wit h th e wor k o f FefTerman-Graha m [26] , i n connectio n wit h thei r study o f conforma l invariant s o f Riemannia n metrics . Recen t mathematica l wor k in thi s are a ha s bee n significantl y influence d b y th e AdS/CF T (o r gravity-gauge ) correspondence i n strin g theory , introduce d b y Maldacen a [36] . W e will only com ment briefl y her e on aspect s o f the AdS/CFT correspondence , an d refe r t o [2,7,42 ] for genera l surveys . In thi s paper , w e discus s recen t mathematica l progres s i n thi s area , focusin g mainly o n globa l aspect s o f conformall y compac t Einstei n metric s an d th e globa l existence questio n fo r th e Diriclile t problem . On e reaso n fo r thi s i s tha t i t no w appears tha t th e beginning s o f a genera l existenc e theor y fo r suc h metric s ma y b e emerging, a t least in dimension 4. O f course to date there is no general theory for th e existence o f complet e Einstei n metric s o n manifolds , wit h tw o notabl e exceptions ; the existenc e theor y fo r Kahler-Einstei n metric s du e t o Calabi , Yau , Aubi n an d others, an d th e existenc e theor y i n dimensio n 3 , du e t o Perelman , Hamilto n an d Thurston. I n contras t t o the situation fo r compac t 4-manifolds , a n existence theor y for conformall y compac t Einstei n metric s ma y no t b e tha t fa r beyon d th e curren t horizon. We discus s numerou s ope n problem s o n thi s topic ; som e ne w result s ar e als o presented, cf . i n particular Theore m 3. 4 an d th e discussio n an d result s i n Section s 4 and 5 . 2000 Mathematics Subject Classification. Primary : 53C25 : Secondary : 58J60 . Partially supporte d b y NS F Gran t DM S 0305865 . I woul d lik e t o than k Davi d Calderbank , To m Farrell , Lowel l Jones , Claud e LeBrun , Rafe Mazze o an d Michae l Singe r fo r discussion s relate d t o variou s issue s i n th e paper . Thank s also t o Vestisla v Apostolo v an d colleague s fo r organizin g a n interestin g worksho p a t th e CRM , Montreal i n July , 04 . This i s th e fina l for m o f th e paper . ©2006 America n Mathematica l Societ y 1

2 MICHAE

L T . ANDERSO N

In brief , th e content s o f the pape r ar e a s follows. Th e groundwor k i s laid i n §2, where w e discus s th e modul i spac e o f conformall y compac t Einstei n metric s an d the boundar y ma p t o the spac e of conformal infinities . Th e genera l situatio n i s also illustrated b y th e discussio n o f a simple bu t importan t clas s of examples, th e stati c AdS blac k hol e metrics . Sectio n 3 deals wit h th e genera l asymptoti c behaviou r o f the metric s nea r conforma l infinity , an d th e contro l o f the asymptoti c behaviou r b y the metri c a t infinity . I t will b e see n that , a t leas t i n eve n dimensions , thi s issu e is no w quit e wel l understood . The n i n Sectio n 4 w e tur n t o th e analysi s o f th e behaviour o f th e metric s o n compac t regions , awa y fro m infinity , mostl y i n dimen sion 4 wher e th e possibl e degeneration s ca n b e describe d i n term s o f orbifol d an d cusp degenerations . I n Sectio n 5 , w e conclud e wit h a discussio n o f th e possibilit y of actuall y finding example s wher e orbifol d o r cus p degeneration s occur . 2. Conformall y Compac t Einstei n Metric s Let M b e the interior of a compact (n-hl)-dimensiona l manifol d M wit h bound ary dM . A complet e Riemannia n metri c g o n M i s C nha conformall y compac t i f there i s a definin g functio n p o n M suc h tha t th e conformall y equivalen t metri c = P 29

(2-1) 9

extends t o a C m,a metri c o n th e compactificatio n M. Her e p i s a smooth , non negative functio n o n M wit h p - 1 (0) = dM an d dp ^ 0 on dM. Th e induce d metri c 7 = g\ dM i s the boundar y metri c associate d t o th e compactificatio n g. Sinc e ther e are man y possibl e definin g functions , ther e ar e man y conforma l compactification s of a give n metri c g , an d s o onl y th e conforma l clas s [7 ] o f 7 o n dM, calle d con formal infinity , i s uniquel y determine d b y (M,g). Clearl y an y manifol d M carrie s many conformall y compac t metric s but w e are mainly concerne d her e with Einstei n metrics g, normalize d s o tha t (2.2) Ric

p

= -ng.

A simpl e computatio n fo r conforma l change s o f metri c show s tha t i f g i s a t leas t C2 conformall y compact , the n th e sectiona l curvatur e K g o f g satisfie s (2.3) \K

g

+ l\=0(p

2

).

Thus, th e loca l geometr y o f (M , g) approache s tha t o f hyperboli c space , an d con formally compac t Einstei n metric s ar e frequentl y calle d asymptoticall y hyperboli c (AH), o r als o Poincare-Einstein . Al l thes e notion s wil l b e use d her e interchange ably. Th e natura l "threshol d level " fo r smoothnes s i s C 2 , sinc e eve n i f g i s C m ' Q conformally compact , m > 2 , (2.3 ) canno t b e improve d t o \K g + 1 | = o(p 2) i n general. Mathematically, a n obviousl y basi c issu e i n thi s are a i s th e Dirichle t proble m for conformall y compac t Einstei n metrics : give n th e topologica l dat a (M,dM), and a conforma l clas s [7 ] on dM , doe s ther e exis t a conformall y compac t Einstei n metric g o n M , wit h conforma l infinit y [7] ? I n on e for m o r another , thi s questio n is th e basi c leitmoti v throughou t thi s paper . A s will b e see n later , uniquenes s o f solutions wit h a give n conforma l infinit y fail s i n general . To set th e stage, we first examine the structure o f the moduli space of PoincareEinstein metric s on a given (n+1 )-manifol d M. Le t E m,a b e the space of PoincareEinstein metric s o n M whic h admi t a C 2 conforma l compactificatio n g a s i n (2.1 ) , with C m,a boundar y metri c 7 o n dM. Her e 0 < a < l , r a > 2 , an d w e allo w

TOPICS I N CONFORMALL Y COMPAC T EINSTEI N METRIC S 3

77i = o c or m = LU, th e latte r correspondin g t o real-analytic . Th e topolog y o n E rn'a is given by a weighted Holder norm, cf. (2.7 ) below; briefly, th e topology is somewhat stronger tha n th e C 2 topolog y on metrics o n M unde r a conformal compactiflcatio n g a s i n (2.1 ) . Le t £ m>a = E m>aJT>iSY+1 >a{M), wher e Diff™ +1 'Q(M) i s th e grou p of C m + l a diffeomorphism s o f M inducin g th e identit y o n dM, actin g o n E i n th e usual way by pullback. Next , le t Met m,Q ( C w ' a , II[0

