Titles i n Thi s Serie s

27 Sun-Yun g A . Chang , Pau l C . Yang , Karste n Grove , an d Jo n G . Wolfson , Conformal, Riemannia n an d Lagrangia n geometry , Th e 200 0 Barret t Lectures , 200 2

26 Susum u Ariki , Representation s o f quantu m algebra s an d combinatoric s o f Youn g

tableaux, 200 2

25 Wil l ia m T . Ros s an d Harol d S . Shapiro , Generalize d analyti c continuation , 200 2

24 Victo r M . Buchstabe r an d Tara s E . Panov , Toru s action s an d thei r application s i n

topology an d combinatorics , 200 2

23 Lui s Barreir a an d Yako v B . Pesin , Lyapuno v exponent s an d smoot h ergodi c theory ,

2002

22 Yve s Meyer , Oscillatin g pattern s i n imag e processin g an d nonlinea r evolutio n equations ,

2001

21 Bojk o Bakalo v an d Alexande r Kirillov , Jr. , Lecture s o n tenso r categorie s an d

modular functors , 200 1

20 Aliso n M . Etheridge , A n introductio n t o superprocesses , 200 0

19 R . A . Minlos , Introductio n t o mathematica l statistica l physics , 200 0

18 Hirak u Nakajima , Lecture s o n Hilber t scheme s o f point s o n surfaces , 199 9

17 Marce l Berger , Riemannia n geometr y durin g th e secon d hal f o f th e twentiet h century ,

2000

16 Harish-Chandra , Admissibl e invarian t distribution s o n reductiv e p-adi c group s (wit h

notes b y Stephe n DeBacke r an d Pau l J . Sally , Jr.) , 199 9

15 Andre w Mathas , Iwahori-Heck e algebra s an d Schu r algebra s o f th e symmetri c group , 199 9

14 Lar s Kadison , Ne w example s o f Frobeniu s extensions , 199 9

13 Yako v M . Eliashber g an d Wil l ia m P . Thurston , Confoliations , 199 8

12 I . G . Macdonald , Symmetri c function s an d orthogona l polynomials , 199 8

11 Lar s Garding , Som e point s o f analysi s an d thei r history , 199 7

10 Victo r Kac , Verte x algebra s fo r beginners , Secon d Edition , 199 8

9 S tephe n Gelbart , Lecture s o n th e Arthur-Selber g trac e formula , 199 6

8 Bern d Sturmfels , Grobne r base s an d conve x polytopes , 199 6

7 A n d y R . Magid , Lecture s o n differentia l Galoi s theory , 199 4

6 D u s a McDuf f an d Die tma r Salamon , J-holomorphi c curve s an d quantu m cohomology ,

1994

5 V . I . Arnold , Topologica l invariant s o f plan e curve s an d caustics , 199 4

4 Davi d M . Goldschmidt , Grou p characters , symmetri c functions , an d th e Heck e algebra ,

1993

3 A . N . Varchenk o an d P . I . Etingof , Wh y th e boundar y o f a roun d dro p become s a

curve o f orde r four , 199 2

2 Frit z John , Nonlinea r wav e equations , formatio n o f singularities , 199 0

1 Michae l H . Freedma n an d Fen g Luo , Selecte d application s o f geometr y t o

low-dimensional topology , 198 9

http://dx.doi.org/10.1090/ulect/027

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Conformal, Riemannia n and Lagrangia n Geometi y

The 200 0 Barret t Lecture s

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University

LECTURE Series

Volume 2 7

Conformal, Riemannia n and Lagrangia n Geometr y

The 200 0 Barret t Lecture s Sun-Yung A. Chan g

Paul C . Yan g Karsten Grov e

Jon G . Wolfso n

Alexandre Freire , Edito r

American Mathematica l Societ y Providence, Rhod e Islan d

E D I T O R I A L C O M M I T T E E

Jer ry L . Bon a (Chair ) Nicola i Reshet ikhi n Nigel J . Hi t chin

2000 Mathematics Subject Classification. P r imar y 53C20 , 53C21 , 53D12 ; Secondary 53C55 , 57S15 , 35B65 .

Library o f Congres s Cataloging-in-Publicatio n Dat a

Conformal, Riemannia n an d Lagrangia n geometr y : th e 200 0 Barret t lecture s / Sun-Yun g A . C h a n g . . . [e t al.] .

p. cm . — (Universit y lectur e series , ISS N 1047-399 8 ; v. 27 ) Includes bibliographica l references . ISBN 0-8218-3210- 7 (alk . paper ) 1. Conforma l geometry . 2 . Geometry , Riemannian . 3 . Geometry , Differential . I . Chang ,

Sun-Yung A. , 1948 - II . Universit y lectur e serie s (Providence , R.I. ) ; 27.

