A max inequality for spectral invariants of disjointly supported Hamiltonians
AA max inequality for spectral invariants of disjointlysupported Hamiltonians.
Shira TannyFebruary 16, 2021
Abstract
We study the relation between spectral invariants of disjointly supported Hamil-tonians and of their sum. On aspherical manifolds, such a relation was establishedby Humili`ere, Le Roux and Seyfaddini. We show that a weaker statement holds in awider setting, and derive applications to Polterovich’s Poisson bracket invariant andto Entov and Polterovich’s notion of superheavy sets.
Contents U . . . . . . . . . . . . . . . . . . . . . 19 a r X i v : . [ m a t h . S G ] F e b Uniform bounds on spectral invariants. 29 C ( U ) in special cases. 37 Hamiltonian spectral invariants on closed symplectic manifolds were introduced by Ohand Schwartz [13, 19]. These invariants assign to each Hamiltonian H : M × S → R and a non-zero quantum homology class α ∈ QH ∗ ( M ) a real number, denoted by c ( H ; α ).In this paper we consider only spectral invariants with respect to the fundamental class,and therefore abbreviate c ( · ) := c ( · ; [ M ]). Spectral invariants have been widely studiedand have many applications in symplectic geometry. One relevant application concernslower bounds for Polterovich’s Poisson bracket invariant, which was introduced in [15, 16].Given a finite open cover { U i } Ni =1 of a closed symplectic manifold, the Poisson bracketinvariant of { U i } is defined by pb ( { U i } ) := inf { f i } max | x j | , | y k |≤ (cid:88) j x j f j , (cid:88) k y k f k , where the infimum is taken over all smooth partitions of unity that are subordinate tothe cover { U i } . Polterovich explained the relation of this invariant to quantum mechanicsand conjectured a lower bound for it, in terms of the displacement energies of the sets.Moreover, he showed how upper bounds for the spectral invariants of sums of disjointlysupported Hamiltonians can be used to establish lower bounds for pb . This inspired severalworks studying upper bounds for the spectral invariant of a sum of disjointly supportedHamiltonians: In [16], Polterovich produced upper bounds for Hamiltonians supported incertain domains on symplectically aspherical manifolds, namely when both the symplecticform ω and the first Chern class c vanish on π ( M ). Later, in [20], Seyfaddini constructedso called spectral killers and bounded the spectral invariant of a sum of Hamiltonianssupported in disjoint small balls on monotone manifolds, i.e., when ω is proportionalto c . In [11], Ishikawa considered Hamiltonians supported in symplectic embeddings ofstrongly convex sets in R n , into monotone manifolds. Finally, in [10], Humili`ere, Le Rouxand Seyfaddini proved that, on symplectically aspherical manifolds, the spectral invariantof a sum of Hamiltonians supported in certain disjoint domains, is equal to the maximumover the spectral invariants of the Hamiltonians: Theorem (Humili`ere-Le Roux-Seyfaddini, [10]) . Let H , . . . , H N be Hamiltonians sup-ported in disjoint incompressible Liouville domains in a symplectically aspherical manifold. A partition of unity is a collection of non-negative functions that sum up to 1. We say that { f i } issubordinate to { U i } if supp ( f i ) ⊂ U i for each i . hen, c ( H + · · · + H N ) = max { c ( H ) , . . . , c ( H N ) } . This result is referred to as the “max formula” for spectral invariants. An alternativeproof for the max formula, as well as an inequality for spectral invariants with respectto a general homology class, were given in [6]. Humili`ere, Le Roux and Seyfaddini alsoshowed that the max formula does not hold on the sphere, by constructing Hamiltonians H and H , supported in disjoint disks on S , for which c ( H + H ) < max { c ( H ) , c ( H ) } .A natural question is whether an inequality holds in general. In what follows, we considerdisjointly supported Hamiltonians H , . . . , H N on a closed connected symplectic manifold( M, ω ), and show that under certain conditions one has c ( H + · · · + H N ) ≤ max { c ( H ) , . . . , c ( H N ) } . (1)The main ingredient of the proof is the construction of spectral killers, in the spirit ofSeyfaddini [20]. We change Seyfaddini’s construction in order to prove a max inequality,as well as extend it to a more general setting. Let (
M, ω ) be a closed symplectic manifold. Throughout the paper, we consider Hamil-tonians supported in domains satisfying certain conditions. These domains include, forexample, symplectic embeddings into M of star-shaped domains in R n with smoothboundaries and such that the radial vector field is transverse to the boundary (following[8], we call such domains “nice star-shaped domains”). In order to describe the class ofrelevant domains in full generality, let us recall a few standard notions. A domain U ⊂ M has a contact type boundary if there exists a vector field Y , called the Liouville vector field ,that is defined on a neighborhood of the boundary ∂U , satisfies L Y ω = ω , is transverseto the boundary, and points outwards. In this case, λ := ι Y ω is a primitive of ω and itsrestriction to ∂U is called the contact form associated to Y . The Reeb vector field R on ∂U is defined by the equations ω ( Y, R ) = 1 , ω ( R, · ) | T ∂U = 0 . The flow ϕ tR : ∂U → ∂U of R is called the Reeb flow , and we denote the set of itscontractible periodic orbits (of any period) by P ( ∂U ). The action of a periodic Reeb orbit γ ∈ P ( ∂U ) is given by (cid:82) γ λ . The action spectrum of ∂U isSpec( ∂U ) := (cid:26)(cid:90) γ λ : γ ∈ P ( ∂U ) (cid:27) . Finally, the boundary ∂U ⊂ M is called incompressible if the map π ( ∂U ) → π ( M )induced by the inclusion is injective. In particular, when ∂U is simply connected, it isincompressible.We prove that the max inequality (1) holds for Hamiltonians supported in disjointdomains with incompressible contact type boundaries, under additional conditions onthe Hamiltonians and the domains, which depend on whether the symplectic manifold3 U ψ log(1+ σ ) ∂U U Figure 1:
An illustration of a σ -extendable domain with a contact type boundary. is positively monotone, negatively monotone or rational. For the sake of convenience weassume from now on that dim M = 2 n is greater than 2, unless stated otherwise. The maxformula proved by Humili`ere, Le Roux and Seyfaddini holds for all symplectic surfacesother than the sphere. We discuss the max inequality on the sphere in Section 1.1.3 whichconcerns positively monotone manifolds. Let us start by describing the results on rationalsymplectic manifolds. Let (
M, ω ) be a closed rational symplectic manifold, namely ω ( π ( M )) = κ Z for some κ ∈ R . It is simpler to establish a max inequality if the disjoint supports are “far enough”from one another. In order to make this condition precise consider the following definition. Definition 1.1.
Let U ⊂ M be a domain with a contact type boundary. We say that U is σ -extendable , for σ >
0, if the flow ψ τ of the Liouville vector field Y exists for all time0 < τ < log(1 + σ ). The σ -extension of such a domain U is defined to be(1 + σ ) U := U ∪ (cid:91) τ ∈ [0 , log(1+ σ )] ψ τ ∂U . See Figure 1 for an illustration of a σ -extendable domain. We remark that for everydomain U with a contact type boundary, there exists ε > U is ε -extendable,see Section 2.3. Example 1.2.
Suppose that the ball B of radius r around the origin in R n (endowed withthe standard symplectic form) embeds into M . Then the restriction of this embedding tothe ball of radius r/ √ B is a 1-extendable domain.The following theorem asserts that the max inequality holds for Hamiltonians whichare supported in extendable domains with disjoint extensions (see Figure 2), and whosespectral invariants are small compared to the “size” of the extensions. Theorem 1.
Let U i be σ i -extendable domains with incompressible contact type boundaries,such that the extensions { (1 + σ i ) U i } are pairwise disjoint. Then, for Hamiltonians H i supported in U i , such that c ( H i ) < min { κ, σ i · min Spec( ∂U i ) } it holds that c ( H + · · · + H N ) ≤ max { c ( H ) , . . . , c ( H N ) } . H Figure 2:
An illustration of Hamiltonians supported in extendable domains, such that the exten-sions are disjoint.
When the domains containing the supports are not necessarily “far” from each other,we assume that the action spectrum of the contact boundaries, Spec( ∂U ), is contained in alattice T Z , such that T divides κ . Examples for such domains are symplectic embeddingsof balls of radius r in R n such that πr divides κ . Theorem 2.
Let U i be disjoint domains with incompressible contact type boundaries suchthat Spec( ∂U i ) ⊂ T i Z and T i | κ for all i . Then, for Hamiltonians H i supported in U i suchthat c ( H i ) < T i , it holds that c ( H + · · · + H N ) ≤ max { c ( H ) , . . . , c ( H N ) } . (2) On negatively monotone manifolds, namely when ω | π ( M ) = κ · c | π ( M ) for κ ≤
0, weshow that the max inequality (1) holds for Hamiltonians supported in disjoint domainswith incompressible contact type boundaries, if we assume in addition that the contactConley-Zehnder index of every Reeb orbit is non-negative. The contact Conley-Zehnderindex assigns an integer, which we denote by CZ R ( γ, u ), to every periodic Reeb orbit γ ∈ P ( ∂U ) and a capping disk u ⊂ ∂U . This index is well defined when the Reebflow is non-degenerate , and is reviewed in Section 2 together with other preliminariesfrom symplectic and contact geometry. It is well known that when U is a symplecticembedding of a strictly convex domain in R n into M , the CZ index of every Reeb orbitis non-negative . Theorem 3.
Let ( M, ω ) be a negatively monotone symplectic manifold and suppose U i ⊂ M are disjoint domains with incompressible contact type boundaries, such that the contactConley-Zehnder index of every Reeb orbit is non-negative. Then, for any collection ofHamiltonians H i : M × S → R supported in U i respectively, c ( H + · · · + H N ) ≤ max { c ( H ) , . . . , c ( H N ) } . We remark that one can perturb the Liouville vector field to make the Reeb flow non-denegerate, seeSection 2. In [9], Hofer, Wysocki and Zehnder proved that for every strictly convex domain in R n with a smoothboundary, the contact CZ index of every Reeb orbit is at least n + 1, and in particular is positive (thedefinitions and proofs are written for n = 2, see the remark on p.222 for the general case). .1.3 Positively monotone manifolds. When the symplectic manifold is positively monotone, namely ω | π ( M ) = κ · c | π ( M ) for κ ≥
0, we need to impose additional assumptions on the domains U i and the Hamil-tonians H i , in order prove the max inequality (1). The first requirement is that U i are dynamically convex , namely, that the contact Conley-Zehnder index of every Reeb or-bit (with respect to a capping disk that is contained in the boundary) is at least n + 1,where n is half the dimension of M . It is known that every strictly convex domain in R n with a smooth boundary is dynamically convex, see, e.g., [9]. Secondly, we require thatthe spectral invariants of the Hamiltonians are smaller than the monotonicity constant,namely, c ( H i ) < κ . Finally, we assume that the domains U i are “not too big” comparedto the monotonicity constant κ . The size of a domain is measured by maximizing theaction-index ratio over all Reeb orbits on the boundary: Definition 1.3.
Let U ⊂ M be a domain with an incompressible contact type boundary,such that the Reeb flow is non-degenerate. We define C ( U ) := sup (cid:26) (cid:82) λ γ CZ R ( γ, u ) − n + 1 : γ ∈ P ( ∂U ) , u ⊂ ∂U (cid:27) ∈ R ∪ { + ∞} . (3) Remark 1.4.
The above definition can be extended to disjoint unions of domains U = (cid:116) i U i . In this case, the invariant will be equal to the maximum over the invariants ofthe connected components, C ( U ) = max i C ( U i ). Definition 1.3 can be also extended todomains with degenerate Reeb flows. This is done in Section 6, together with estimatesof the invariant C on certain classes of domains: • Using results from [8], we show that for concave toric domains (and, in particular,ellipsoids) C ( U ) coincides with the Gromov width , c G ( U ). For convex toric do-mains, c G ( U ) ≤ C ( U ) ≤ c G ( B ) for every ball B whose image under the momentmap contains the image of U , namely µ ( B ) ⊃ µ ( U ). We review the definitions ofconvex and concave toric domains in Section 6. • Using a result by Ishikawa from [11], we show that for strictly convex domains, C ( U )can be bounded in terms of the curvature of the boundary ∂U . Theorem 4.
Let ( M, ω ) be a positively monotone symplectic manifold of dimension greaterthan 2, and with monotonicity constant κ > . Suppose that U i ⊂ M are disjoint domainswith incompressible dynamically convex boundaries, such that C ( U i ) ≤ κ for all i . ForHamiltonians H i : M × S → R supported in U i respectively, such that c ( H i ) < κ , we have c ( H + · · · + H N ) ≤ max { c ( H ) , . . . , c ( H N ) } . The condition c ( H i ) < κ in the above theorem can be guaranteed if, for example,the supports are displaceable with small displacement energy, as follows from the energy-capacity inequality. This inequality, as well as the definitions for displaceability and In fact, it is sufficient to assume that the contact CZ index of every Reeb orbit is at least n . The Gromov width of U is defined to be the supremum of πr over all radii r such that the ball ofradius r in R n (equipped with the standard symplectic form) can be symplectically embedded into U . c ( H i ) < κ holds if we assume in addition that U i are portableLiouville domains : Definition 1.5 (Following [16]) . • A domain U with a contact type boundary is calleda Liouville domain if the Liouville vector field Y extends to U and satisfies L Y ω = ω there. • The core of a Liouville domain U is defined to be Q := ∩ s ∈ (0 , ψ log s U , where { ψ τ } τ ≤ is the flow of the Liouville vector field. • A Liouville domain U is called portable if Q is displaceable in U . • The portability number of U is defined to be p ( U ) := lim s → e ( ψ log s U ; U ) /s, (4)where e ( ψ log s U ; U ) is the displacement energy of ψ log s U inside U . Example 1.6.