] = [7] ,

takes a conformally compac t Einstei n metric g on M t o its conformal infinit y o n dM. Thus, globa l existenc e fo r th e Dirichle t proble m i s equivalent t o th e surjectivit y o f II, whil e uniquenes s i s equivalent t o th e injectivit y o f II . The followin g resul t describe s th e genera l structur e o f £ an d th e ma p II , build ing o n previou s wor k o f Graham-Le e [29 ] an d Biquar d [1 5] . Theorem 2. 1 (Manifol d structur e [5,6]) . Let M be a compact, oriented (n + 1 )-manifold with boundary dM with n > 3 . If £ m,a is non-empty, then £' m-a is a smooth infinite dimensional manifold. Further, the boundary map TT. cm,a r>m,ot

is a C°° smooth Fredholm map of index 0 . When rn < oc , £ m,a ha s th e structur e o f a Banac h manifold , whil e £°° ha s the structur e o f a Freche t manifold . Fo r n = 3 , one expect s tha t Theore m 2. 1 als o holds fo r rn > 2 , bu t thi s i s a n ope n problem . Theorem 2. 1 shows that th e moduli space £ ha s a very satisfactory globa l struc ture. I n particula r i f M carrie s som e Poincare—Einstei n metric , the n i t als o carrie s a larg e se t o f them , mappin g unde r I I t o a t leas t a variet y o f finite codimensio n i n C. Recal l tha t a metri c g £ £ i s a regula r poin t o f I I i f D gIl i s surjective . Sinc e II i s Fredhol m o f inde x 0 , D gIl i s injectiv e a t regula r points ; hence , b y th e invers e function theorem , I I i s a loca l diffeomorphis m i n a neighbourhoo d o f eac h regula r point. Remark 2.2 . Not e tha t Theore m 2. 1 doe s no t hold , a s stated , whe n n = 1 , i.e. i n dimensio n 2 . I n thi s case , th e spac e £ a s define d abov e i s infinit e dimen sional, bu t i t become s finite dimensiona l whe n on e divide s ou t b y th e large r grou p of diffeomorphism s isotopi c t o th e identit y o n M. Thi s spac e o f conformall y com pact (geometricall y finite) hyperboli c metric s o n a surfac e E i s a smooth , finite dimensional manifold , bu t th e conforma l infinit y i s unique . Th e boundar y 1 ont o a poin t 7 G C ? At thi s point , i t i s usefu l t o illustrat e th e discussio n o n th e basi s o f som e concrete examples . Example 2. 3 (Stati c Ad S blac k hol e metrics) . Le t iV n _ 1 b e an y close d (n — l)-dimensiona l manifold , whic h carrie s a n Einstei n metri c QN satisfyin g (2.10) Ric

5iV

=k(n-2)g

N,

where k = +1 , 0 o r — 1. W e assum e n > 3 . Conside r th e metri c g m o n l defined b y (2.11) g

m

= V' 1 dr

2

+ Vd0 2+ r

2

2

xiV

gN,

where (2.12) V(r)

= k+r

2

-

. n~2' Here r G [r + ,oc), wher e r + i s th e larges t roo t o f V , an d th e circula r paramete r 0 e [0,/3] , wher e (2.13) (3

= 4nr+/(nrl +

fe(n - 2)) .

This choice of/3 i s required s o that th e metric g rn i s smooth a t th e locus { r = r+} ; if j3 i s arbitrary, th e metri c wil l hav e con e singularities norma l t o th e locu s {r = r+}, although th e metri c i s otherwis e smooth . Sinc e thi s locu s i s th e fixed poin t se t o f the isometri c 5 1 actio n give n b y rotatio n i n 0, the se t {r = r + } i s diffeomorphic t o N an d i s totally geodesic ; i t correspond s t o th e horizo n o f the blac k hole . A simpl e computation show s that th e metric s g rn ar e Einstein , satisfyin g (2.2) . Further , i t i s easy t o se e these metric s ar e smoothl y conformall y compact ; th e conforma l infinit y of g ni i s given b y th e conforma l clas s o f th e produc t metri c o n 5 x(/3) x (A T, gjy). We discus s th e case s fc > 0 , k = 0. k < 0 in turn .

6

MICHAEL T . ANDERSO N

I. Suppose k = - f 1. As a function o f m £ (0 , oo), observe that / ? has a maximu m -I Icy

value o f PQ = 27r(( n — 2)/n) , and fo r ever y ?? i ^ rag, ther e ar e tw o value s m ± o f 777 givin g the same value of (3. Thus two metrics have the sam e conformal infinity ; i n particular, th e boundary ma p II in (2.4 ) i s not 1 — 1 along this curve. Thi s behaviou r is the first exampl e o f non-uniqueness fo r th e Dirichle t problem , an d wa s discovere d in [32 ] i n th e contex t o f th e Ad S Schwarzschil d metrics , wher e N = S 2{1 ). The ma p I I i s a fol d map , (o f th e for m x »— > x 2), i n a neighbourhoo d o f th e curve g m nea r m — mo. Th e loca l degre e at g m,0 i s 0 and I I i s not locall y surjective . In fact , Theore m 2. 4 belo w implie s tha t I I i s globall y no t surjective , i n tha t th e conformal clas s o f 5 2 (L) x (N.gw), fo r L > /3o, i s not i n Imll , cf . [5] . Observ e tha t this resul t require s globa l smoothnes s o f th e Einstei n metrics ; i f on e allow s con e singularities alon g th e horizo n A r = { r = r+} , i.e . i f j3 is allowe d t o b e arbitrary , then on e ca n g o past th e "wall " throug h S 1 (f3o) x (N,gi\r). Thi s clearl y illustrate s the globa l natur e o f th e globa l existenc e o r surjectivit y problem . II. Suppos e k = 0 . I n thi s cas e / ? = A-nr +/(nr\) i s a monoton e functio n o f r + or m , s o tha t i t assume s al l value s i n R + a s 77 7 £ (0 , oc). O n th e curv e g m , I I i s 1-1. However, th e actua l situatio n i s somewhat mor e subtl e tha n this . Suppos e fo r instance tha tT V = T n _ 1 , s o tha t M = R 2 x T n~l i s a soli d torus . Topologically , the dis c D 2 — R2 ca n b e attache d ont o any simpl e close d curv e i n th e boundar y dM = T n instea d o f just th e "trivial " S 1 facto r i n th e produc t T n = S 1 x T71'1. The resultin g manifold s ar e al l diffeomorphic . Thi s ca n als o b e don e metrically , preserving th e Einstei n condition , cf . [4] , an d lead s t o th e existenc e o f infinitel y many distinc t Einstei n metric s o n R 2 x T n _ 1 wit h th e sam e conforma l infinit y (T n , [go]), wher e go is an y fla t metric . Each of these metrics lies in a distinct componen t o f the modul i space £, s o tha t £ ha s infinitel y man y components . Thi s situatio n i s closely relate d t o th e mappin g class grou p SL(n,Z ) o f T n, i.e . th e grou p o f diffeomorphism s o f T n modul o thos e nomotopic t o th e identit y map , (so-calle d "larg e diffeomorphisms") . An y elemen t of SL(n , Z) extend s t o a diffeomorphis m o f th e soli d toru s R 2 x T n~l, an d whil e SL(n, Z) act s triviall y o n th e modul i spac e o f flat metric s o n T n , th e actio n o n £ i s highly non-trivial, givin g rise to the distinc t component s o f £. Simila r construction s can obviousl y b e carrie d ou t fo r manifold s N o f the for m N = T k x JV 7, fc > 1 , bu t it woul d b e interestin g t o investigat e th e mos t genera l versio n o f this phenomenon . III. Suppos e k = — 1. Again j3 i s a monoton e functio n o f m, an d s o takes o n al l values i n R + ; th e boundar y ma p I I i s 1 - 1 o n th e curv e g m. Furthe r aspect s o f thi s case ar e discusse d late r i n §5 .

