FIBERWISE VOLUME GROWTH VIA LAGRANGIAN INTERSECTIONS

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1 FIBERWISE VOLUME GROWTH VIA LAGRANGIAN INTERSECTIONS URS FRAUENFELDER AND FELIX SCHLENK Abstract. We study certain topological entropy-type growth characteristics of Hamiltonian flows of a special form on the cotangent bundle T M over a closed Riemannian manifold M, g. More precisely, the Hamiltonians generating the flows have to be of the form Ht, q, p = f p for p, and the growth characteristics reflect how the Riemannian volumes of the unit balls in the fibers of T M grow with the flow. Using a version of Lagrangian Floer homology and recent work of Abbondandolo and Schwarz, we obtain uniform lower bounds for the growth characteristics that depend only on M, g or, more precisely, on the homology of the sublevel sets of the energy functional defined by the Riemannian metric g on the space of based loops of Sobolev class W,2 in M. These lower bounds, in turn, can be estimated from below by purely topological characteristics of M. Our results in particular refine previous results of Dinaburg, Gromov, Paternain, and Paternain Petean on the topological entropy of geodesic flows. As an application, we obtain that for Riemannian manifolds all of whose geodesics are closed so-called P-manifolds, the fiberwise volume growth of every symplectomorphism in the symplectic isotopy class of the Dehn Seidel twist is at least linear. This extends the main result of [30] from the class of all currently known P-manifolds to all P-manifolds.. Introduction and main results.. Topological entropy and volume growth. The topological entropy h top ϕ of a compactly supported C -diffeomorphism ϕ of a smooth manifold X is a basic numerical invariant measuring the orbit structure complexity of ϕ. There are various ways of defining h top ϕ, see [39]. If ϕ is C -smooth, a geometric way was found by Yomdin and Newhouse in their seminal works [76] and [54]: Fix a Riemannian metric g on X. For j {,...,dimx} denote by Σ j the set of smooth compact not necessarily closed j-dimensional submanifolds of X, and by µ g σ the volume of σ Σ j computed with respect to the measure on σ induced by g. The j th volume growth of ϕ is defined as v j ϕ = sup lim inf σ Σ j m m log µ g ϕ m σ, Date: October, This work was supported by the Japanese Society for Promotion of Science UF, and the Deutsche Forschungsgemeinschaft FS Mathematics Subject Classification. Primary 53D35, Secondary 37B40, 53D40.

2 2 Fiberwise volume growth via Lagrangian intersections and the volume growth of ϕ is defined as vϕ = max v jϕ. j dim X Newhouse proved in [54] that h top ϕ vϕ, and Yomdin proved in [76] that h top ϕ vϕ provided that ϕ is C -smooth, so that h top ϕ = vϕ if ϕ is C -smooth. The topological entropy measures the exponential growth rate of the orbit complexity of a diffeomorphism. It therefore vanishes for many interesting dynamical systems. Following [40, 30] we thus also consider the j th slow volume growth s j ϕ = sup lim inf σ Σ j m log m log µ g ϕ m σ and the slow volume growth sϕ = max s jϕ. j dimx It measures the polynomial volume growth of the iterates of the most distorted smooth j-dimensional family of initial data. Note that v j ϕ, vϕ, s j ϕ, sϕ do not depend on the choice of g, and that v dim X ϕ = s dim X ϕ = 0. The aim of this paper is to give uniform lower estimates of localized versions of vϕ and sϕ for certain symplectomorphisms of cotangent bundles. We consider a smooth closed connected d-dimensional Riemannian manifold M, g and the cotangent bundle T M over M endowed with the induced Riemannian metric g and the standard symplectic form ω = d j= dp j dq j. We abbreviate Dr = {q, p T M p r} and D q r = T q M Dr. Let ϕ be a C -smooth symplectomorphism of T M, ω which preserves Dr. If ϕ is C -smooth, says that the maximal orbit complexity of ϕ Dr is already contained in the orbit of a single submanifold of Dr. Usually, lower estimates of the topological entropy do not give any information on the dimension or the location of such a submanifold. An attempt to localize such submanifolds for symplectomorphisms was made in [30], where we considered Lagrangian submanifolds only. In this paper we further localize and consider for ϕ as above the fiberwise volume growth ˆv fibre ϕ; r = sup lim inf q M m and the slow fiberwise volume growth ŝ fibre ϕ; r = sup lim inf q M m m log µ g ϕ m D q r log m log µ g ϕ m D q r. In fact, we shall give uniform lower estimates of the slow volume growth of each fibre by considering the uniform fiberwise volume growth ˇv fibre ϕ; r = inf lim inf q M m m log µ g ϕ m D q r

