An Arnold-type principle for non-smooth objects
aa r X i v : . [ m a t h . S G ] S e p An Arnold-type principle for non-smooth objects
Lev Buhovsky, Vincent Humili`ere, Sobhan SeyfaddiniSeptember 17, 2019
Abstract
In this article we study the Arnold conjecture in settings whereobjects under consideration are no longer smooth but only continuous.The example of a Hamiltonian homeomorphism, on any closed sym-plectic manifold of dimension greater than 2, having only one fixedpoint shows that the conjecture does not admit a direct generalizationto continuous settings. However, it appears that the following Arnold-type principle continues to hold in C settings: Suppose that X is anon-smooth object for which one can define spectral invariants. If thenumber of spectral invariants associated to X is smaller than the num-ber predicted by the (homological) Arnold conjecture, then the set offixed/intersection points of X is homologically non-trivial, hence it isinfinite.We recently proved that the above principle holds for Hamiltonianhomeomorphisms of closed and aspherical symplectic manifolds. In thisarticle, we verify this principle in two new settings: C Lagrangians incotangent bundles and Hausdorff limits of Legendrians in 1-jet bundleswhich are isotopic to 0-section.An unexpected consequence of the result on Legendrians is that theclassical Arnold conjecture does hold for Hausdorff limits of Legendri-ans in 1-jet bundles.
Contents C Lagrangians and proof of Theorem 1.1 12 C Lagrangians . . . . . . . . . . . . . 124.2 Proof of Theorem 1.1 . . . . . . . . . . . . . . . . . . . . . . . 17
The Arnold conjecture states that a Hamiltonian diffeomorphism of a closedand connected symplectic manifold (
M, ω ) must have at least as many fixedpoints as the minimal number of critical points of a smooth function on M . The classical Lusternik-Schnirelmann theory shows that this minimalnumber is always at least the cup length of M , a topological invariant of M defined as cl( M ) := max { k + 1 : ∃ a , . . . , a k ∈ H ∗ ( M ) , ∀ i, deg( a i ) = dim( M )and a ∩ · · · ∩ a k = 0 } . Therefore, a natural interpretation of the Arnold conjecture, sometimesreferred to as the homological Arnold conjecture, is that a Hamiltonian dif-feomorphism of (
M, ω ) must have at least cl( M ) fixed points. Successfulefforts at resolving this conjecture were pioneered by Floer [7, 8, 10] and ledto the development of what is now called Floer homology. The original ver-sion of the Arnold conjecture has been proven on symplectically asphericalmanifolds [31], [9], [12] while the homological version has been proven on alarger class of manifolds, e.g. C P n by Fortune-Weinstein [11], and symplecticmanifolds which are negatively monotone by Lˆe-Ono [21].The Arnold conjecture admits reformulations for symplectic objects otherthan Hamiltonian diffeomorphisms: For example, a Lagrangian version ofthe conjecture states that in a cotangent bundle T ∗ N , a Lagrangian sub-manifold which is Hamiltonian isotopic to the zero section must have at leastcl( N ) intersection points with the zero section O N (See [12, 20]). Here is aLegendrian reformulation of this last statement: a Legendrian submanifoldin a 1-jet bundle J N = T ∗ N × R , which is isotopic to the zero sectionthrough Legendrians, must have at least cl( N ) intersections with the 0-wall O N × R . The goal of this article is to understand the Arnold conjecture in settingswhere objects under consideration are no longer smooth but only continuous. Here, ∩ refers to the intersection product in homology. The cup length can be equiv-alently defined in terms of the cup product in cohomology. Note that we do not make any assumptions regarding non-degeneracy of Hamiltoniandiffeomorphisms here. Sandon has recently presented a reformulation of the Arnold conjecture for contacto-morphisms; see [32, 33]. C settings. These re-formulations, which involve counting fixed/intersection points and certain“homologically essential” critical values of the action ( i.e. spectral invari-ants ), are inspired by the following statement from Lusternik–Shnirelmantheory: Let f be a smooth function on a closed manifold M . If the number ofhomologically essential critical values of f is smaller than cl( M ) , then theset of critical points of f is homologically non-trivial. The above statement can be deduced from Proposition 3.1. Homologi-cally essential critical values, which are usually referred to as spectral invari-ants in the symplectic literature, are defined in Section 3.1. A subset A ⊂ M is homologically non-trivial if for every open neighborhood U of A the map i ∗ : H j ( U ) → H j ( M ), induced by the inclusion i : U ֒ → M , is non-trivial forsome j >
0. Clearly, homologically non-trivial sets are infinite.The reformulations of the Arnold conjecture which continue to hold in C settings may be summarized as follows: Principle 1.
Suppose that X is a non-smooth object for which one candefine spectral invariants. If the number of spectral invariants associated to X is smaller than the number predicted by the homological Arnold conjecture,then the set of fixed/intersection points of X is homologically non-trivial,hence it is infinite. In our recent article [2], we established the above principle for Hamilto-nian homeomorphisms of symplectically aspherical manifolds: Suppose that(
M, ω ) is closed, connected, and symplectically aspherical. In Theorem 1.4of [2] we prove that if φ is a Hamiltonian homeomorphism of ( M, ω ) withfewer spectral invariants than cl( M ), then the set of fixed points of φ is ho-mologically non-trivial. A variant of this statement for negative monotonesymplectic manifolds and for complex projective spaces has been proven byY. Kawamoto in [17].The main results of this article establish Principle 1 in two more contexts: C Lagrangians in cotangent bundles and Hausdorff limits of Legendriansin 1-jet bundles. C Lagrangians:
Consider the cotangent bundle T ∗ N of a closed manifold N and denote by O N its zero section. As we will see in Section 4, (La-grangian) spectral invariants can be defined for a C Lagrangian of the form L = φ ( O N ) where φ is a compactly supported Hamiltonian homeomorphismof T ∗ N ; this is proven in Theorem 4.1. We call such a C Lagrangian “a3 Lagrangian Hamiltonian homeomorphic to the zero section”. It is notdifficult to see that in this setting our principle translates to the followingstatement.
Theorem 1.1.
Let φ denote a compactly supported Hamiltonian homeo-morphism of T ∗ N and suppose that L = φ ( O N ) . If the number of spectralinvariants of L is smaller than cl( N ) , then L ∩ O N is homologically non-trivial, hence it is infinite. Remark 1.2.