These simpl e example s alread y sho w a numbe r o f subtl e feature s o f th e globa l behaviour o f the boundar y ma p II . Wit h regar d t o th e globa l surjectivit y question , the basi c propert y tha t on e need s t o mak e progres s i s t o understan d whethe r I I is a prope r map ; i f I I i s no t proper , i t i s importan t t o understan d exactl y wha t possible degeneration s o f Poincare-Einstein metric s ca n o r d o occur wit h controlle d conformal infinity . Recal l tha t I I i s prope r i f an d onl y i f I I - 1 (if) i s compac t i n £ , whenever K i s compac t i n C. If I I i s proper , the n on e ha s a well-define d Z2-value d degree , cf . [41 ] . I n fact , since the space s £ an d C can b e given a well-defined orientation , on e has a Z-value d

TOPICS I N CONFORMALL Y COMPAC T EINSTEI N METRIC S

7

degree, give n b y (2.14) d e

g

n = J2

inds

(-l)

S

Pi€n-i[7l

where [7 ] is a regular valu e o f II an d ind^ . i s the L 2 inde x o f D giU, i.e . th e numbe r of negativ e eigenvalue s o f th e operato r L i n (2.8 ) a t g^ acting o n L 2 , cf . [5] . O f course, i f degl l ^ 0 , the n I I i s surjective; (i f degl l = 0 , the n I I ma y o r ma y no t b e surjective). Not e tha t degl l i s define d o n eac h componen t £Q of £ an d ma y diffe r on differen t components . Let M = M 4 b e a 4-manifold , satisfyin g (2.15) H

2(dM,R)

- + H 2(M,R) - + 0 .

It i s proved i n [5 ] that I I i s then proper , whe n restricte d t o th e spac e £° o f Einstei n metrics whose conforrnal infinit y i s of non-negative scala r curvature . Mor e precisely, (2.16) n ° : £ ° - ^ e

0

is proper, wher e C° is the space of conforrnal classe s having a non-flat representativ e of non-negativ e scala r curvatur e an d £ ° = I I - 1 (C°); i n particula r ther e ar e onl y finitely man y component s o f £° ; compar e wit h Exampl e 2.3 , case I I above . In situation s wher e I I i s proper , th e degre e ca n b e calculate d i n a numbe r o f concrete situation s b y th e following : Theorem 2. 4 (Isometr y Extension , [5]) . Let ( M n + 1 , g) be a C 2 conforrnally compact Einstein metric with C°° boundary metric 7 . n > 3 . Then any connected group G of conforrnal isornetries of (dM, 7 ) extends to a group G of isornetries of(M.g). This resul t ha s a numbe r o f immediat e consequences . Fo r instance , i t implie s that th e Poincar e (o r hyperbolic ) metri c i s th e uniqu e C 2 conformall y compac t Einstein metri c o n a n (n - f l)-manifol d wit h conforrna l infinit y give n b y th e roun d metric on S n: se e also [1 2,39 ] fo r previou s specia l cases of this result. I n particular , one ha s o n (B 4, S 3 ), degn° = 1 , so that I I i s surjectiv e ont o C°. O n th e othe r hand , o n (M 4 , S 3 ), M 4 ^ JB 4 , degn0 = 0 , since I I canno t b e surjectiv e i n thi s case . Anothe r applicatio n o f Theore m 2. 4 i s the following : Corollary 2.5 . Let M be any compact (n + l)-manifold with boundary dM, n > 3 . and let M = M UQM M be the closed manifold obtained by doubling M across its boundary. Suppose dM admits an effective S 1 action, but M admits no effective S 1 action. Then I I = II(M ) is not surjective: in fact ImnnMet5i(0M) = 0 , where ]\let5i (dM) is the space ofS1 -invariant metrics on dM. The space Metsi (dM) is of infinite dimension and codimension in Me t (dM). As a simpl e example , le t M = E p x A 7, wher e E i s an y surfac e o f genu s g > 1 an d A T i s an y K(TT, 1 ) manifol d wit h n havin g n o center ; e.g . N ha s a metri c of non-positiv e curvature . Le t a b e a close d curv e i n T, g whic h disconnect s E ^

8

MICHAEL T . ANDERSO N

into tw o diffeomorphi c component s E + an d E ~ wit h commo n boundar y cr , and le t M — E + x N. B y [22] , M doe s no t admi t a n effectiv e S 1 action , bu t o f cours e dM = S 1 x N admit s suc h actions . Hence , Corollar y 2. 5 hold s fo r suc h M. On th e othe r hand , i f £ = S 2 i s o f genu s 0 , the n M = R 2 x N doe s admi t l S -invariant Poincare-Einstei n metrics , a s discusse d i n Exampl e 2.3 . A basic issu e is to exten d th e theor y describe d abov e beyond boundar y metric s of non-negativ e scala r curvatur e C°. Thi s wil l b e on e o f th e theme s discusse d below. W e begin with th e analysi s of Poincare-Einstein metric s nea r th e boundary , i.e. conforma l infinity . 3. Behaviou r Nea r th e Boundar y In this section, w e study th e behaviou r o f Poincare-Einstein metric s i n a neighbourhood o f conforma l infinit y (dM, 7) . For man y purposes , th e mos t natura l compactification s ar e thos e define d b y geodesic definin g functions . Thus , a compactificatio n g = p 2g a s i n (2.1 ) i s calle d geodesic i f p(x) = dist^(x,9Af) . Eac h choic e of boundary metri c 7 G [7] determine s a unique geodesic defining functio n p. Fo r a geodesic compactification, on e typicall y loses one derivative in the possible smoothness, bu t thi s w rill not b e of major concer n here, cf . als o [1 1 , App. B ] on restorin g los s o f derivatives . The Gaus s Lemm a give s th e splittin g = dp 2+gp, g

(3.1) 9

= p-

2

(dp2+gp),

where g p i s a curv e o f metric s o n dM. A simpl e an d natura l ide a t o examin e th e behaviour o f g near infinit y i s to expan d th e curv e o f metrics g p o n dM i n a Taylo r series i n p. Surprisingl y (a t first), thi s turn s ou t no t alway s t o b e possible , a s discovered i n [26] . I t turn s ou t tha t th e exac t for m o f th e expansio n depend s o n whether n i s odd o r even . I f n i s odd , i.e . M i s even-dimensional, the n (3.2) g

p

~ 0 (O) + p 2g{2) + • • • + P n"1 9(n-i) +

P n9(n) + p n + 1 #( n +i) + • • • .

