Viterbo conjecture for Zoll symmetric spaces
We prove a conjecture of Viterbo from 2007 on the existence of a uniform bound on the Lagrangian spectral norm of Hamiltonian deformations of the zero section in unit cotangent disk bundles, for bases given by compact rank one symmetric spaces We discuss generalizations and give applications, in particular to symplectic topology. Our key methods, which are of independent interest, consist of a reinterpretation of the spectral norm via the asymptotic behavior of a family of cones of filtered morphisms, and a quantitative deformation argument for Floer persistence modules, that allows to excise a divisor.
- 1 Introduction and main results
- 2 Preliminary notions
- 3 Proof of Theorem B
- 4 Applications to symplectic topology, and quasi-morphisms
- 5 Discussion
1. Introduction and main results
In 2007 Viterbo (see [102, Conjecture 1]) has conjectured that the spectral norm  of each exact Lagrangian deformation of the zero section in the unit co-disk bundle of the closed manifold with respect to a Riemannian metric is uniformly bounded by a constant The spectral norm is given by the difference of two homological minimax values in suitable generating function homology, and can be recast in terms of Lagrangian Floer homology . This conjecture has since been completely open.
In this paper we start with the observation that the conjecture of Viterbo makes sense, and is of interest for arbitrary closed manifolds Our main theorem proves it for all belonging to the four infinite families of compact rank one symmetric spaces. In particular we prove the original conjecture of Viterbo for We observe the curious fact that while it is known that for the Lagrangian Hofer metric , even in the case of it is easy to see by an argument of Khanevsky  that has infinite diameter.
We remark that all homological notions and computations in this paper depend a priori on the choice of coefficients. We work with the ground field throughout the paper, and write or if we want to emphasize the symplectic manifold wherein it is computed, for the corresponding spectral norm.
Let and equipped with a Riemannian metric Then there exists a constant such that
for all exact Lagrangian deformations of the zero section in
We note that since bounds from above the difference between each two spectral invariants of the same bound applies to each such difference. To slightly strengthen this result, and to reflect on its proof, we recall that manifolds in are precisely those compact symmetric spaces that admit a Riemannian metric with all prime geodesics closed and of the same length
The Zoll cut construction , which is a special case of the symplectic cut construction , allows us to embed each manifold in as a monotone Lagrangian submanifold of a closed monotone symplectic manifold in such a way as to exhibit the open unit cotangent disk bundle as the complement of a symplectic Donaldson divisor The manifolds up to scaling of symplectic forms, belong to the four infinite families respectively, where is endowed with the standard Fubini-Study symplectic form denotes is the complex Grassmannian of two-planes in and is a smooth complex quadric in Record the dimensions of the minimal Maslov numbers of as a Lagrangian submanifold of and set
The normalization of the symplectic form on that is naturally obtained from the Zoll cut construction is such that the minimal symplectic area of a disk in with boundary on is
Let be a Lagrangian submanifold of Let be an exact Lagrangian submanifold of that is an exact Lagrangian deformation of the zero section considered as a Lagrangian submanifold of That is, there exists a Hamiltonian isotopy of with Then for
One key topological property of Zoll manifolds is that their homology algebra is isomorphic to a truncated polynomial ring on a homogeneous element The class is given by in the case of respectively. This enables one to prove [12, 91, 60] that for the self-Floer homology of also known as the Lagrangian quantum homology , with coefficients in the Novikov field with quantum variable of degree satisfies, as a ring
In patricular for all and is a quantum root of unity: This allows us to provide a uniform bound  on the spectral norm and boundary depth of as computed in that is smaller than the minimal area of a pseudo-holomorphic disk in with boundary on . In fact, there is an algebraic version of the spectral norm, that is easy to see to be sufficient for our purposes, and to require only the above algebraic properties. Reinterpreting the spectral norm in terms of cones of filtered morphisms depending on a large parameter, and proving an algebraic deformation argument for the suitable persistence modules allows us to deduce from the above upper bound our desired result, by means of symplectic field theory .