3 3 and the uniform slow fiberwise volume growth š fibre ϕ; r = inf lim inf q M m Writing vϕ; r = v ϕ Dr and so on, we clearly have 2 3 vϕ; r sϕ; r log m log µ g ϕ m D q r. v d ϕ; r ˆv fibre ϕ; r ˇv fibre ϕ; r, s d ϕ; r ŝ fibre ϕ; r š fibre ϕ; r. We shall obtain lower estimates of ˇv fibre ϕ; r and š fibre ϕ; r in terms of the growth of certain homotopy-type invariants of M. We introduce these concepts of growth right now..2. Energy hyperbolic manifolds. Let M, g be a closed Riemannian manifold. We assume throughout that M is connected. We fix q 0 M and denote by Ω M, q 0 the space of all paths q: [0, ] M of Sobolev class W,2 such that q0 = q = q 0. This space has a canonical Hilbert manifold structure, [44]. The energy functional E = E g : Ω M, q 0 Ê is defined as Eq := qt 2 dt 2 0 where qt 2 = g qt qt, qt. For a > 0 we consider the sublevel sets E a q 0 := { q Ω M, q 0 Eq a }. Let È be the set of non-negative integers which are prime or 0, and set p = p and 0 = É. Throughout, H denotes singular homology. Let ι k : H k E a q 0 ; p H k Ω M, q 0 ; p be the homomorphism induced by the inclusion E a q 0 Ω M, q 0. It is well-known that for each a the homology groups H k E a q 0 ; p vanish for all large enough k, cf. [5], and so the sums in the following definition are finite. Definition. The Riemannian manifold M, g is energy hyperbolic with exponent CM, g if CM, g := sup lim inf p È m m log dim ι k H k E 2 m2 q 0 ; p > 0. k 0 We also set cm := sup lim inf p È m log m log k 0 dim ι k H k E 2 m2 q 0 ; p. Remarks. a The numbers CM, g and cm are less mysterious than they may look at first sight: They just measure the maximal exponential resp. polynomial homological growth of the energy sublevels E 2 m2 q 0 viewed as subsets of the full loop space Ω M, q 0. b Since M is connected, CM, g and cm do not depend on q 0.

4 4 Fiberwise volume growth via Lagrangian intersections c Since M is closed, neither the property energy hyperbolic nor cm depend on g. We say that the closed manifold M is energy hyperbolic if M, g is energy hyperbolic for some and hence any Riemannian metric g on M. d For all λ > 0 we have CM, λ g = λ CM, g. Examples of energy hyperbolic and non energy hyperbolic Riemannian manifolds are given in Sections.4,.5 and Main result. Consider again a closed Riemannian manifold M, g, and let H : [0, ] T M Ê be a C 2 -smooth Hamiltonian function meeting the following assumption: There exists r H > 0 and a function f : [0, Ê with f r H 0 such that 4 Ht, q, p = f p for p r H. The Hamiltonian flow ϕ t H of the time-dependent vector field X H given by ω X Ht, = dh t is defined for all t [0, ]. We abbreviate ϕ H = ϕ H. Theorem. Consider a closed Riemannian manifold M, g, and let H : [0, ] T M Ê be a C 2 -smooth Hamiltonian function satisfying 4. i If M, g is energy hyperbolic, then ii In general, ˇv fibre ϕ H ; r H 2 f r H r H CM, g > 0. š fibre ϕ H ; r H cm. Remarks. a As the identity mapping illustrates, the assumption f r H 0 in 4 is essential. b Theorem extends well-known results from the study of geodesic flows, see [56, Corollary 3.9 and Chapter 5]: These results imply Theorem if there exists an ǫ > 0 such that H = 2 p 2 on Dr H \ Dr H ǫ. In this situation, these results as well as Theorem itself imply i with ˇv fibre ϕ H ; r H replaced by the uniform spherical volume growth ˇv sphere ϕ H ; r H = inf lim inf q M m m log µ g ϕ m H D q r H. c Assume that M is energy hyperbolic, and let GM be the set of Riemannian metrics g on M with VolM, g =. In view of Theorem i one may ask whether inf {CM, g g GM} > 0. This is not so in general: Denote by ϕ g the geodesic flow on the unit sphere bundle D. Then h top g := h top ϕ g = vϕ g ˇv sphere ϕ g ; 2 CM, g > 0. As shown in [58], many energy hyperbolic manifolds M carry a sequence of Riemannian metrics {g n } GM such that h top g n 0 and thus CM, g n 0.