This expansio n i s eve n i n power s o f p u p t o orde r n . Th e coefficient s g(2k)i 2fc < (n — 1), ar e locall y determine d vi a th e Einstei n equation s (2.2 ) b y th e boundar y metric 7 = g( 0). The y ar e explicitl y computabl e expression s i n th e curvatur e o f 7 and it s covarian t derivatives , althoug h thei r complexit y grow s rapidl y wit h A: . Th e term g^ i s transverse-traceless, i.e . (3.3) tr

7

g (n) = 0 , £

7#(n)

= 0,

but i s otherwise undetermine d b y 7 an d th e Einstei n equations ; i t depend s o n th e particular structur e o f the A H Einstein metri c (M , g) nea r infinity . I f n i s even, on e has (3-4) g p - g

{0)

+ p 2g{2) + • • • + p n " 2 S(n-2) + p n9(n) + p n lo g p H + p n + 1 P( n +i) + • • • •

Again (vi a th e Einstei n equations ) th e term s g^k) U P t o orde r n — 2 ar e explicitl y computable fro m th e boundar y metri c 7 , a s i s th e coefficien t H o f th e first lo g p term. Th e ter m 7i i s transverse-traceless. Th e ter m g^ satisfie s (3.5) tr

7

g {n) = r , £

7 # (n )

= 5,

where agai n r an d 6 ar e explicitl y determine d b y th e boundar y metri c 7 an d it s derivatives; how rever, a s befor e g^ i s otherwis e undetermine d b y 7 . Ther e ar e (\ogp)k term s tha t appea r i n th e expansio n a t orde r > n.

TOPICS I N CONFORMALL Y COMPAC T EINSTEI N METRIC S 9

Note als o that thes e expansion s (3.2 ) an d (3.4 ) depen d o n th e choic e of bound ary metric . Transformatio n propertie s o f th e coefficient s g^. i < n , unde r con formal change s hav e bee n explicitl y studie d i n th e physic s literature , cf . [24] . A s discovered b y Fefferman-Graham [26] , the term H i s conformally invariant , o r mor e precisely, covariant : i f 7 — 0 2 7, the n H = (jr~ nH. Remark 3.1 . Analogou s t o th e Fefferman-Graha m expansio n above , ther e i s a forma l expansio n o f a vacuu m solutio n t o th e Einstei n equation s nea r nul l infin ity, althoug h thi s ha s bee n carrie d ou t i n detai l onl y i n dimensio n 3 + 1 , cf . [1 6] . This expansio n i s closel y relate d t o th e propertie s o f th e Penros e conforma l com pactification. Mor e recently , a s discusse d i n [20] , logarithmic term s appea r i n th e expansion i n general , an d thes e pla y a n importan t rol e i n understandin g th e globa l structure o f th e space-time . Mathematically, i t i s o f som e importanc e t o kee p i n min d tha t th e expansion s (3.2), (3.4 ) ar e onl y formal , obtaine d b y conformall y compactifyin g th e Einstei n equations an d takin g iterate d Li e derivative s o f g a t p = 0 :

(3.6) g

{k)

= ±rt£ ]g,

where T — Vp. I f g G C nua(M), the n th e expansion s hol d u p t o orde r m - f a . However, boundar y regularit y result s ar e neede d t o ensur e tha t i f a n A H Einstei n metric g with boundar y metri c 7 satisfies 7 G Crn,ol(dM), the n th e compactificatio n geCm>Q(M) o r C m'>a'(M). In bot h case s n od d o r even , th e Einstei n equation s determin e al l highe r orde r coefficients g^.) (an d coefficient s o f th e lo g terms) , i n term s o f g^ an d #( n ), s o that a n A H Einstei n metri c i s formally determine d b y p( 0) an d g^ nea r dM. Th e term p( 0) correspond s t o Dirichle t boundar y dat a o n dM , whil e

9(n)

is analogous to the Dirichlet-to-Neumann ma p for harmonic functions. However , th e map (3.7 ) is only well-defined i f there is a unique AH Einstein metri c with boundar y data 7 = 2 , the n b y definition g ha s a C 2 conforma l compactificatio n t o a C m , Q boundar y metri c 7 . I n [4], i t i s prove d tha t ther e i s a C m,a conforma l compactificatio n g G Crn,Q(M) o f g. cf . als o [6] . Thi s resul t als o hold s i f rn = o c o r m = UJ. It i s prove d usin g th e fact tha t 4-dimensiona l Einstei n metric s satisf y th e Bac h equations , cf . [1 4] , which are conformall y invariant . I n suitabl e gauges , the Bac h equatio n ca n b e recas t a s a non-degenerate ellipti c syste m o f equations fo r a conformal compactificatio n g, an d the resul t follow s fro m ellipti c boundar y regularity .

10

MICHAEL T . ANDERSO N

In dimensio n 4 , th e Bac h tenso r i s th e Fefferman-Graha m obstructio n tenso r H above . I n an y eve n dimension , th e syste m o f equation s (3.8) H

