0521400015 cambridge university press geometry of low dimensional manifolds vol 2 symplectic manifolds and jones witten theory jan 1991

More specifically, the lrincipal developments concern: 1 geometric structures on manifolds, 2 symplectic topology and geometry, 3 applications of Yang-Mills theory to three- and four-dim

104 Elliptic structures on 3-manifolds, C.B THOMAS

105 A local spectral theory for closed operators, I ERDELYI & WANG SHENGWANG

106 Syzygies, E.G EVANS & P GRIFFITH

107 Compactification of Siegel moduli schemes, C-L CHAI

108 Some topics in graph theory, H.P YAP

109 Diophantine Analysis, J LOXTON & A VAN DER POORTEN (eds)

110 An introduction to surreal numbers, H GONSHOR

III Analytical and geometric aspects of hYPerbolic space, D.B.A.EPSTEIN (ed)

112 Low-dimensional topology and Kleinian groups, D.B.A EPSTEIN (ed)

113 Lectures on the asymptotic theory of ideals, D REES

114 Lectures on Bochner-Riesz means, K.M DAVIS & Y-C CHANG

115 An introduction to independence for analysts, H.G DALES & W.H WOODIN

116 Representations of algebras, P.I WEBB (ed)

117 Homotopy theory, E REES & J.D.S JONES (eds)

118 Skew linear groups, M SHIRVANI & B WEHRFRITZ

119 Triangulated categories in the representation theory of finite-dimensional algebras, D HAPPEL

121 Proceedings of Groups - St Andrews 1985, E ROBERTSON& C CAMPBELL (eds)

122 Non-classical continuum mechanics, R.I KNOPS & A.A LACEY (eds)

124 Lie groupoids and Lie algebroids in differential geometry, K MACKENZIE

125 Commutator theory for congruence modular varieties, R FREESE & R MCKENZIE

126 Van der Corput's method for exponential sums, S.W GRAHAM & G KOLESNIK

127 New directions in dynamical systems, T.I BEDFORD & J.W SWIFf (eds)

128 Descriptive set theory and the structure of sets of uniqueness, A.S KECHRIS & A LOUVEAU

129 The subgroup structure of the finite classical groups, P.B KLEIDMAN & M.W.LIEBECK

130 Model theory and modules, M PREST

131 Algebraic, extremal & metric combinatorics, M-M DEZA, P FRANKL & LG ROSENBERG (eds

132 Whitehead groups of finite groups, ROBERT OLIVER

133 Linear algebraic monoids, MOHAN S PUTCHA

134 Number theory and dynamical systems, M DODSON & J VICKERS (eds)

135 Operator algebras and applications, 1, D EVANS & M TAKESAKI (eds)

136 Operator algebras and applications, 2, D EVANS & M TAKESAKI (eds)

137 Analysis at Urbana, I, E BERKSON, T PECK, & J UHL (eds)

138 Analysis at Urbana, II, E BERKSON, T PECK, & J UHL (eds)

139 Advances in homotopy theory, S SALAMON, B STEER & W SUTHERLAND (eds)

140 Geometric aspects of Banach spaces, E.M PEINADOR and A RODES (eds)

141 Surveys in combinatorics 1989, J SIEMONS (ed)

142 The geometry of jet bundles, D.J SAUNDERS

143 The ergodic theory of discrete groups, PETER J NICHOLLS

144 Introduction to uniform spaces, I.M JAMES

145 Homological questions in local algebra, JAN R STROOKER

146 Cohen-Macaulay modules over Cohen-Macaulay rings, Y YOSHINO

147 Continuous and discrete modules, S.H MOHAMED & B.I MULLER

148 Helices and vector bundles, A.N RUDAKOV et al

149 Solitons, nonlinear evolution equations and inverse scattering, M.A ABLOWITZ & P.A CLARKSON

150 Geometry of low-dimensional manifolds 1, S DONALDSON & C.B THOMAS (eds)

151 Geometry of low-dimensional manifolds 2, S DONALDSON & C.B THOMAS (eds)

152 Oligomorphic permutation groups, P CAMERON

153 L-functions in Arithmetic, J COATES & MJ TAYLOR

154 Number theory and cryptography, J LOXTON (ed)

155 Classification theories of polarized varieties, TAKAO FUJITA

156 Twistors in mathematics and physics, T.N BAILEY &R.l BASTON (eds)

London Mathematical Society Lecture Note Series 151

Geometry of Low-dimensional Manifolds

Proceedings of the Durham Symposium, July 1989

II'U~I:rll""'dby H("Ir,' VJ/linUJ4

Thl' Uni1'f!rsilY ho.r prmud

40 West 20th Street, New York, NY 10011, USA

10, Stamford Road, Oakleigh, Melbourne 3166, Australia

© Cambridge University Press 1990

First published 1990

Printed in Great Britain at the University Press, Cambridge

Library ofCongress cataloguing in publication data.available British Library cataloguing in publication data av.ailable

ISBN 0 521 40001 5

New results in Chern-Simons theory

Edward Witten, notes by Lisa Jeffrey

( icometric quantization of spaces of connections

(~valuationsof the 3-manifold invariants of Witten and

Reshetikhin-Turaev forsl(2,C)

Robion Kirby and Paul Melvin

Representations of braid groups

M.F.Atiyah, notes by S.K Donaldson

PART 3: THREE-DIMENSIONAL MANIFOLDS

Int roduction

1\11 introduction to polyhedral metrics of non-positivecurvatur~.on 3-manifolds

I,'illitegroups of hyperbolic isometries

('.H. Thomas

!'i,/structures on low-dimensional manifolds

I{.('. Kirby and L.R Taylor

viviiviiixixiv

13

7153545

69

7173

97

101115

123

125127

163

177

Andreas Floer, notes by Dieter Kotschick

Ronald Fintushel and Ronald J Stern

On theFIoerhomology of Seifert fibered homology 3-spheres 149Christian Okonek

Mikio Furuta

PART 3: DIFFERENTIAL GEOMETRY AND MATHEMATICAL PHYSICS 175

N.S Manton

Representations of braid groupsandoperators coupled to monopoles 191Ralph E Cohen and John D.S Jones

D.H Hartley andR.W.Tucker

I R Aitchison, Department of Mathematics, University of Melbourne, Melbourne, Australia

M F Atiyah, Mathematical Institute, 24-29 St Giles, Oxford OXl 3LB, UK

F E Burstall, School of Mathematical Sciences, University of Bath, Claverton Down, Bath, UKRalph E Cohen, Department of Mathematics, Stanford University, Stanford CA 94305, USA

S K Donaldson, Mathematical Institute, 24-29 S1 Giles, Oxford OXl 3LB, UK

Yakov Eliashberg, Department of Mathematics, Stanford University, Stanford CA 94305, USARonald Fintushel, Department of Mathematics, Michigan State University, East Lansing,

MI 48824, USA

A Floer, Department of Mathematics, University of California, Berkeley CA 94720, USAMikio Furuta, Department of Mathematics, University of Tokyo, Hongo, Tokyo 113, Japan,and,

Mathematical Institute, 24-29 St Giles, Oxford OXl 3LB, UK

A.B Givental, Lenin Institute for Physics and Chemistry, Moscow, USSR

Robert E Gompf, Department of Mathematics, University of Texas, Austin TX, USA

() H Hartley, Department of Physics, University of Lancaster, Lancaster, UK

N.J Hitchin, Mathematical Institute, 24-29 St Giles, Oxford OXl 3LB, UK

II.Hofer, FB Mathematik, Ruhr Universitat Bochum, Universitatstr 150, D-463 Bochurn, FRG(jsa Jeffrey, Mathematical Institute, 24-29 St Giles, Oxford OX} 3LB, UK

I:.A E Johnson, Department of Mathematics, University College, London WCIE 6BT, UK

J D S Jones, Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK

Robion Kirby, Department of Mathematics, University of California, Berkeley CA 94720, USA

I)icter Kotschick, Queen's College, Cambridge CB3 9ET, UK,and, The Institute for Advanced

Study, Princeton NJ 08540, USA

Matthias Kreck, Max-Planck-Institut rUr Mathematik, 23 Gottfried Claren Str., Bonn, Gennany

N.S Manton, Department of Applied Mathematics and Mathematical Physics, University of( 'ambridge, Silver St, Cambridge CB3 9EW, UK