We expect that by adapting the construction of the coun-terexample in [3], one should be able to construct L as in the above theoremsuch that L ∩ O N is a singleton. Of course, as a consequence of the theorem,such L must have at least cl( N ) distinct spectral invariants. ◭ Remark 1.3.
It is reasonable to ask if in the above theorem the hypothesis L = φ ( O N ) could be weakened to L being the Hausdorff limit of a sequence L i , where each L i is Hamiltonian isotopic to the zero section. This is relatedto a conjecture of Viterbo; see also Remark 4.4 below. ◭ Hausdorff limits of Legendrians:
As we will show in Section 5, one canassociate spectral invariants to the Hausdorff limit of a sequence of Legen-drians which are contact isotopic to the zero section in a 1-jet bundle J N .The interpretation of our principle in this case turns out to be particularlyinteresting for the following reason: Consider an intersection point ( q, , z )between such a Legendrian L and the 0-wall. This point corresponds to acritical point of the action and the associated critical value is z . In otherwords, the critical value can be read directly from the intersection point. Itfollows that in this context Principle 1 implies the Arnold conjecture itself!As explained above, for a smooth Legendrian L the action spectrum isgiven by spec( L ) = π R ( L ∩ ( O N × R )), where π R : J N = T ∗ N × R → R is the natural projection. By analogy, we will define the spectrum of anysubset L ⊂ J N to be spec( L ) = π R ( L ∩ ( O N × R )) . Theorem 1.4.
Let L i be a sequence of Legendrian submanifolds in J N which are contact isotopic to the zero section O N × { } . Suppose that thissequence has a limit L for the Hausdorff distance, where L ⊂ J N is acompact subset.Assume that the cardinality spec( L ) is strictly less than cl( N ) . Then,there exists λ ∈ spec( L ) such that L ∩ ( O N × { λ } ) is homologically non-trivial in O N × { λ } . In particular, L ∩ ( O N × R ) is infinite. L . Infact, we do not even require L to be a C submanifold of J N . Remark 1.5.
A careful examination of the proof of Theorem 1.4 revealsthat the assumption of Hausdorff convergence of L i to L can be relaxed tothe following: any neighborhood of L contains L i for i large. ◭ Remark 1.6.
In an ongoing project [16], the second author and N. Vicheryshow that Principle 1 can also be established for singular supports of sheaves(belonging to a certain subcategory of sheaves introduced by Tamarkin).These singular supports can be seen as (singular) generalizations of Legen-drian submanifolds. ◭ Organization of the paper
In Section 2, we recall some basic notions from symplectic geometry. InSection 3, we introduce preliminaries on Lusternik-Schnirelmann theory andspectral invariants.Section 4 is dedicated to establishing Principle 1 for C LagrangiansHamiltonian homeomorphic to the zero section. The main technical stepfor doing so, which is of independent interest, consists of proving that La-grangian spectral invariants can be defined for such C Lagrangians. Thisis achieved in Section 4.1; see Theorem 4.1 therein. Theorem 1.1 is provenin Section 4.2. Lastly, Theorem 1.4 is proven in Section 5.
Aknowledgments
Lemma 4.5 was proven jointly with R´emi Leclercq. We are grateful to himfor generously sharing his ideas with us. Our proofs of Theorems 1.1 and1.4 were inspired by the paper of Wyatt Howard [14].The first author was partially supported by ERC Starting Grant 757585and ISF Grant 2026/17. The second author was partially supported bythe ANR project “Microlocal” ANR-15-CE40-0007. This material is basedupon work supported by the National Science Foundation under Grant No.DMS-1440140 while the third author was in residence at the Mathemati-cal Sciences Research Institute in Berkeley, California, during the Fall 2018semester. The third author greatly benefited from the lively research atmo-sphere of the MSRI and would like to thank the members and staff of theMSRI for their warm hospitality.
For the remainder of this section (
M, ω ) will denote a connected symplec-tic manifold. Recall that a symplectic diffeomorphism is a diffeomorphism5 : M → M such that θ ∗ ω = ω . The set of all symplectic diffeomorphisms of M is denoted by Symp( M, ω ). Hamiltonian diffeomorphisms constitute animportant class of examples of symplectic diffeomorphisms. These are de-fined as follows: A smooth Hamiltonian H ∈ C ∞ c ([0 , × M ) gives riseto a time-dependent vector field X H which is defined via the equation: ω ( X H ( t ) , · ) = − dH t . The Hamiltonian flow of H , denoted by φ tH , is bydefinition the flow of X H . A compactly supported Hamiltonian diffeomor-phism is a diffeomorphism which arises as the time-one map of a Hamiltonianflow generated by a compactly supported Hamiltonian. The set of all com-pactly supported Hamiltonian diffeomorphisms is denoted by Ham c ( M, ω );this forms a normal subgroup of Symp(
M, ω ). We equip M with a Riemannian distance d . Given two maps φ, ψ : M → M, we denote d C ( φ, ψ ) = max x ∈ M d ( φ ( x ) , ψ ( x )) . We will say that a sequence of compactly supported maps φ i : M → M , C –converges to φ , if there is a compact subset of M which contains thesupports of all φ i ’s and if d C ( φ i , φ ) → i → ∞ . Of course, the notion of C –convergence does not depend on the choice of the Riemannian metric. Definition 2.1.
A homeomorphism θ : M → M is said to be symplecticif it is the C –limit of a sequence of symplectic diffeomorphisms. We willdenote the set of all symplectic homeomorphisms by Sympeo(
M, ω ) . The Eliashberg–Gromov theorem states that a symplectic homeomor-phism which is smooth is itself a symplectic diffeomorphism. We remarkthat if θ is a symplectic homeomorphism, then so is θ − . In fact, it is easyto see that Sympeo( M, ω ) forms a group.
Definition 2.2.