= 0

is conformall y invariant , an d i s satisfie d b y metric s conforma l t o Einstei n metrics . Thus, on e migh t expec t tha t th e metho d usin g th e Bac h equatio n i n [4 , 6] whe n n = 3 can b e extende d t o al l n odd . Thi s i s i n fac t th e case , an d ha s bee n worke d out i n detai l b y Helliwel l [31 ] . Thus , essentiall y th e sam e regularit y result s hol d for n odd . When n i s even, s o that di m M i s odd, thi s typ e o f boundary regularit y canno t hold o f course , du e t o th e presenc e o f th e logarithmi c term s i n th e F G expansion . A result o f Lee [35 ] show s that i f g £ E m'a an d m < n, the n g is C m , a conformall y compact. Thi s i s optimal , bu t doe s no t reac h th e importan t threshol d leve l m — n, wher e logarithmi c term s an d th e importan t g^ ter m firs t appear . Recently , Chrusciel e t al . [21 ] hav e prove d tha t whe n g £ E°°, i.e . g ha s a C°° boundar y metric 7 , the n g ha s a C°° polyhomogeneou s conforma l compactification , s o tha t the expansio n (3.4 ) exist s a s a n asymptoti c series . Moreover , i f 7 £ C m,Q(dM), then th e expansio n exist s u p t o orde r k, wher e k ca n b e mad e larg e b y choosin g ra sufficiently larg e (i n general m mus t b e much large r tha n k). Finally , i t ha s recentl y been prove d b y Kichenassam y [34 ] tha t w rhen g £ E^ an d g^ i s real-analytic , th e formal serie s (3.4 ) exists , i.e . i t i s summable, an d i t converge s t o g p. These result s hav e th e followin g immediat e consequence . Suppos e n i s odd . Given an y real-analyti c symmetri c bilinea r form s p( 0) a n d g^) o n dM, satisfyin g (3.3), ther e exist s a uniqu e C" conformall y compac t Einstei n metri c g define d i n a thickening dM x [0 , e) o f dM. I f instea d n i s even , give n an y analyti c symmetri c bilinear form s p( 0) a n d 9{ n) o n dM, satisfyin g (3.5) , ther e exist s a uniqu e C° ° polyhomogeneous conformall y compac t Einstei n metri c g define d i n a thickenin g dM x [0 , e) o f dM. I n bot h cases , th e expansion s (3.2 ) o r (3.4 ) converg e t o th e metric g p. Thes e result s follo w fro m th e wor k i n [4,6,31 ] whe n n i s odd, an d [34 ] when n i s even . Sinc e analyti c dat a #(0 ) a n d 9{ n) m a v b e specifie d arbitraril y an d independently o f eac h other , subjec t onl y t o th e constrain t (3.3 ) o r (3.5) , t o giv e "local" A H Einstein metrics , defined i n a neighbourhood o f dM, thi s shows that th e correspondence (3.7 ) mus t depen d highl y o n globa l properties o f Poincare-Einstei n metrics. On the other hand , i t is well known that th e use of analytic data to solve elliptictype problem s i s misleading . Whil e th e Dirichle t o r Neuman n proble m i s formall y well-posed, th e Cauch y proble m i s not. Standar d example s involvin g Laplac e oper ator an d harmoni c function s sho w that eve n if Cauchy dat a o n a boundary converg e smoothly t o limit Cauch y dat a o n the boundary , th e correspondin g solution s d o no t converge t o a limi t i n an y neighbourhoo d o f th e boundary . To pas s fro m analyti c t o smoot h boundar y data , on e need s a prior i estimate s or equivalentl y a stabilit y result . I n thi s respect , on e ha s th e following : Theorem 3. 2 (Loca l Stability , [4,6]) . Let g be a C 2 conformally compact Einstein metric, defined in a region ft = [0 , po] x dM containing dM, where p is a geodesic compactification. Suppose there exists a compactification g with C m,a boundary metric 7 . such that (3.9)

||$llci.«(n) < K.

T O P I C S I N C O N F O R M ALLY C O M P A C T E I N S T E I 1 N METRIC S 1

If n — 3, then there is a (possibly different) compactification, also called g, such that, in Q' => [0, po/2] x dM. one has the estimate (3.10) ||sl|c-»(fi
n, provide d th e boundar y metrics ar e clos e i n a stron g norm . I t woul d b e ver y interestin g i f a simila r resul t can b e prove d whe n n i s eve n (i.e . i n od d dimensions) . A direc t generalizatio n i s of cours e no t possible , du e agai n t o th e logarithmi c terms . Redefinin g th e Holde r norms t o tak e suc h logarithmi c term s int o account , i t woul d b e ver y surprisin g i f such a stabilit y resul t di d no t hold ; however , a proo f remain s t o b e established . In eve n dimensions , th e loca l stabilit y theore m allow s on e t o pas s t o limit s i n the analyti c dat a proble m above . Thus , suppos e 7 = g^ an d g^ ar e arbitrar y Qm.oc dat a o n QJ^/T^ s u k j e c t t o th e constrain t (3.3) . Le t 7 ^ and (g( n))i b e a sequenc e of analyti c dat a satisfyin g (3.3 ) convergin g t o 7 an d #( n) i n th e C m'a topology , (such sequence s alway s exist) , an d le t g- t be th e correspondin g sequenc e o f confor mal compactification s o f Poincare-Einstei n metric s define d i n region s f^ . I f th e hypothesis (3.9 ) (wit h C l a replace d b y C n'Q fo r n > 3 ) hel d o n th e sequenc e {#«}, i.e. Qj = Q i s uniform , the n i t follow s tha t {&} converge s i n th e C m , Q topolog y on ft t o a limi t g G C m,Q (f2). Th e metri c g i s a conforma l compactificatio n o f a Poincare-Einstein metri c g, define d a t leas t o n Q. I n othe r words , i t woul d the n follow tha t arbitrar y smoot h 7 an d #( n) ca n b e realize d a s loca l boundar y data . However, th e followin g resul t show s tha t thi s canno t b e th e case : Theorem 3. 3 (Uniqu e Continuation , [8]) . Let data (9(0), 9 (n)) be arbitrarily given, satisfying the constraints (3.3 ) or (3.5) , in som,e open set U C dM, with (9(o)i9(n)) £ C m,Q(U), for ? n > n and any n > 3 . If g is a C m , Q conformally compact Einstein metric realizing the data (g(o),9(n))> defined in a neighbourhood Q with Q fl dM = U. then g is the unique such metric, up to local isometry. This resul t implie s in particular tha t loca l Cauchy dat a i n a n ope n se t U C dM determine th e globa l behaviou r o f the metric , an d th e topolog y o f the manifold , u p to coverin g spaces ; her e w e us e th e fac t tha t Einstei n metric s ar e real-analyti c in th e interior , an d s o triviall y satisf y a uniqu e continuatio n property . I t follow s that ( C i s th e boundary map , the n w e conjecture tha t fo r al l i large , 1 1 ^ is surjective ont o a fixe d neighbourhood V of II(^ ) = (T n ? 7T"). In thi s case , ever y conforma l clas s [7 ] G V, for som e ope n se t V C C containin g gx*, i s th e conforma l infinit y o f a n infinit e sequenc e o f Poincare-Einstei n metric s on M, limitin g o n a Poincare-Einstei n cus p metri c o n N. Thi s indicate s tha t Conjecture 4. 2 i s fals e i f th e assumptio n tha t £ 0 i s connected i s dropped . We poin t ou t tha t exactl y th e sam e discussio n hold s wit h R x T n replace d b y any conformally compac t hyperboli c manifol d iV , with a collection o f cusp ends. A s shown i n [23] , the cus p ends ca n b e Dehn fille d wit h soli d tori t o produce Poincare Einstein metric s (Mi.gi) wit h a fixe d conforma l infinity . I n thi s case , instea d o f having infinitel y man y component s £i = £j.(M) o f £ o n a fixe d manifol d M , on e has a collectio n o f components £{ = £{Mi) o n infinitel y man y topologicall y distinc t manifolds Mj , wit h commo n boundar y dN. The discussio n abov e present s som e speculativ e evidenc e tha t cusp s d o no t form withi n £$, fo r an y componen t £Q of £ . Next , w e presen t a constructio n o f (connected) familie s o f conformall y compac t metric s whic h ar e ver y clos e t o bein g Einstein, an d whic h d o limi t o n cusps . Thi s seem s t o b e th e simples t possibl e construction o f such metrics , sinc e it i s based o n the formatio n o f cusps on surfaces . However, w e argue that , rathe r surprisingly , i t i s unlikely tha t thes e metric s ca n b e perturbed t o nearb y Poincare-Einstei n metrics . To begi n th e construction , retur n t o th e stati c Ad S blac k hol e metric s (2.1 1 ) on R 2 x A [, with k — — 1. I n thi s situation , g m i s well-defined fo r negativ e value s of