I)usa McDuff, Department of Mathematics, SUNY, Stony Brook NY, USA

I)aulMelvin, Department of Mathematics, Bryn Mawr College, Bryn Mawr PA 19010, USA( 1hristian Okonek, Math Institut der Universitat Bonn, Wegelerstr 10, 0-5300 Bonn I, FRG.J II.Rubinstein, Department of Mathematics, University of Melbourne, Melbourne, Australia,

(/11£1,The Institute for Advanced Study, Princeton NJ 08540, USA

R()nald J Stem, Department of Mathematics, University of California, Irvine CA 92717, USA

II R Taylor, Department of Mathematics, Notre Dame University, Notre Dame IN 46556, USA

( c. B Thomas, Department of Pure Mathematics and Mathematical Statistics, University of(1~Hnbridge,16, Mill Lane, Cambridge CB3 9EW, UK

K P.Tad, Mathematical Institute, 24-29 St Giles, Oxford OXI 3LB, UK

I~. W Tucker, Department of Physics, University of Lancaster, Lancaster, UK

I':dward Witten, Institute for Advanced Study, Princeton NJ 08540, USA

'()hn (~.Wood, Department ofPureMathematics, University of Leeds, Leeds, UK

S Wang (Oxford)

R Ward (Durham)P.M.H Wilson (Cambridge)

E Witten (lAS, Princeton)

J Wood (Leeds)

In the past decade there have been a number of exciting new developments in an

area lying roughly between manifold theory and geometry More specifically, the

l)rincipal developments concern:

(1) geometric structures on manifolds,

(2) symplectic topology and geometry,

(3) applications of Yang-Mills theory to three- and four-dimensional manifolds,(4) new invariants of 3-manifolds and knots

Although they have diverse origins and roots spreading out across a wide range

( ) f mathematics and physics, these different developments display many commonf(~atures-somedetailed and precise and some more general Taken together, thesedevelopments have brought about a shift in the emphasis of current research onluanifolds, bringing the subject much closer to geometry, in its various guises, and)hysics

()neunifying feature of these geometrical developments, which contrasts with some

~(\ometrical trends in earlier decades, is that in large part they treat phenomena inspecific, low, dimensions This mirrors the distinction, long recognised in topology,I)ptween the flavours of "low-dimensional" and "high-dimensional" manifold theory(n.lthough a detailed understanding of the connection between the special roles of

t1)(~dimension in different contexts seems to lie some way off) This feature explainst.he title of the meeting held in Durham in 1989 anq in turn of these volumes ofPl'oeeedings, and we hope that it captures some of the spirit of these different

cI(-velopments

It, tnay be interesting in a general introduction to recall the the emergence of some

of t.hese ideas, and some of the papers which seem to us to have been landmarks

(We postpone mathematical technicalities to the specialised introductions to theHix separate sections of these volumes.) The developments can be said to have

1.(~~t1nwith the lectures [T] given in Princeton in 1978-79 by W.Thurston, in which1)(' developed his "geometrisation" programme for 3-manifolds Apart from theillll)(~tusgiven to old classification problems, Thurston's work was important forfoil<' way in which it encouraged mathematicians to look at a manifold in terms ofvarious concomitant geometrical structures For example, among the ideas exploitedill I'f]the following were to have perhaps half-suspected fall-out: representations ofliuk groups as discrete subgroups ofPSL 2 (C),surgery compatible with geometrictdol'tlet.ure, rigidity, Gromov's norm with values in the real singular homology, and1l1oHt important of all, use of the theory of Riemann surfaces and Fuchsian groups

to develop a feel for what might be true for special classes of manifolds in higher

fIililt-llsions

f\1(·H.llwhile, another important signpost for future developments was Y Eliashberg'sproofin 1981 of "symplectic rigidity"- the fact that the group of symplectic diffeo-IllOl'phisrns of a symplecti(~ l'unJ1ifo]<! isCfo-elosedin the full diffeomorphism group

This is perhaps a rather technical result, but it had been isolated by Gromov in

1970 as the crux of a comprehensive "hard versus soft" alternative in tic topology": Gromov showed that if this rigidity result was not true then anyproblem in symplectic topology (for example the classification of symplectic struc-tures) would admit a purely algebro-topological solution (in terms of cohomology,characteristic classes, bundle theory etc.) Conversely, the rigidity result shows theneed to study deeper and more specifically geometrical phenomena, beyond those

"symplec-of algebraic topology

Eliashberg's original proof of symplectic rigidity was never fully published but thereare now a number of proofs available, each using new phenomena in symplecticgeometry as these have been uncovered The best known of these is the "Arnol'dConjecture" [A] on fixed points of symplectic diffeomeorphisms The original form

of the conjecture, for a torus, was proved by Conley and Zehnder in 1982 [CZ]and this established rigidity, since it showed that the symplectic hypothesis forcedmore fixed points than required by ordinary topological considerations Anotherdemonstration of this rigidity, this time for contact manifolds, was provided in 1982

by Bennequin with his construction [B] of "exotic" contact structures on R3.

Staying with symplectic geometry, but moving on to 1984, Gromov [G) introduced

"pseudo-holomorphic curves" as a new tool, thus bringing into play techniquesfrom algebraic and differential geometry and analysis He used these techniques

to prove many rigidity results, including some extensions of the Arnol'd conjectureand the existence of exotic symplectic structures on Euclidean space ( Our "low-dimensional" theme may appear not to cover these developments in symplecticgeometry, which in large part apply to symplectic manifolds of all dimensions: whatone should have in mind are the crucial properties of the two-dimensional sunaces,

or pseudo-holomorphic curves, used in Gromov's theory Moreover his results seem

to be particularly sharp in low dimensions.)

We turn now to 4-manifolds and step back two years At the Bonner Arbeitstagung

in June 1982 Michael Atiyah lectured on Donaldson's work on smooth 4-manifoldswith definite intersection,form , proving that the intersection form of such a manifoldmust be "standard" This was the first application of the "instanton" solutions ofthe Yang-Mills equations as a tool in 4-manifold theory, using the moduli space ofsolutions to provide a cobordism between such a 4-manifold and a specific union

of Cp2,s [D] This approach again brought a substantial amount of analysis and ,differential geometry to bear in a new way, using analytical techniques which were 'developed shortly before Seminal ideas go back to the 1980 paper [SU] of Sacks andUhlenbeck They showed what could be done with non-linear elliptic problems forwhich, because of conformal invariance, the relevant estimates lie on the borderline

of the Sobolev inequalities These analytical techniques are relevant both in: theYang-Mills theory and also to pseudo-holomorphic curves Other important andinfluential analytical techniques, motivated in part by Physics, were developed byC.Taubes [Tal

Introduction xi

Combined with the topological h-cobordism theorem of M Freedman, proved shortlybefore, the result on smooth 4-manifolds with definite forms was quickly used todeduce, among other things, that R4admits exotic smooth structures Many differ-ent applications of these instantons, leading to strong differential-topological con-clusions, were made in the following years by a number of mathematicians; theother main strand in the work being the definition of new invariants for smooth4-manifolds, and their use to detect distinct differentiable structures on complexalgebraic surfaces (thus refuting the smooth h-cobordism theorem in four dimen-sions)

From an apparently totally different direction the Jones polynomial emerged in a

series of seminars held at the University of Geneva in the summer of 1984 This was

a.new invariant of knots and links which, in its original form [J], is defined by thetraces of a series of representations of the Braid Groups which had been encountered

in the theory of von Neumann algebras, and were previously known in statisticalluechanics For some time, in spite of its obvious power as an invariant of knotsand links in ordinary space, the geometric meaning of the Jones invariant remainedrather mysterious, although a multitude of connections were discovered with (amongother things) combinatorics, exactly soluble models in statistical physics and con-rormal field theories

[n the spring of the next year, 1985, A Casson gave a series of lectures in Berkeley

()1) a new integer invariant for homology 3-spheres which he had discovered ThisCasson invariant "counts" the number of representations of the fundamental group

ill SU(2) and has a number of very interesting properties On the one hand it gives

an integer lifting of the well-established Rohlin Z/2 J-t-invariant On the other handCasson's definition was very geometric, employing the moduli spaces of unitaryr<~presentationsof the fundamental groups of surfaces in an essential way (Theseluoduli spaces had been extensively studied by algebraic geometers, and from thepoint of view of Yang-Mills theory in the influential 1982 paper of Atiyah and Bott

lAB].) Since such representations correspond to flat connections it was clear that(~asson'stheory would very likely make contact with the more analytical work onYang-Mills fields On the other hand Casson showed, in his study 9f the behaviour

and more familiar techniques in geometric topology For a very readable account of

(~assonswork see the survey by A Marin [M].