5 In the following three sections we describe lower bounds for CM, g and cm by homotopy-theoretical invariants of M. This will provide more computable lower bounds for fibre growth, will show that most closed manifolds are energy hyperbolic, and will reveal that Theorem refines results of Dinaburg, Gromov, Paternain and Paternain Petean on the topological entropy of geodesic flows. In general, our lower bounds for CM, g and cm will be the sum of two growths, namely the homological growth of a part of the based loop space Ω M of the universal cover of M, and the growth of the fundamental group π M. A direct approach combining both of these growths is given in Section.6. For pedagogical reasons, we first look at the case of finite π M, where the lower bounds come from Ω M alone, and then describe quantitative lower bounds provided by π M alone..4. Finite fundamental group. Consider a closed manifold M with finite fundamental group π M. Let ΩM, q 0 be the space of continuous paths q: [0, ] M with q0 = q = q 0 endowed with the compact open topology. Since M is connected, the homotopy type of ΩM, q 0 does not depend on q 0 and is denoted ΩM. For each m Æ and p È set m r m M; p := dim H k ΩM; p. k=0 The manifold M is called hyperbolic if for some p È the sequence r m M; p grows exponentially in m i.e., there exists C > such that r m M; p C m for all large enough m, and M is called elliptic if for each p È the sequence r m M; p grows at most polynomially in m i.e., for each p È there exists c p < such that r m M; p m cp for all m. We set 5 ρm := sup lim inf p È m log m log r m M; p. If M is elliptic, the supremum over p È is attained and ρm is finite, [50]. It is conjectured that every closed manifold M with finite π M is either elliptic or hyperbolic. This is known if dim M 5, see [57, Section 3]. Moreover, the dichotomy r m M; p grows either exponentially or polynomially is known for p = 0 and p > dim M, and if r m M; p grows faster than polynomially, it grows at least like C m for some C >, see [9]. Examples. a It is easy to see that M is elliptic resp. hyperbolic if and only if its universal cover M is, and that ρm = ρ M, see Lemma 2. a below. We thus assume that M is simply connected. In dimensions 2 and 3, the standard sphere is the only such manifold up to diffeomorphism in view of the proof of the Poincaré conjecture; it is elliptic. In higher dimensions, most simply connected manifolds are hyperbolic. In dimension 4, the simply connected elliptic manifolds up to homeomorphism are S 4, P 2, S 2 S 2, P 2 # P 2, P 2 # P 2, 5