20

MICHAEL T . ANDERSO N

ra; i n fact , g m i s well-defined fo r m € [ra_ , oc), wher e (5.3) m

_= ,

1 /n-2\ n/2 / n - 2 \ 1 wit h r + = ii — 2 \ n J \ n J

/ 2

For th e extrema l valu e 7?i _ o f ra, V ( r + ) = V'(r+ ) = 0 , an d a simpl e calculatio n (cf. (5.6 ) below ) show s tha t th e horizo n {r = r+ } occur s a t infinit e distanc e t o an y given poin t i n (R 2 x N.g m_)] th e horizo n i n thi s cas e i s calle d degenerat e (wit h zero surfac e gravity) . Not e tha t /3(m_ ) = oc , s o tha t th e ^-circle s ar e i n fac t line s R. A s in decrease s t o ra_, th e horizo n diverge s to infinit y (i n the opposite directio n from th e conforma l infinity) , whil e th e lengt h o f the ^-circle s expands t o oc . Thus , the metri c g rn_ i s a complet e metri c o n th e manifol d R x R x N = R 2 x TV , but i s no longe r conformall y compact : th e conforma l infinit y i s R x (N.gw). However, on e ma y divid e th e infinit e 0-facto r R o f th e metri c g m_ b y Z t o obtain a complet e metri c gE o n C x A r = R x S 1 x A r of th e for m (5.4) g

E

= V~ ldr2 + Vd6 2 + r 2gN,

where V(r) — — 1 - f r 2 — 2 m _ / r n " 2 , wit h m _ an d r + give n b y (5.3) . Th e lengt h / ? of th e ^-paramete r i n (5.4 ) i s no w arbitrary . Th e metri c gE i s calle d a n extrema l black hol e metric, an d i s Poincare-Einstein wit h a single cusp end; g& has a smoot h conformal compactificatio n t o th e boundar y metri c 5 a(/3) x (N.gx). To understan d th e behaviou r o f th e metri c gE i n th e cus p region , w e conver t to geodesi c coordinates . Le t ds — V~xl2 dr, s o that (u p t o a n additiv e constant )

(5.5) s= Thus, a s r ^ r form

J V~ +,

1 /2

(r)dr.

s — > — oo, w rhile a s r — > oc , s — > oc . Th e metri c (5.4 ) take s th e

(5.6) g

E

= ds 2 + V( S)d02 +

2

r

(s)gN,

and th e integra l curve s o f V s ar e geodesies . A simpl e calculatio n show s tha t (5.7) V(s)

= ne 2V"s(l +

e{s)), a

ss

-+ -oc ,

where e(s) — > 0 as s — > — oc. As s — > —oc, th e lengt h o f th e ^-circle s o f cours e goe s t o 0 , i.e . on e ha s a collapse. However , th e collaps e ca n b e unwrappe d b y passin g t o larg e coverin g spaces o f th e S 1 factor ; o n an y sequenc e o f bas e point s xi wit h S(XJ) —> — oc, on e may choos e covering s s o tha t th e lengt h o f th e S 1 facto r a t xi i s approximatel y 1 . One ma y the n pas s t o a pointe d limi t t o obtai n th e metri c = ds 2 4 - e2^s dO

(5.8) 9oo

2

+ r%g N.

The metri c (5.8 ) i s a produc t o f th e constan t curvatur e metri c o n th e cus p C = R x 5 1 w rith a rescalin g o f th e metri c ^ v o n N. B y (2.1 0) , th e Ricc i curvatur e o f r\gN equal s —r+ 2(n — 2 ) = —ng^. whil e th e curvatur e o f th e cus p metri c i s als o —n. Th e metri c (5.8 ) (o f course ) ha s Ricc i curvatur e R i e ^ = —ng^- Th e limi t (5.8) i s unique , u p t o rescaling s o f th e lengt h o f th e S 1 factor . The curvatur e o f the extrema l metri c gE converge s t o tha t o f g^ exponentiall y fast i n s. Straightforwar d computatio n fro m th e estimate s abov e show s tha t (5.9) \\R

gE

- R

9oc

| | < Ce^\ a

ss

- -oc ,

TOPICS I N CONFORMALL Y COMPAC T EINSTEI N METRIC S 2 1

where th e nor m i s the L°° norm . Th e sam e estimat e hold s fo r al l covarian t deriva tives o f thes e curvatures . We now mak e QE conformall y compact , b y closing of f th e cus p end. T o do this, glue E x A r, w4ier e E i s an y hyperboli c surfac e wit h a n ope n expandin g end , ont o the cus p en d o f QE> Fo r simplicity , assum e tha t E ha s a singl e end : i t i s eas y t o generalize th e constructio n belo w t o an y finit e numbe r o f ends . Thus, le t (E,gs ) b e an y conformally compac t Rieman n surfac e wit h connected , non-empty boundar y o f constan t negativ e curvature , normalize d s o tha t (5.10) Kz

= Ric E = - n ,

and wit h 7Ti(E ) ^ 0 . Le t a b e the uniqu e close d geodesi c in the fre e homotop y clas s of th e en d E o f E , an d le t a b e th e lengt h o f a. W e wil l onl y conside r metric s g^ in th e modul i spac e M, discusse d i n §4 , for whic h (5.11) a

The metri c gr> i s Einstein, o f Ricc i curvatur e — n. We now truncate th e tw o metrics g& and g£ an d glu e them together . T o begin, topologically, se t M = E x N. While th e produc t metri c gjj o n M i s Einstein, i t i s not conformall y compact . Thi s metric ha s on e en d E o f th e for m C x N, wher e C i s a n expandin g cusp . I n th e r tubular neighbourhoo d o f the geodesi c a C E C E , th e metri c ha s th e for m (5.1 2) . Choose R larg e (t o be determine d below) , an d le t DR b e th e regio n i n E x N wher e t < R, i n th e en d E. Thus , dD R = S 1 ^-) x ( r 2 . ^ ) wher e (5.14) L~

=cosh(y/nR)a.

Next, tak e the conformall y compac t extrem e metri c g£ o n C x N. an d truncat e it t o th e regio n E R wher e s > —R. Th e lengt h o f th e boundar y circl e dC i s the n (5.15) L

+ - V^

2

(-R)P.

To perfori n th e gluing , w e requir e tha t th e length s o f th e circle s agree , L~ = L + . Since th e lengt h j3 is fixed, give n th e lengt h a , thi s impose s th e relatio n (5.16) a

- {cosh(

1 1 /2 y/7iR)}~ V {-R)p.