A symplectic homeomorphism φ is said to be a Hamilto-nian homeomorphism if it is the C –limit of a sequence of Hamiltonian dif-feomorphisms. We will denote the set of all Hamiltonian homeomorphismsby Ham(
M, ω ) . It is not difficult to see that Ham(
M, ω ) forms a normal subgroup ofSympeo(
M, ω ). It is a long standing open question whether a smooth Hamil-tonian homeomorphism, which is isotopic to identity in Symp(
M, ω ), is aHamiltonian diffeomorphism; this is often referred to as the C Flux con-jecture; see [19, 36, 1].We should add that alternative definitions for Hamiltonian homeomor-phisms do exist within the literature of C symplectic topology. Most no-table of these is a definition given by M¨uller and Oh in [29]. A homeomor-phism which is Hamiltonian in the sense of [29] is necessarily Hamiltonian6n the sense of Definition 2.2 and thus, the results of this article apply tothe homeomorphisms of [29] as well. We will denote the Hofer norm on C ∞ c ([0 , × M ) by k H k = Z (cid:18) max x ∈ M H ( t, · ) − min x ∈ M H ( t, · ) (cid:19) dt. The Hofer distance on Ham(
M, ω ) is defined via d Hofer ( φ, ψ ) = inf k H − G k , where the infimum is taken over all H, G such that φ H = φ and φ G = ψ .This defines a bi-invariant distance on Ham( M, ω ).Given B ⊂ M , we define its displacement energy to be e ( B ) := inf { d Hofer ( φ, Id) : φ ∈ Ham(
M, ω ) , φ ( B ) ∩ B = ∅} . Non-degeneracy of the Hofer distance is a consequence of the fact that e ( B ) > B is an open set. This fact was proven in [13, 30, 18]. We fix a ground field F , e.g. Z , Q , or C . Singular homology, Floer homologyand all notions relying on these theories depend on the field F . Let f ∈ C ∞ ( M ) a smooth function on a closed and connected manifold M .For any a ∈ R , let M a = { x ∈ M : f ( x ) < a } . Let α ∈ H ∗ ( M ) be a non-zerosingular homology class and define c LS ( α, f ) := inf { a ∈ R : α ∈ Im( i ∗ a ) } , where i ∗ a : H ∗ ( M a ) → H ∗ ( M ) is the map induced in homology by the naturalinclusion i a : M a ֒ → M . The number c LS ( α, f ) is a critical value of f andsuch critical values are often referred to as homologically essential criticalvalues.The function c LS : H ∗ ( M ) \ { } × C ∞ ( M ) → R is called a min-max criti-cal value selector. In the following proposition [ M ] denotes the fundamentalclass of M and [ pt ] denotes the class of a point. Proposition 3.1.
The min-max critical value selector c LS possesses thefollowing properties. . c LS ( α, f ) is a critical value of f ,2. c LS ([ pt ] , f ) = min( f ) c LS ( α, f ) c LS ([ M ] , f ) = max( f ) ,3. c LS ( α ∩ β, f ) c LS ( α, f ) , for any β ∈ H ∗ ( M ) such that α ∩ β = 0 ,4. Suppose that deg( β ) < dim( M ) and c LS ( α ∩ β, f ) = c LS ( α, f ) . Then,the set of critical points of f with critical value c LS ( α, f ) is homologi-cally non-trivial. The above are well-known results from Lusternik-Schnirelmann theoryand hence we will not present a proof here. For further details, we refer thereader to [24, 6, 40].
Let N be a closed manifold. The canonical symplectic structure on thecotangent bundle T ∗ N is induced by the form ω = − dλ where λ = p dq .We will denote by Lag the space of Lagrangian submanifolds of T ∗ N whichare Hamiltonian isotopic to the zero section, i.e. Lag := { φ ( O N ) : φ ∈ Ham c ( T ∗ N, ω ) } .Consider φ ∈ Ham c ( T ∗ N, ω ) and let L = φ ( O N ). We will briefly ex-plain how one may associate Lagrangian spectral invariants to the Hamil-tonian diffeomorphism φ . Pick a compactly supported Hamiltonian H ∈ C ∞ c ([0 , × T ∗ N ) such that φ = φ H . The action functional associated to H is defined by A H : Ω( T ∗ N ) → R , z Z H t ( z ( t )) dt − Z z ∗ λ where Ω( T ∗ N ) = { z : [0 , → T ∗ N | z (0) ∈ O N , z (1) ∈ O N } . The criticalpoints of A H are the chords of the Hamiltonian vector field X H which startand end on O N . Note that such chords are in one-to-one correspondencewith L ∩ O N . The spectrum of A H consists of the critical values of A H . It isa nowhere dense subset of R which turns out to depend only on the time–1map φ H , hence we will denote it by Spec( L ; φ ).Now, using Lagrangian Floer homology, in a manner similar to what wasdone in the previous section, one can define a mapping ℓ : H ∗ ( N ) \ { } × Ham c ( T ∗ N, ω ) → R which associates to a homology class a ∈ H ∗ ( N ) \ { } a value in Spec( L ; φ ).These numbers are often referred to as the Lagrangian spectral invariantsof φ . They were first introduced by Viterbo in [40] via generating functiontechniques. The Floer theoretic approach was carried out by Oh [27]. La-grangian spectral invariants have many properties some of which are listedbelow. For a more comprehensive list of their properties, as well as a survey8f their construction, we refer the reader to [26]; see for example Theorems2.11 and 2.17 in [26]. Proposition 3.2.
The map ℓ : H ∗ ( N ) \ { } × Ham c ( T ∗ N, ω ) → R , satisfiesthe following properties:1. ℓ ( a, φ ) ∈ Spec( L ; φ ) ,2. | ℓ ( a, φ H ) − ℓ ( a, φ G ) | k H − G k ,3. ℓ ( a ∩ b, φψ ) ℓ ( a, φ ) + ℓ ( b, ψ ) ,4. ℓ ([ pt ] , φ ) ℓ ( a, φ ) ℓ ([ N ] , φ ) ,5. ℓ ([ N ] , φ ) = − ℓ ([ pt ] , φ − ) ,6. If φ ( O N ) = ψ ( O N ) , then ∃ C ∈ R such that ℓ ( a, φ ) = ℓ ( a, ψ ) + C forall a ∈ H ∗ ( N ) \ { } ,7. Suppose that f : N → R is a smooth function and define the La-grangian L f := { ( q, ∂ q f ( q )) : q ∈ N } . Denote by F any compactlysupported Hamiltonian of T ∗ N which coincides with π ∗ f = f ◦ π ona ball bundle T ∗ R N of T ∗ N containing L f . Then, ℓ ( a, φ F ) = c LS ( a, f ) for all a ∈ H ∗ ( N ) \ { } .