Every nice star-shaped domain in ( R n , ω ) is a portable Liouville domain,and its portability number is equal to its displacement energy, p ( U ) = e ( U ; R n ).The next corollary follows from Theorem 4 together with Theorem 5 below, whichgives an upper bound for the spectral invariants of Hamiltonians supported in portableLiouville domains with dynamically convex incompressible boundaries. Corollary 1.7.
Let ( M, ω ) be a positively monotone manifold with monotonicity constant κ , of dimension greated than 2. Suppose that U i ⊂ M are disjoint portable Liouvilledomains with incompressible dynamically convex boundaries, such that C ( U i ) ≤ κ for all i . Then, for any collection of Hamiltonians H i supported in U i respectively, the maxinequality holds: c ( H + · · · + H N ) ≤ max { c ( H ) , . . . , c ( H N ) } . The last statement of this section establishes the max inequality for Hamiltonianssupported in certain disks on the sphere.
Claim 1.8.
Let ( S , ω ) be the sphere with area normalized to 1. Let H i be Hamiltonianssupported on disjoint disks D i ⊂ S such that area ( D i ) / ∈ (1 / , / for each i . Then, c ( H + · · · + H N ) ≤ max { c ( H ) , . . . , c ( H N ) } . The method of our proof of the claim does not apply when the condition on the areaof the disks is not satisfied. To deduce the max inequality we need to bound the actionsof certain Reeb orbits on the boundary of the domain away from the spectral invariantsof the considered Hamiltonians. The area of the disk determines the actions of the Reeborbits on its boundary. We remark that in [20], Seyfaddini considered balls in monotonemanifolds whose displacement energy is bounded by half the monotonicity constant. On S (with total area normalized to 1) this amounts to disks of area less than 1 /
4. In thissetting Seyfaddini proved that the spectral invariant of a sum of Hamiltonians supported in7uch balls is bounded by the maximal capacity of these balls. Corollary 1.7 and Claim 1.8together with Theorem 5 can be thought of as an extension of the results from [20] forpositively monotone manifolds, where Theorems 3 and 6 extend the results to the settingof negatively monotone manifolds.
The main application of the max inequality (1) concerns the Poisson bracket invariantof covers, which was defined by Polterovich in [16]. As explained above, this invariantassigns a non-negative number, pb ( U ), to a finite open cover U = { U i } of a closed sym-plectic manifold. The Poisson bracket invariant is known to be strictly positive when thecover consists of displaceable sets. Polterovich conjectured a lower bound for the Poissonbracket invariant: Conjecture 1.9 (Polterovich, [16]) . Let (
M, ω ) be a closed symplectic manifold. Thereexists a constant c M , depending only on the symplectic manifold ( M, ω ), such that forevery finite open cover U = { U i } of M , pb ( U ) ≥ c M e ( U ) , where e ( U ) := max i e ( U i ) is the maximal displacement energy of a set from U .This conjecture was proved for the case where M is a surface in [2], and for surfacesother than the sphere in [14]. In higher dimensions the conjecture is still open and allknown lower bounds for pb decay with the degree of the cover [16, 20, 11]. The degree ofan open cover U := { U i } Ni =1 is defined to be the maximal number of sets intersected by asingle set: d ( U ) := max i { j : ¯ U i ∩ ¯ U j (cid:54) = ∅} . In [16], Polterovich proved that on symplectically aspherical manifolds pb ( U ) ≥ c M / ( d ( U ) · max i p ( U i ))for every cover U consisting of portable Liouville domains. Here p ( U i ) is the portabilitynumber of U i , from Definition 1.5. Later, Seyfaddini in [20] proved that on monotonemanifolds, i.e. when ω | π ( M ) = κc | π ( M ) , one has pb ( U ) ≥ / (2 d ( U ) · max i c G ( U i )) ≥ / (2 d ( U ) · e ( U )) for every cover U by balls that are displaceable with energy smaller than | κ | /
2. Finally, in [11], Ishikawa gave a lower bound for covers consisting of embeddings ofstrictly convex sets into monotone manifolds, which decays quadratically in the degree anddepends on the curvature of the boundaries. The max inequality yields lower bounds interms of the displacement energies of the sets, still decaying with the degree. The followingcorollary follows from Theorems 3 and 4 together with Corollary 1.7, by arguments thatappear in [16, 20, 17]. More specifically, we refer the reader to the proof of [20, Theorem9].
Corollary 1.10.
Let ( M, ω ) be a monotone symplectic manifold with monotonicity con-stant κ , and let U := { U i } Ni =1 be a finite open cover of M by domains with incompressibledynamically convex boundaries. Assume in addition that one of the following holds: κ ≤ • κ > , C ( U i ) ≤ κ for all i and, for each i , either e ( U i ) < κ or U i is a portableLiouville domain.Then, pb ( U ) ≥ · d ( U ) · e ( U ) . (5) Remark 1.11.
When the cover U consists of portable Liouville domains with incompress-ible dynamically convex boundaries, the maximal displacement energy, e ( U ), in the lowerbound (5) can be replaced by max i p ( U i ) if κ ≤
0, and by max i C ( U i ) otherwise. Thisfollows from the proof of [20, Theorem 9] together with Theorems 5 and 6 below, whichgive uniform bounds for the spectral invariants of Hamiltonians supported in such sets.In this case, one obtains a positive lower bound for the Poisson bracket invariant whenthe cover does not necessarily consist of displaceable sets.We can use Theorem 1 to deduce a lower bound for pb for certain covers on rationalmanifolds. Following the notations of Definition 1.1 above, assume that U is a cover by1-extendable balls and notice that 2 U := { U i } is also a cover of M by symplecticallyembedded balls. When the symplectic manifold ( M, ω ) is rational and the capacity ofeach ball in the cover is not greater than the rationality constant κ , the Poisson bracketinvariant of U can be bounded from below using the degree of the cover 2 U . Corollary 1.12.
Let ( M, ω ) be a rational manifold with rationality constant κ , and let U be a cover by 1-extendable balls, such that c G ( U i ) ≤ κ . Then, pb ( U ) ≥ · d (2 U ) · e ( U ) . Corollary 1.12 can be deduced from Theorem 1 in the same way that [20, Theorem9] is deduced from Theorem 2 there, together with the following observation. Every 1-extendable ball U is displaceable with energy e ( U ) = c G ( U ). Every Hamiltonian H thatis compactly supported in U is also supported in a slightly smaller ball U (cid:48) , of capacitystrictly less than c G ( U ). By the energy capacity inequality, c ( H ) ≤ e ( U (cid:48) ) = c G ( U (cid:48) ) Let ( M, ω ) be a positively monotone symplectic manifold and suppose U ⊂ M is a disjoint union of portable Liouville domains with dynamically convex incompressibleboundaries. If C ( U ) ≤ κ , then for every Hamiltonian H : M × S → R supported in U , c ( H ) < C ( U ) . Theorem 6. Let ( M, ω ) be a negatively monotone symplectic manifold and suppose that U ⊂ M is a disjoint union of portable Liouville domains with dynamically convex incom-pressible boundaries. Then for every Hamiltonian H : M × S → R supported in U , itsspectral invariant is bounded by the portability number of U , namely c ( H ) ≤ p ( U ) . Here the portability number of a disjoint union of portable Liouville domains is de-fined to be the maximal portability number of a connected component, namely p ( U ) :=max i p ( U i ). Theorems 5 and 6 can be seen as versions of Ishikawa’s result, [11, Proprosi-tion 4.4], with different upper bounds and constraints. Ishikawa proved that, on negativelymonotone manifolds, the spectral invariant of every Hamiltonian supported in a disjointunion of embeddings of strictly convex sets into M , is bounded by a constant dependingon the curvature of the boundary. On positively monotone manifolds, he gave a differentupper bound which also depends on the curvature, under the assumption that the mini-mal curvature is not too small, compared to the monotonicity constant. As explained in[11], an immediate corollary of Theorems 5 and 6 concerns the notion of a superheavy set,which was introduced by Entov and Polterovich in [4]: A closed subset X ⊂ M is calledsuperheavy if lim k →∞ c ( kH ) k ≤ sup X × S H, ∀ H ∈ C ∞ ( M × S ) . (6) Corollary 1.13. Let ( M, ω ) be a monotone symplectic manifold with monotonicity con-stant κ . Let U ⊂ M be a portable Liouville domain with a dynamically convex incom-pressible boundary, and assume in addition that either κ ≤ or C ( U ) ≤ κ . Then, M \ U is superheavy. Organization of the paper. Section 2 contains an overview of the necessary preliminaries and fixes some notations.In Section 3, we construct “spectral killers” for Hamiltonians supported in a domainwith incompressible contact type boundary, under a certain condition involving the Reebdynamics on the boundary and the spectral invariant of the Hamiltonian. We also explainhow the existence of spectral killers implies the max inequality and prove Theorem 1. InSection 4 we show that the aforementioned condition holds in various settings, and thusprove Theorems 2, 3 and 4. Section 5 concerns uniform bounds on spectral invariants andcontains the proofs of Theorems 5 and 6. Finally, in Section 6 we estimate the invariant C ( U ) on certain classes of domains. Acknowledgements. I am very grateful to my advisors Lev Buhovsky and Leonid Polterovich for their guid-ance and insightful inputs. I also thank Yaniv Ganor, Vincent Humili`ere, R´emi Leclercq10nd Sobhan Seyfaddini for useful discussions. It was recently brought to my attentionthat some of the arguments in this paper were known to Matthew Strom Borman, includ-ing the idea to use spectral killers in order to prove a max inequality. I am grateful tohim for kindly sharing his previous findings and ideas with me. The research leading tothese results was partially funded by the Israel Science Foundation, grants 1102/20 and2026/17, as well as by the Levtzion Scholarship. Let us review the necessary preliminaries and fix some notations. Note that throughoutthe paper we assume ( M, ω ) to be a rational closed symplectic manifold, namely, ω ( π ( M ))is a discrete subgroup of Z . Given a Hamiltonian H : M × S → R , its symplectic gradient is the vector fielddefined by the equation ω ( X H , · ) = − dH and the flow ϕ tH of this vector field is calledthe Hamiltonian flow of H . The set of 1-periodic orbits of ϕ tH is denoted by P ( H ). TheHamiltonian H is called non-degenerate if the graph of dϕ H is transversal to the diagonalin T M × T M . Equivalently, H is non-degenerate if every γ ∈ P ( H ) is non-degenerate,that is, if 1 is not an eigenvalue of dϕ H ( γ (0)) for every γ ∈ P ( H ).We denote by L M the space of contractible loops in M . A capping disk of γ ∈ L M is a map u : D → M from the unit disk to M , satisfying u | ∂D = γ . Two capping disks u , u of γ are equivalent if [ u − u )] ∈ ker ω ∩ ker c . We denote by (cid:103) L M the space ofequivalence classes of capped loops, ( γ, u ). The action functional corresponding to H isdefined on the space (cid:103) L M by A H ( γ, u ) = (cid:90) H ( γ ( t ) , t ) dt − (cid:90) u ω. The critical points of the action functional are (equivalence classes of) capped 1-periodicorbits of ϕ tH and the set of their values is denoted by Spec( H ). For a non-degenerateHamiltonian H and a generic ω -compatible almost complex structure J , the Floer chaincomplex CF ∗ ( H, J ) is generated by these critical points and its differential is defined bycounting certain negative gradient flow lines of A H (with respect to a metric induced by J on (cid:103) L M ). For more details, see, e.g., [12, 17]. The chain complex CF ∗ ( H, J ) is graded bythe Conley-Zehnder (abbreviated to CZ) index, whose definition is recalled below. Moreformally, CF k ( H, J ) is generated by (equivalence classes of) capped 1-periodic orbits whoseindex , CZ H ( γ, u ), is equal to − k . For k ∈ Z , we denote by Spec k ( H ) the set of actionvalues of the generators of CF k ( H, J ). We add the subscript H to the notation in order to distinguish this index from the contact CZ index,which will be discussed later. .1.1 The Conley-Zehnder and Robbin-Salamon indices. The Conley-Zehnder index is defined for non-degenerate capped 1-periodic