The paramete r t i n (5.1 2 ) i s related wit h th e paramete r s in (5.6 ) b y setting t — R = s -f R. Given thes e choices , on e ma y easil y construc t a conformall y compac t approxi mate Einstei n metri c g on M b y attachin g th e truncate d metric s DR an d E R alon g their boundarie s an d smoothin g th e sea m i n a neighbourhoo d U o f radiu s 1 of

22

MICHAEL T . ANDERSO N

the boundaries . Th e metri c (M , g) i s smoothl y conformall y compact , an d Einstei n outside th e gluin g regio n U. Conforma l infinit y i s give n b y th e conforma l clas s (dM, [70(/?)]) , where dM = S 1 x TV , and 7o (/3) - /3 2 d« 2 + &v , wit h 0 e [0,1 ] . Let $ = $ $ b e a s i n (2.6)-(2.7) . Then , b y construction , §~ 9(g) = 0 outside U. An elementar y computation , usin g (5.9 ) an d simpl e estimate s fo r th e 2 n d funda mental form s o f th e boundarie s o f (DR. go) an d (E R,QE) give s th e estimat e (5.17) |$5(5)

R

| < Ce-^

,

inside [/ , wher e C i s independen t o f R. This gives the construction o f the approximat e solution s g. I n fact th e construc tion give s a smoot h modul i spac e M £ o f approximat e solution s o n M, naturall y diffeomorphic t o th e modul i spac e M E o n E . On e ha s a natura l boundar y ma p (5.18) fl.Me^C, which i s th e constan t ma p t o th e conforma l clas s [7 0 (/?)]; not e tha t dim.A/ f = m = 6 9 i s a submersio n a t g, s o tha t th e image o f & contain s a n ope n bal l B(/i) C C about $ 9(g). However , on e need s t o obtain a lowe r boun d o n th e radiu s / / o f suc h a ball , independen t o f i? , o r a t leas t prove tha t (5.19) fi

> e'^

R

,

for R large . Now i t i s clea r tha t th e linearize d operato r L i n (2.8 ) canno t b e uniforml y invertible a t g E SA. I n fact , ther e i s an approximat e kerne l K o f L actin g o n form s in S 72n'a, induce d b y form s K tangent t o th e modul i spac e M. o f hyperboli c metric s on E an d extende d t o M i n a natura l way , s o that, i f ||£||L° C = 1 , the n (5.20) ||L(£)||

L oc->0,

as th e gluin g radiu s R — » 00. Not e tha t form s K tangen t t o M (equivalen t t o holomorphic quadrati c differential s o n E ) deca y t o 0 o n th e en d E outsid e th e closed geodesi c a. Suppose tha t th e Einstei n manifol d (N^gx) i n (2.1 0 ) ha s th e propert y tha t (5.21) K

N

< 0,

i.e. gjsi has non-positiv e curvature . The n i t i s no t particularl y difficul t t o show , although w e wil l no t giv e th e detail s here , tha t L i s uniforml y invertibl e o n th e orthogonal complemen t K 1 - o f K i n S^' ^ wit h respec t t o th e L 2 metric . On e ha s then di m K — m = 6g — 3 . There ar e no w tw o method s t o tr y t o establis h (5.1 9 ) an d thu s obtai n a Poincare-Einstein metri c g nea r g. First , on e ca n tr y t o arrang e tha t (5.22) **(£

) € J?- \

possibly b y modifyin g o r perturbin g g, an d iterativel y solv e $> 9(g) — 0 withi n th e space §2 l ' a . Thi s woul d lea d t o th e existenc e o f a Poincare-Einstei n metri c g wit h the sam e boundar y metri c a s g. However , thi s i s not possible :

TOPICS I N CONFORMALL Y COMPAC T EINSTEI N METRIC S

23

P r o p o s i t i o n 5.3 . There is no Einstein metric on M with conforrnal infinity given by S l{i3) x (N,g N). PROOF. Thi s i s an immediate consequenc e o f Corollary 2.5 . • Thus, t o obtai n a n Einstei n metri c g on M clos e t o g require s changin g the boundary metri c o f g, i.e. working outsid e th e space S™' 0. Thi s mean s on e must use the dependence o f ^ o n the boundary metric s t o try to kill the cokernel o f L, as in the proof o f Theorem 2.1 . In turn, thi s require s provin g tha t th e pairing (2.9 ) is non-degenerat e an d bounded below . Mor e precisely , fo r any k € K, ther e mus t exist a variatio n 7 o f the boundary metri c 70 , wit h H&HL 00 < 1 a nd UTIIL 00 < 1 ? such tha t (5.23 )

1/ (D&( M \JM I

where // ' also satisfie s (5.1 9) . We hav e no t been abl e t o verify (5.23) , an d expect i t i s not true. First , th e choice o f K i s not canonical, an d to establish (5.23 ) on e needs a precise definitio n or choice . I f K i s defined a s forms tangen t t o the moduli spac e M e o n £, naturall y extended t o forms on M an d whic h hav e compac t support , the n K, = 0 near infinity . However, b y the definition give n i n §2, the support o f D T 1 , 0 ; th e correspondin g sectio n o f CTQH is< £ + S 2 , and 7 i s th e restrictio n o f th e standar d metri c o f S s t o th e contac t distribution , actually 7(- , •) = d(i]/2)(-, J-). I n pola r coordinates , wit h p = tanh(r) , on e get s (2.1) g

= dr 2 + sinh 2 (2r) (^ J + sinh 2 (r)7.

One ca n recove r th e C R structur e o n th e spher e a t infinit y fro m th e metri c b y taking th e limi t o n concentri c sphere s S^: 7 = \im(l - p 2)g\ . The limi t i s finite onl y o n th e contac t distributio n H. Th e conforma l facto r (1 — p 2) coul d b e change d b y multiplicatio n b y a positiv e functio n o n S 3 , s o onl y the conforma l clas s of 7 i s well defined, tha t is , the metri c induce s the C R structur e J o n th e boundar y a t infinity . Cheng-Yau metric. Mor e generally , a strictly pseudoconve x domai n i n C n car ries a uniqu e complet e Kahler-Einstei n metri c [1 5] . Moreover , th e asymptoti c be havior o f thi s Cheng-Ya u metri c i s simila r t o tha t o f th e Bergma n metric , tha t is, (2 2)

-

*~—40*—

+ 20 '

or, equivalently , (2.3) 9

~ g 0 = dr 2 + \e 4rV2 +

\e* 7,

where 0 = 2e~ 2 r i s a definin g functio n o f th e boundary , rj a contac t for m o n th e boundary, 7(- , •) = drj(-, J-). In vie w o f th e asymptotic s o f th e Bergma n metric , o r mor e generall y o f th e Cheng-Yau metric , i t i s natural t o mak e th e followin g definition . Definition 4 . Le t X 3 b e a strictl y pseudoconve x C R manifold , an d M 4 a manifold wit h dM = X. Identif y a neighborhood o f X i n M wit h (0 , oo) x X, wit h coordinate r o n th e first facto r goin g to infinity . A metric g on M i s asymptoticall y complex hyperboli c (i n shor t ACH ) nea r th e boundar y X i f 9 = 90 + O(e-

r

),

where g 0 i s defined b y formul a (2.3) , an d 0(e~ r) i s wit h respec t t o #o-