6 6 Fiberwise volume growth via Lagrangian intersections and in dimension 5 the simply connected elliptic manifolds up to diffeomorphism are S 5, S 2 S 3, S 2 S 3, SU3/ SO3, where S 2 S 3 is the nontrivial S 3 -bundle over S 2, see [57]. If M is a sphere or P 2, then ρm = see Section.7 below, and for the six other manifolds above, ρm = 2. For the above 5-manifolds, a proof is given in [57, Section 3], and a variation of their proof also applies to the above 4-manifolds. b Examples with ρm < 2 are rare in view of McCleary s theorem from [49]: If ρm < 2, then ρm =. Equivalently, the reduced integral cohomology ring of the universal cover of M is generated by one element and hence agrees with the cohomology ring of a compact rank one symmetric space CROSS. This implies further restrictions on the topology of M, see Section 3 below. c Fix a prime number p È, and let M p be a simply connected 5-manifold with H 2 M p ; = p p, see [68]. Then the sequence r m M p ; p grows exponentially, while for all p È \ {p} the sequence r m M p ; p grows linearly, see [57, Section 3]. We refer to [8, 9, 20, 45, 56, 57] and the references therein for more information on elliptic and hyperbolic manifolds. Using results of Serre, Gromov and Benci we shall prove Proposition. Consider a closed manifold M with π M finite. i M is hyperbolic if and only if M is energy hyperbolic. ii In general, cm = ρm. Suppose that the conjectured dichotomy M is either elliptic or hyperbolic is true. Proposition then implies that a closed manifold M with finite π M is either energy hyperbolic or elliptic with cm finite. Discussion. Theorem and Proposition show that for a closed Riemannian manifold M, g with finite π M and H as in Theorem, i ˇv fibre ϕ H ; r H > 0 if M is hyperbolic, ii š fibre ϕ H ; r H cm = ρm = in general. ad i: If H is C -smooth, then h top ϕ H ; r H ˇv fibre ϕ H ; r H by Yomdin s Theorem and 2. Together with i we find that the topological entropy h top ϕ H ; r H is positive. This implies and is implied by the positivity of the topological entropy h top g of the geodesic flow of a C -smooth Riemannian metric g on a hyperbolic manifold, a result found by Gromov and Paternain see [56, Corollary 5.2]. ad ii: There exist Riemannian manifolds and Hamiltonians H with š fibre ϕ H ; r = for all large enough r, see Discussion 4 a below. The estimate in ii is thus sharp..5. Infinite fundamental group. Assume now that π M is infinite. Since M is closed, π M is then an infinite finitely presented group. Consider, more generally, an infinite finitely generated group Γ. The growth function γ S associated with a finite set S of generators of Γ is defined as follows: For each positive integer m, let γ S m be the number

7 of distinct group elements which can be written as words with at most m letters from S S. As is easy to see, [53], the limit log γ S m νs := lim [0, m m exists, and the property νs > 0 is independent of the choice of S. In this case, the group Γ is said to have exponential growth. Example 2. If a closed manifold M admits a Riemannian metric of negative sectional curvature, then π M has exponential growth, [53]. For d = 2, the converse to this statement holds true, while for d 3 there are closed d-dimensional manifolds for which π M has exponential growth and which carry no Riemannian metric with negative sectional curvature, see [48, Corollaire III.0] and [6, p. 90]. If Γ does not have exponential growth, Γ is said to have subexponential growth. In this case, the degree of polynomial growth σγ := lim inf m log γ S m log m [0, ] is independent of the choice of S. The group Γ has intermediate growth if σγ =, and polynomial growth if σγ <. While there are finitely generated groups of intermediate growth, [32, 6], it is still unknown whether there are finitely presented groups of intermediate growth; we shall thus not consider a more refined notion of growth for such groups. According to a theorem of Gromov, [34], the group Γ has polynomial growth if and only if Γ has a nilpotent subgroup of finite index. Let k k be its lower central series defined inductively by = and k+ = [, k ]. Then σγ = k k dim k / k+ É, 7 see [74, 36, 4, 69]. We in particular see that σγ is a positive integer. Examples 3. a For the fundamental group of the torus, σ d = d = d. b For the Heisenberg group Γ = 0 0 x 0 x, y, z z y we have = Γ and 2 = {Mx, y, z Γ x = y = 0} = and k = {e} for k 3, so that σγ = = 4. Much more information on growth of finitely generated groups can be found in [6]. Consider now a closed Riemannian manifold M, g such that π M has exponential growth. Given a finite set S of generators let ls, g be the smallest real number such that