Let and a closed manifold equipped with a Riemannian metric Then there exists a constant such that
for all exact Lagrangian submanifolds
This conjecture and hence Theorem B would follow from Theorem A for , if the nearby Lagrangian conjecture (see ) were true. In particular it holds for by a folklore argument, for by a combination of  and , and for by . We expect it to be possible to prove this generalized conjecture for all by verifying more algebraically the conditions of Proposition 20 for all exact In the generality of Conjecture 1, the best known result is currently an upper bound of that is linear in the boundary depth [97, 98] of the Floer complex of with a Lagrangian fiber of . Finally, while in this paper we work with coefficients in we expect the same statement for to hold with arbitrary choice of ground field
The first application of Theorem B is the following -continuity statement for the Hamiltonian spectral norm, inspired by [87, Remark 1.9]. Considering the distance function on coming from a Riemannian metric, we define the following distance function on the group for set
We call the topology induced by the topology on Recall that in  following [83, 101], a spectral norm on any closed symplectic manifold was introduced and shown to be non-degenerate. Moreover, provides a lower bound on the celebrated Hofer norm [55, 61].
The spectral norm is continuous with respect to the -topology on In fact, when comes from the Zoll metric, we obtain for all the inequality
for the constant for
This statement implies that the spectral norm is continuous in the topology. This latter fact was known for being a closed symplectic surface . During the preparation of this paper, this was also shown in  for closed symplectically aspherical manifolds In the case the linear bound (2) in improves upon the Hölder bound of exponent in .
Similarly to the -continuity result of , Theorem C has further applications in symplectic topology, extending results that were previously known for the most part in dimension or for open symplectic manifolds, to closed higher-dimensional symplectic manifolds. First, a partial answer to a question of Le Roux  for follows. Statements of this kind first appeared in  for and for a class of closed aspherical symplectic manifolds containing and in  for certain additional open symplectic manifolds.
Let Then for all the interior of in is non-empty.
For all by [60, Theorem F] there exists such that and hence Moreover, by continuity of there is an open -ball around in with and hence ∎
Second, the displaced disks problem of Béguin, Crovisier, and Le Roux, solved for closed surfaces in , follows for
Let be a Hamiltonian homeomorphism that displaces a symplectic closed ball of radius Then
As first observed in , to a Hamiltonian normalized by the zero-mean or the compact support condition, one can, via the theory of persistence modules, associate a multi-set of intervals in called a barcode. This map is Lipschitz with respect to the -distance on and the bottleneck distance on the space of barcodes. This observation was used in , in [99, 105, 78, 3, 92, 40] and more recently in [60, 66, 18, 93, 95, 31] to produce various quantitative results in symplectic topology. Set for the quotient space of with respect to the isometric -action by shifts.
Denoting for by its barcode of index with coefficients in the Novikov field with quantum variable of degree considered up to shifts. By [60, Corollary 6], and Theorem C, the barcode depends only on the time-one map of and we immediately obtain the following statement.
The map is continuous, and hence extends to completions:
In fact, one may take coefficients in with quantum variable of degree in which case the same statement holds for being the image in of either the index or the index barcode of .
In the case of surfaces, a similar statement was proven in  using different tools, while the same statement was proven in [60, Remark 8] using [60, Corollary 6]. It was also shown in  for closed symplectically aspherical manifolds, using [60, Corollary 6].
Following , we use Corollary 6, and the conjugation invariance property (15) of to establish that is constant on weak conjugacy classes. Two elements of a topological group are called weakly conjugate if for all continuous conjugacy-invariant maps to a Hausdorff topological space This is an equivalence relation, that was studied in ergodic theory and dynamical systems, see [50, 51, 68] and references therein. We consider this notion for
It is important to remark the following. For an element of a topological group denote by the conjugacy class of in Then are weakly conjugate given that there exist with and with for all the closures of the conjugacy classes being taken in In particular if lies in the closure then are weakly conjugate. We refer to  for further discussion of this notion.
The barcode for is a weak conjugacy invariant.
Theorem D ().
Let the fixed points of in the contractible class be a finite set. Then the barcode consists of a finite number of bars, and the number of endpoints of these bars equals
This result together with Corollary 7 implies that does not possess a dense conjugacy class, that is - it is not a Rokhlin group. The same consequence for surfaces of higher genus was known by [45, 46]. The case of the two-torus, as well as that of was settled in  (the former case building on , see also ), and that of the sphere in , while  shows it for closed symplectically aspherical manifolds.