8. For any other manifold N ′ , the spectral invariants on T ∗ ( N × N ′ ) satisfy ℓ ( a ⊗ a ′ , φ × φ ′ ) = ℓ ( a, φ ) + ℓ ( a ′ , φ ′ ) , for all φ ∈ Ham c ( T ∗ N, ω ) , φ ′ ∈ Ham c ( T ∗ N ′ , ω ) , a ∈ H ∗ ( N ) \ { } and a ′ ∈ H ∗ ( N ′ ) \ { } . Note that the sixth property above tells us that spectral invariants ℓ ( a, φ )are essentially invariants of the Lagrangian L := φ ( O N ). As a consequenceof this property, the set of spectral invariants of L is well-defined upto ashift by a constant. In particular, we can make sense of the total numberof spectral invariants of any Lagrangian L which is Hamiltonian isotopic tothe zero section. Similarly, we see that γ : Lag → R , defined by γ ( φ ( O N )) := ℓ ([ N ] , φ ) − ℓ ([ pt ] , φ ) (1)is well-defined, i.e. it only depends on the Lagrangian φ ( O N ) and not on φ .Viterbo showed in [40] that γ induces a non-degenerate distance on Lag.Finally, we should mention that Lagrangian spectral invariants havebeen constructed in settings more general than what is described aboveby Leclercq [22] and Leclercq-Zapolsky [23]. Hamiltonian Spectral Invariants:
In order to prove that Lagrangianspectral invariants can be defined for C Lagrangians Hamiltonian home-omorphic to the zero section, that is to prove Theorem 4.1 below, we will9eed to use certain results from the theory of Hamiltonian spectral invari-ants. Here, we will briefly recall the aspects of this theory which will beneeded below. For further details on the construction of these invariantssee [34, 28]. The specific result used here, which compares Lagrangian andHamiltonian spectral invariants, was proven in [26].Given φ ∈ Ham c ( T ∗ N, ω ) and a ∈ H ∗ ( N ) \ { } , using HamiltonianFloer homology, one can define the Hamiltonian spectral invariant c ( a, φ );this is a real number which belongs to the (Hamiltonian) action spectrumof φ , i.e. there exists a fixed point of φ whose action is the value c ( a, φ ).These spectral invariants satisfy a list of properties similar to those listed inProposition 3.2. We will be needing the following property which is provenin [26]: For any φ ∈ Ham c ( T ∗ N, ω ) and any a ∈ H ∗ ( N ) \ { } we have c ([ pt ] , φ ) ℓ ( a, φ ) c ([ N ] , φ ) . (2)See Proposition 2.14 and item iv of Theorem 2.17 in [26].Similarly to Equation (1), we define γ : Ham c ( T ∗ N, ω ) → R via γ ( φ ) := c ([ N ] , φ ) − c ([ pt ] , φ ) . (3)Like its Lagrangian cousin, γ induces a non-degenerate distance on Ham c ( T ∗ N, ω ). We will need the following properties:1. Comparison Inequality:
As an immediate consequence of Equation2, the Lagrangian version of γ is smaller than the Hamiltonian version.More precisely, for any φ ∈ Ham c ( T ∗ N, ω ) we have γ ( φ ( O N )) γ ( φ ) . (4)2. Conjugacy Invariance:
For any φ ∈ Ham c ( T ∗ N, ω ) and any sym-plectic diffeomorphism ψ of T ∗ N , we have γ ( φ ) = γ ( ψφψ − ) . (5)3. Triangle Inequality:
For any φ, ψ ∈ Ham c ( T ∗ N, ω ), we have γ ( φψ ) γ ( φ ) + γ ( ψ ) . (6)4. Energy-Capacity Inequality:
Suppose that the support of φ canbe displaced, then γ ( φ ) e (supp( φ )) , (7)where e (supp( φ )) is the displacement energy of supp( φ ).10 .3 Spectral invariants for Legendrians via generating func-tions Once again let N be a closed manifold. The standard contact structureon the 1-jet bundle J N = T ∗ N × R is induced by the contact form α = dz − λ , where z is the coordinate on R . We will denote by Leg the spaceof Legendrian submanifolds of J N which are contact isotopic to the zerosection. It was proven by Chaperon [4] and Chekanov [5] that for every L ∈ Leg there exists a generating function quadratic at infinity (gfqi) S : N × E → R , where E is some auxiliary vector space, such that L = (cid:26)(cid:18) q, ∂S∂q ( q, e ) , S ( q, e ) (cid:19) : ∂S∂e ( q, e ) = 0 (cid:27) . Observe that critical points of S correspond to the intersection pointsof L with the zero wall O N × R : ( q, e ) is a critical point of S if and onlyif ( q, , S ( q, e )) is a point on L . Note that one can obtain the critical valueof a given critical point of S by simply reading the z –coordinate of thecorresponding intersection point of L with the zero wall.By applying a min-max construction similar to that of Section 3.1 to thegfqi S , one can define Legendrian spectral invariants of the Legendrian L : ℓ : H ∗ ( N ) \ { } × Leg → R . The fact that ℓ ( a, L ) does not depend on the choice of the gfqi S is a con-sequence of the uniqueness theorem of Th´eret and Viterbo [39, 40]. Forfurther details on the construction see [41].We will now state those properties of Legendrian spectral invariantswhich will be used below. Proposition 3.3. [See [41]] The map ℓ : H ∗ ( N ) \ { } × Leg → R , satisfiesthe following properties:1. ℓ ( a, L ) is a critical value of the corresponding gfqi S ,2. The map ℓ ( a, · ) : Leg → R is continuous with respect to the C ∞ topol-ogy,3. ℓ ( a ∩ b, L + L ′ ) ℓ ( a, L ) + ℓ ( b, L ′ ) , for all L, L ′ ∈ Leg such that L + L ′ := { ( q, p + p ′ , z + z ′ ) : ( q, p, z ) ∈ L, ( q, p ′ , z ′ ) ∈ L ′ } is a smoothLegendrian submanifold contact isotopic to the -section.4. Suppose that f : N → R is a smooth function and define the Legen-drian L f := { ( q, ∂ q f ( q ) , f ( q )) : q ∈ N } . Then, ℓ ( a, L f ) = c LS ( a, f ) for all a ∈ H ∗ ( N ) \ { } . Remark 3.4.