8 8 Fiberwise volume growth via Lagrangian intersections for each q M each generator s S of π M = π M, q can be represented by a smooth loop based at q of length no more than ls, g. Then set νm, g = sup νs ls, g where the supremum is taken over all finite sets S generating π M. A simple argument will prove Proposition 2. Consider a closed Riemannian manifold M, g with π M infinite. i If π M has exponential growth, M, g is energy hyperbolic with CM, g νm, g. ii In general, cm σ π M. In small dimensions, the converse to Proposition 2 i holds true: Assume that dim M 3, and if dim M = 3 also assume that the orientation cover of M is geometrizable. Then π M has exponential growth if and only if M is energy hyperbolic. We refer to [2] and [3] for a proof. Discussion 2. Theorem and Proposition 2 show that for a closed Riemannian manifold M, g with infinite π M and H as in Theorem, i ˇv fibre ϕ H ; r H 2 f r H r H νm, g if π M has exponential growth, ii š fibre ϕ H ; r H cm σπ M in general. ad i: Yomdin s Theorem and i yield the positive lower bound h top g 2 νm, g for the topological entropy of the geodesic flow of a C -smooth Riemannian metric g on a manifold whose fundamental group has exponential growth. This is a version of Dinaburg s Theorem, which holds for C 2 -smooth g, see [5] or [56, Theorem 5.8]. ad ii: For the flat torus T d and H = 2 p2 we have š fibre ϕ H ; r = σ d = d for all r > 0, so that ii is sharp..6. Arbitrary fundamental group. In Sections.4 and.5, lower bounds for CM, g and cm have been obtained via the homological growth of ΩM if π M is finite, and via the growth of π M if π M is infinite. If π M is infinite but of subexponential growth, the bound cm σ π M from Proposition 2 ii is often far from optimal. Indeed, recent work of Paternain and Petean, [58], implies that in this situation a better lower bound for cm or even energy hyperbolicity of M can often be established by adding to the growth of π M the homological growth coming from the based loop space of a suitable simply connected complex in M. Thurston s Geometrization Conjecture says that any closed orientable 3-manifold is geometrizable. This conjecture has been proven in many cases, see [0, Section 4], and Perelman s work presumably implies it in general.

9 Given a continuous map f : X, x Y, fx between path-connected pointed spaces, let Ωf: ΩX ΩY be the induced map between based loop spaces and let H Ωf; p : H ΩX; p H ΩY ; p be the map induced in homology. We say that a closed connected manifold M is hyperbolic if there exists a finite simply connected CW complex K and a continuous map f : K M such that for some p È the sequence m r m M, K, f; p := rank H k Ωf; p grows exponentially in m. In general we set 6 ρm := sup K,f sup lim inf p È m k=0 log m log r m M, K, f; p, where the latter supremum is taken over all finite simply connected CW complexes K and continuous maps f : K M. Notice that in the case of finite π M, the term hyperbolic and ρm were defined in Section.4 in a different way. An elementary argument shows that these definitions are equivalent, cf. Lemma 2. a below. Examples 4. Let M be a closed 4- or 5-manifold with π M infinite and of subexponential growth. As shown in [38, 58], most such manifolds are hyperbolic: If dim M = 4, then M is hyperbolic unless M is finitely covered by S 3 S or by a manifold s-cobordant to S 2 T 2 or unless M is aspherical. In particular, T 4 # P 2 is hyperbolic. If dim M = 5, then M is hyperbolic unless its universal cover has the rational homotopy type of a finite simply connected elliptic CW-complex. The papers [38, 58] contain many further restrictions on the topology of non-hyperbolic manifolds with infinite π M. Using an extension of Gromov s theorem from [33] proved in [58] we shall obtain Proposition 3. Consider an arbitrary closed connected manifold M. i If M is hyperbolic, then M is energy hyperbolic. ii In general, cm ρm + σπ M. Discussion 3. Theorem and Proposition 3 show that for a closed Riemannian manifold M, g and for H as in Theorem, i ˇv fibre ϕ H ; r H > 0 if M is hyperbolic, ii š fibre ϕ H ; r H cm ρm + σπ M in general. ad i: Again, Yomdin s Theorem and i imply the positivity of the topological entropy of the geodesic flow of a C -smooth Riemannian metric g on a hyperbolic manifold with π M infinite, a result found by Paternain and Petean in [58]. 9