Around 1986 A Floer introduced important new ideas which applied both to pl(~cticgeometry and to Yang-Mills theory, providing a prime example of the in-t.~~raction between these two fields Floer's theory brought together a number ofI)()we:r£ul ingredients; one of the most distinctive was his novel use of ideas fromMorse theory An important motivation for Floer's approach was the 1982 pa-I)('r by E Witten [WI] which, among other things, gave a new analytical proof of

sym-I.hc' Morse inequalities and explained their connection with instantons, as used in

<.Jl1a.ntum Theory

In symplectic geometry one of Floer's main acheivements was the proof of ageneralised form of the Arnol'd conjecture [FI] On the Yang-Mills side, Floerdefined new invariants of homology 3-spheres, the instanton homology groups [F2].

By work of Taubes the Casson invariant equals one half of the Euler characteristic

of these homology groups Their definition uses moduli spaces of instantons over

a 4-dimensional tube, asymptotic to flat connections at the ends, and these areinterpreted in the Morse theory picture as the gradient flow lines connecting criticalpoints of the Chern-Simons functional

Even more recently (1988), Witten has provided a quantum field theoretic tation of the various Yang-Mills invariants of 4-manifolds and, in the other direction,has used ideas from quantum field theory to give a purely 3-dimensional definition

interpre-of the Jones link invariants (W2] Witten's idea is to use a functional integral volving the Chern-Simons invariant and holonomy around loops, over the space ofall connections over a 3-manifold The beauty of this approach is illustrated by thefact that the choices (quantisations) involved in the construction of the represen-tations used by Jones reflect the need to make this integral actually defined Inaddition Witten was able to find new invariants for 3-manifolds

in-It should be clear, even (roln this bald historical summary, how fruitful the fertilisation between the various theories has been When the idea of a Durhamconference on this area was first mooted, in the summer of 1984, the organiserscertainly intended that it should cover Yang-Mills theory, symplectic geometry andrelated developments in theoretical physics However the proposal was left va.gueenough to allow for unpredictable progress, sudden shifts of interest, new insights,and the travel plans of those invited We believe that the richness of the contribu-tions in both volumes has justified our approach, but as always the final judgementrests with the reader

cros-References

[A] Arnold, V.I Mathematical Methods of Classical Mechanics Springer,

Grad-uate Texts in Mathematics, New York (1978)

[AB] Atiyah, M.F and Batt, R The Yang-Mills equations over Riemann surfaces

Phil Trans Roy Soc London, Sere A 308 (1982) 523-615

[B] Bennequin, D Entrelacements et equations de Pfaff Asterisque 107-108

1983) 87-91

[CZ] Conley, C and Zehnder, E The Birkhoff-Lewis fixed-point theorem and a conjecture of V.I Arnold Inventiones Math.73 (1983) 33-49

[D) Donaldson, S.K An application ofgauge theozy to four dimensional topology

Jour Differential Geometry 18 (1983) 269-316

[Fl] Floer, A Morse Theozy for Lagrangian intersections Jour DifferentialGeometry 28 (1988) 513-547

[M] Marin, A (after A Casson) Un nouvel invariant pour les spheres d'homologie

de dimension troisSem Bourbaki, no 693,fevrier 1988 (Asterisque161-162(1988)151-164 )

[SU] Sacks, J and Uhlenbeck, I{.K The existence of minimal immersions of

2-spheres Annals of Math 113(1981) 1-24

[T] Thurston, W.P The Topology and Geometry of 3-manifolds Princeton versity Lecture Notes, 1978

Uni-[Ta] Taubes, C.H Self-dual connections on non-sell-dual lour manifolds Jour.Differential Geometry 17 (1982) 139-170

[WI] Witten, E Supersymmetry and Morse TheoryJour Differential Geometry

17 (1982) 661-692

[W2] Witten, E Some geometrical applications of Quantum Field Theo:cy Proc

IXth International Congress on Mathematical Physics, Adam Hilger (Bristol) 1989,

pp 77-110

We should like to take this opportunity to thank the London Mathematical ciety and the Science and Engineering Research Coun,cil for their generous support

So-of the Symposium in Durham We thank the members So-of the Durham ics Department, particularly Professor Philip Higgins, Dr John Bolton and Dr.Richard Ward, for their work and hospitality in putting on the meeting, and Mrs

Mathemat-S Nesbitt and Mrs J Gibson who provided most efficient organisation We alsothank all those at Grey College who arranged the accommodation for the partici-pants Finally we should like to thank Dieter Kotschick and Lisa Jeffrey for writing

up notes on some of the lectures, which have made an important addition to thesevolumes

PART 1 SYMPLECTIC GEOMETRY

In this section we gather together papers on symplectic and contact geometry.Ilecall that a symplectic manifold(M, w)is a smooth manifoldMof even dimension2n with a closed, nondegenerate, 2-formw i.e dw = 0 andw n is nowhere zero Acontact structure is an odd-dimensional analogue; a contact manifold(V, H)is a pair(O()llsisting of a manifoldV of odd dimension2n+1 with a fieldH of 2n-dimensionalHuhspaces of the tangent bundleTVwhich is maximally non-integrable, in the sensethat ifa is a I-form defining H, then dan 1\ a is non-zero (i.e da is non-degenerate

(HlH).

III t.heir different ways, all the articles in this section are motivated by the work

ofM Gromov, and in particular by his paper [G2] on pseudo-holomorphic curves.11(~rcthe idea is to replace a complex manifold by an almost-complex manifold with

I\.("ompatible symplectic structure, and to study the generalisations of the complex(,Ilrvcs-defined by this almost-complex structure The paper of McDuff belowv;iV(~Ha direct application of this method by showing that a minimal 4-dimensionalHylllplectic manifold containing an embedded, symplectic, copy of 82 = cpt isC'ither Cp2 or an S2 bundle over a Riemann surface, with the symplectic forml-e'inp; non-degenerate on fibres The uniqueness of the structure in the minimal4'U.:'H\can be thought of as an example of rigidity

AIlother important symplectic notion investigated by Gromov is that of "squeezing"

11(0proves for example that the polycylinderD2(1)x··· xD 2 (1) (nfactors) cannot1"0 ~ymplectically embedded in D 2 (R) X R2n-2 if the radius R of the disc is less

t.han 1 H Hofer (see below) approaches this and other questions from the point of

vic'wof a symplectic capacity: we can summarise his definition as follows

Acapacity c is a function defined on all sub8ets of symplectic manifold8 of a given

tli17u~nsion2n taking values in the positive real numbers augmented by 00 and ' •Iying the axioms;

8at-( ~()) Iff is a symplectic DEM defined in a neighbourhood of a subset S C (M, w) ,It"fl c(S,w) =c(f(5), f*(w)).

( ~I )(Conformality) If A>0 then c(S, AW) = AC( S,w).

(~2)(Monotonicity)If (S,w) C (T,w) then c(S,w):5 c(T,w).

(~:J)(Normalisation)c(D2n(1) =c(D 2 (1)xD 2n-2(r))= 7rfor allr ~ 1, with respect

/" lh(~standard symplectic form on en =C X Cn- 1

All ('xample of a capacity is provided by the "displacement energy"-heuristically,

#J.iv(IU two disjoint bounded subsets S, S' ofR2

n how much energy does one need

t u dpform 8 into 5'? As Hofer shows, this capacity can be used to prove the'Ic(lH"('hing theorem above, and furthermore Axioms CO-C2 suffice to recover the

t11,.idit,ytheorem (mentioned in the general introduction) namely that the symplectic

J)11:M p;roup isCo closed in 4Diff(M).

'1'11(' ('xi~tence of periodic solutions for Hamiltonian systems leads to a whole family

•If independent capacities, which provide a framework in which to discuss the

ex-4 Introduction

istence of a closed integral curves for the vector field associated with an arbitrarycontact structure on S2n-l. More generally still there is the possibility of usingFloer's symplectic instanton homology groups to define sequence-valued capacities.These homology groups were the tool used by Floer to prove Arnol'd's conjecture

on the fixed points of symplectic diffeomorphisms for a wide class of manifolds Werefer to the notes by Kotschick in the first volume of these proceedings for the def-inition of the symplectic instanton homology groups, and their relation to Floer'shomology groups for 3-manifolds Gromov's paper provides many parallels betweenthe theory of Yang-Mills instantons and pseudo-holomorphic curves, and betweenthe results derived from the two For example, the existence of a symplectic form

onR2n which is not the restriction of the standard form under some embedding ofR2n in itself is reminiscent of the exotic smooth structures on R4•

The Arnol'd conjecture can also be discussed in the framework of the "NonlinearMaslov Index" developed by A Givental and described in his article below Thedefinition can be summarised as follows: the linear Maslov index of a loop'Y in themanifoldAnof linear Legendrian subspaces ofRp2n-l is its class in the7rl(An) = z.