orbits,through an index of a path of symplectic matrices which is obtained from the linearizedflow after trivializing the tangent bundle. In [18], Robbin and Salamon defined a Maslov-type index for possibly degenerate orbits, that coincides with the Conley-Zehnder indexon non-degenerate ones. Since we will consider degenerate Hamiltonians as well, we givehere the definition of the Robbin-Salamon index, following the exposition in [7]. • Let Φ = { Φ( t ) } t ∈ [0 ,T ] ⊂ Sp(2 n ) be a path of symplectic matrices. A number t ∈ [0 , T ]is called a crossing if det(Φ( t ) − crossing form Γ t = Γ t (Φ) is thequadratic form obtained by restricting the symmetric matrix S ( t ) := − J ˙Φ( t )Φ − ( t )to ker(Φ( t ) − t is called regular if the crossing form Γ t is non-degenerate. • For a path Φ = { Φ( t ) } t ∈ [0 ,T ] having only regular crossings, the Robbin-Salamonindex (abbreviated to RS) is defined to beRS(Φ) := 12 sign (Γ ) + (cid:88) T M along u . Then, the differential of the flow ϕ tH along the loop γ is identified with a path of symplectic matrices, dϕ tH ( γ (0)) (cid:55)→ Φ( t ) ∈ Sp(2 n ). TheRS index of the capped orbit is given by RS( γ, u ) := RS(Φ). Throughout the paper,we use the notation CZ for non-degenerate orbits and RS for degenerate ones. Auseful property of the CZ index is that the indices with respect to different cappingdisks differ by twice the first Chern class of the connected sum. More formally,let γ ∈ P ( H ) be a non-degenerate 1-periodic orbit of a Hamiltonian H and let u, v : D → M be two different capping disks for γ . Then, for A := u − v ) ∈ π ( M ),CZ H ( γ, u ) = CZ H ( γ, v A ) = CZ H ( γ, v ) + 2 c ( A ) , (8)12here c denotes the first Chern class of M , see, e.g., [12, Sections 2.6-7].We remark that there are texts choosing an opposite sign for the CZ index, such as [20].In our sign convention, the index of a critical point p of a C -small Morse function witha constant capping disk, u p ( D ) = { p } , is related to the Morse index via CZ H ( p, u p ) = n − i Morse ( p ). The Floer complex admits a natural filtration by the action value. Let CF a ∗ ( H, J )be the sub-complex generated by (equivalence classes of) capped 1-periodic orbits whoseaction is bounded by a from above. Since the Floer differential is action decreasing, itrestricts to the sub-complex CF a ∗ ( H, J ) and the homology HF a ∗ ( H, J ) is well defined. Thespectral invariant with respect to the fundamental class is defined to be the smallest valueof a for which the fundamental class appears in HF a ∗ ( H, J ), namely, c ( H ) := inf { a : [ M ] ∈ Im ( ι a ∗ ) } , (9)where ι a ∗ : HF a ∗ ( H, J ) → HF ∗ ( H, J ) is the map induced by the inclusion ι a : CF a ∗ ( H, J ) (cid:44) → CF ∗ ( H, J ). We remark that spectral invariants are defined for general quantum homologyclasses, but we consider only the spectral invariant with respect to the fundamental class.Spectral invariants have several useful properties, let us state the relevant ones: • (stability) For any Hamiltonians H and G , (cid:90) min x ∈ M ( H ( x, t ) − G ( x, t )) dt ≤ c ( H ) − c ( G ) ≤ (cid:90) max x ∈ M ( H ( x, t ) − G ( x, t )) dt. In particular, c : C ∞ ( M × S ) → R is a continuous functional and is extendedby continuity to degenerate Hamiltonians. Moreover, this implies that the spectralinvariant is monotone: If G ( x, t ) ≤ H ( x, t ) for all ( x, t ) ∈ M × S , then c ( G ) ≤ c ( H ). • (spectrality) c ( H ) ∈ Spec( H ). Moreover, if H is non-degenerate, c ( H ) ∈ Spec n ( H ). • (subadditivity) For every Hamiltonians H and G , one has c ( H G ) ≤ c ( H ) + c ( G ),where H G := H + G ◦ ( ϕ tH ) − . In particular, if H and G are disjointly supportedthen c ( H + G ) ≤ c ( H ) + c ( G ). • (energy-capacity inequality) If the support of H is displaceable, its spectral invariantis bounded by the displacement energy of the support, namely, c ( H ) ≤ e ( supp ( H )).We remind that a subset X ⊂ M is displaceable if there exists a Hamiltonian G suchthat ϕ G ( X ) ∩ X = ∅ . In this case, the displacement energy of X is given by e ( X ) := inf G : ϕ G ( X ) ∩ X = ∅ (cid:90) (cid:18) max M G ( · , t ) − min M G ( · , t ) (cid:19) dt. (10)For a wider exposition see, for example, [12, 17]. A standard method for estimatingthe spectral invariant of a Hamiltonian H is through a bifurcation diagram . Given a13ontinuous deformation { H τ } τ of H , the corresponding bifurcation diagram is the set ∪ τ ( { τ } × Spec( H τ )) ⊂ R . By the spectrality and stability properties, the spectral in-variant of H τ moves continuously in the diagram as τ varies. Therefore, if we constructa deformation such that the value of the spectral invariant is known at a certain point ofthe deformation, we can study the bifurcation diagram in order to estimate c ( H ). Thisapproach was used by Polterovich in [16], by Seyfaddini in [20] and by Ishikawa in [11],and is frequently used in the present paper as well. As mentioned earlier, a domain U ⊂ M has a contact type boundary if there existsa vector field Y , called the Liouville vector field , that is defined on a neighborhood ofthe boundary ∂U , satisfies L Y ω = ω , is transverse to the boundary and points outwards.The flow ψ s of Y defines a radial coordinate, called the Liouville coordinate , on a tubularneighborhood of the boundary, by ∂U × (1 − (cid:15), (cid:15) ) ∼ = N ( ∂U ) , ( y, s ) (cid:55)→ x = ψ log( s ) y. (11)Note that the Liouville flow expands the symplectic form, namely (cid:0) ψ log( s ) (cid:1) ∗ ω = s · ω .The 1-form λ := ι Y ω is a primitive of ω and the kernel of its restriction to T ∂U , namely ξ := ker λ | T ∂U , is called the contact distribution . We denote by R the Reeb vector field ,which is defined on a neighborhood of ∂U by λ ( R ) = 1 , R ( y,s ) ∈ ker dλ ( y,s ) | T ψ log( s ) ∂U . (12)We remark that the Liouville vector field is not unique (and hence so are the 1-form λ and the Reeb vector field). In fact, for every C -small Hamiltonian H defined near ∂U , Y (cid:48) := Y − X H is also a Liouville vector field. If the vector filed Y extends to U (andsatisfies L Y ω = ω there), we say that U is a Liouville domain .Denote by ϕ tR : ∂U → ∂U the flow of the Reeb vector field, and let P ( ∂U ) be theset of all contractible periodic orbits of ϕ tR (of any period). The action of such orbits isdefined to be the integral of the 1-form λ along the orbit, and coincides with the period.The set of action values is called the Reeb spectrum of ∂U and is denoted by Spec( ∂U ).We say that the Reeb flow is non-degenerate if the graph of the restriction of dϕ tR to thecontact distribution ξ intersects the diagonal in ξ × ξ transversely. In this case, for eachorbit γ ∈ P ( ∂U ) and a capping disk u ⊂ ∂U , one can assign an integer, which we call thecontact Conley-Zehnder index and denote by CZ R ( γ, u ), in the following way. Trivializingthe contact distribution ξ along the disk u , the restriction of the linearized flow dϕ tR to ξ along γ is identified with a path of symplectic matrices, dϕ tR | ξ ( γ (0)) (cid:55)→ Φ( t ) ∈ Sp(2 n − R ( γ, u ) is defined to be the RS index of the path Φ, as in (7). As mentioned earlier, a Liouville domain U is called portable if its core, which is givenby Q := ∩ s ∈ (0 , ψ log( s ) U , is displaceable in U . In this case, the displacement energy of We add the subscript R to the notation in order to distinguish the contact CZ index from the Hamil-tonian CZ index that appeared earlier. log( s ) U is arbitrarily small as s approaches zero, as explained in [16, p.499]. This fact willbe used in the proof of Theorem 5 which asserts a uniform bound for spectral invariantsof Hamiltonians supported in portable Liouville domains with incompressible dynamicallyconvex boundaries on positively monotone manifolds, see Section 5.1. In [20], Seyfaddini presented a construction of certain functions, called spectral killers ,which can be used to produce upper bounds for the spectral invariant of a sum of disjointlysupported Hamiltonians. A spectral killer for a Hamiltonian H supported in a domain U is a function K : M → R supported in U such that c ( H + K ) = 0. Claim 3.1 (Seyfaddini) . Let H , . . . , H N be Hamiltonians supported in pairwise disjointdomains U , . . . , U N ⊂ M and suppose there exist Hamiltonians K i supported in U i , suchthat c ( H i + K i ) = 0 . Then, c ( H + · · · + H N ) ≤ max i (cid:107) K i (cid:107) C . (13) Proof. The following argument is taken from [20]. Using the stability and subadditivityproperties of spectral invariants and noticing that { H i + K i } are all disjointly supported,we have c ( H + · · · + H N ) ≤ c (cid:32)(cid:88) i ( H i + K i ) (cid:33) + (cid:13)(cid:13)(cid:13) − (cid:88) i K i (cid:13)(cid:13)(cid:13) C ≤ (cid:88) i c ( H i + K i ) + (cid:13)(cid:13)(cid:13) (cid:88) i K i (cid:13)(cid:13)(cid:13) C = (cid:88) i i (cid:107) K i (cid:107) C . In [20], Seyfaddini constructed spectral killers for Hamiltonians supported in displace-able balls having small displacement energies in monotone manifolds. The norms of thespectral killers are bounded by the capacities of the balls.In order to obtain the max inequality (1) we will construct spectral killers whose normsare equal to the spectral invariants of the Hamiltonians { H i } whenever c ( H i ) > 0. Notethat for Hamiltonians with non-positive spectral invariants the max inequality followsfrom the subadditivity property of spectral invariants. In this section we show that undera certain condition on the domain U and the spectral invariant c ( H ) of H , there existssuch a spectral killer K . We begin with a simpler construction of what we call a “slowspectral killer”, which is supported on a domain larger than U . In this section we use the notion of σ -extendable domains from Definition 1.1. Ourgoal is to prove the following statement. 15 roposition 3.2. Let ( M, ω ) be a rational symplectic manifold, namely ω ( π ( M )) = κ · Z ,and let U ⊂ M be a σ -extendable domain with an incompressible contact type boundary.Then, for every Hamiltonian H supported in U such that ≤ c ( H ) < min { κ, σ · min Spec( ∂U ) } (14) there exists K : M → R supported in (1+ σ ) U with (cid:107) K (cid:107) C = c ( H ) such that c ( H + K ) = 0 . The above proposition guarantees that the max inequality (1) holds for Hamiltonianssupported in disjoint extendable domains that satisfy (14), only if we assume in additionthat the supports (1 + σ i ) U i of the spectral killers are disjoint. In particular, Theorem 1is an immediate consequence of Proposition 3.2 and Claim 3.1: Proof of Theorem 1. Let { H i } be Hamiltonians supported in domains { U i } with incom-pressible contact type boundaries such that U i is σ i -extendable, the sets { (1 + σ i ) U i } arepairwise disjoint, and c ( H i ) < min { κ, σ i · min Spec( ∂U i ) } . By Proposition 3.2 there exist K i : M → R supported in (1 + σ i ) U i such that (cid:107) K i (cid:107) C = c ( H i ) and c ( H i + K i ) = 0.Applying Claim 3.1 to { H i } and { K i } with the disjoint domains { (1 + σ i ) U i } , we concludethat the max inequality (1) holds for { H i } .In order to prove Proposition 3.2 we need some preliminary notations and calculations.We use the notations of Section 2 and, in particular, the Liouville coordinate s definedin (11), using the Liouville vector field. The spectral killer K will be a function of theLiouville coordinate. This will enable us to relate its 1-periodic orbits to the Reeb orbitson ∂U . Definition 3.3. We say that an autonomous Hamiltonian H is radial if there exists χ : R → R such that H = χ ( s ) wherever the the Liouville coordinate is defined, and islocally constant elsewhere.Radial Hamiltonians and their periodic orbits were studied in [9, 11] for the case where U is a strictly convex domain in R n . The following lemma relates between the 1-periodicorbits of radial Hamiltonians and the Reeb orbits on ∂U . Lemma 3.4. Let H = χ ( s ) be a radial Hamiltonian, then the Hamiltonian flow of H isconjugated to the Reeb flow up to a time reparametrization: ϕ tH = ψ log s ◦ ϕ χ (cid:48) ( s ) · tR ◦ ψ − log s . (15) In particular, every non-constant 1-periodic orbit γ of H is contained in a level set of theLiouville coordinate s = s ( γ ) and is conjugated to a periodic Reeb orbit, ˆ γ ∈ P ( ∂U ) , via ˆ γ ( χ (cid:48) ( s ) · t ) = ψ − log s γ ( t ) . Moreover, the Reeb action of ˆ γ is equal to the absolute value ofthe derivative of χ at s = s ( γ ) : (cid:90) ˆ γ λ = | χ (cid:48) ( s ) | . (16) In particular, | χ (cid:48) ( s ) | belongs to the Reeb spectrum of ∂U . roof of Lemma 3.4. Let us prove the following relation between the Hamiltonian vectorfield X H of H and the Reeb vector field: dψ − log s X H ◦ ψ log s = χ (cid:48) ( s ) · R. (17)Note that (15) will follow from uniqueness of solutions of ODEs. Let us show that theLHS of (17) satisfies the equations defining the Reeb vector field with a factor of χ (cid:48) ( s ).Given any vector v ∈ T x M for x ∈ ∂U , ω x ( v, dψ − log s X H ◦ ψ log s ( x )) = ( ψ − log sx ) ∗ ω ψ log s x ( dψ log s v, X H )= s − · ω ψ log s x ( dψ log s v, X H )= s − dH ψ log s x ( dψ log s v ) . (18)When v ∈ T ∂U , its image under the linearized Liouville flow, dψ log s v , is tangent to a levelset of the Liouville coordinate s and hence to a level set of H . In this case, ω x ( v, dψ − log s X H ◦ ψ log s ( x )) = s − dH ψ log s x ( dψ log s v ) = 0On the other hand, taking v to be the Liouville vector field, equation (18) implies: ω x ( Y, dψ − log s X H ◦ ψ log s ( x )) = s − dH ψ log s x ( dψ log s Y )= s − · ddτ (cid:12)(cid:12)(cid:12) τ =0 H ◦ ψ log( s ) ψ τ ( x )= s − · ddτ (cid:12)(cid:12)(cid:12) τ =0 H ◦ ψ log( s · e τ ) ( x )= s − · ddτ (cid:12)(cid:12)(cid:12) τ =0 χ ( s · e τ ) = χ (cid:48) ( s ) . Having established the conjugation of the Hamiltonian and the Reeb flows (15), weturn to prove the relation between the Reeb action and the derivative of χ stated inequation (16). For a 1-periodic orbit γ of H lying in level s , let ˆ γ ⊂ ∂U be the curvedefined by ˆ γ ( χ (cid:48) ( s ) · t ) = ψ − log s γ ( t ). Then, using (15) we find ddt ˆ γ ( t ) = ddt (cid:16) ψ − log s γ ( t/χ (cid:48) ( s )) (cid:17) = ddt (cid:16) ψ − log s ϕ t/χ (cid:48) ( s ) H γ (0) (cid:17) = ddt (cid:16) ϕ t · χ (cid:48) ( s ) /χ (cid:48) ( s ) R ψ − log s γ (0) (cid:17) = R. We conclude that ˆ γ is a periodic Reeb orbit whose period, and therefore action, is equalto | χ (cid:48) ( s ) | .We are now ready to prove Proposition 3.2. Proof of Proposition 3.2. Let U be a σ -extendable domain with an incompressible contacttype boundary and let H be a Hamiltonian supported in U . Let N ( ∂U ) ∼ = ∂U × (1 − (cid:15), σ ) be a tubular neighborhood of the boundary on which the Liouville coordinate isdefined, and assume that (cid:15) is small enough so that H | N ( ∂U ) = 0. Consider the autonomous17 = 1 s = 1 − (cid:15) H τ s H = H s = 1 + σ − (cid:15)s = 1 + σ Figure 3: An illustration of the graphs of H τ (solid line) and H (dashed line) in the radialcoordinate s . radial Hamiltonian defined by K ( x ) := − x ∈ U \ (cid:0) ∂U × (1 − (cid:15), (cid:1) ,χ ( s ( x )) x ∈ ∂U × (1 − (cid:15), σ − (cid:15) ) , χ : R → R is a smooth approximation of the continuous piecewise linear functiontaking the value − s ≤ − (cid:15) and 0 for s ≥ σ − (cid:15) . We choose χ such that itsderivative is bounded by 1 /σ and that, for s > − (cid:15) , its derivative vanishes only in the“flat region”, i.e. where χ itself vanishes. For each τ ∈ [0 , c ( H )], set K τ := τ · K , then { K τ } τ is a continuous family of radial Hamiltonians. Denoting χ τ := τ · χ , its derivativeis bounded by τ /σ , and therefore K τ has no non-constant 1-periodic orbits as long as τ /σ < min Spec( ∂U ). This follows from Lemma 3.4, which states that non-constant 1-periodic orbits of radial Hamiltonians appear only for s such that | χ (cid:48) ( s ) | ∈ Spec( ∂U ).Assumption (14) in Proposition 3.2 states that c ( H ) < σ · min Spec( ∂U ) and therefore K τ has no non-constant 1-periodic orbits for all τ ∈ [0 , c ( H )]. We will show that K c ( H ) ,which is supported in (1 + σ ) U , is a spectral killer for H , namely, that c ( H + K c ( H ) ) = 0.Consider the deformation of the Hamiltonian H given by { H τ = H + K τ } τ ∈ [0 ,c ( H )] , which isillustrated in Figure 3, and let us study the corresponding bifurcation diagram. We remindthat by the stability and spectrality properties, the spectral invariant moves continuouslyin this diagram. Let us show that the bifurcation diagram consists of lines with slope − U , and of horizontal lines with values in κ Z , corresponding toconstant orbits outside of U , as illustrated in Figure 4. Indeed: • Orbits in { s > − (cid:15) } : In this region H τ coincides with K τ which has only constant1-periodic orbits. Their actions are given by χ τ ( s ) − ω ( A ) for some A ∈ π ( M ) and s such that χ (cid:48) τ ( s ) = 0. When s > − (cid:15) , the derivative of χ τ vanishes if and only if s > σ − (cid:15) , in which case χ τ ( s ) = 0. Overall, the 1-periodic orbits of H τ in thisregion all have actions lying in κ Z and do not change with τ .18 pec ( H τ ) τ c ( H ) κ κ − κ Figure 4: An illustration of the n -spectrum of a non-degenerate perturbation of H τ . The dashedlines correspond to actions of orbits in { s < − (cid:15)/ , } while the solid lines correspond to actionsof orbits in { s > − (cid:15)/ } . • Orbits in { s ≤ − (cid:15) } : Here H τ = H − τ . Therefore, the 1-periodic orbits of H τ donot change with τ but their actions decrease linearly: A H τ ( γ, τ ) = (cid:90) H τ ( γ ( t ) , t ) dt − (cid:90) u ω = (cid:90) H ( γ ( t ) , t ) dt − τ − (cid:90) u ω. We conclude that the bifurcation diagram consists of decreasing lines of slope − κ Z . By assumption (14), c ( H ) < κ and thus there are nohorizontal lines between 0 and c ( H ). It follows that the spectral invariant c ( H τ ) movesalong a single decreasing line and hence c ( H + K τ ) = c ( H τ ) = c ( H ) − τ for τ ∈ [0 , c ( H )].In particular, c ( H + K c ( H ) ) = 0 and K c ( H ) is the required spectral killer. U . In order to construct spectral killers that are supported in U , rather than on a largerdomain, we must consider radial Hamiltonians with arbitrarily large slopes. Such Hamil-tonians may have a lot of non-constant 1-periodic orbits which correspond to Reeb orbitson the boundary of U , as stated in Lemma 3.4. In this case, we need to impose certain as-sumptions on the spectral invariant of the Hamiltonian considered and the Reeb dynamicson the boundary. Let us fix a domain U with an incompressible contact type boundarysuch that the Reeb flow is non-degenerate. The non-degeneracy of the Reeb flow can beachieved by a small perturbation of the Liouville vector field (or, equivalently, the contactform), see, e.g., [1, p.47]. For a non degenerate domain we define the relative n -spectrum of the boundary ∂U in M : Definition 3.5. • For k ∈ Z , we denote bySpec k ( ∂U ) := (cid:26)(cid:90) γ λ : γ ∈ P ( ∂U ) , ∃ u : D → ∂U, u | ∂D = γ, CZ R ( γ, u ) = − k (cid:27) the action spectrum of ∂U of index − k .19 Denote Spec n (0) := (cid:110) − ω ( A ) : A ∈ π ( M ) , c ( A ) ∈ {− n, . . . , } (cid:111) . Morally speaking, Spec n (0) is the “ n -spectrum of the zero function”, namely, theset actions that can be attained by index − n capped orbits of a non-degenerateperturbation of the zero function. • We define the relative n -spectrum of ∂U inside M to beSpec n ( ∂U ; M ) := Spec n (0) ∪ (cid:91) k ∈ Z , A ∈ π ( M ) ,c ( A ) = (cid:100) k − n (cid:101) {− Spec k ( ∂U ) − ω ( A ) } . We will show that the relative n -spectrum Spec n ( ∂U ; M ) is related to the n -spectrum ofradial Hamiltonians (see Definition 3.3). The sets Spec k ( ∂U ) and Spec n ( ∂U ; M ) dependon the choice of a Liouville vector field (or, equivalently, a contact form). We omit theLiouville vector field from the notation for the sake of brevity. Our main goal for thissection is to prove the following statement. Proposition 3.6. Let H : M × S → R be a Hamiltonian supported in a domain U with an incompressible contact type boundary such that the Reeb flow is non-degenerate.Assume that c ( H ) > and Spec n ( ∂U ; M ) ∩ (0 , c ( H )] = ∅ . (20) Then, there exists a Hamiltonian K : M → R supported in U such that (cid:107) K (cid:107) C = c ( H ) and c ( H + K ) = 0 . Remark 3.7. The non-degeneracy of the Reeb flow on ∂U is required for the contactCZ to be defined. However, for possibly degenerate domains one can define a relativespectrum bySpec( ∂U ; M ) := {− T − ω ( A ) : T ∈ { } ∪ Spec( ∂U ) , A ∈ π ( M ) } . and construct spectral killers for Hamiltonians satisfying Spec( ∂U ; M ) ∩ (0 , c ( H )] = ∅ . Theproof in this case is simpler than that of Proposition 3.6 and does not require Lemma 3.9below. This observation will be used in the proof of Theorem 2 which concerns the maxinequality on rational manifolds.We start by stating a simple corollary of Lemma 3.4 from the previous section. Corollary 3.8. For a radial Hamiltonian H = χ ( s ) , the action of a capped non-constant1-periodic orbit ( γ, u ) is given by A H ( γ, u ) = χ ( s ) − s · χ (cid:48) ( s ) − ω ( A ) , (21) where s = s ( γ ) is the Liouville coordinate of the level set containing γ and A ∈ π ( M ) issuch that u = u A for a capping disk u ⊂ { s = s ( γ ) } . roof. Recall that Lemma 3.4 states that every non-constant 1-periodic orbit γ of a radialHamiltonian H = χ ( s ) corresponds to a Reeb orbit ˆ γ ∈ P ( ∂U ) whose action is (cid:82) ˆ γ λ = | χ (cid:48) ( s ) | . Notice that ˆ γ and ψ − log s γ have the same orientation if χ (cid:48) ( s ) > χ (cid:48) ( s ) < 0. Let ˆ u be a capping disk of ˆ γ whose image coincides with ψ − log s u .Then, A H ( γ, u ) = H ( γ ) − (cid:90) u ω = χ ( s ) − (cid:90) u ω − ω ( A )= χ ( s ) − (cid:90) ψ − log s u (cid:16) ψ log s (cid:17) ∗ ω − ω ( A ) = χ ( s ) − s · (cid:90) ψ − log s u ω − ω ( A )= χ ( s ) − s · (cid:90) ∂ ( ψ − log s u ) λ − ω ( A ) = χ ( s ) − s · sign χ (cid:48) ( s ) · (cid:90) ˆ γ λ − ω ( A )= χ ( s ) − s · χ (cid:48) ( s ) − ω ( A ) , where in the last equality we used equation (16) from Lemma 3.4.Since radial Hamiltonians are degenerate, we will perturb them into non-degenerateHamiltonians. After the perturbation, there may appear several periodic orbits, { γ i } ,in a small neighborhood of every degenerate orbit γ . In this case, the CZ index of thenon-degenerate perturbed orbits is close to the RS index of the original degenerate orbit: | CZ H ( γ i , u i ) − RS( γ, u ) | ≤ 12 dim ker(( dϕ H ) γ (0) − , (22)see, for example, section 3 of [11]. The next lemma relates between the RS indices of1-periodic orbits of radial Hamiltonians and the CZ indices of Reeb orbits. Moreover, itshows that generically, the kernel of ( ϕ H ) γ (0) − 1l is 1-dimensional. Lemma 3.9. Let H = χ ( s ) be a radial Hamiltonian. For every non-constant 1-periodicorbit γ of H and a capping u ⊂ { s = s ( γ ) } of γ , RS( γ, u ) = sign ( χ (cid:48) ( s )) · CZ R (ˆ γ, ˆ u ) + 12 sign (cid:0) χ (cid:48)(cid:48) ( s ) (cid:1) (23) where ˆ γ ( t ) := ψ − log s γ ( t/χ (cid:48) ( s )) ∈ P ( ∂U ) is the Reeb orbit conjugate to γ , ˆ u is a cappingdisk of ˆ γ whose image coincides with ψ − log s u ⊂ ∂U and the sign of zero is considered tobe zero. Moreover, if χ (cid:48)(cid:48) ( s ) (cid:54) = 0 for s = s ( γ ) , then dim ker(( dϕ H ) γ (0) − .Proof. In order to relate the two different indices we show that the restrictions of thelinearized flows to the contact distribution are conjugated. This can be done by differen-tiating the conjugation of the Hamiltonian and Reeb flows, given in (15). Extending thecontact distribution ξ to a neighborhood of ∂U using the Liouville flow ψ log s , we have dϕ tH | ξ = d (cid:16) ψ log s ◦ ϕ χ (cid:48) ( s ) · tR ◦ ψ − log s (cid:17) (cid:12)(cid:12)(cid:12) ξ = dψ log s dϕ χ (cid:48) ( s ) · tR dψ − log s | ξ , where the last equality follows from the contact distribution ξ is tangent to level sets ofthe Liouville coordinate s . We therefore conclude that the linearized Hamiltonian and21eeb flows are conjugated on the contact distribution. By the conjugation property ofthe RS index for paths of matrices, the RS index of the restriction of dϕ tH to ξ is equal tosign ( χ (cid:48) ( s )) · CZ R (ˆ γ, ˆ u ). By the product property of the RS index, it remains to computethe index of the restriction of the linearized Hamiltonian flow to span { Y, R } . We remindthat the Reeb vector field is defined wherever the Liouville vector field is, by equations (12).Since X H is