A proof of item 3 in Proposition 3.3 is based on the followingobservation: If
S, S ′ are gfqi’s for L, L ′ , respectively, then S ⊕ S ′ : N × E × E ′ → R defined by S ⊕ S ′ ( q, e, e ′ ) := S ( q, e ) + S ′ ( q, e ′ ) is a gfqi for theLegendrian L + L ′ . ◭ C Lagrangians and proof of Theorem 1.1
In this section, we give a proof of Theorem 1.1. We begin by giving a precisedefinition of compactly supported Hamiltonian homeomorphisms of T ∗ N .Equip N with a Riemannian metric and denote by T ∗ r N := { ( q, p ) ∈ T ∗ N : k p k < r } the cotangent disc bundle of radius r >
0. We defineHam c ( T ∗ r N, ω ) to be the set of Hamiltonian diffeomorphisms whose supportis contained in T ∗ r N . A compactly supported Hamiltonian homeomorphismis a homeomorphism which belongs to the uniform closure of Ham c ( T ∗ r N, ω )for some r >
0; we will denote their collection by Ham c ( T ∗ N, ω ). C Lagrangians
We will now prove that Lagrangian spectral invariants can be defined for C Lagrangians of the form L = φ ( O N ) where φ ∈ Ham c ( T ∗ N, ω ). Belowis the continuity result which allows us to define spectral invariants for such C Lagrangians.
Theorem 4.1.
Lagrangian spectral invariants satisfy the following two prop-erties:1. For any homology class a ∈ H ∗ ( N ) \ { } , the map ℓ ( a, · ) : Ham c ( T ∗ N, ω ) → R is continuous with respect to the C topology on Ham c ( T ∗ N, ω ) andextends continuously to the closure Ham c ( T ∗ N, ω ) .2. If φ ( O N ) = ψ ( O N ) , then ∃ C ∈ R such that ℓ ( a, φ ) = ℓ ( a, ψ ) + C forall a ∈ H ∗ ( N ) \ { } and for any φ, ψ ∈ Ham c ( T ∗ N, ω ) . Note that as a consequence of the second item, we can define the spec-tral invariants of a C Lagrangian Hamiltonian homeomorphic to the zerosection, upto shift. In particular, it makes sense to speak of the number ofspectral invariants of such a C Lagrangian.The first part of the above theorem follows from techniques which haveby now become rather standard in C symplectic topology and hence, wewill only sketch a proof of this part of the theorem. The second part of thestatement, however, is based on a trick which was recently introduced in ourarticle [2] in the course of proving C continuity of spectral invariants forHamiltonian diffeomorphisms; see Theorem 1.1 therein. Proof of Theorem 4.1.
We begin with the proof of the first statement. Wewill be needing the following claim.
Claim 4.2.
For every r > , there exist constants C, δ > , depending on r , such that for any ψ ∈ Ham c ( T ∗ r N, ω ) , if d C (Id , ψ ) δ , then | ℓ ( a, ψ ) | Cd C (Id , ψ ) . roof of Claim 4.2. As a consequence of Inequality (2), it is sufficient toprove the result for the Hamiltonian spectral invariants. This is provedin [35] in the case of symplectically aspherical closed manifolds; see seeTheorem 1 therein. The proof given in [35] easily adapts to our settings.Claim 4.2 proves continuity of our map at the identity. Next, we considerId = φ ∈ Ham c ( T ∗ r N, ω ). We leave it to the reader to check that Properties3, 4 and 5 in Proposition 3.2 yield the following: | ℓ ( a, φψ ) − ℓ ( a, φ ) | max {| ℓ ([ N ] , ψ ) | , | ℓ ([ pt ] , ψ ) |} . Combining this with Claim 4.2 we conclude that for any φ, ψ ∈ Ham c ( T ∗ r N, ω ) d C (Id , ψ ) δ = ⇒ | ℓ ( a, φψ ) − ℓ ( a, φ ) | Cd C (Id , ψ ) . This proves that ℓ ( a, · ) : Ham c ( T ∗ r N, ω ) → R is locally Lipschitz continuous.Hence, it extends continuously to the closure Ham c ( T ∗ r N, ω ). This finishesthe proof of the first statement of the theorem.We now turn our attention to the second statement of the theorem. Webegin with the following apriori weaker statement. Theorem 4.3.
Let φ ∈ Ham c ( T ∗ N, ω ) be a Hamiltonian homeomorphism.If φ ( O N ) = O N , then there exists a constant C such that ℓ ( a, φ ) = C for all a ∈ H ∗ ( N ) \ { } . Note that in the case where φ is a smooth Hamiltonian diffeomorphism,the above theorem reduces to Property 6 in Proposition 3.2. Remark 4.4.
It can be checked that Theorem 4.3 is a consequence of thefollowing conjecture of Viterbo: If L i ⊂ T ∗ N is a sequence of LagrangiansHamiltonian isotopic to the zero section, which Hausdorff converges to thezero section O N , then γ ( L i ) →
0. This conjecture has been established inseveral case by Shelukhin, e.g. N = S n , C P n , T n and others; See [37, 38]. ◭ Let us prove that the result follows from the above theorem. Supposethat φ ( O N ) = ψ ( O N ). First, note that, as a consequence of the third itemin Proposition 3.2, we have the following inequality: − ℓ ([ N ] , φ − ψ ) ℓ ( a, φ ) − ℓ ( a, ψ ) ℓ ([ N ] , ψ − φ ) . Hence, it is sufficient to show that ℓ ([ N ] , ψ − φ ) = − ℓ ([ N ] , φ − ψ ). Now,by the fifth item of Proposition 3.2, − ℓ ([ N ] , φ − ψ ) = ℓ ([ pt ] , ψ − φ ) and byTheorem 4.3 we have ℓ ([ pt ] , ψ − φ ) = ℓ ([ N ] , ψ − φ ).It remains to prove Theorem 4.3. The proof we present below relies onan idea similar to what was used in the proof of Theorem 1.1 of [2].13 roof of Theorem 4.3. Pick a sequence φ i in Ham c ( T ∗ ρ N, ω ) which con-verges uniformly to φ (for some ρ > C such that ℓ ( a, φ i ) → C for any a ∈ H ∗ ( N ) \ { } . Denote L i := φ i ( O N ) and observe that, as a consequenceof the fourth property in Proposition 3.2, it is sufficient to show that γ ( L i )converges to zero.As we will now explain, we may assume without loss of generality that φ admits a fixed point on the zero section O N . Indeed, fix p ∈ O N and pick aHamiltonian G which vanishes on the zero section such that φ ◦ φ G ( p ) = p .For all i , we have γ ( φ i ◦ φ G ) = γ ( φ i ), by the sixth item of Proposition 3.2.Thus, we can replace φ i by φ i ◦ φ G and φ by φ ◦ φ G .Observe that the Lagrangians L i converge in Hausdorff topology to thezero section, i.e. for any δ > L i ⊂ T ∗ δ N for i sufficiently large. Wewill reduce the theorem to the following lemma which was obtained jointlywith R. Leclercq. A variant of this lemma was established in [15]; see Lemma8 therein.Given B ⊂ N , we denote T ∗ B := { ( q, p ) ∈ T ∗ N : q ∈ B } and O B := { ( q,
0) : q ∈ B } . Lemma 4.5.