The nonlinear invariant is obtained by replacing An by the infinite-dimensionalhomogeneous space of all Legendrian embeddings ofRpn-l in Rp2n-l

The various fixed-point theorems now in the literature illustrate the "hard" aspect

of symplectic geometry (for this terminology see [G3]) This rests on the Gromov rigidity theorem to which we have referred above, and in the general in-troduction In another striking parallel with Coo theory Gromov has shown that4-dimensional symplectic theory has some of the flavour of smooth surfaces Forexample, ifM = 82X 82 has the standard symplectic structurewEBw coming fromthe Kahler forms on the factors, then DiffW(jjw(M) contracts onto the isometrygroup ofM, which is a Z/2Z extension ofSO(3) X SO(3) Question: does rigidity

Eliashberg-give rise to similar results for contact manifolds in dimension 5, for example 85 or

of the difficulties which arise are illustrated as follows Let E be a 2-dimensional face embedded in the three-dimensional boundary of an almost-complex manifold

sur-M 4 , and consider the natural foliations of the 3-dimensional cylinderZ = D 2X[0,1]

(or 83 minus two poles) by holomorphic discs If E is diffeomorphic to8Zor8D 3 ,

and if the embedding of E in8Mcan be extended to an embedding of Z or D 3 in

A.J4 which is holomorphic on the leaves of the foliation, then we say thatE is fillable

I)y holomorphic discs Under certain conditions, explained below in the article of11~liashberg,these extensions exist and this can be used, for example, to provide alu'cessary condition (not "overtwisted") for a contact structure on8M to bounda

!Iylllplectic structure onM.

W(, conclude this introduction with a few remarks about contact manifolds Here

lH one reason for believing that contact manifolds may be "softer" than

symplec-tiC": ifVi andV2 are contact, then their connected sum admits a contact structure

"~J'(~cingwith the original forms on VI, V 2 outside small discs This cannot ('III"for symplectic manifolds-the basic difference being that the odd-dimensional,'pltcreS2n+l admits a contact forma such that da is the pull-back of the standard

oc-nylllplectic form on cpn It follows that O-surgeries can be performed on contactIlIH.uifolds, and2n surgeries seem to fit into the same framework The situation for

:~u+1surgeries is more delicate at least under certain conditions a I-surgery canI ,(°a.rried out, as Thurston and Winkelnkemper sho\ved in dimension 3 And by

tHiiIJp;a rag-bag of special tricks it is possible to prove the soft realisation conjecture,ill thecontact case, for a large class of n - I-connected 2n+I-manifolds Thus atI,ll(' time of writing the situation is tantalisingly similar to that for codimension-I

I'c)lint.ionsbefore the major contribution of Thurston; see [T] forasummary of what

WItS known a few years back

AN li~liashberg's paper shows, much work has been done in dimension3, stimulated

111)t()nly by Gromov but also by the work of Bennequin It would be very interesting

t I ~ S('Cif contact geometry can be applied to classification problems in 3-dimensional

topology For example, A Weinstein conjectures that ifHI (V 3 ;Z)is finite (and in

I'llrt.icular if V 3 has universal cover diffeomorphic to S3), then the characteristicfoliat.ion of an s-fillable contact form contains at least one closed orbit Under what,·,,"dit.ions is it possible to use the existence of such orbits to construct a Seifert

Ii I "ring of V? Such a manifold would then be elliptic, completing part of the

~J.''.)1)letrisation programme

1(: 1]Gromov, M Partial differential relations Springer Berlin-Heidelberg (1986)

1<;2] Gromov,M Pseudoholomorphic curves in symplectic manifolds Inventiones

~1"tl1 82 (1985) 307-347

1< ;:~] Gromov, M Soft and hard Symplectic Geometry Proc Int Congress

Math-.<tllilticians, Berkeley 1986, Vol I, 81-98

IIIIIIHellCentre Publications 18 (1986) 254-270

I'his note describes the structure of compact symplectic 4-manifolds (V, w) which contain

nsYlnplectically embedded copy C of S2 with non-negative self-intersection number

(Suchcurves C are called "rational curves" by Gromov: see [G].) It turns out that there is

IIconcept of minimality for symplectic 4-manifolds which mimics that for complex

"lIrl~lces. Further, a minimal manifold (V,(0) which contains a rational curve C is eitherNYlnplectomorphic to CP2 with its usual Kahler structure T,or is the total space of attPliynlplectic ruled surface" i.e an S2-bundle over a Riemann surface M, with a symplectic

f.»nn which is non-degenerate on the fibers It follows that if a (possibly non-minimal)(V,w) containsamtionalculVe C with C·C>0, then (V,w) maybe blown down either

tu S2x S2 with a product form or to (CP2, T),and hence is birationally equivalent to

G' P 2 in Guillemin and Sternberg's sense: see [GS] (In analogy with the complex case,

we will call such manifolds rationaL) Moreover,if V contains a rational curve C with( ' (' =0, then V may be blown down to a symplectic ruled surface Thus, symplectic 4-lIutnifolds which contain rational curves of non-negative self-intersection behave very much

Itkl-rational or ruled complex surfaces

It Isnatural to ask about the uniqueness of the symplectic structure on the manifolds under

• nnsideration: more precisely, if000 and WI are cohomologous symplectic forms on V

which both admit rational curves of non-negative self-intersection, are theyIIYlnplectomorphic? We will see below that the answer is "yes" if the manifolds in

.,u("stionare minimal In the general case, the most that is known at present is that any two

- rh forms may be joined by a family· Wt, 0s ts 1, of (possibly non-cohomologous)

"vlllpicctic forms on V Since the cohomology class varies here, this does not imply thatIhc'fonns WQ and WI are symplectomorphic: cf[McD 1] Similarly, all the symplectic

( +) partially supported by NSF grant no: DMS 8803056

I tlKO Mathematics Subject Classification (revised 1985): 53 C 15, 57 R 99

~fV words: symplectic manifold, 4-manifolds, pseudo-holomorphic curves, almost plcx manifold, blowing up

fonns under consideration are Kihler for some integrable complex structure J on V,provided that V is minimal In the general case, we know only that wmay be joined to aKIDder fonn by a family as above Note also that there might be some completely differentsymplectic forms on these manifolds which do not admit rational curves.

The present work was inspired by Gromov's result in [G] that if (V,w) is a compactsymplectic 4-manifold whose second homology group is generated by a symplecticallyembedded 2-sphere of self-intersection +1, then V is Cp2 with its usual Kahlerstmcture Our proofs rely heavily on his theory of pseudo-holomorphic curves The maininnovation is a homological version of the adjunction fonnula which is valid for almostcomplex 4-manifolds (See Proposition 2.9 below.) This gives a homological criterionfor a pseudo-holomorphic curve in an almost-complex 4-manifold to be embedded, and is apowerful mechanism for relating the homological properties of a symplectic manifold V tothe geometry of its pseudo-holomorphic curves We also use some new cutting andpasting techniques to reduce the ruled case to the rational case

Proofs of the results stated here appear in [McD 3,4] I wish to thank Ya Eliashberg formany stimulating discussions about the questions studied here I am also grateful to MRSIfor its hospitality and support during the initial stages of this work

2 STATEMENT OF RESULTS

We will begin by discussing blowing up and blowing down All manifolds considered will

be smooth, compact and, unless specific mention is made to the contrary, withoutboundary

By analogy with the theory of complex surfaces, we will say that (V,w) is minimal if itcontains no exceptional CUNes, that is, symplectically embedded 2-spheres I: with self-intersection number r r.= -1 We showed in [McD 2] Lemma 2.1 that every exceptionalcurve E has a neighbourhood NEwhose boundary (ONE'(0) may be identified with theboundary (OB4(A+e),(00) of the ball of radius A+e in CP2,where 11"A2=w(E) and

e> 0 is sufficiently small Hence E can be blown down by cutting out NE and gluing inthe ball B4(A+e),with its standard form 000. It is easy to check that the resultingmanifold is independent of the choice of f" so that there is a well-defined blowing downoperation, which is inverse to symplectic blowing up

The following result is not hard to prove:itsmain point is that one blowing down operationsuffices

McDuff: Rational and ruled symplectic 4-manifolds 9

2.1 Theorem

I!'vcry symplectic 4-manifold(V,00) covetsaminimal symplectic manifold(V',(a)')which

'lilly be obtained from V by blowing downa finite collection of disjoint exceptional ('lIlves Moreover, the induced symplectic form 00' on V' is unique up to isotopy

There is also a version of Theorem 1 for manifold pairs (V, C) where C is aNynlplectically embedded compact 2-manifold in V We will call such a pair minimal if

V C contains no exceptional curves

2.2 'rheorem

I,'very symplectic pair (V, C, (0) coversaminimal symplectic pair (V', C, 00') which

"'/I.Ybe obtained by blowing downafinite collection ofdisjoint exceptional CULVes inV-C

Alol"cover, the induced symplectic form 00' on V' is unique up to isotopy (reI C).