proportional to R , and since H is autonomous, the linearized Hamiltonianflow preserves R : dϕ tH ( R ) = dϕ tH ( χ (cid:48) ( s ) X H ) = χ (cid:48) ( s ) X H ◦ ϕ tH = R ◦ ϕ tH . In order tocompute the linearized Hamiltonian flow on the Liouville vector field, let us first computethe conjugation of ϕ tH under the Liouville flow. Let x ∈ U and abbreviate s = s ( x ). Usingthe conjugation between the Hamiltonian and the Reeb flows (15) from Lemma 3.4, wehave ψ − τ ◦ ϕ tH ◦ ψ τ ( x ) = ψ − τ ψ log( se τ ) ϕ χ (cid:48) ( se τ ) · tR ψ − log( se τ ) ψ τ ( x )= ψ log s ϕ χ (cid:48) ( se τ ) · tR ψ − log s ( x )= ϕ χ (cid:48) ( seτ ) χ (cid:48) ( s ) · tH ( x ) . (24)Using the above, dϕ tH ( Y ) = ddτ (cid:12)(cid:12)(cid:12) τ =0 ϕ tH ◦ ψ τ (24) = ddτ (cid:12)(cid:12)(cid:12) τ =0 ψ τ ◦ ϕ χ (cid:48) ( seτ ) χ (cid:48) ( s ) · tH = Y ◦ ϕ tH + dψ τ X H · se τ χ (cid:48)(cid:48) ( se τ ) χ (cid:48) ( s ) · t (cid:12)(cid:12) τ =0 = Y ◦ ϕ tH + t · sχ (cid:48)(cid:48) ( s ) χ (cid:48) ( s ) · X H = Y ◦ ϕ tH + t · sχ (cid:48)(cid:48) ( s ) · R. Fix a trivialization ˆ T : Im (ˆ u ) × R n − → ξ | Im (ˆ u ) of the contact distribution ξ alongIm (ˆ u ) ⊂ ∂U , then dψ log s ˆ T is a trivialization of ξ along the image of u . We can completeit to a trivialization T : Im ( u ) × R n → T M | Im ( u ) of T M along the image of u , usingthe vector fields Y and R , namely T ( x, e n − ) = Y ( x ), T ( x, e n ) = R ( x ). It follows fromthe above computations that in this trivialization the matrices Φ H ( t ) representing dϕ tH along γ are Φ H ( t ) = Φ R ( sχ (cid:48) ( s ) · t ) 0 00 1 00 t · sχ (cid:48)(cid:48) ( s ) 1 , where Φ R ( t ) are the matrices representing the restriction of dϕ tR to ξ in T . By the productproperty of the RS index, we haveRS(Φ H ) = sign ( χ (cid:48) ( s )) · RS(Φ R ) + RS(Ψ) , (25)where Ψ( t ) := (cid:18) t · sχ (cid:48)(cid:48) ( s ) 1 (cid:19) . = 1 − (cid:15)s = 1 − (cid:15) H τ s H = H s = 1 Figure 5: An illustration of the graphs of H τ (solid line) and H (dashed line) in the radialcoordinate s . The path of symplectic matrices Ψ( t ) is degenerate. By Proposition 4.9 from [7], the RSindex of Ψ is RS(Ψ) = sign (cid:0) sχ (cid:48)(cid:48) ( s ) (cid:1) / χ (cid:48)(cid:48) ( s )) / . Together with equation (25), this implies that RS( γ, u ) = sign ( χ (cid:48) ( s )) · CZ R (ˆ γ, ˆ u ) + sign ( χ (cid:48)(cid:48) ( s )). This proves formula (23). It remains to prove that, if χ (cid:48)(cid:48) ( s ) (cid:54) = 0, thekernel of ( dϕ H ) γ (0) − 1l is one-dimensional. Indeed, since ˆ γ is non-degenerate, the kernel ofΦ H (1) − 1l coincides with that of Ψ(1) − χ (cid:48)(cid:48) ( s ) (cid:54) = 0.Having established Lemma 3.9, we are now ready to prove Proposition 3.6. Proof of Proposition 3.6. Let H be a Hamiltonian supported in U . In what follows, weconstruct a continuous deformation H τ of the Hamiltonian H and, after perturbing intonon-degenerate Hamiltonians, follow the corresponding bifurcation diagram of the n -spectrum (that is, the actions of capped orbits of index − n ). We remind that by thespectrality property for non-degenerate Hamiltonians, the spectral invariant lies in the n -spectrum.Let N ( ∂U ) ∼ = ∂U × (1 − (cid:15), (cid:15) ) be a small enough neighborhood of the boundaryon which the Liouville coordinate is defined and such that H | N ( ∂U ) = 0. Consider theautonomous radial Hamiltonians defined by K ( x ) := − x ∈ U \ (cid:0) ∂U × (1 − (cid:15), (cid:1) ,χ ( s ( x )) x ∈ ∂U × (1 − (cid:15), − (cid:15) ) , . (26) More accurately, one should apply Proposition 4.9 from [7] to the path of transposed matrices Ψ T anduse the inverse property of the RS index. χ : R → R is a smooth approximation of the continuous piecewise linear functiontaking the value − s ≤ − (cid:15) and 0 for s ≥ − (cid:15) , and which coincides with thepiecewise linear function outside of a neighborhood of the corners at 1 − (cid:15) and 1 − (cid:15) . Wealso require that χ (cid:48)(cid:48) ( s ) will be strictly negative near the corners 1 − (cid:15) and 1 − (cid:15) . Considerthe family of radial Hamiltonians given by { K τ := τ · K } τ ∈ [0 ,c ( H )] and set χ τ := τ · χ .Note that since the Reeb spectrum is a discrete set, for a generic τ ∈ [0 , c ( H )] non-constant 1-periodic orbits of K τ appear only near s = 1 − (cid:15) and s = 1 − (cid:15) . We definethe deformation of the Hamiltonian H to be { H τ := H + K τ } [0 ,c ( H )] , see Figure 5. Letus follow the change of the n -spectrum of a non-degenerate perturbation of { H τ } τ ≥ .We will show that the corresponding bifurcation diagram is (approximately) composed ofhorizontal lines, corresponding to orbits appearing for s > − (cid:15)/ 2, and of lines with slope − 1, corresponding to capped orbit with s < − (cid:15)/ 2, see Figure 6. Let us split into fourregions: • In M \ U : Here the H τ is zero for all τ . After perturbing into non-degenerateHamiltonians, the actions of capped orbits with CZ index − n in this region arecontained in a small neighborhood of Spec n (0) ⊂ Spec n ( ∂U ; M ) and do not changewith τ . • In U \ N ( ∂U ): Here H τ = H − τ , and therefore the periodic orbits remain the sameas τ changes, and the action of each orbit decreases linearly: A H τ ( γ, u ) = (cid:90) H τ ◦ γ dt − (cid:90) u ω = (cid:90) H ◦ γ dt − τ − (cid:90) u ω = A H ( γ, u ) − τ. Therefore, the action spectrum of H τ in this region changes linearly in τ , with slope − 1. After perturbing H into a non-degenerate Hamiltonian in this region H τ = H − τ are non-degenerate as well, and the action spectrum consists of lines with slope − • Near s = 1 − (cid:15) : Here H τ coincides with the radial Hamiltonian K τ = χ τ ( s ).Lemma 3.4 states that every non-constant 1-periodic orbit γ of K τ corresponds toa Reeb orbit of action χ (cid:48) τ ( s ( γ )). Since χ (cid:48) τ ( s ) takes values between 0 and τ /(cid:15) in thisregion, H τ may admit more 1-periodic orbits as τ grows. The action of each orbit,once it appeared, decreases approximately linearly in τ , since the value of H τ does.Indeed, by Corollary 3.8 the action with respect to a radial Hamiltonian is given by A H τ ( γ, u ) = χ τ ( s ) − χ (cid:48) τ ( s ) · s − ω ( A ) ≈ − τ − (1 − (cid:15) ) · (cid:90) ˆ γ λ − ω ( A ) , where s = s ( γ ), ˆ γ ( t ) = ψ − log s γ ( t/χ (cid:48) ( s )) ∈ P ( ∂U ) and A := u u ∈ π ( M ). Afterperturbing H τ into non-degenerate Hamiltonians, the actions attained by orbits inthis region remain in a neighborhood of lines with slope − τ . • Near s = 1 − (cid:15) : Here again H τ coincides with the radial Hamiltonian K τ = χ τ ( s ) andby Lemma 3.4, non-constant 1-periodic orbits correspond to Reeb orbits of action χ (cid:48) τ ( s ), which takes values between 0 and τ /(cid:15) . As before, H τ may admit more 1-periodic orbits as τ grows, but this time the action of each orbit, once it appeared,24 pec n ( H τ ) τc ( H ) Figure 6: An illustration of the n -spectrum of a non-degenerate perturbation of H τ . The dashedlines correspond to actions of orbits in { s < − (cid:15)/ , } while the solid lines correspond to actionsof orbits in { s > − (cid:15)/ } . remains approximately constant when τ varies: A H τ ( γ, u ) = χ τ ( s ) − χ (cid:48) τ ( s ) · s − ω ( A ) ≈ − (1 − (cid:15) ) · (cid:90) ˆ γ λ − ω ( A ) , where again s = s ( γ ), ˆ γ ( t ) = ψ − log s γ ( t/χ (cid:48) ( s )) ∈ P ( ∂U ) and A := u u ∈ π ( M ).Let us now show that, after perturbing H τ into non-degenerate Hamiltonians, theactions of capped orbits with index − n lie in a small neighborhood of the relative n -spectrum Spec n ( ∂U ; M ). For this end, we need to compare the RS index of 1-periodicorbits of H τ with the contact CZ indices of the corresponding Reeb orbits. By ourchoice of χ τ , its first derivative is strictly decreasing in this region, namely, χ (cid:48)(cid:48) τ < γ, u ) and the contactCZ index of (ˆ γ, ˆ u ) are related by RS( γ, u ) = CZ R (ˆ γ, ˆ u ) − . After perturbing H τ into a non-degenerate Hamiltonian, every non-degenerate capped orbit ( γ (cid:48) , u (cid:48) )must have CZ index close to the RS index of ( γ, u ). More accurately, since χ (cid:48)(cid:48) τ (cid:54) = 0,Lemma 3.9 guarantees that the kernel of dϕ tH ( γ (0)) − 1l is one-dimensional and, by(22), | CZ H ( γ (cid:48) , u (cid:48) ) − RS( γ, u ) | ≤ . We therefore conclude that capped orbits ofindex − n could appear only for a capping u (cid:48) = u (cid:48) A for A ∈ π ( M ) such that12 ≥ | CZ H ( γ (cid:48) , u (cid:48) ) − RS( γ, u ) | = | − n − c ( A ) − RS( γ, u ) | = (cid:12)(cid:12)(cid:12) − n − c ( A ) − CZ R (ˆ γ, ˆ u ) + 12 (cid:12)(cid:12)(cid:12) . Since c and CZ R take integer values, this is equivalent to c ( A ) = (cid:108) − CZ R (ˆ γ, ˆ u ) − n (cid:109) , which implies that the action of the index − n orbit ( γ (cid:48) , u (cid:48) ) is approximately − (cid:82) ˆ γ λ − ω ( A ) ∈ Spec n ( ∂U ; M ). 25his analysis shows that the bifurcation diagram corresponding to the n -spectrum ofa non-degenerate perturbation of { H τ } τ is composed of (neighborhoods of) decreasinglines of slope − 1, corresponding to orbits in { s < − (cid:15)/ } , and of (neighborhoods of)horizontal lines with values in Spec n ( ∂U ; M ), corresponding to orbits in { s > − (cid:15)/ } ,as illustrated in Figure 6.The condition that Spec n ( ∂U ; M ) does not intersect (0 , c ( H )], namely, (20), guaranteesthat there are no horizontal lines in the action window (0 , c ( H )]. As a consequence,the bifurcation diagram contains no intersections in this window. By the continuity of c ( H τ ) (with respect to the parameter τ ) the spectral invariant c ( H τ ) must move alonga single line, which implies that c ( H τ ) = c ( H ) − τ for τ ≤ τ := c ( H ). In particular, c ( H τ ) = 0. This proves the proposition for K := K τ , as c ( H + K ) = c ( H τ ) = 0 and (cid:107) K (cid:107) C = τ = c ( H ). Proposition 3.6, together with Claim 3.1, imply that the max inequality (1) holds forany collection of Hamiltonians H , . . . , H N , supported in disjoint domains U , . . . , U N withincompressible contact type boundaries, respectively, provided that condition (20) is satis-fied for each pair ( H i , U i ). Namely, that for each i , the relative n -spectrum Spec n ( ∂U i ; M )does not intersect the interval (0 , c ( H i )]. In this section we prove Theorems 2, 3 and 4,by showing that condition (20) is satisfied under the assumptions stated in the theorems.Before that, let us show that condition (20) always holds on symplectically asphericalmanifolds. When ( M, ω ) is symplectically aspherical, namely, ω | π ( M ) = 0, the max inequality (1)holds for every Hamiltonians H , . . . , H N , supported in disjoint domains U , . . . , U N withincompressible contact type boundaries. Indeed, given a domain U with incompressiblecontact type boundary, the fact that ω vanishes on π ( M ) implies thatSpec n ( ∂U ; M ) ⊂ − Spec( ∂U ) ∪ { } ⊂ ( −∞ , n ( ∂U ; M ) does not intersect the interval (0 , c ( H )] for any Hamilto-nian H with c ( H ) > 0. Therefore condition (20) is satisfied with no additional assumptionson U .We remind that the max inequality is a weaker statement than the max formula provedby Humili`ere, Le Roux and Seyfaddini in [10], which states that the spectral invariant of thesum of Hamiltonians supported in disjoint incompressible Liouville domains on symplec-tically aspherical manifolds is equal to the maximal spectral invariant of the summands. Let ( M, ω ) be a rational symplectic manifold, namely ω ( π ( M )) = κ Z for some κ ∈ R .In this section we prove that the max inequality (1) holds for Hamiltonians H i supported26n disjoint domains U i with incompressible contact type boundaries, if Spec( ∂U i ) ⊂ T i Z for some T i | κ and c ( H i ) < T i , respectively. Proof of Theorem 2. Let U ⊂ M be a domain with an incompressible contact type bound-ary and suppose there exists T | κ such that Spec( ∂U ) ⊂ T Z . In order to be able toapply Proposition 3.6 and Remark 3.7, we need to show that the relative spectrumSpec( ∂U ; M ) does not intersect the interval (0 , c ( H )] for every Hamiltonian H supportedin U with c ( H ) < T . The max inequality for such Hamiltonians will then follow fromClaim 3.1. When Spec( ∂U ) ⊂ T Z , the relative spectrum Spec( ∂U ; M ) is contained in theset −{ } ∪ Spec( ∂U ) + κ Z ⊂ T Z + κ Z . Since T | κ , the relative spectrum is contained in thelattice T Z , and the intersection Spec( ∂U ; M ) ∩ (0 , c ( H )] is empty for every Hamiltonianwith c ( H ) < T . Let ( M, ω ) be a negatively monotone symplectic manifold, namely, ω = κ · c on π ( M ) for some κ ≤ 0. We now prove Theorem 3, which states that the max inequality(1) holds for Hamiltonians supported in disjoint domains with incompressible contact typeboundaries, such that the contact CZ indices of the Reeb orbits are all non-negative. Proof of Theorem 3. Let U ⊂ M be a domain with an incompressible contact type bound-ary, such that the contact CZ index of every Reeb orbit on ∂U is non-negative, namelyCZ R ( γ, u ) ≥ γ ∈ P ( ∂U ) and u ⊂ ∂U. In what follows we show that the relative n -spectrum Spec n ( ∂U ; M ) is non-positive and inparticular does not intersect the interval (0 , c ( H )] for every Hamiltonian H supported in U .This will establish condition (20) and will enable us to conclude the max inequality fromProposition 3.6 and Claim 3.1. The relative n -spectrum Spec n ( ∂U ; M ) contains termscoming from the action spectrum of the zero function, Spec n (0), and terms coming fromactions of Reeb orbits. Starting with Spec n (0), it is composed of − ω ( A ) for A ∈ π ( M )such that c ( A ) ∈ {− n, . . . , } . In this case, − ω ( A ) = − κc ( A ) ≤ 0, and the spectrumis indeed non-positive. The rest of the elements in the relative n -spectrum are of theform − (cid:82) γ λ − ω ( A ), where A is such that c ( A ) = (cid:6) k − n (cid:7) and k = − CZ R ( γ, u ) for someReeb orbit γ ∈ P ( ∂U ) with respect to a capping disk u ⊂ ∂U . Since we assumedthat the contact CZ index of every capped Reeb orbit is non-negative, k is non-positive.Therefore, the Chern class of A is non-positive as well, i.e., c ( A ) ≤ 0. We conclude that − (cid:82) γ λ − ω ( A ) = − (cid:82) γ λ − κ · c ( A ) ≤ Let ( M, ω ) be a positively monotone symplectic manifold, namely, ω = κ · c on π ( M )for some κ > 0, and assume in addition that its dimension is greater than 2. We now proveTheorem 4, which states that the max inequality (1) holds for Hamiltonians H i supportedin disjoint domains U i with incompressible dynamically convex boundaries, such that foreach i , C ( U i ) ≤ κ and c ( H i ) < κ . 27 roof of Theorem 4. Let U ⊂ M be a domain with an incompressible dynamically convexboundary, such that C ( U ) ≤ κ , where C ( U ) is the invariant from Definition 1.3. Inwhat follows we show that Spec n ( ∂U ; M ) does not intersect the interval (0 , κ ). As aconsequence, condition (20) will follow for every Hamiltonian H supported in U , suchthat c ( H ) < κ . Starting with Spec n (0), which is the first component of Spec n ( ∂U ; M ),we see that − ω ( A ) = − κc ( A ) ∈ { , κ, . . . , nκ } , and therefore Spec n (0) ∩ (0 , κ ) = ∅ . Therest of the elements of the relative spectrum Spec n ( ∂U ; M ) are of the form − (cid:82) γ λ − ω ( A ),where γ is a periodic Reeb orbit and A ∈ π ( M ) is such that c ( A ) = (cid:108) − CZ R ( γ, u ) − n (cid:109) , for a capping disk u ⊂ ∂U of γ . Recalling that the invariant C ( U ) was defined to be thesupremum over ratios of the form 2 (cid:82) γ λ/ (CZ R ( γ, u ) − n + 1), we can use it to bound theabsolute value of c ( A ) from below: − c ( A ) ≥ CZ R ( γ, u ) + n − 12 = CZ R ( γ, u ) − n + 12 + n − ≥ (cid:82) γ λC ( U ) + n − , where, in the last inequality, we used our assumption that ∂U is dynamically convex, andhence CZ R ( γ, u ) − n + 1 > 0. In light of this observation, the action can be bounded asfollows: − (cid:90) γ λ − ω ( A ) = − (cid:90) γ λ − c ( A ) · κ ≥ − (cid:90) γ λ + κ · (cid:32) (cid:82) γ λC ( U ) + n − (cid:33) = − (cid:90) γ λ + κC ( U ) · (cid:90) γ λ + ( n − κ ≥ ( n − κ, where the last inequality follows from our assumption that C ( U ) ≤ κ . Since the dimensionof M is 2 n and is assumed to greater than 2, we conclude that the action − (cid:82) γ λ − ω ( A )is bounded from below by κ . As a result, the relative spectrum, Spec n ( ∂U ; M ), doesnot intersect the interval (0 , κ ), which, by our assumption, contains (0 , c ( H )]. Havingestablished condition (20), Proposition 3.6 together with Claim 3.1 guarantee that themax inequality holds in this setting.We end this section with a proof of Claim 1.8, which states that the max inequalityholds for Hamiltonians supported in certain disjoint disks on the sphere. Proof of Claim 1.8. We assume that the area form ω is normalized so that the total areaof the sphere is 1. In this case, the monotonicity constant is κ = . Let U ⊂ S be a diskof area a and let H be a Hamiltonian supported in U . In what follows we show that if a / ∈ (1 / , / H . Claim 3.1 will then guaranteethat the max inequality (1) holds for such Hamiltonians. Consider the family of radialHamiltonians K τ := τ · K that was constructed in the proof of Proposition 3.6. Recall that K is supported in U , constant and equal to − s < − (cid:15) , and is approximately linearlyincreasing for s ∈ (1 − (cid:15), − (cid:15) ). Here and in what follows s is the Liouville coordinate28n the disk U . In order to conclude that K c ( H ) is a spectral killer for H , we need toshow that the actions of index − n capped periodic orbits of K τ for s ≥ − (cid:15)/ , c ( H )]. After perturbing K τ into a non-degenerate Hamiltonian,its 1-periodic orbits for s ≥ − (cid:15)/ { γ k , γ k } k and p , where γ ki rotates k times around ∂U and p ∈ U c is a maximum point of action approximately zero. When paired withcapping disks u ki, ⊂ U , the actions of γ ki are approximately − k · area ( U ) = − k · a , andtheir CZ indices are CZ H ( γ k , u k , ) = 2 k, CZ H ( γ k , u k , ) = 2 k − . See, for example, [20, p.11] . Therefore, there exists a capping disk u ki such that CZ H ( γ ki , u ki ) = − n = − i = 2 and u k = u k , A k , for A k ∈ π ( S ) such that c ( A k ) = − k .In this case, the action of the index − − k · a − ω ( A k ) = − k · a − κ · c ( A k ) = − k · a − / · c ( A k ) = k · (1 / − a ) . Overall, the actions of non-degenerate orbits of index − { , k · (1 / − a ) } , and wecan construct a spectral killer for H if these actions do not intersect the interval (0 , c ( H )].Let us show that this holds whenever a / ∈ (1 / , / a of U is greaterthan or equal to 1 / 2, then k (1 / − a ) is non-positive and does not lie in the interval(0 , c ( H )]. Any open disk in S of area less than 1 / H is compactly supportedin U , and therefore is also supported in a slightly smaller disk. Applying the energy-capacity inequality to the smaller disk we conclude that c ( H ) < a . Therefore, (0 , c ( H )] ⊂ (0 , / 3) when a ≤ / 3. On the other hand, recalling that the minimal Chern number on S is 2 and that k = − c ( A k ), we conclude that k (1 / − a ) ≥ / − a ) ≥ · (1 / − / 3) = 1 / k (1 / − a ) does not lie in (0 , c ( H )]. In this section we prove Theorems 5 and 6, which state uniform bounds for the spectralinvariants of Hamiltonians supported in portable Liouville domains with dynamically con-vex incompressible boundaries, in monotone manifolds. When the manifold is positivelymonotone, one has to add a condition regarding the “size” of the domain containing thesupport, as seen in the following example. Example 5.1. Let ( M, ω ) be the two-dimensional sphere, endowed with an area form.It is known that the equator E ⊂ S is superheavy, see, e.g., [17]. In [3], Entov andPolterovich proved that the spectral invariant of any Hamiltonian is not smaller than theminimal value the Hamiltonian attains on a superheavy set. Therefore, when U is a largedisk containing the equator, there is no uniform upper bound for the spectral invariant ofHamiltonians supported in U .In order to bound the spectral invariant of a general Hamiltonian supported in aLiouville domain U , it is enough to consider simple, arbitrarily large radial Hamiltonians Note that the sign choice for the CZ index in [20] is opposite to ours. Let ( M, ω ) be a positively monotone symplectic manifold, namely, ω = κ · c on π ( M )for κ > 0. Let us prove Theorem 5, which states that for every Hamiltonian supported ina disjoint union U of portable Liouville domains with incompressible dynamically convexboundaries such that C ( U ) ≤ κ , the spectral invariant is smaller than C ( U ). In whatfollows we concern the Liouville coordinate of U , by which we mean the coordinate oneach connected component. Proof of Theorem 5. As mentioned above, in order to prove upper bounds for spectralinvariants of Hamiltonians supported in U , it is enough to consider (non-degenerate per-turbations of) non-increasing radial Hamiltonians. The claim for general Hamiltonianswill then follow from the monotonicity property of spectral invariants. Let H := χ ( s )where s is the Liouville coordinate on U and χ : R → R is a smooth approximation of apiecewise linear function that is constant for s ≤ ε , vanishes for s ≥ − δ (here δ > χ such that, outside ofa neighborhood of the “corners” s = ε and s = 1 − δ , its derivative is constant and doesnot lie in the Reeb spectrum of ∂U . We also assume that χ (cid:48)(cid:48) is strictly positive outside ofthe intervals in which χ (cid:48) is constant (and, in particular, near the “corners”). Finally, wechoose ε to arbitrarily smaller than max χ − δ , so that ε · χ (cid:48) is arbitrarily small. Let us provethat c ( H ) < C ( U ) in two steps. First we use a continuous deformation of H to show that c ( H ) is bounded by the maximal action that can be possibly attained by index − n cappedorbits near s = 1 − δ for Hamiltonians of this shape, and then we show that these actionsare all smaller than C ( U ).Step 1: In this step we consider a continuous deformation of the Hamiltonian H , anddescribe the bifurcation diagram of its spectrum. The deformation is given by composing H on the inverse Liouville flow: H τ := (cid:40) H ◦ ψ − log τ on ψ log τ U, ψ log τ U ) c (27)for τ ∈ (0 , H τ = χ τ ( s ) := χ ( s/τ ) is a radialHamiltonian for each τ , its non-constant 1-periodic orbits are in correspondence with theReeb orbits on ∂U and we can use Lemmas 3.4, 3.9 to relate their actions and indices.Abbreviating s = s ( γ ), Lemma 3.4 states that ˆ γ ( t ) := ψ − log s γ ( t/χ (cid:48) ( s )) ⊂ ∂U is a Reeborbit of action − χ (cid:48) τ ( s ) ∈ Spec( ∂U ). Recalling that the Reeb spectrum is a discrete set,we conclude that for generic τ ∈ (0 , H τ appear onlynear the “corners” s = τ ε and s = τ (1 − δ ). As stated in Corollary 3.8, the action of acapped 1-periodic orbit ( γ, u ) of H τ is given by A H τ ( γ, u ) = χ τ ( s ) − s · χ (cid:48) τ ( s ) − ω ( A ) , − δH = χ ( s ) H τ = χ τ ( s ) τ (1 − δ ) sε Figure 7: The deformation of the radial Hamiltonian H is given by shrinking its support, usingthe Liouville flow. Generically, non-constant 1-periodic orbits appear near the “corners”, namelynear s = τ ε and s = τ (1 − δ ). where A ∈ π ( M ) is such that u = u A and u ⊂ { s = s ( γ ) } . Let us describe thebifurcation diagram of the spectrum of H τ as τ varies. When τ decreases, more 1-periodicorbits may appear near the corners s = τ ε and s = τ (1 − δ ). The action of a capped orbitwith s = s ( γ ) near τ ε , once it appeared, is approximately χ τ ( τ ε ) − ετ χ (cid:48) τ ( s ) − ω ( A ) = χ ( ε ) − ετ τ χ (cid:48) ( s/τ ) − ω ( A ) ≈ χ (0) − ω ( A ) , where the last approximation is due to our assumption that ε is arbitrary small com-pared to the derivative of χ . In particular, the action of orbits near s = τ ε remainsapproximately constant as τ varies. This is also the case for capped constant 1-periodicorbits (namely, pairs of a critical point of H τ and a sphere A ∈ π ( M )) in the region s ≥ τ (1 − δ ), in which case the action is approximately − ω ( A ). On the other hand, theaction of a non-constant 1-periodic orbit near s = τ (1 − δ ), once it appeared, is approx-imately 0 − τ (1 − δ ) · χ (cid:48) τ ( s ) − ω ( A ) and, in particular, changes linearly in τ with slope − (1 − δ ) χ (cid:48) τ ( s ) ∈ (1 − δ ) · Spec( ∂U ). We conclude that the bifurcation diagram is (ap-proximately) composed of horizontal lines, corresponding to orbits near s = τ ε , as well asconstant orbits, and of lines with positive slopes, corresponding to orbits near s = τ (1 − δ ),see Figure 8. When τ is very small, H