Let L i denote a sequence of Lagrangians in T ∗ N which areHamiltonian isotopic to O N . Suppose that there exists a ball B ⊂ N suchthat L i ∩ T ∗ B = O B . If the sequence L i Hausdorff converges to O N , then γ ( L i ) → .Proof. Pick φ i ∈ Ham c ( T ∗ N, ω ) such that φ i ( O N ) = L i . We begin withthe following observation: Since L i ∩ T ∗ B is connected, any two points( q , , ( q , ∈ L i ∩ T ∗ B have the same action. Let C i denote this value.For any given ε >
0, pick a smooth function f : N → R whose criticalpoints are all contained in B and such that max( f ) − min( f ) < ε . Denoteby π : T ∗ N → N the natural projection and define F = β π ∗ f where β : T ∗ N → [0 ,
1] is compactly supported and β = 1 on T ∗ R N where R ≫ φ tF ( q, p ) = ( q, p + t df ( q )) for t ∈ [0 ,
1] and ( q, p ) ∈ T ∗ N .Therefore, φ F φ i ( O N ) = L i + L f where L i + L f := { ( q, p + df ( q )) : ( q, p ) ∈ L i } .The Hausdorff convergence of the sequence L i to O N and the fact that L i ∩ T ∗ B = O B combine together to imply that ( L i + L f ) ∩ O N = { ( q,
0) : df ( q ) = 0 } for i large enough.It is easy to see that the action of ( q, ∈ ( L i + L f ) ∩ O N is given by C i + f ( q ) where C i is the constant introduced above. Therefore, γ ( L i + L f ) max( f ) − min( f ) < ε. On the other hand, by the second property from Proposition 3.2, we have | γ ( L i + L f ) − γ ( L i ) | f ) − min( f )) < ε . Combining this with theprevious inequality we obtain γ ( L i ) < ε for i large enough which provesthe lemma. 14he end of the proof of Theorem 4.3 will consist in reducing to Lemma4.5. We will assume from now on that N has even dimension. The case where N has odd dimension reduces to the even dimensional case by replacing N with N × S and all φ i ’s by φ i × Id S .We introduce for that the auxiliary mapsΦ i = φ i × φ − i : T ∗ N × T ∗ N → T ∗ N × T ∗ N, ( x, y ) ( φ i ( x ) , φ − i ( y )) , where we endow T ∗ N × T ∗ N with the symplectic form ω ⊕ ω ; observethat this is canonically symplectomorphic to T ∗ ( N × N ) equipped with itscanonical symplectic structure.Denote L i := φ − i ( O N ) and note that Φ i ( O N × N ) = L i × L i . The mapΦ i is a Hamiltonian diffeomorphism which is not compactly supported. Toobtain a compactly supported Hamiltonian diffeomorphism, we cut off thegenerating Hamiltonian of Φ i far away from O N × N and obtain a new Hamil-tonian diffeomorphism which we will continue to denote by Φ i . It is notdifficult to see that Φ i remains unchanged on a large enough neighborhoodof the zero section and so Φ i ( O N × N ) continues to be L i × L i .Properties 8 and 5 of Proposition 3.2 yield γ ( L i × L i ) = γ ( L i ) + γ ( L i ) = 2 γ ( L i ) . (8)Our proof crucially relies on the following lemma. Lemma 4.6.
Fix ε > . We can find a ball B ⊂ N , and Ψ i ∈ Ham c ( T ∗ N × T ∗ N, ω ⊕ ω ) such that the following properties hold :(i) γ (Ψ i ( O N × N )) < ε for i sufficiently large,(ii) Ψ i Φ i ( O N × N ) converges in Hausdorff topology to O N × N ,(iii) Ψ i Φ i ( O N × N ) ∩ T ∗ ( B × B ) = O B × B for i sufficiently large. We now explain why this lemma implies that γ ( L i ) →
0. Fix ε > B and Ψ i be as provided by Lemma 4.6. Using (8), the triangleinequality and the fifth property in Proposition 3.2, we get γ ( L i ) = γ ( L i × L i ) = γ (Φ i ( O N × N )) γ (Φ i ◦ Ψ i ( O N × N )) + γ (Ψ − i ( O N × N )) < γ (Φ i ◦ Ψ i ( O N × N )) + ε . The second and the third items of Lemma 4.6 allow us to apply Lemma4.5 and conclude that γ (Φ i ◦ Ψ i ( O N × N )) →
0. This implies that γ ( L i ) → roof of Lemma 4.6. Fix ε >
0. Pick a non-empty open ball B in N ≃ O N containing a fixed point p of φ and such that the displacement energy of U := T ∗ B in T ∗ N is less than ε . Note that the displacement energy of U × U inside T ∗ ( N × N ) is also less than ε .The following claim asserts the existence of a convenient Hamiltoniandiffeomorphism which switches coordinates on a small open set. Claim 4.7.