QA609.C67 200 2 516.3'62—dc21 200207165 0

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Contents

Preface

Chapter 1 . Partia l Differential Equation s Related to the Integrand o n 4-manifold s Sun-Yung A. Chang and Paul C. Yang

Introduction 1. Finitenes s o f conformally fla t structure s 2. Backgroun d o n a 2 3. Deformin g a 2 to a positiv e functio n 4. Deformin g a2 to a constan t Bibliography

Chapter 2 . Geometr y of , an d via , Symmetrie s Karsten Grove

Introduction 1. Geometr y o f isometr y group s 2. Structur e an d classificatio n progra m 3. Construction s an d example s 4. Emergenc e o f isometrie s 5. Ope n problem s Bibliography

Chapter 3 . Lagrangia n Cycle s an d Volum e Jon G. Wolfson

Introduction 1. Th e variationa l proble m 2. Existenc e 3. Regularity : cone s an d monotonicit y Bibliography

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Preface

The presen t volum e contain s thre e survey s i n differentia l geometry , base d o n the thirtieth Joh n H . Barrett Memoria l Lecture s delivered a t th e Universit y o f Ten-nessee, Knoxvill e i n May 2000 . Directe d a t researcher s an d advance d graduat e stu -dents, th e survey s introduc e th e background , contex t an d mai n technique s o f ver y recent development s i n thre e distinc t area s o f geometry : (i ) conformall y invarian t curvatures an d operator s i n fou r dimensions , an d th e associate d partia l differentia l equations; (ii ) th e us e o f isometri c grou p action s an d metri c geometr y technique s to understan d example s an d th e classificatio n o f classe s o f Riemannia n manifolds , especially of positive curvature; (iii ) variational problems for lagrangian immersion s in symplectic manifolds , i n particular th e lagrangia n volum e minimization proble m in a homolog y class .

The mai n them e o f Chapte r 1 , writte n b y S.-Y.A . Chan g an d Pau l Yang , i s conformal geometr y i n four dimensions . Th e integran d i n the Chern-Gauss-Bonne t formula fo r th e Eule r characteristi c o f a close d four-manifol d involve s th e confor -mally invarian t Wey l tensor an d a fourth-order curvatur e invarian t Q. I n section 1 , the author s describ e thei r wor k (join t wit h J.Qing ) o n conforma l compactificatio n of complete , non-compac t locall y conformall y fla t four-manifold s wit h integrabl e Q, includin g a Chern-Gauss-Bonne t formul a wit h a 'defect ' ter m relate d t o th e isoperimetric profil e o f the end s (Theore m 1.2) . The y the n tur n t o a discussio n o f the geometr y o f the developin g ma p imag e o f a locally conformall y fla t manifol d o f positive scalar curvatur e ( a domain on the sphere whose complement i s the limi t se t of a Kleinia n group) , i n particula r thei r recen t wor k relatin g geometri c finitenes s and Hausdorf f dimensio n o f the limi t set .

The nex t thre e section s dea l wit h th e secon d elementar y symmetri c functio n <72{A)1 wher e A i s the 'conforma l Ricc i tensor' ; th e integra l o f <J2(A) i s conformall y invariant. Th e mai n theore m i n sectio n 3 (Theore m 2.1 , join t wit h M.Gursky ) states tha t a compac t four-manifol d wher e bot h thi s invarian t an d th e Yamab e invariant ar e positive admits a conformally relate d metric for which <J2{A) i s positive pointwise, an d i n particula r ha s pinche d positiv e Ricc i curvature . Thi s involve s a PDE o f Monge-Amper e type , th e authors ' wor k o n functiona l determinants , an d at a crucia l ste p th e Yamab e flow . Th e las t sectio n o f thi s chapte r explain s th e main step s i n the difficul t proo f o f a very recen t result : unde r th e sam e hypothesis , a furthe r conforma l deformatio n yield s a metri c o f constan t (72(A). Th e proo f involves degre e theor y fo r full y non-linea r equations , a s wel l a s delicat e continuit y and blowing-u p arguments .