τ is supported in ψ − log τ U which is displaceable withsmall displacement energy, as explained in Section 2.3.1 above. By the energy-capacityinequality, c ( H τ ) is bounded by the displacement energy of the support and hence is verysmall. Following the bifurcation diagram of the spectrum of H τ , it is clear that c ( H ) can-not be larger than the maximal point on a non-horizontal line. After perturbing { H τ } intonon-degenerate Hamiltonians, their spectral invariants lie in the n -spectrum. Repeatingthe arguments above for the n -spectrum we see that c ( H ) is not greater than the maximalpoint on a non-horizontal line in the diagram corresponding to the n -spectrum.31 ← τ τ = 1 Figure 8: The bifurcation diagram corresponding to the n -spectrum of H τ . Since the spectralinvariant starts close to zero and moves continuously on the diagram, its value is bounded by themaximal height attained by a non-horizontal line. Step 2: In this step we show that, for every τ ∈ (0 , − n capped orbit of a non-degenerate perturbation of H τ that appears near s = τ (1 − δ ) isnot greater than C ( U ) − δ · min Spec( ∂U ). Together with the previous step, this willimply that c ( H ) < C ( U ) as required. Let ( γ, u ) be a 1-periodic orbit of H τ such that oneof the non-degenerate capped orbits, ( γ (cid:48) , u (cid:48) ), appearing after perturbing H τ into a non-degenerate Hamiltonian, is of CZ index − n . In what follows we compute the Chern classof A ∈ π ( M ) for which u = u A , where u is a capping disk of γ that is contained inthe level set { s = s ( γ ) } . Since the manifold M is monotone, we will use this computationwhen estimating the action of ( γ, u ). As mentioned above, ˆ γ ( t ) := ψ − log s γ ( t/χ (cid:48) ( s )) ⊂ ∂U is a Reeb orbit of action − χ (cid:48) τ ( s ) ∈ Spec( ∂U ). Let ˆ u ⊂ ∂U be a capping disk of ˆ γ whoseimage coincides with ψ − log s u . Due to our assumption that χ (cid:48)(cid:48) ( s ) > s that contains 1-periodic orbits, Lemma 3.9 guarantees thatRS( γ, u ) = − CZ R (ˆ γ, ˆ u ) + 12 . (28)Having a non-degenerate orbit ( γ (cid:48) , u (cid:48) ) of index − n appearing after the perturbation, we canestimate RS( γ, u ) using inequality (22), which states that the index difference betweenthe perturbed and original orbits is bounded by dim ker( dϕ H ( γ (0)) − dϕ H ( γ (0)) − 1l is 1-dimensional if χ (cid:48)(cid:48) ( s ) (cid:54) = 0.Therefore, | CZ H ( γ (cid:48) , u (cid:48) ) − RS( γ, u ) | ≤ , where u (cid:48) is a capping disk of γ (cid:48) that is containedin a small neighborhood of u . We conclude that − n = CZ H ( γ (cid:48) , u (cid:48) ) = CZ H ( γ (cid:48) , u (cid:48) ) + 2 c ( A ) ≤ RS( γ, u ) + 12 + 2 c ( A ) , and, together with (28), this yields that c ( A ) ≥ CZ R (ˆ γ, ˆ u ) − n − . A , it remains to use the monotonicity of M to bound the action of the index − n capped orbit ( γ (cid:48) , u (cid:48) ). Since the action ( γ (cid:48) , u (cid:48) ) is closeto A H τ ( γ, u ), we will estimate the latter. Recalling the formula for the action of orbits ofradial Hamiltonians, which was established in Corollary 3.8, we have A H τ ( γ, u ) = χ τ ( s ) − s · χ (cid:48) τ ( s ) − ω ( A ) ≈ − τ (1 − δ ) · χ (cid:48) τ ( s ) − ω ( A )= τ (1 − δ ) · (cid:90) ˆ γ λ − ω ( A ) ≤ (1 − δ ) · (cid:90) ˆ γ λ − ω ( A )= (1 − δ ) · (cid:90) ˆ γ λ − κ · c ( A ) ≤ (1 − δ ) · (cid:90) ˆ γ λ − κ · (cid:18) CZ R (ˆ γ, ˆ u ) − n − (cid:19) . Our assumption that ∂U is dynamically convex implies that CZ R (ˆ γ, ˆ u ) ≥ n + 1. Since C ( U ) is assumed to be not-greater than κ , this yields A H τ ( γ, u ) ≤ (1 − δ ) (cid:90) ˆ γ λ − C ( U ) · (cid:18) CZ R (ˆ γ, ˆ u ) − n − (cid:19) . The next step is to bound the contact CZ index by the Reeb action divided by C ( U ).Recall that C ( U ) was defined as the supremum over ratios 2 (cid:82) ˆ γ λ/ (CZ R (ˆ γ, ˆ u ) − n + 1).Therefore, CZ R (ˆ γ, ˆ u ) − n − ≥ (cid:82) ˆ γ λ/C ( U ) − 2, and we can bound the action by A H τ ( γ, u ) ≤ (1 − δ ) (cid:90) ˆ γ λ − C ( U ) · (cid:18) CZ R (ˆ γ, ˆ u ) − n − (cid:19) ≤ (1 − δ ) (cid:90) ˆ γ λ − C ( U ) · · (cid:32) (cid:82) ˆ γ λC ( U ) − (cid:33) = (1 − δ ) (cid:90) ˆ γ λ − (cid:90) ˆ γ λ + C ( U ) ≤ C ( U ) − δ min Spec( ∂U ) . This finishes the proof, as the first step guaranteed that the spectral invariant c ( H ) isnot-greater than the maximal action of such a capped orbit. In particular, we concludethat c ( H ) < C ( U ) as required. In [16, Proposition 5.4] Polterovich proved that on an aspherical manifold M , the spec-tral invariant of every Hamiltonian supported in a disjoint union U of portable Liouvilledomains is bounded by the portability number p ( U ). Theorem 6 asserts that this holdstrue for negatively monotone manifolds, if one demands in addition that ∂U is dynamicallyconvex. As before, the Liouville coordinate of the disjoint union is given be the Liouvillecoordinate on each connected component. Proof of Theorem 6. Following [16], the idea is to use symplectic contraction and follow thespectral invariant in the corresponding bifurcation diagram. The symplectic contraction33 H = χ ( s ) H τ = χ τ ( s ) s Figure 9: Symplectic contraction of a radial non-increasing Hamiltonian H . of a Hamiltonian H supported in the Liouville domain U is defined to be H τ := (cid:40) τ · H ◦ ψ − log τ on ψ log τ U, M \ ψ log τ U for τ ∈ [0 , H τ is the composition of the Liouville flow on flow of H , namely, ϕ tH τ = ψ log τ ϕ tH . Inthe aspherical case there are no intersections in the bifurcation diagram corresponding tothis deformation, and the spectral invariant changes linearly, c ( H τ ) = τ · c ( H ). However,on monotone manifolds, there could be a lot of intersections and the change of c ( H τ ) isin general more complicated. To simplify the situation, we consider only non-increasingradial Hamiltonians. The claim for general Hamiltonians will follow from the monotonicityproperty of spectral invariants. Let H := χ ( s ) where s is the Liouville coordinate on U and χ : R → R is a non-increasing function that vanishes on s ≥ s = 0. We assume that the second derivative of χ does not vanish on level sets containing1-periodic orbits, i.e., whenever | χ (cid:48) ( s ) | ∈ Spec( ∂U ). Let us compute the change in theaction spectrum when symplectically contracting H . The non-constant 1-periodic orbitsof H and H τ are in bijection: for γ ∈ P ( H ), its image ψ log τ γ under the Liouville flow isa 1-periodic orbit of H τ . Given a capping disk u ⊂ { s = s ( γ ) } of γ , its image ψ log τ u isa capping disk of ψ log τ γ . The action of ( ψ log τ γ, ψ log τ u ) with respect to H τ is A H τ ( ψ log τ γ, ψ log τ u ) = H τ ( ψ log τ γ ) − (cid:90) (cid:16) ψ log τ u (cid:17) ∗ ω = τ · H ( γ ) − τ · (cid:90) u ∗ ω = τ · A H ( γ, u ) . In addition, the RS indices of these capped orbits are equal. Indeed, given a trivialization T : u ∗ T M → D × R n , we can define a trivialization along ψ log τ u by T ◦ dψ − log τ . Under34 spec n ( H τ ) −| κ | | κ | | κ | | κ | c ( H τ ) Figure 10: An illustration of the bifurcation diagram for symplectic contraction of a non-increasingradial Hamiltonian on a dynamically convex domain. When all slopes and starting points are non-negative, the spectral invariant c ( H τ ) can only move along lines of slope not greater than the initialslope. these trivializations, the differentials of the flows coincide and thus they have the same RSindex. More generally, if u is any capping of γ , we may write u = u A for A ∈ π ( M )and then ( ψ log τ u ) A is a capping of ψ log τ γ . In that case, A H τ ( ψ log τ γ, ( ψ log τ u ) A ) = A H τ ( ψ log τ γ, ψ log τ u ) − ω ( A )= τ · A H ( γ, u ) − ω ( A ) . If follows that the corresponding bifurcation diagram consists of lines whose slopes equalto A H ( γ, u ) for γ ∈ P ( H ) and u ⊂ { s = s ( γ ) } and whose starting points (that is, thevalues at τ = 0) are − ω ( A ). After perturbing H into a non-degenerate Hamiltonian,it is possible to choose non-degenerate perturbations of H τ such that 1-periodic orbitswill still be in bijection with the same relations for the actions and indices. After thisperturbation the spectral invariant will lie in the spectrum of index − n capped orbits.Our next goal is to show that the slopes and the starting points are all non-negative, asillustrated in Figure 10. This will allow us to estimate the path of the spectral invariant inthe bifurcation diagram. We start with the slopes, which corresponds to actions of orbits γ ∈ P ( H ) with capping disks u that are contained in the Liouville level set: • A H ( γ, u ) ≥ 0: Since we assumed χ to be non-increasing, the non-negativity of theslopes in the bifurcation diagram follows immediately from the formula for actionsof orbits of radial Hamiltonians, that was stated in Corollary 3.8, when applied for u = u , A = 0: A H ( γ, u ) = χ ( s ) − s · χ (cid:48) ( s ) ≥ . • − ω ( A ) ≥ 0: The non-negativity of the stating points requires index computations.We start by recalling that the index difference between the perturbed and origi-nal orbits is bounded by half the dimension of ker( dϕ H ( γ (0)) − χ , its second derivative χ (cid:48)(cid:48) ( s ) does not vanish whenever − χ (cid:48) ( s ) ∈ Spec( ∂U ) and Lemma 3.9 guarantees that the kernel of dϕ H ( γ (0)) − 1l is1-dimensional. Thus | CZ H ( γ (cid:48) , u (cid:48) ) − RS( γ, u ) | ≤ , where u (cid:48) is a capping disk of γ (cid:48) , obtained as a perturbation of u ⊂ { s = s ( γ ) } . Wetherefore conclude that capped orbits ( γ (cid:48) , u (cid:48) ) of index − n can appear only for ( γ, u )such that12 ≥ | RS( γ, u ) − CZ H ( γ (cid:48) , u (cid:48) ) | = | RS( γ, u ) − CZ H ( γ (cid:48) , u (cid:48) ) + 2 c ( A ) | = | RS( γ, u ) + n + 2 c ( A ) | , (29)where A ∈ π ( M ) is such that u (cid:48) = u (cid:48) A . Next, we use Lemma 3.9 in orderto replace the RS index of ( γ, u ) by the contact CZ index of the correspondingcapped Reeb orbit (ˆ γ, ˆ u ). Here, ˆ γ ( t ) := ψ − log s γ ( t/χ (cid:48) ( s )) ∈ P ( ∂U ) and ˆ u is acapping disk whose image coincides with that of ψ − log s u . Recalling that χ (cid:48) ( s ) ≤ γ, u ) of theradial Hamiltonian H is given byRS( γ, u ) = − CZ R (ˆ γ, ˆ u ) + 12 sign ( χ (cid:48)(cid:48) ( s )) ≤ − CZ R (ˆ γ, ˆ u ) + 12 . Therefore, capped 1-periodic orbits of CZ index − n can appear only for ( γ, u ) with u = u A such that c ( A ) ≥ − RS( γ, u ) − n − ≥ CZ R (ˆ γ, ˆ u ) − n − , for some Reeb orbit ˆ γ ∈ P ( ∂U ) with a capping ˆ u ⊂ ∂U . Since ∂U is dynamicallyconvex, CZ R (ˆ γ, ˆ u ) ≥ n +1 for every capped Reeb orbit. As a consequence, c ( A ) ≥ M is negatively monotone, − ω ( A ) = − κc ( A ) ≥ c ( H ) ≤ p ( U ) follows. We follow c ( H τ ) along the diagram as τ grows from 0 to 1. When τ = 0, H = 0 and by the stability property of spectral invariants, c ( H ) = 0. When τ > c ( H τ )moves along a single line, (cid:96) , that starts at zero and whose slope we denote by a ≥ c ( H τ ) = a · τ for τ < (cid:15) . We claim that c ( H τ ) ≤ a · τ for all τ ≤ 1. Indeed, since theline (cid:96) starts at the lowest possible point (zero), every line (cid:96) (cid:48) of slope bigger than a iscompletely contained in the upper region bounded by (cid:96) , namely (cid:96) (cid:48) ⊂ { y > a · τ } . Thus,when starting on the line (cid:96) , the spectral invariant c ( H τ ) can only move along lines ofslope ≤ a , see Figure 10 for an illustration. Therefore, c ( H τ ) ≤ a · τ for all τ ≤ 1. Toconclude a bound for the spectral invariant of H = H , it remains to bound the slope a of (cid:96) . This argument is identical to that in the aspherical case, presented by Polterovichin [16]: The contracted Hamiltonian H τ is supported in ψ log τ U . When τ is very small,this set is displaceable in U with energy e ( ψ log τ U ; U ). By the energy-capacity inequality,36 ( H τ ) ≤ e ( ψ log τ U ; M ) ≤ e ( ψ log τ U ; U ) for τ close enough to zero. Since c ( H τ ) lies on (cid:96) there, we conclude that the slope is bounded by the ratio a ≤ e ( ψ log τ U ; U ) /τ . Taking thelimit τ → c ( H ) ≤ a · ≤ p ( U ). C ( U ) in special cases. In this section we provide upper bounds for the invariant C from Definition 1.3 onseveral classes of domains. The domains considered here are all topological balls, whichmeans that their boundaries are aspherical, π ( S n − ) = 0. As a consequence, the contactCZ index is independent of the choice of a capping disk, and we will use the notationCZ R ( γ ). Let us start with the simplest example, a generic ellipsoid. Example 6.1. Let U = E ( a , . . . , a n ) = { z ∈ C n : (cid:80) nj =1 π | z j | a j ≤ } be an ellipsoidsuch that a i /a j is irrational when i (cid:54) = j . The periodic Reeb orbits on ∂E are γ k,(cid:96) := { e πit/a k z k } t ∈ [0 ,a k (cid:96) ] for k = 1 , . . . , n and (cid:96) ∈ N > . The action (or period) of γ k,(cid:96) is a k (cid:96) andits CZ index (with respect to any capping disk u ⊂ ∂E ) is CZ R ( γ k,(cid:96) ) = n − n (cid:88) j =1 (cid:106) (cid:96) · a k a j (cid:107) , see e.g. [8, p.16]. A simple computation shows that in this case C ( E ) = min j a j = c G ( E ).Wider classes of examples that are considered below are convex and concave toricdomains. Since the Reeb flow on toric domains is degenerate (unless the domain is anellipsoid), we need to extend Definition 1.3 of the invariant C ( U ) to degenerate domains. Definition 6.2. Let U ⊂ M be a domain with an incompressible contact type boundary. • Assume that the Reeb flow on ∂U is non-degenerate. For every T > 0, define C T ( U ) := sup (cid:40) (cid:82) γ λ CZ R ( γ, u ) − n + 1 : γ ∈ P ( U ) , (cid:90) γ λ ≤ T, u ⊂ ∂U (cid:41) , and notice that C ( U ) = sup T > C T ( U ). • When the Reeb flow on ∂U is degenerate, we define C T ( U ) := lim inf U (cid:48) → U C T ( U (cid:48) ) , C ( U ) := sup T > C T ( U ) , where the limit is over domains U (cid:48) with incompressible contact type boundaries andnon-degenerate Reeb flows that C -converge to U .37 .1 Convex and concave toric domains. Denote by R n ≥ the set of points x ∈ R n with non-negative entries, x i ≥ 0. Considerthe moment map µ : C n → R n ≥ defined by µ ( z , . . . , z n ) = π ( | z | , . . . , | z n | ). We say thata domain U ⊂ M with a contact type boundary is a toric domain if it is symplectomorphicto X Ω := µ − (Ω) ⊂ C n , for some domain Ω ⊂ R n ≥ (that is, a connected set that is openin the topology of R n ≥ ). Moreover, • If ˆΩ := { ( x , . . . , x n ) ∈ R n : ( | x | , . . . , | x n | ) ∈ Ω } is convex, we say that X Ω is a convex toric domain . • If ¯Ω is compact and R n ≥ \ Ω is convex, we say that X Ω is a concave toric domain .In [8], Gutt and Hutchings estimate actions and Conley-Zehnder indices of periodic Reeborbits for convex and concave toric domains, in order to compute the capacities c k comingfrom positive S -equivariant symplectic homology. Using their calculations, we give upperbounds for the invariant C ( U ) from Definition 1.3 when U is a concave or convex toricdomain. Claim 6.3. Suppose that U ⊂ M is a domain with an incompressible contact type boundarythat is symplectomorphic to a concave toric domain. Then, C ( U ) = c G ( U ) . Claim 6.4. Suppose that U ⊂ M is a domain with an incompressible contact type boundarythat is symplectomorphic to a convex toric domain, U ∼ = X Ω . Then, c G ( U ) ≤ C ( U ) ≤ c G ( B ) , for every ball B that contains X Ω . The lower bounds in the above claims actually hold for general nice star-shaped do-mains, and follow from a comparison between the invariant C ( U ) and certain invariantsdefined by Gutt and Hutchings in [8]. Lemma 6.5. Let U ⊂ R n be a nice star shaped domain, then C ( U ) ≥ c G ( U ) . Proof. The proof relies on the capacities c k coming from positive S -equivariant symplectichomology , which were introduced by Gutt and Hutchings in [8]. These are numericalinvariants of nice star-shaped domains which admit several useful properties. Let us statethe properties that will be of use for us (see [8, Theorem 1.1]):1. c ( U ) ≥ c G ( U ) for every nice star-shaped domain U .2. c k are continuous with respect to the Hausdorff metric.3. For a non-degenerate nice star-shaped domain U and k ∈ N , c k ( U ) belongs to theaction spectrum of Reeb orbits with contact Conley-Zehnder index equal to 2 k + n − C ( U ) ≥ sup k c k ( U ) /k ≥ c ( U ) ≥ c G ( U ). For degenerate domains,the inequality C ( U ) ≥ c G ( U ) now follows from the fact that c G is continuous in theHausdorff metric. Alternatively, one can use the capacity coming from symplectic homology [5]. Similar arguments implythat C ( U ) is greater or equal to this capacity as well. C fromabove. Let us start with convex toric domains. Proof of Claim 6.4. By Definition 6.2, it is enough to prove the upper bound for C T ( U ),for arbitrary T > 0. In [8, p.18-20], Gutt and Hutchings show that one can perturb anyconvex toric domain U into a domain with a contact type boundary and non-degenerateReeb flow, such that, after the perturbation, every Reeb orbit γ ∈ P ( ∂U ) whose action isnot greater than T corresponds to a vector with non-negative integer entries v ∈ N n (here N denotes the set of natural numbers including zero) satisfying (cid:90) γ λ ≈ (cid:107) v (cid:107) ∗ Ω and Z ( v ) + 2 n (cid:88) i =1 v i ≤ CZ R ( γ ) ≤ n − n (cid:88) i =1 v i , (30)where (cid:107) v (cid:107) ∗ Ω := sup {(cid:104) v, w (cid:105) : w ∈ Ω } is the support function associated to Ω and Z ( v ) isthe number of elements of v that are equal to zero. As a consequence, C T ( U ) ≤ sup (cid:40) (cid:107) v (cid:107) ∗ Ω + ε (cid:80) ni =1 v i − n − − Z ( v )2 (cid:12)(cid:12)(cid:12) v ∈ N n (cid:41) , (31)where ε > ε , by sup w ∈ Ω (cid:107) w (cid:107) ∞ . Indeed, (cid:107) v (cid:107) ∗ Ω (cid:80) ni =1 v i − n − − Z ( v )2 = sup w ∈ Ω (cid:104) v, w (cid:105) (cid:80) ni =1 v i − n − − Z ( v )2 = sup w ∈ Ω (cid:80) ni =1 v i w i (cid:80) ni =1 v i − n − − Z ( v )2 ≤ sup w ∈ Ω (max i w i ) · (cid:80) ni =1 v i (cid:80) ni =1 v i − n − − Z ( v )2 ≤ sup w ∈ Ω (cid:107) w (cid:107) ∞ . (32)Let B = B ( a ) ⊂ R n be a ball of capacity a that contains X Ω , then µ ( B ) ⊃ Ω. As aconsequence, sup w ∈ Ω (cid:107) w (cid:107) ∞ ≤ sup w ∈ µ ( B ) (cid:107) w (cid:107) ∞ = a . Remark 6.6. In fact we proved a stronger upper bound for the invariant C ( U ) thanthe one stated in Claim 6.4. Inequality (32) implies that C ( X Ω ) ≤ sup w ∈ Ω (cid:107) w (cid:107) ∞ . Inparticular, the invariant C ( X Ω ) can be bounded by the minimal a such that the polydisk D ( a ) n contains X Ω .The following example shows that invariant C is not continuous with respect to theHausdorff metric and is not monotone with respect to inclusion. Example 6.7. Given an ellipsoid E = E ( a , a ) ⊂ C with a < a , then it is a convextoric domain. As mentioned in the proof of Claim 6.4, Gutt and Hutchings showed in [8,p.18-20] that there exists a perturbation of E ( a , a ) for which the periodic Reeb orbitcorrespond to vectors v ∈ N whose actions and indices are given by (30). Let us describethis perturbation. Let X Ω be a convex toric domain such that Ω is Hausdorff-close to µ ( E ),and such that the curve ∂ Ω is almost perpendicular to the axes, see Figure 11. Denoteby a (cid:48) the intersection point of ∂ Ω with the x -axis and note that a (cid:48) is close to a and,in particular, is greater than a . The next step is to perturb X Ω into a non-degenerate39 a ∂ Ω ∂µ ( E ) a Figure 11: The dashed line is the image of the boundary of E under the moment map and thesolid line is ∂ Ω, which intersects the axes at almost right angles. domain. After this perturbation there exists a periodic Reeb orbit γ corresponding to v = (1 , ∈ N . Recalling (30), the action of γ is approximately (cid:107) v (cid:107) ∗ Ω = a (cid:48) and its indexis CZ R ( γ ) = 1 + 2 = 3. As a result, we obtain a domain U with a contact type boundaryand non-degenerate flow, that is contained in E and is Hausdorff-close to it, such that C ( U ) ≥ (cid:82) γ λ CZ R ( γ ) − n + 1 ≈ a (cid:48) − a (cid:48) ≈ a . By Example 6.1, C ( E ) = a and we conclude that C is neither monotone nor Hausdorff-continuous.We turn to prove Claim 6.3. Proof of Claim 6.3. By Definition 6.2, it is enough to prove the upper bound for C T ( U ),for arbitrary T > 0. In [8, p.22-23], Gutt and Hutchings show that one can perturb U into a domain with a contact type boundary and non-degenerate flow, such that, after theperturbation, every Reeb orbit γ ∈ P ( ∂U ) either has a large contact CZ index (in whichcase, either the action is larger than T , or the ratio in the definition of C ( U ) is small) orcorresponds to a vector v ∈ N n> , such that (cid:90) γ λ ≤ [ v ] Ω + ε and 1 − n + 2 n (cid:88) i =1 v i ≤ CZ R ( γ ) ≤ n (cid:88) i =1 v i , where [ v ] Ω := inf {(cid:104) v, w (cid:105) : w ∈ R n ≥ \ Ω } and ε > C T ( U ) ≤ sup (cid:26) [ v ] Ω + ε (cid:80) ni =1 v i − n + 1 (cid:12)(cid:12)(cid:12) v ∈ N n> (cid:27) . (33)Fix ˆ v ∈ N n> such that C T ( U ) ≤ [ˆ v ] Ω (cid:80) ni =1 ˆ v i − n +1 + 2 ε . Consider the ellipsoid defined by E := µ − (cid:0) { x ∈ R n ≥ : (cid:104) x, ˆ v (cid:105) < [ˆ v ] Ω } (cid:1) , E ⊂ U and [ˆ v ] µ ( E ) = [ˆ v ] Ω . Let 0 < a ≤ · · · ≤ a n such that E = E ( a , . . . , a n ), then C T ( U ) ≤ [ˆ v ] Ω (cid:80) ni =1 ˆ v i − n + 1 + 2 ε = [ˆ v ] µ ( E ) (cid:80) ni =1 ˆ v i − n + 1 + 2 ε = min i a i ˆ v i (cid:80) ni =1 ˆ v i − n + 1 + 2 ε ≤ min i a i ˆ v i max i ˆ v i + 2 ε ≤ a + 2 ε = c G ( E ) + 2 ε ≤ c G ( U ) + 2 ε. In this section, we show that when U is strictly convex, C ( U ) < ∞ . This follows fromcomputations carried by Ishikawa in [11]. On strictly convex domains, Ishikawa boundedthe ratio between the action and CZ index of 1-periodic orbits of radial Hamiltonians interms of the sectional curvatures of the boundary. Definition 6.8 ([11, Definition 4.1]) . Let U ⊂ R n be a strictly convex open set witha smooth boundary, such that 0 ∈ U . Denote by f : R n → R the squared semi-normassociated to U , namely f ( ty ) = t for y ∈ ∂U . Defineˆ C ( U ) := inf V ⊂ R n ,a> πa where the infimum is taken over all one-dimensional complex subspaces V ⊂ C n ∼ = R n and a > D f ( x ) > a | V × V ⊕ | V ⊥ × V ⊥ , for every x ∈ R n \ { } .Roughly speaking, ˆ C ( U ) is 2 π over the maximum over all complex planes of theminimal sectional curvature in ∂U , restricted to the plane. In particular, it is finite forevery strictly convex domain. Claim 6.9. Let U ⊂ R n be a strictly convex open set with a smooth boundary, such that ∈ U . Then, C ( U ) ≤ ˆ C ( U ) .Proof. Let ˆ γ ∈ P ( ∂U ) be a periodic Reeb orbit and denote its action by α := (cid:82) ˆ γ λ > (cid:82) ˆ γ λ/ (CZ R (ˆ γ ) − n + 1) in the definition of C ( U )is bounded by ˆ C ( U ). The argument goes through radial Hamiltonians in order to usea result from [11]. Consider a radial Hamiltonian H that is linear with slope − α withrespect to the Liouville coordinate near the boundary of U , i.e., H | N ( ∂U ) = − α · s = − α · f ,where f is the squared semi-norm associated to U , as in Definition 6.8. Lemma 3.4 statesthat γ ( t ) = ˆ γ ( χ (cid:48) ( s ) · t ) = ˆ γ ( − α · t ) is a 1-periodic orbit of the Hamiltonian flow of H . Thelinearized flow Φ( t ) := dϕ tH ( γ (0)) of H along γ satisfies the equation ddt Φ( t ) = J D H ( γ ( t )) · Φ( t ) = − αJ D f ( γ ( t )) · Φ( t ) . V ⊂ C n and a path { Φ( t ) } of symplectic matrices satisfying ddt Φ( t ) = J S ( t )Φ( t ) for S ( t ) ≤ − c | V × V ⊕ | V ⊥ × V ⊥ , itsRS index is bounded by RS (Φ) ≤ − n − c/ π ] < − , where [ x ] < stands for the largest integer that is smaller than x . Applying this statementto S ( t ) = − αD f ( γ ( t )) and c = α · a , it follows that the RS index of the orbit γ , withrespect a capping disk u ⊂ { s = s ( γ ) } is bounded byRS( γ, u ) ≤ − n − (cid:104) α · a π (cid:105) < − ≤ − n − (cid:104) α ˆ C ( U ) (cid:105) < − . (34)By Lemma 3.9, the RS index of ( γ, u ) is equal to minus the contact CZ index of ˆ γ .Therefore, inequality (34) yieldsCZ R (ˆ γ ) ≥ n + 2 (cid:104) α ˆ C ( U ) (cid:105) < + 1 = n + 2 (cid:104) (cid:82) ˆ γ λ ˆ C ( U ) (cid:105) < + 1 . As the above inequality holds for every Reeb orbit, we can use it to bound the invariant C ( U ) as follows: C ( U ) = sup ˆ γ (cid:82) ˆ γ λ CZ R (ˆ γ, ˆ u ) − n + 1 ≤ sup ˆ γ (cid:82) ˆ γ λn + 2 (cid:104) (cid:82) ˆ γ λ ˆ C ( U ) (cid:105) < + 1 − n + 1 ≤ sup ˆ γ (cid:82) ˆ γ λ · (cid:82) ˆ γ λ ˆ C ( U ) − C ( U ) , where the last inequality follows from the fact that [ x ] < ≥ x − References [1] Fr´ed´eric Bourgeois. A survey of contact homology. 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