There exist an open ball B ⊂ B containing the fixed point p , < r < and a Hamiltonian diffeomorphism f of T ∗ N × T ∗ N such that: • f ( O N × N ) = O N × N , • f is the time-1 map of a Hamiltonian supported in U × U , • for all ( x, y ) ∈ U × U , we have f ( x, y ) = ( y, x ) , where U := T ∗ r B .Proof. Since N is assumed even dimensional, there is an identity isotopy,say ϕ t , of N × N which is supported in B × B with the following property:there exists a ball B ⊂ B containing p such that ϕ ( q , q ) = ( q , q ) on B × B .Let ˜ ϕ t denote the canonical lift of this isotopy to T ∗ N × T ∗ N . Theisotopy ˜ ϕ t is symplectic, it preserves O N × N , it is supported in T ∗ B × T ∗ B ,and it can be checked that ˜ ϕ ( x, y ) = ( y, x ) on T ∗ B × T ∗ B . Furthermore,the isotopy is Hamiltonian. Let H denote a generating Hamiltonian of theisotopy which is supported in T ∗ B × T ∗ B .To construct our desired Hamiltonian diffeomorphism f , we simply re-place H by βH where β is a smooth cut-off function on T ∗ ( N × N ) suchthat β = 1 on T ∗ − δ ( N × N ), where δ is a small positive number, and β = 0outside T ∗ ( N × N ). We set f to be the time-1 map of the Hamiltonian flowof βH and leave it to the reader to check that it satisfies the requirementsof the claim.We can now complete the proof of Lemma 4.6. Since p ∈ B , thereexists a ball B ⊂ B and 0 < r < r such that φ ( U ) ⋐ U (i.e., φ ( U ) iscompactly contained in U ), where U := T ∗ r B .Let Υ i = φ i × Id T ∗ N and letΨ i = Υ − i ◦ f − ◦ Υ i ◦ f. We will first show that γ (Ψ i ( O N × N )) < ε . Note that by Equation (2),we have γ (Ψ i ( O N × N )) γ (Ψ i ), where γ (Ψ i ) is the Hamiltonian γ whichwas introduced above in Equation (3). Hence, it is sufficient to show that γ (Ψ i ) < ε . The triangle inequality for γ (Equation (6)) and its conjugacyinvariance (Equation (5)) yield γ (Ψ i ) γ ( f ). Lastly, γ ( f ) < ε becausethe displacement energy of its support is smaller than ε ; see Equation (7).This implies Property (i) in Lemma 4.6.16ext, we will verify the second property in Lemma 4.6. Define Ψ :=Υ − ◦ f − ◦ Υ ◦ f , where Υ := φ × Id T ∗ N , and let Φ := φ × φ − . Since f, Υ and Φ preserve O N × N , we conclude that Φ ◦ Ψ also preserves O N × N .Now, there exists a neighborhood of O N × N where the sequences Ψ i and Φ i converge uniformly to Ψ and Φ, respectively. It follows that Φ i ◦ Ψ i ( O N × N )converges in Hausdorff topology to O N × N .It remains to verify the third property from the lemma. We leave itto the reader to check that Φ i ◦ Ψ i ( x, y ) = ( x, y ) for all ( x, y ) ∈ U × U ,when i is large enough. This relies crucially on the following observations: f ( x, y ) = ( y, x ) on U × U and Υ i ( U × U ) ⊂ U × U for i large enough.The last statement is a consequence of the fact that φ ( U ) ⋐ U .Let B = B × B and r = r , so that T ∗ r B = U × U . As we have seen,for i large, Φ i ◦ Ψ i coincides with the identity on T ∗ r B . We claim that thisimplies the third property. Indeed, it clearly implies O B ⊂ Φ i ◦ Ψ i ( O N × N ) ∩ T ∗ B . Furthermore, it also implies that if Φ i ◦ Ψ i ( O N × N ) ∩ T ∗ B containsa point which is not in O B , then such a point is in T ∗ B \ T ∗ r B . But ofcourse this cannot happen for i large because of the Hausdorff convergenceof Φ i ◦ Ψ i ( O N × N ) to O N × N . This establishes the third property in Lemma4.6. By the assumptions of the theorem, one can find some r > φ i ∈ Ham c ( T ∗ r N, ω ) such that φ i converges uniformly to φ . Since the num-ber of Lagrangian spectral invariants of φ is assumed to be less than cl( N ),there exist some α, β ∈ H ∗ ( N ) with deg α, deg β < dim N and α ∩ β = 0, suchthat ℓ ( α, φ ) = ℓ ( α ∩ β, φ ) =: λ . By the continuity of spectral invariants ( i.e. the first item of Theorem 4.1), we have lim ℓ ( α, φ i ) = lim ℓ ( α ∩ β, φ i ) = λ ,when i → ∞ .Let U ⊂ O N be any neighbourhood of L ∩ O N in O N . It is enough toshow that the closure U is homologically non-trivial in O N . For doing this,pick a smooth function f : N → R such that f = 0 on U and f < N \ U . Denote by π : T ∗ N → N the natural projection and define F = βπ ∗ f where β : T ∗ N → R is compactly supported and β = 1 on T ∗ R N where R istaken to be large in comparison to r . Claim 4.8.
There exists an integer i such that for any i > i , and forsufficiently small values of ε > , ℓ ( α ∩ β, φ εF φ i ) = ℓ ( α ∩ β, φ i ) . Proof.