30

OLIVIER BIQUAR D

The C R structur e i s calle d th e conforma l infinity , o r th e boundar y a t infinit y of th e metri c g. A s fo r th e Bergma n metric , i t ca n b e recovere d fro m th e metri c g b y takin g a conforma l limi t o f g restricte d t o th e slice s MR — {r — R) whe n R goes t o infinity . It i s eas y t o chec k tha t a n AC H metri c ha s it s curvatur e tenso r clos e a t orde r 0(e~r) t o th e curvatur e tenso r o f comple x hyperboli c space . Thi s fac t migh t b e seen a s a justification fo r th e terminolog y "asymptoticall y comple x hyperbolic" . Example 5 . Le t E b e a hyperboli c Rieman n surface , the n th e tota l spac e o f the dis k bundl e y/TH carrie s a complet e AC H comple x hyperboli c metri c (induce d by a representatio n 7Ti(E )— > SU1 4 C SUi^) Asymptotically real hyperbolic 'metrics. O f cours e th e theor y her e i s parallel t o the theor y i n th e rea l case . Th e rea l hyperboli c space , wit h curvatur e —1 , ca n b e written i n th e mode l o f th e bal l a s

and ther e i s a conforma l limi t o n th e spher e a t infinity : 7 = lim \g\ . Asymptotically hyperboli c (AH ) metric s o n M n wit h boundar y a conforma l metri c 7 o n dM — {0 = 0 } ar e define d a s metric s asymptoti c t o

#2 +7 they hav e sectiona l curvatur e goin g t o — 1 at th e boundary . 3. Ellipti c Operator s an d AC H Metric s In this section , w e do no t stat e an y precis e result , bu t tr y t o giv e a roug h ide a of th e analysi s o f ellipti c operator s o n AC H metrics , mainl y throug h th e exampl e of th e scala r Laplacian . So loo k a t a C R 3-manifol d X 3 , an d fix a contac t structur e 77 , and therefor e a metric 7 o n th e contac t distribution . W e shal l denot e th e Laplacia n i n horizonta l directions b y D : w w 2

u = -^T(v

ei,e.;")

1

where e ?; is an orthonormal basi s of the contact distribution , an d V w i s the Tanaka Webster connection . Thi s formul a make s sens e als o on tenso r bundle s a s well a s on functions. W e shal l als o us e th e Ree b vecto r field R an d th e derivativ e V)^ . In complex hyperbolic space , with polar coordinate s (2.1 ) , one has the followin g formula fo r th e scala r Laplacian : 2 w - 1^ - 2(coth(2r ) + c o t h ( r ) ) ^ - \ (V^) K-U . 2v v v 2 V R> 2 dr ' "dr sinh (2r) sinh r On a genera l AC H manifol d M 4 wit h conforma l boundar y X, th e leadin g orde r terms ar e the same . I f one look s only a t smoot h functions , the n th e leadin g ter m i s

A=

C A U C H Y - R I E M A N N 3-MANIFOLD S A N D E I N S T E I N F I L L I N G S 3 1

This i s the indicia l operato r associate d t o th e Laplacian . Solution s fo r th e indicia ! operator ar e 1 and e~ 4 r , leadin g t o indicia ! roots 0 and 4 . Actuall y th e Laplacia n i s well-known t o b e a n isomorphis m betwee n Sobole v space s H 2— > L 2 , an d th e fac t that th e indicia ! root s ar e 0 an d 4 implie s tha t i t i s stil l a n isomorphis m betwee n weighted space s correspondin g t o a deca y e~ Xr fo r an y 0 < A < 4 , tha t i s betwee n spaces (3.1) H

2

= e 6rH2^L2 =

e

6r 2

L

for an y — 2 < S < 2 (th e transitio n betwee n A and 5 conies fro m th e fac t tha t th e limit deca y rat e fo r Lr function s i s e" 2 r ). This resul t ca n b e use d t o solv e th e Dirichle t proble m o n M: give n a functio n (£> o n X , fin d a harmoni c functio n / i n th e interio r wit h boundar y valu e (j). Formal solution. Firs t on e ca n tr y t o fin d th e asymptoti c developmen t o f / near th e boundary . Construc t a functio n / o b y extendin g (j) alon g rays , the n fro m the formul a fo r th e Laplacia n i t i s clea r tha t A/ o — 0 ( e ~ 2 r ) , actuall y A/o = 02e- 2r ' + O ( e - 3 r ) , where 2^~2r b y J2 — je~2r(t>2 gives a n approximat e solutio n fo + fa satisfyin g A(/o + / 2 ) = ^ 3 e - 3 r + 0 ( e - 4 r ) . This i s the beginnin g o f a n inductio n procedur e whic h stop s a t th e indicia l roo t 4 : the indicia l proble m {—d 2/dr2 — 4d/dr)f4 — 4>4e~Ar i s solved b y F 4 = — | r e ~4 r 0 4 , and the n A(/o + -- - + / 3 + F 4 ) = : 0 ( r e - 5 ' ' ) ; but ther e i s a possibl e undetermine d ter m o f typ e After that , th e forma l resolutio n b y function s r^e~ kr ( upolyhomogeneous expan sion" ) continue s withou t furthe r obstruction . Global solution. Th e undetermine d ter m ca n be fixed onl y after findin g a global solution: on e ca n loo k a t a finit e numbe r o f term s o f th e forma l resolution , s o one get s a n approximat e solutio n / wit h A / = 0 ( e " A r ) . The n w e appl y th e isomorphism theore m (3.1 ) t o correc t / b y a g £ H 2 suc h tha t A ( / + g) = 0 (actually g £ H$° b y ellipti c regularity) . Ther e ar e tw o cases : • i f A < 4 one ca n tak e S = A — 2 — e; • i f A > 4 one ca n tak e onl y S = 4 — e: thi s reflect s th e fac t tha t g contain s the undetermine d term , necessaril y a t orde r 0 ( e _ 4 r ) . An importan t fac t i s that th e solutio n usin g th e isomorphis m theore m (3.1 ) i s smooth, bu t wit h a ver y wea k contro l i n th e direction s o f X: indee d g £ H$° onl y means, fo r a horizonta l vecto r h £ TX, tha t on e ha s e~ kr(\/}^)kg £ L 2, an d thi s is very fa r fro m sayin g tha t th e coefficient s o f th e developmen t nea r X ar e smoot h functions o n X. Thi s mean s tha t th e isomorphis m theore m ca n b e refined , s o tha t extra regularit y o f A g alon g X wil l giv e extr a regularit y o f g alon g X. In particular , on e ca n prov e tha t i f th e dat a