There exists a homeomorphism that is not weakly conjugate to any diffeomorphism in
It was recently proven in [48, Corollary 5.2] that for each pseudo-rotation that is a Hamiltonian diffeomorphism with precisely periodic points of all positive integer periods, there exists an increasing integer sequence such that Moreover, under a certain strong irrationality assumption on the vector given by the mean indices of the periodic points, it is shown in [48, Theorem 1.4] that there is a sequence such that in topology. By Theorem C, the latter result, whenever it holds, implies the former result on
Quasimorphisms on the Hamiltonian group of cotangent disk bundles
The second application of Theorem A is to the symplectic topology of unit cotangent disk bundles These applications were anticipated in . We start with the notion of a quasi-morphism on a group and refer to  for further exposition.
A quasimorphism on a group is a function satisfying the bound
The number is called the defect of the quasimorphism. If then the quasimorphism is called trivial: in this case it is in fact a homomorphism For each quasimorphism there exists a unique homogeneous, that is additive on each abelian subgroup of quasimorphism such that is a bounded function. This homogeneization is given by the formula
Quasi-morphisms on the (universal cover of) the Hamiltonian group of closed symplectic manifolds were constructed in  and many subsequent works (we refer to  for a review of the literature). However, not many examples are known in the case of open symplectic manifolds [62, 15]. One can construct such quasimorphisms by pulling them back by a conformally symplectic embedding of an open symplectic manifold into a closed symplectic manifold In our case, endowing with the standard Zoll metric, we observe that each gives a symplectic embedding of into the respective in the complement of Now, as shown in , there exists a non-trivial homogeneous Calabi quasimorphism on the universal cover of enjoying the following additional property. If is a Hamiltonian with zero mean, with for then for the class generated by the Hamiltonian path Now, via we obtain a natural homomorphism giving a homogeneous quasi-morphism on Looking at Hamiltonians with zero mean, and constant on it is easy to check that is non-trivial. However, it was hitherto unknown whether quasimorphisms can be constructed intrinsically from the symplectic geometry of itself. We resolve this question below for Consider the invariant
Define by the spectral invariants being computed inside These maps were defined, and shown to enjoy various properties in [71, Theorems 1.3 and 1.8, Propositions 1.4 and 1.9]. In particular, depends only on and defines a map We prove, via Theorem A, the following new properties of these maps.
The map is a non-zero homogeneous quasimorphism. Moreover vanishes on each element such that is displaceable. For the map satisfies
where is the Poisson bracket of In particular, whenever we obtain
I thank Peter Albers, Paul Biran, Octav Cornea, Asaf Kislev, Leonid Polterovich, Vukašin Stojisavljević, Dmitry Tonkonog, Renato Vianna, and Frol Zapolsky for fruitful collaborations during which I learnt many of the tools that I apply in this paper. I thank Sobhan Seyfaddini and Georgios Dimitroglou Rizell for useful conversations. This work was initiated and was partially carried out during my stay at the Institute for Advanced Study, where I was supported by NSF grant No. DMS-1128155. It was partially written during visits to Tel Aviv University, and to Ruhr-Universität Bochum. I thank these institutions and Helmut Hofer, Leonid Polterovich, and Alberto Abbondandolo, for their warm hospitality. At the University of Montréal, I am supported by an NSERC Discovery Grant and by the Fonds de recherche du Québec - Nature et technologies.
2. Preliminary notions
We briefly describe the pertinent part of the standard package of filtered Floer homology in the context of monotone symplectic manifolds, and their monotone Lagragnian submanifolds. We refer to [73, 64, 60] for more details and a review of the literature. However, we emphasize two points. Firstly, in Section 2.1.5 we describe how filtered relative Hamiltonian Floer homology of a Hamiltonian and a Lagrangian is isomorphic to the filtered Lagrangian Floer homology of the pair of Lagrangian submanifolds and with appropriate choices of additional data called anchors [73, Chapter 14]. Second, in Section 2.4 we recall and describe a few ways to determine the collection of bar-lengths of the barcodes of persistence modules associated to filtered Floer homology.