J J Note

If (., is a closed subset of V, two symplectic fonns 000 and 001 are said to be isotopic(1"("1 <:) if they can be joined by a family of cohomologous symplectic forms whose

Irstrictions to C are all equal If C is symplectic, Moser's theorem then implies that there

INun isotopy gt of V which is the identity on C and is such that gl*((01) = 000.

ItIswell-known that the diffeomorphism type of V' isnot uniquely detennined by that of

V For example, because (S2x82)#Cp2 is diffeomorphic to CP2# CP2# CP2, the.nunifold V = (S2 x 82 ) #CP2 may be reduced to Cp2 as well as to S2 x S2.Ilowcver,this is essentially the only ambiguity, and V'is detennined up to diffeomorphism

Ifwe fix the homology classes of the curves which are blown down

( 'onvcrsely, one can ask to what extent the minimal manifold (V',w') determines itshlowing up (V, (0) Since each exceptional curve L in (V, (0) corresponds to an

r.uhcdded ball in V' of radius A,wherew(L)=ll' A 2, this question is related to properties

tt'-tilespace of symplectic embeddings of llB(Ai) into (V',00'),where liB(Ai) is the.U!ijoint union of the symplectic 4-balls B()q) of radius Ai We discussed the

• «.rresponding question for manifold pairs in [McD 2] We showed there that, if C·C= 1,fUld if V is diffeomorphic to CP2 with k points blown up, there is a unique symplecticIlI.lIctureon (V, C) in the cohomology class a if and only if the space of symplectic'-Ulht·ddings of liB(Ai) into Cp2 - CP1 is connected, where 1l' A1 2, •~ ,ll'Ak2 are thevuhJt~sof a on the exceptional curves in V Unfortunately nothing is known about this'1pUC:t·of embeddings per see In fact, the information we have goes the other way: weprovedin [McD 2] that the structure on (V, C) is unique when k 1, which implies that

the corresponding space of embeddings is connected (Because CP2#CP2 is ruled, thisuniqueness statement is closely related to the results in Theorem 2.4 below.) It is not clearwhat happens when k~2 However, because any two embeddings of liB()\j) into(V',oo') are isotopic when restricted to the union llB(Ei) of suitably small subballs, anytwo forms on V which blow-down to (a)' may be joined by a family of non-cohomologous forms.

Thus, the problem is essentially reduced to understanding the minimal case The nextingredient isaresult on the structure of symplectic52-bundles (symplectic ruled surfaces).2.4 TheoremLet V be an oriented5 2 -bundle 1£: V -+ M overa COllJpact oriented surfaceMwith fiber F

(i) The cohomology class a of any symplectic [onn on V which is non-degenerate on each fiber of 1t' satisfies the conditions:

(a) a(F) and a2(V) are positive, and

(b) a2(V) > (a(F))2 ifthe bundle is non-trivial.

(ii) Any cohomology class a E H2(V;Z) which satisfies the above conditions may be represented by a symplectic (onn tL) which is non-degenerate on each fiber of 1T.

Moreover, this {ann is unique up to isotopy.

The existence statement in (ii) above is well-known It is obvious if the bundle is trivial If

itis non-trivial, one can think of V as the suspension of a circle bundle of Euler class I

with the corresponding Sl-action, and can then provide V with an invariant symplecticform in any class a which satisfies (i) (a,b) since these conditions correspond to requiringthat a be positive on each of the two fixed point sets of the 5 I-action (See [Au].) Theother statements are more delicate Consider first the case when the base manifold M is

52 Gr~movshowed in [G] that any symplectic fonn on 52x 52 which admitssymplecticallyembedded spheres in the classes [52xpt] and [ptx52] is isotopic to aproduct (or split) form (In fact, Gromov assumed that the form has equal integrals over thetwo spheres, butit is not hard to remove this condition.) A corresponding uniquenessresult when V=CP2#CP2 (which is the non-trivial SLbundle over 52) was proved in[McD 2J Here one requires the existence ofjust one symplectically embedded sphere, but

itmustbein the class of a section of self-intersection 1, not of the fiber Gromov showedthat this hypothesis also implies that there must be a symplectically embedded sphere in theclass of the blown-up point, which is equivalent to condition (i)(b) In the presentsituation, we have less infonnation since we start with only one symplectically embeddedsphere, the fiber Following an idea of~liashberg's,we can construct a symplectic section

of1T (i.e a section on which the symplectic form 00 does not vanish) and so reduce to thepreviously considered case In the process, we have t~change the form w by adding

11'*(0) where 0is a 2-fooo on M such that o(M)>o. The argument is then completed

by the following lemma, the proof of which uses the theory of hoiomorphic curves

McDuff: Rational and ruled symplectic 4-manifolds 11

1.S Lemma

I,et V be an S2-bundJe overaRiemann surface M and suppose that oot,0s t s 1, isa

lill"ily of(non-cohomologous) symplectic forms onV which are non-degenerate on one

I1bl·( ofV Then, if001 admits a symplectic section, so does 000.

II'(hegeneral case, one cuts the fibration open over the I-skeleton of M in order to reduce

S.' 1>ulldJe, or (if M is S2) to a section ofthis bundle.

I hus, jfp is odd and~3, (V,(0)is symplectomorphic to the Kahler manifold Cp2# CP2;

II P== 1, (V, (0) is Cp2 with its standard Kahler form; if p is even and ~2, (V,(0)I,.theproduct S2 x S2 with a product symplectic form (or, if p= 4,it could be CP2);Iliul ifp=0, (V,(0) is a symplectic S2-bundle From this, itis easy to prove:

J / (~orollary

( I) II' (V, C,(0) is as above, the diffeomorphism type ofthe pair (V, C) is detennined by

II ,.rovided that p :1=.0, 4

(II) When p= 4,there are two possibilities for (V, C): it can be either (CP2,Q) or

(SJ xS2, [2), where Q is a quadric and where [2 is the graph ofa holomorphic

self-""'1'of S2 ofdegree 2 When p=0, C is a fiber ofa symplectic S2-bundle.

(Ill) (V) C,(0) is determined up to symplectomolphism by the cohomology class of00.

J.H<:orollary

, '" ill illlaJ symplectic 4-manifold (V,w) which contains a rational curve C with C· C>0

i~ "flip/ectomorphic either to CP2 or to S2 xS2 with the standard form.

Ill(' Illain tool in the proof of Theorem 2.6 is the following version of the adjunction

'ullllula We will suppose that J is an almost complex structure on V with first Chern

.I,I.\S cl,andthat f:S2~ V isaJ-holomorphicmap (ie dfoJo=Jodf,where Jo is

tlu· usualalmost complex structureon S2) which represents the homology class A E

H2(V;I). The assumption that f is somewhere injective lUles out the multiply-coveredcase, and implies that f is an embedding except for a fmite number of multiple points and afinite number of "critical points" , i.e points where dfz vanishes.

2.9 Proposition

If f is somewhere injective, then

A·A~Cl(A)-2

with equality ifand only if f is an embedding.