The purpose of Chapter 2 , written by Karsten Grove , is to present the 'fairly un -explored territory ' o f geometry an d topolog y tha t symmetr y group s o f Riemannia n manifolds bot h posses s an d reveal . Th e firs t sectio n sketche s th e know n structur e of th e quotien t o f a Riemannia n manifol d vi a a symmetr y group . Th e discussio n

ix

x P R E F A C E

includes the Principa l Orbi t Theore m an d the structure o f such quotients when con-sidered a s Alexandro v spaces . Sectio n 2 lays ou t a genera l classificatio n program , namely tha t o f classifyin g al l positivel y o r non-negativel y curve d manifold s wit h large isometr y groups . O f cours e ther e ar e man y possibl e meaning s o f 'large, ' five of whic h ar e considere d i n mor e detail : hig h degre e o f symmetry , hig h symmetr y rank (thes e tw o refe r t o hig h dimensio n an d ran k o f the isometr y grou p G) , smal l dimension o f the manifol d M relativ e to tha t o f G, smal l dimension o f the quotien t M/G (cohomogeneity) , an d fixed poin t se t o f dimensio n clos e t o th e dimensio n o f M/G (fixe d poin t cohomogeneity) .

Prom thi s poin t o n th e chapte r follow s tw o themes : classificatio n result s an d examples tha t aris e whe n on e trie s t o find classificatio n results , especiall y i n con -nection wit h non-negativ e curvature . I n fact , al l know n method s fo r constructin g manifolds o f non-negativ e curvatur e ar e describe d i n thi s chapter , togethe r wit h several o f the mos t importan t classification s o f manifolds tha t aris e fro m someho w restricting geometr y an d symmetr y groups . Th e chapte r conclude s wit h a lis t o f open problem s an d conjectures .

The geometr y o f lagrangia n immersion s int o symplecti c manifold s i s the topi c of Chapter 3 , by Jon Wolfson . Sectio n 1 introduces th e definition s an d example s of lagrangian submanifolds , an d th e basi c topologica l result s underlyin g th e theory . The perio d an d Maslo v inde x o f a lagrangian immersio n ar e defined , an d th e inter -play wit h Ricc i curvatur e i s discussed, i n particular fo r Kahler-Einstei n manifolds . The proble m o f minimizin g volum e amon g lagrangia n cycle s i n a lagrangia n ho -mology clas s i s introduced , wit h a brie f discussio n o f potentia l application s t o th e Strominger-Yau-Zaslow mirro r symmetr y conjectur e an d t o problems i n non-linea r elasticity.

The next two sections are devoted to Wolfson's recent work (joint wit h R.Schoen ) on th e lagrangia n volum e minimizatio n problem . Sectio n 2 deal s wit h existence , including th e geometri c measur e theor y approac h an d th e mappin g problem . Th e former yield s an integral curren t whic h is mass-minimizing amon g al l integral cycle s in a given lagrangia n homolog y clas s (Theore m 2.5) . Thi s i s followed b y a detaile d discussion o f the 'bubblin g behavior ' fo r minimizin g sequence s o f maps o f surfaces , including th e importan t distinction s wit h th e classica l (unconstrained ) problem . Section 3 describe s thre e aspect s o f th e partia l regularit y theory : hamiltonian -stationary lagrangia n cones , Holder continuit y an d a new monotonicity formul a fo r lagrangian stationar y surface s (whic h involves contact geometry) . Thi s leads to th e main existenc e an d partia l regularit y result , Theore m 3.7 .

The Joh n H . Barret t Memoria l Lecture s wer e establishe d i n 1970 , i n memor y of a professo r an d mathematic s departmen t hea d a t th e Universit y o f Tennessee , through whos e influenc e a n activ e grou p i n differentia l equation s evolved . Eac h year fro m on e t o fou r leadin g researcher s delive r surve y lecture s i n a n activ e are a of mathematics . Th e 200 0 Barret t Lecture s receive d financial suppor t fro m th e National Scienc e Foundatio n an d Scienc e Alliance . Informatio n o n th e Barret t lecture series may be found i n the University of Tennessee Mathematics Departmen t web site : http://www.math.utk.edu .

Alexandre Freire , B o Guan , an d Conra d Plau t February 200 2