2.1. Filtered Floer homology.
All Lagrangian submanifolds we consider in this paper shall be weakly homologically monotone that is the class of the symplectic form in cohomology relative to and the Maslov class are positively proportional
for Moreover, when is closed, we require that for an integer called the minimal Maslov number of in In this case will be weakly homologically monotone, that is in When is not closed, we require it be exact, that is for a one-form and to have symplectically convex boundary. This means that the vector field on defined by is transverse to and points outwards at In this open case, we shall consider exact Lagrangian submanifolds, that is for We denote by the space of time-dependent Hamiltonians on where in the closed case is normalized to have zero mean with respect to and in the non-closed case, it is normalized to vanish near the boundary. The time-one maps of isotopies generated by time-dependent vector fields are called Hamiltonian diffeomorphisms and form the group For we call the Hamiltonians For we have while the isotopy viewed as a path in is homotopic to with fixed endpoints. Since homotopic Hamiltonian isotopies give naturally isomorphic graded filtered Floer complexes, we shall identify the two operations and In particular we will identify between and the two Hamiltonians Similarly, for we set to generate the flow in other words A homotopic path is generated by for surjective monotone non-decreasing reparametrizations such that Finally, let be the space of -compatible almost complex structures on
In each case below, Floer theory, first introduced by A. Floer [37, 38, 39], is a way to set up Morse-Novikov homology for an action functional defined on a suitable cover of a path or a loop space determined by the geometric situation at hand. We refer to  and references therein for details on the constructions described in this subsection.
Absolute Hamiltonian case.
Consider Let be the space of contractible loops in Let be the surjection given by Let be the cover of associated to The elements of can be considered to be equivalence classes of pairs of and its capping The symplectic action functional
is given by
that is well-defined by monotonicity: Assuming that is non-degenerate, that is the graph intersects the diagonal transversely, the generators over the base field of the Floer complex are the lifts to of -periodic orbits of the Hamiltonian flow These are the critical points of and we denote by the set of its critical values. Choosing a generic time-dependent -compatible almost complex structure and writing the asymptotic boundary value problem on maps defined by the negative formal gradient on of the count of isolated solutions, modulo -translation, gives a differential on the complex This complex is graded by the Conley-Zehnder index , with the property that the action of the generator of has the effect Its homology does not depend on the generic choice of Moreover, considering generic families interpolating between different Hamiltonians and writing the Floer continuation map, where the negative gradient depends on the -coordinate we obtain that in fact does not depend on either. While is finite-dimensional in each degree, it is worthwhile to consider its completion in the direction of decreasing action. In this case it becomes a free graded module of finite rank over the Novikov field with being a variable of degree
Moreover, for the subspace spanned by all generators with forms a subcomplex with respect to and its homology does not depend on Arguing up to one can show that a suitable continuation map sends to for
It shall also be useful to define Finally, one can show that for each depends only on the class of the path in the universal cover of the Hamiltonian group of
We mention that it is sometimes beneficial to consider the slightly larger covers defined via the evident inclusions This corresponds to extending coefficients to with and respectively.
In case when is degenerate, we consider a perturbation with such that is non-degenerate, and is generic with respect to and define the complex generated by and filtered by the action functional
Relative Hamiltonian case.
Consider and a monotone Lagrangian as above. Let be the space of path from to in contractible relative to Let be the composition of the Hurewicz map with the Maslov class. Let be the cover of associated to The elements of can be considered to be equivalence classes of pairs of and its capping The symplectic action functional
is given by
that is well-defined by monotonicity: Let be the Hamiltonian flow of Assuming is non-degenerate, that is intersects transversely, the generators over of the Floer complex are the lifts to consisting of integral trajectories of with endpoints in These are the critical points of and we denote by the set of its critical values. Choosing a generic and writing the asymptotic boundary value problem on maps defined by the negative formal gradient on of the count of isolated solutions, modulo -translation, gives a differential on the complex This complex is graded by the Conley-Zehnder (or Robbin-Salamon) index , with the property that the action of the generator of has the effect Its homology does not depend on the generic choice of Moreover, considering generic families interpolating between different Hamiltonians and writing the Floer continuation map, where the negative gradient depends on the -coordinate we obtain that in fact does not depend on either. While is finite-dimensional in each degree, it is worthwhile to consider its completion in the direction of decreasing action. In this case it becomes a free graded module of finite rank over the Novikov field with being a variable of degree
Moreover, for the subspace spanned by all generators with forms a subcomplex with respect to and its homology does not depend on Arguing up to one can show that a suitable continuation map sends to for Finally, one can show that depends only on the class