This is well-known if J is integrable: the quantity 1/2 (A A - cl(A)+2) is knownasthe "virtual genus" of the curve C= 1m f It is also easy to prove if f is an immersion.For in this case cl(A)=2+Cl(VC) where vc is the nonnal bundle to C, and Cl(VC) S

A·A, with equality ifand only if f isanembedding In the general case, one has to showthat each singularity ofC contributes positively to A·A This is not hard to show for thesimplest kind of singularities, and, using the techniques of [NW], one can reduce to these

by a rather delicate perturbation argument

With this in hand, we prove Theorem 2.6 by showing that V must contain an embeddedI-simple curve of self-intersection+1 or o. (J-simple curves do not decompose, so thattheir moduli space is compact.) It then follows by arguments of Gromov that V is CP2

in the former case and a symplectic S2-bundleinthe latter

2.10 Note

Given an arbitrary symplectic 4-manifold one can always blow up some points to create amanifold (W, 00) which contains a symplectically embedded 2-sphere with an arbitrarynegative self-intersection number Hence, the existence of such a 2-sphere gives noinfonnation on the structure of (W,w).

Corollary 2.7 maybeunderstood as a statement about the uniqueness of symplectic fillings

of certain contact manifolds Indeed, consider an oriented (2n-l)-dimensional manifold Iiwith closed 2-fonn o We will say that (A,o) has contact~if there is a positivelyoriented contact form a on !J such that da=o It is easy to check that the contactstructure thus defined is independent of the choice of a. Following Eliashberg [E], wesay that the symplectic manifold (Z,oo) fills (Ii,o) if there is a diffeomorphism f:oZ-+

A such that f*(o)=wloZ Further the filling (Z,(0) is said to be minimal ifZ contains

no exceptional curves in its interior

As Eliashberg points out, information on symplectic fillings provides a way to distinguishbetween contact structures: if one constructs a filling of (A,02) which does nothave a

McDuff: Rational and ruled symplectic 4-manifolds 13

certain property which one knows must be possessed by all fillings of (a,ol), then thecontact structures on a defined by 01 and 02 must be different In particular, it isinteresting to look for manifolds of contact type which have unique minimal fillings.()bvious candidates are the lens spaces Lp ,p > 1, which are obtained as the quotients of

S3c C2 by the standard diagonal action of the cyclic subgroup rp c SI of order p on

4~2,and whose 2-form 0 is induced by 000

Itis not hard to see that if (Z,oo) fills (Lp,a) we may quotient outaz=Lp by the HopfInapto obtain a rational curve Cp with self-intersection p in a symplectic manifold (V,0)

without boundary Hence Corollary 2.7 implies:

2.11 Theorem

The lens spaces Lp,p~1, all have minimal symplectic fillings Ifp:/=.4,minimal fillings (Z,oo) of (Lp,a) are unique up to diffeomorphism, and up to symplectomorphism if one lixes the cohomology class [00] However, (L4, 0) has exactly two non-diffeomorphic 11linimaJ fillings.

Inhigher dimensions, one cannot hope for such precise results However, in dimension 6there are certain contact-type manifolds (such as the standard contact sphere S5) whichimpose conditions on any filling (Z,oo), even though they may not dictate thediffeomorphism type of minimal fillings In dimensions> 6, one must restrict to "semi-positive" fillings to get analogous results See [McD5]

IMcD 2] McDuff, D.: Blowing up and symplectic embeddings in dimension 4, to appear

Symplectic Capacities

H.HOFER

FB Mathematik Ruhr Universitat Bochum Federal Republic of Germany

August 3, 1990

In 1985 M Gromov proved in his seminal paper Pseudoholomorphic Curves in Symplectic Geometry,[1],a striking rigidity result, the so-called squeezing theorem.Consider the symplectic vectorspace R2n:= (R2)n with the coordinates z =

(Zl' • ,Zn),Zi=(Xi,Yi), and the symplectic form( j , defined by

Gromov proved, that given a symplectic embedding

we necessarily have the inequality r :5 r'. His proof of this fact relied on his existencetheory for pseudoholomorphic curves, [1]

Coming from the variational theory of Hamiltonian Dynamics, I Ekeland andthe author observed in [2,3], that using Hamiltonian dynamics the squeezing theo-rem not only can be proved as well, but in fact many interesting symplectic invari-

ants, so-called Symplectic Capacities can be constructed.

In this way a fruitful hook up between Symplectic Geometry and Hamiltonian

Dynamics - with their different methods - has been achieved This merger forexample allows to see the nonobvious but very deep relationship between certainaspects of Hamiltonian Dynamics and Symplectic rigidity The aim of this paper is

to survey this relationship We start with an axiomatic approach, then we surveyseveral constructions of a symplectic capacity

Acknowledgement

I would like to thank 1 Ekeland for many stimulating discussions

Consider the category S ymp 2n., consisting of all (2n )-dimensional symplectic

man-ifolds, together with the morphisms being the symplectic embeddings We denote

byS a subcategory ofSymp 2n. We do not(!) assume that S is a full subcategory

The following scaling axiom is imposed:

(8) If (M,w) E8 then also(M,OlW) ES fora ER \ {O}

Denote by r = (0,+00)U{+oo} the extended positive half line The obviousordering onr is denoted by ~. In the usual way we consider(r,S) as a category

Definition 2.1 A symplectic capacity for 8 is a covariant functor c :S ~r satis fying the axioms

(N) C(Z2n(1),u) < +00

Here(N) stands for nontriviality and(C) for conformality

Several remarks are in order

Remark 2.2 1 The map (M,w) (fwn)'k sati8fie8 (C) but not (N) ifn 2:: 2

So if n ~2the nontriviality axiom excludes volume-relatedinvari~nts. However, if

n = 1, then (M,w) vol(M,w) is a symplectic capacity.

2 Besides S = S ymp 2n there are several other interesting subcategories FOT

example let (V, w) be a symplectic manifold Denote by Op (V, w) the collection of all symplectic manifolds of the form (U, OlWIU) where U is a nonempty open subset

of V and a ER \ {OJ. As morphisms we take the symplectic embeddings Clearly

S =Op (V,w) is an admissible category Next assume G is a compact Lie group.

Hofer: Symplectic capacities 17

l)(~note by Sa the collection of all smooth (2n )-dimensional symplectic G-spaces, 'where G acts Jymplectically As morphisms we take the symplectic G-embeddings.

3 One might look for functors c : S ~ r which transform differently For

t~xamplec(M,aw) =1a Ik c(M,w) for some k f:.1. To exclude certain pathologies

tine could impose the following axiom system Let k E {l, ,n} A symplectic

~~.capacityis a covariant functorc :S-+r satisfying

{

c(B2(k-l)(1) X R 2(n-k+l),u) = +00

(N)k C(B2k(1) XR2(n-k),u) <+00

We note that for k=1 we precisely recover our definition of a symplectic capacity

[nfact the first part of(N)! follows from (C)l and the fact that c(B2n(1), 0') > o.

l~or k = n the volume map is an example of a n-capacity So far no examplesn.re known for intermediate capacities (1 < k < n). It is quite possible that they

<10 not exist Some evidence for this possibility are given by the fact that there

iH an enormous amount of flexibility for symplectic embeddings M C- + N withclimM~ dimN+2, see Gromov's marvellous book [4] For example, the existence

()fa symplectic embedding

- 2 (1) X R2

for some e > 0, would immediately contradict the axioms for a k-capacity 1 <

k < n As a consequence of results in [3J, concerning symplectic capacities, the

inequality£-2 ~n-1is a necessary condition for the existence of such a symplectic(\lnbedding

Next we derive some easy consequences of the axioms Set S = S ymp 2n andassume c :S ~r is a symplectic capacity

Lemma 2.3 Assume there ezists a symplectic embedding 'I1 : Z2n(rl) ~ Z2n(r2) Then rl ~r2.

PROOF: First we note that the map

is a symplectic diffeomorphism Hence

HereT E (0,+00) as a consequence of the axioms Ifnow W :z2n(rl) ~ Z2n(r2)

we obtain, using the axioms, Tr~ $ rri.

An even stronger result is

Lemma 2.4 There exists a constant R oE(0,+00) such that there is no symplectic ' embedding

PROOF: IfB 2 n(R)embeds symplectica11y into z2n(1) we obtain

So far we have only studied trivial consequences The following corollaries aresubstantially deeper We call a subsetE C R2nan ellipsoid if there exists a positivedefinite quadratic form q such that E = {z E R2

n Iq(z) < I} Assume c is acapacity defined on S = Op(R2n, u). We extend c to a map 2R2n X (R \ {O}) -+

[0,+00] by defining

•

Hence R o= 1 is the best possible choice.