Let L i = φ i ( O N ) and L εf = φ εF ( O N ). Note that φ tF ( q, p ) = ( q, p + t df ( q )) for t ∈ [0 ,
1] and ( q, p ) ∈ T ∗ r N . Therefore, we have L εf = { ( q, εdf ( q )) : q ∈ N } and φ εF φ i ( O N ) = φ εF ( L i ) = L i + L εf where L i + L εf := { ( q, p + εdf ( q )) : ( q, p ) ∈ L i } . 17ince L ∩ π − ( O N \ U ) is compact and does not intersect O N , and sincethe sequence φ i converges uniformly to φ , we conclude that for small enough ε and large enough i , ( L i + L εf ) ∩ π − ( O N \ U ) does not intersect O N as well.On the other hand, since f = 0 on U , we get that ( L i + L εf ) ∩ π − ( U ) = L i ∩ π − ( U ). Therefore, for small enough ε > i , theLagrangians L i and L i + L εf have the same intersection points with the zerosection O N . Moreover, it is easy to see that for each such intersection point,the two action values corresponding to φ i and φ εF φ i coincide. Therefore,by fixing i and ε >
0, and considering the family of Lagrangians L i + L sεf when s ∈ [0 , L i + L sǫf , φ sεF φ i ) do notdepend on s . Also, recall that the action spectrum has an empty interiorin R . As a result, since the value ℓ ( α ∩ β, φ εF φ i ) depends continuously on s , we conclude that it in fact does not depend on s ∈ [0 , ℓ ( α ∩ β, φ i ) = ℓ ( α ∩ β, φ εF φ i ).The triangle inequality of Proposition 3.2 implies that, for all i , ℓ ( α ∩ β, φ εF φ i ) − ℓ ( α, φ i ) ℓ ( β, φ εF ). Using the above claim, for i large and ε smallenough, we have ℓ ( α ∩ β, φ i ) − ℓ ( α, φ i ) ℓ ( β, φ εF ). Taking limit as i → ∞ ,and recalling that lim ℓ ( α, φ i ) = lim ℓ ( α ∩ β, φ i ) = λ , we obtain 0 ℓ ( β, φ εF ).We can now conclude our proof as follows. On the one hand, by Proposi-tion 3.2.7, we have ℓ ( β, φ εF ) = c LS ( β, εf ) = c LS ([ M ] ∩ β, εf ). On the otherhand, Propostion 3.1.2 gives c LS ([ M ] , εf ) = 0. Thus, using Proposition3.1.3, we obtain the equality c LS ([ M ] ∩ β, εf ) = c LS ([ M ] , εf ). By Proposi-tion 3.1.4 it follows that the zero level set of f , that is U , is homologicallynon-trivial. This section is dedicated to the proof of Theorem 1.4. Recall that we con-sider a sequence L i of Legendrian submanifolds, contact isotopic to the zerosection in J N = T ∗ N × R , which has a Hausdorff limit L . Denote by π R : J N = T ∗ N × R → R the natural projection.We have not been able to verify whether it is possible to define Leg-endrian spectral invariants for the Hausdorff limit L . However, as we willnow explain, it is still possible to view Theorem 1.4 as an incarnation ofPrinciple 1: Let K be a (smooth) Legendrian submanifold of J N which iscontact isotopic to the zero section. Then, as was explained in Section 3.3,the set spec( K ) = π R ( K ∩ ( O N × R )) is the set of critical values of the gfqiassociated to K . Hence, if the cardinality of spec( K ) is smaller than cl( N ),then so is the total number of spectral invariants of K . Therefore, despitethe fact that we cannot define spectral invariants for the Hausdorff limit L ,we can interpret the cardinality of the set spec( L ) = π R ( L ∩ ( O N × R )) being maller than cl( N ) to mean that L has fewer spectral invariants than cl( N ) .Proof of Theorem 1.4. Observe that the Hausdorff convergence of L i ’s to L implies that the set L i ∩ ( O N × R ) is contained in an arbitrarily smallneighbourhood of L ∩ ( O N × R ) for large i . Because ℓ ( a, L i ) corresponds toan intersection point of L i with the zero wall, we conclude that the set oflimit points of { ℓ ( a, L i ) : a ∈ H ∗ ( N ) \ { } , i ∈ N } is contained in spec( L ).Assume that spec( L ) has less than cl( N ) points. It follows from theabove discussion that there exist α, β ∈ H ∗ ( N ) \ { } and λ ∈ spec( L ) suchthat for a subsequence ( i k ) of indices we have ℓ ( α, L i k ) → λ and ℓ ( α ∩ β, L i k ) → λ as k → ∞ . By passing to this subsequence, we may furtherassume that ℓ ( α, L i ) → λ and ℓ ( α ∩ β, L i ) → λ as i → ∞ . Let us show that L ∩ ( O N × { λ } ) is homologically non-trivial in O N × { λ } .Pick any neighbourhood V of L ∩ ( O N × { λ } ) in J N . Denote U := π N ( V ), where π N : J N → N is the natural projection, and pick a smoothfunction f : N → R such that f = 0 on U and f < N \ U . Claim 5.1.
There exists an integer i such that for any i > i , and forsufficiently small values of ε > , ℓ ( α ∩ β, L i + L εf ) = ℓ ( α ∩ β, L i ) . Proof.
By the Hausdorff convergence of L i to L , there exists some δ > i large enough and ε > L i + L εf ) ∩ ( O N × ( λ − δ, λ + δ )) ⊂ V. Furthermore, for any ( q, p, z ) ∈ V , we have that q ∈ U and thus f ( q ) = 0 and df ( q ) = 0. This implies that ( L i + L εf ) ∩ ( O N × ( λ − δ, λ + δ )) = L i ∩ ( O N × ( λ − δ, λ + δ )), in particular spec( L i + L εf ) ∩ ( λ − δ, λ + δ ) = spec( L i ) ∩ ( λ − δ, λ + δ ).The continuity and spectrality properties of spectral invariants, togetherwith the fact that the spectrum of L i has an empty interior in R and that ℓ ( α ∩ β, L i ) ∈ ( λ − δ, λ + δ ) for i large enough, imply that the spectralinvariant ℓ ( α ∩ β, L i + L εf ) is independent of ε .Now the triangle inequality of Proposition 3.3 implies that, for all i , ℓ ( α ∩ β, L i + L εf ) − ℓ ( α, L i ) ℓ ( β, L εf ). Using the above claim, for i largeand ε small enough, we have ℓ ( α ∩ β, L i ) − ℓ ( α, L i ) ℓ ( β, L εf ). Takinglimit as i → ∞ , and recalling that ℓ ( α ∩ β, L i ) , ℓ ( α, L i ) → λ , we obtain0 ℓ ( β, L εf ).We can now conclude our proof as follows. On the one hand, by Proposi-tion 3.3.4, we have ℓ ( β, L εf ) = c LS ( β, εf ). Note that c LS ( β, εf ) = c LS ([ N ] ∩ β, εf ) and by the above paragraph this number is non-negative. On theother hand, Propostion 3.1.2 gives c LS ([ N ] , εf ) = 0. Thus, using Propo-sition 3.1.3, we obtain the equality c LS ([ N ] ∩ β, εf ) = c LS ([ N ] , εf ). ByProposition 3.1.4 it follows that the zero level set of f , that is the closure of19 = π N ( V ), is homologically non-trivial in N . Since our choice of a neigh-bourhood V of L ∩ ( O N ×{ λ } ) was arbitrary, we conclude that L ∩ ( O N ×{ λ } )is homologically non-trivial in O N × { λ } . References [1] L. Buhovsky. Towards the C flux conjecture. Algebr. Geom. Topol. ,14(6):3493–3508, 2014.[2] L. Buhovsky, V. Humili`ere, and S. Seyfaddini. The action spectrumand C symplectic topology. arXiv:1808.09790 .[3] L. Buhovsky, V. Humili`ere, and S. Seyfaddini. A C counterexampleto the Arnold conjecture. Invent. Math. , 213(2):759–809, 2018.[4] M. Chaperon. On generating families. In
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