':!J'

Let us also write c(U) :=c(U,1)for U C R2

n. It has been proved in [2,3]:

(1)

Proposition 2.6 Let T :R2 -+R2 be a linear map_ Assume

~

for every ellipsoid Then T is symplectic or antisymplectic, i.e.

Hofer: Symplectic capacities 19

III [2,3] the normalisation c (B271(1),0') = C (Z2n(1),0') = 1r had been assumedthroughout the paper However, ii had not been used for the above proposition,whieh only depends on (N) and (C). By the definition (1) we have meanwhileobt,ained a map still denoted byc:

Nuchthat for everyq, E Diff (R2 n,0'),t ER:

w(~ call this map again a symplectic capacity Next we obtain the following local

(,fO rigidity result, see [3] or [5]

'I'heorem 2.7 (Local Rigidity) Assume epk : B2n(1) -+ R2n is a sequence of ,'tyrnplectic embeddings, converging uniformly to a continuous map q> : B 2n(1) -+

It2n, which is differentiable at 0 E B2n(1) with derivative tP'(O) Then «p/(D) is

The local rigidity result can be proved only using (2) One shows that <p'(O) :

Il.~n -+ R2n andep'(O)XId R 2 :R2n+2 -+ R2n+2 preserve capacities CR2n andCR2n+2

f(.1' ellipsoids The linear algebra proposition then implies the desired result

So far we surveyed results which can be obtained from the existence of a I)lu~ity. The following results are different Their proofs depend on capacities with,ulclitional features

I)isplacenlent Energy

The first capacity we consider here is very closely related to Hamiltonian IIlUllics Let us introduce some notation We denote by C the vector space of alltHJ)()oth Hamiltonians H : [0,1] R2 -+ R having compact support Denote by

Dy-Implementing the above construction for a disjoint union of small cubes via a able highly oscillatory Hamiltonian we can prove the following.

suit-X Ht the time-dependent Hamiltonian vector field associated toH by the formula

dHt(x)( *)= u(XHt(x), *). Denote by WH the time-I-map associated to the tonian System

Hamil-We define a norm II*II onCby

IIHII = Co'1~':t2"H) - CO.l~t1R2nH) ·

By V we denote the collection of all 'ItH, H Ec. One easily verifies that V is agroup

Let S be a bounded subset of R2 n. We study the importance of energy byanalyzing the following problem: How much energy do I need to deform S into aset S' which is disjoint from 8? To make this precise we define the displacementenergy d( S) by

deS) =inf{IIHIIIWH(S)n S = <p}.

For an unbounded setT c R2n let

d(T)= sup {deS) ISeT, Sbounded}

It is clear that d(~(T))=d(T) for all ~ EV and T C R2n. What is not clear isthatd(T) :IO In order to stress this point considerS= B2(1) XB2n-2(R), R2: 1

To disjoin Sfrom itself it is enough to disjoinB2(1) C R 2 from itself Now B2(1) isvia an element inV symplectomorphic to an almost square with round corners andsidelength approximately Vi. By taking a translation parallel to one side, which

disjoints the square from itself we need a Hamiltonian which has slope f'V Viand

is increasing along lines orthogonal to the above side By cutting this Hamiltonian .~

off smoothly we see that we can separate the square from itself by a Hamiltonian ,I

H with IIHII'"Jr. Hence, a simple corollary: I

So with other words we can displace a set Sfrom itself modulo sets of small measurewith arbitrary small energy However, the surprising fact is

'I'heorem 3.2

In particular the above construction is the best one can do! Clearly, the mapd

("Ul be used to define a capacity c :Op(R2n,u) + r such that in addition to then.xioms(N) and(C) we have

c(B2n(1),u) = c(Z2n(1),u) ='Fr.

'I'his capacity for example implies Gromov's Squeezing Theorem It also has somevt~ryinteresting consequences for the topological group V. Define the energy ofa

IUn.p WEV by

E : V + R:E(w) =in!{I(HIII WH='II}

()uceasily verifies, [7],that

E(w~) ~ E(w)+E(~).

AHHumenow \11 i=ide Then

forsome point Xo ER2n and e>O Hence

Sowe find, summing up

'rheorem 3.3 The map p : V XV -+R defined by

0 «1»

tlt:fines a bi-invariant metric on V Moreover the map

1.11 Lipschitz continuous with Lipschitz constant equal to 1:

II.cmark 3.4 The map H -+ WH extendJ to the completions C + v. Since for

~ ~ 2 the map H + 'II H can be considered as a map Ck + C k - I

J the above theorem

.t4!t.Ol1JS philosophically that this map extends to a map Co +C-I So in some sense

l' d(~finesa C-I-topology.

Remark 3.5 The topology on V is very weak It does for example not even imply pointwise convergence For example we can have

as k -+ 00 for 1 ::; p ~ 00. It would be interesting to have a good model for the completionV.

Remark 3.6 The displacement capacity can be considered - to a certain extent - as

a nonlinear extension of the 2-dimensional Lebesgue measure to higher dimensions.

To see this, note that bounded simply connected domain S with smooth boundary is symplectomorphic to B 2 (R) for some R Hence

• The canonical line boundle£s~S is defined by

.cs = {(x,e) ETS Jw(e,1J) = 0, for all,., ETxS}.

We denote by Ls(x) the leaf through xES and call it a characteristic on S.

Of particular interest are the closed characteristics We denote the set of closedcharacteristics of S by V(S). We give £s an orientation by choosing a section e

of£ s -+ S such that w(e(x), n(x» > 0, where x -+ n( x) is the outward pointingnormal vector field Recall that by Alexander duality R2

n \ S has precisely twocomponents

It is an open problemifV(S) =F4> for every compact hypersurface By a result

of C Viterbo, [8], we have that V(S) =F<PifSis of contact type, [9]

Now let S be a compact connected smooth hypersurface with the orientation of

£ s ~Sjust defined IfP E'D(S) we have

and therefore P inherits on orient!ttion from £s. Define the action ofP by

whereD is a disk inR2n, with aD = P and compatible orientation Clearly A(P)

is well defined

We have the following strong squeezing theorem which follows immediately fromresults in [10]

Hofer: Symplectic capacities 23

'I'heorem 3.7 (Strong Squeezing) Assume S is a smooth compact hypersurface IUHtnding a convez domain C Then

sup {1rr 2 I'11(B2n(r)) C C for some '11 EV}

< inf {1rr2 I'11(C) C z2n(r) for some '11EV}

Jt~I'1'C = B2n(1) we recover Gromov's squeezing theorem.

Itmight be possible that we have in fact equality for the three expressions This1Mfitill an open problem

The following theorem which is a consequence of results in [7], gives a nice aIn'iori estimate for th~displacement energy

'I~heorem3.8 Assume S i3 a smooth compact hyper8urface bounding a convez do

1u(tin C Then

Note that for ellipsoids we have equality That symplectic capacities are related

t.C) closed characteristics on hypersurfaces is no accident and has been detected in{2]. To make it plausible consider the following:

Let C be a subset ofR2n and define its capacity as the infimum of the outer

IIU~fl,sureof the projections of all symplectic images ofC onto the (Zl = (Xl,

Yl))-{I()ordinate-plane

This defines a symplectic capacity, as an easy consequence of Gromov's squeezing

f.ll(~orem. AssumeS=BCis a compact hypersurface Suppose also that the

projec-tiollonto the zl-coordinate has minimal area In order todecr~asethis

projection-IU'('a.further (which isimpossib~eby our assumption) one would like to take a f,{)uianH which is increasing on the leaves ofS,because the associated Hamiltonian

Hamil-vc'ctorfield pushes S into C. Of course such an H does not exist globally, but onr~uhregionsofS. If some region contains a closed characteristic it cannot be the do-

Illn.inof a Hamiltonian which is increasing on leaves So closed characteristics occurh,H obstructions against decreasing the measure of certain projections by pushing4"(tl't.ain subregions ofS intoC.

We already motivated the o·ccurence of periodic Hamiltonian trajectories in the

utudy of symplectic capacities The next construction makes extensive use of tHlie solutions of Hamiltonian Systems and gives a whole sequence of independent.·,q>a,cities

peri-Periodic Hamiltonian Trajectories and Capacities

In [11] I Ekeland and the author studied periodic solutions of HamiltonianSystems on a prescribed energy surface In [11] they associated to convex energysurfacesS symplectic invariants and to their periodic trajectories three independentinvariants and showed that these three quantities are not independent A little bitlater C. Viterbo, [8J, proved the R2n-Weinstein conjecture, [9]. Viterbo's key in-gredient in his proof, meanwhile called Viterbo's trick, was simplified and exploited

by E. Zehnder and the author, [11], in the study of periodic Hamiltonian ries on a prescribed energy surface Roughly speaking one associates to aboundedopen set U C R2n a variational problem in the loop-space of R2n, which givesuseful information about the Hamiltonian flow on the energy surface au. Thesedevelopments were the starting point for the following construction of symplecticcapacities

trajecto-We define a particular class of Hamiltonian systems on R2n, which we denote

by'H,asfollows:

A Hamiltonian H :R2n-+ R belongs to'Iiif

HI There exists an open nonempty setUsuch that H 1u ==O MoreoverH(z) ~ 0forall z ER2 n.

H2 There exist numbers p,a >0, such that a >1r, art?r'Z,and H(z) = aIz 1 2

The critical points of~H are precisely the I-periodic solutions z of the

differ-ential equation -Jz = H'(z), where H' is the gradient ofH with respect to thestandard inner product < ',' >. Alternatively we have of coursez= XH(Z), since

XH =JH' is the Hamiltonian vector field associated to H. Our aim is to findinfinitely many universal minmax-characterisations for critical points of ~H. For

Hofer: Symplectic capacities 25

thiH we take the completion F ofF with respect to the Hi-norm onF. This is

n.llel denote the completion of(F, II · II) by F F has an orthogonal splitting F =

1"- EB FO EI7 F+ by cutting a Fourier series into pieces k<0, k=0and k>O The

Moreover one can show that F embeds compactly into every LP([O,1];R2n) for

1~ p < 00. We have a natural SI-action on F by phase-shift, i.e r E R/Z actsvia

(r * z)(t) = z(t + r).

W(~ define a distinguished subgroup B of homeo(F) the homeomorphism group of /1' bysaying hE B if h : F +F is a homeomorphism admitting the representation

h(z)= exp (i-(Z))Z-+zO+exp (i+(Z))Z+ +K(z),

where,+,,- :F ~ Rare continuous and SI-invariant, mapping bounded sets inF

illt,obounded sets in R MoreoverK : F +F is continuous and Sl equivariant and

IIUtpSbounded sets into precompact sets It is easily verified that usual compositionturns B into a group Next we need a pseudoindex theory in the sense of Benci,113], associated to the Fadell-Rabinowitz index, [14], in order to measure the size

c)f Sl-invariant sets This goes as follows: Given a paracompact SI-space X we

IHlild a free 51-spaceX X Soo by lettingSI act through the diagonal action Here,,",'00 =US2n- l , with S2n-l C R2n ~ en. Taking the quotient with respect to81

weobtain a principal51-bundle

,rile classifying map

f: (X x SOO)/S'~Cpoo

il)<luces a homomorphism

illAlexander-Spanier-Cohomology with rational coefficients HereRS l is the kllown Borel construction of a SI-equivariant cohomology theory We know that

well-H(CpOO) = Q[t], the generator t being of degree 2 We define the index ofX,

denoted bya(X) to be the largest numberk such that

We put a(X) = 00 if f*(t k

- 1 ) =F 0 for all k and a(0) = o. The a-index is thewell-known Fadell-Rabinowitz construction of a Sl-index theory Next we definefor an51-invariant set ec F an index ind(e)by

whereS+ is the unit spherein F+. The key topological result is the following:Lemma 3.9 Let X be an 5 1 -invariant linear subspace of F+ of dimension 2k Then

For a proof see[3].

Finallywedefine for H E1i

Ck,H=inf {sup<PH( e) leeF is 51-invariant and ind(e) ~ k}.

Byconstruction, it is clear that

o <cl,H~C2,H ~ •• ~+00.

For a bounded set 5 c R2n we put

1i(5)= {H E1iIH IU(S)== OJ.

whereU(S) denotes anopen neighbourhood ofSdepending onH. Fora boundedset S let

HE'H(S) ,

For an unbounded set T let

ck(T) = SUp{Ck(S) IS c T bounded}

The first result proved in [3] is

Theorem 3.10 For every k E{I, 2, } the map Ck defines a symplectic capacity Moreover

Cl~C2~.·••

Hofer: Symplectic capacities '27

The Ck have been computed in [3] for certain sets For r = (rt, ,rn) with

() <rl ~r2 ::; rn, define an ellipsoidE(r) by

Forsuch a r define with N*= {1, 2, } :

dk(r)=in!{r >0Icard{(l,j)11EN*, j E {I, , n}, 1rr;1~ r} ~ Ie}.

Forexample for r =(1,1, ,1) we have

where [*] denotes the integer part Moreover observe that

lim dk«1,n, ,n))= k7r

n- oo

It has been proved in [3] that

'I'heorem 3.11 The following equations hold:

Ilere all sets are equipped with the standard structure (1. The capacitiesCk enjoy

n useful representation property explained as follows Call a compact connectedHlllooth hypersurface S to be of restricted contact type provided there exists a

veetor fieldTJ on R2n, such that

'1 is transversal to S, L,,(1= (1.

W(~have, denoting by B s the bounded component of R2n\ s:

'I'heorem 3.12 Let S be as just described Then there exists a sequence (P k ) C

'/)(5) and a sequence nkEN* such that

Finally let us give an application to an embedding problem:

'rheorem 3.13 Assume B 2 (1) x x B 2 (1) admits a symplectic embedding into

"2n(r) Then r 2:: ,;n.

SinceC<1was arbitrary we deducer ~ ;n. We had to introduce the map ~ since

theC/c are by construction only invariant under symplectomorphisms inV •

conjecture

Here we present a more general construction for a symplectic capacity on arbitrarysymplectic manifolds Assume (M,w) E Symp2n. We denote by H(M).the set ofall autonomous Hamiltonian Systems H : M -+R such that

HI There exists a non-empty open setUC M \ 8M such that H IU== o.

H2 There exists a compact subset K C M \ 8M and a constant m(H) > asuchthat

m(H) ifxEN \ im(w)

We call a Hamiltonian H E H(M) admissible provided every T-periodic solution

ofx= XH(x) for T E [0,1] is constant We denote by ?te(M) the collection of alladmissible Hamiltonians inH(M). Clearly w.Ha(M)C Ha(N)and H a(AI) f:. ¢asone easily verifies We define c : Symp2n + r by

c(M,w) =sup{m(H) IH E?ta(M,w)}.

Hofer: Symplectic capacities 29

(ẵnrly c is a covariant functor and c (M, a · w) =1a 1·c(M, w) for a =f ọ Theclif-Hculty lies in proving axiom(N)which is done in [10] Some other results listedh<'loware taken from [15,16]

'Íheorem 4.1 c:S ymp 2n-+r is a symplectic capacity and

Moreover for every every convex bounded domainC with smooth boundary S= 8C

111t~ have

nnd ađitionally for every compact symplectic manifold (N, w) with w 11r2(N)=0

c(NX B 2k (r),wEBu) =7rr2

l)1'óvided k2 1 Also for the standardCpn:

The above capacityis very useful in dealing with the Weinstein conjecture, [9].114't usrecall some notations Consider a compact (2n - 1)-dimensional manifoldSrqllipped with a I-form ,\ such that ,\A(d,\)n-l is a volumẹ One calls ,\ a contactfcn'rn

The manifoldS carries a natural line bundle defined by

£5= {(x,e) ETSId'\(x)(e,'fJ) = 0 forall 'fJET~S}.

1I(~nce£sc TSand sincé\Ăd,\)n-IisavolumeĂX,e) i: afor(x,e)E£5,ife=FỌ

I)(~note by V(S) the closed integral curves for the (integrable) distribution£5 +S.Weinstein conjectured that 1>(8) =1= ¢ at least ifHI(S;R) = Ọ Given (5,'\) we

dt'fine a germ of a symplectic manifold structure on (-~,e) X S by w = d(e t ,\)

where(t, x) E(-e:, e:)xS. We considerS asthe subset {OJ x S in (-€, €)XS. FirstW(~have the following trivial observation

I)roposition 4.2 Assume the previously defined capacity of (-e, e) x S for some

r '>ais finitẹ Then V(S) t ¢

I)ROO F : Fix a smooth map <p : (-e, e) ~ R such that cp == 0 near zero and

(f1(S) = m > 0 for Sclose toeor -ẹ Define a Hamiltonian