10 0 Fiberwise volume growth via Lagrangian intersections ad ii: Choose m,...,m l 2 and n, and endow M = S m S m l T n with the usual product Riemannian metric. For H = 2 p 2 we have š fibre ϕ H ; r = l + n = ρm + σπ M for all r > 0, so that ii is sharp. Given a Riemannian manifold M, g we denote by ϕ g its geodesic flow on T M. The numbers sg := sϕ g ; r and š fibre g := š fibre ϕ g ; r do not depend on r > 0. Note that for M = S we have sg = š fibre g = ρm + σπ M =. Question. Consider a closed Riemannian manifold M, g with infinite π M. a Does ρm + σπ M < 2 imply M = S? b Does sg = or š fibre g = imply M = S? A partial answer to Question in the affirmative is given in [3]..7. Volume growth in the component of the Dehn Seidel twist. A P-manifold is a connected Riemannian manifold all of whose geodesics are periodic. Such manifolds are closed, and except for S have finite fundamental group. As we shall see in Section 3, every P-manifold M different from S is elliptic with cm = ρm =. The known P-manifolds are the CROSSes S d, ÊP d, P n, ÀP n, P 2 with their canonical Riemannian structures, their Riemannian quotients which are all known, and so-called Zoll manifolds, which are modelled on spheres. It is an open problem whether there are other P-manifolds. More information on P-manifolds can be found in [8, 30], Section 0.0 of [7] and Section 3 below. Let M, g be a P-manifold. It is known that the unit-speed geodesics of M, g admit a common period, and we shall assume g to be scaled so that the minimal common period is. We choose a smooth function f : [0, [0, such that 7 fr = 2 r2 near 0 and f r = for r, and following [3, 65, 66] we define the left-handed Dehn Seidel twist ϑ f to be the time- -map of the Hamiltonian flow generated by f p. Since M, g is a P-manifold, ϑ f is the identity on T M \ T M, so that ϑ f is a compactly supported symplectomorphism, ϑ f Symp c T M. We shall write ϑ for any map ϑ f with f satisfying 7. The class [ϑ] in the symplectic mapping class group π 0 Symp c T M clearly does not depend on f. Given ϕ Symp c T M we set š fibre ϕ = š fibre ϕ; r where r > 0 is any number such that ϕ is supported in Dr. In Section 2.5 we shall use Theorem ii to prove Corollary. Let M, g be a P-manifold, and let ϑ be a twist on T M. If ϕ Symp c T M is such that [ϕ] = [ϑ m ] π 0 Symp c T M for some m \ {0}, then š fibre ϕ. Discussion 4. a A computation given in [30, Proposition 2.2 i] shows that s ϑ m f = š fibre ϑ m f = for every m \ {0} and every f. Corollary is thus sharp and shows that twists minimize slow volume growths in their symplectic isotopy class.

11 b Assume that ϕ Symp c T M \ {id} is such that [ϕ] = [ϑ 0 ] = [id]. If d 2, this means that ϕ is a non-identical compactly supported Hamiltonian diffeomorphism of T M. If the support of ϕ misses some fibre, then š fibre ϕ = 0. On the other hand, combining a result in [28] with the arguments in [60] one finds that s ϕ, see [29]. It is not hard to construct examples with sϕ = s j ϕ = for all j {,..., 2d }, see [29]. c It was proved in [66, Corollary 4.5] that the class [ϑ] of a twist generates an infinite cyclic subgroup of π 0 Symp c T M. Corollary yields another proof of this. d Corollary was proved in [30] for all currently known P-manifolds. The proof there only used Lagrangian Floer homology and a symmetry argument for P n, g can. We conclude with a question motivated by results in Section 3. For a closed Riemannian manifold M, g with finite π M, 8 sg š fibre g cm = ρm according to 3 and Discussion ii. For a P-manifold the computation in [30] shows that sg =, so that all numbers in 8 are. Conversely, ρm = implies that H M; is the cohomology ring of a CROSS and hence of a P-manifold in view of McCleary s theorem mentioned in Example b. Question 2. Consider a closed Riemannian manifold M, g with finite π M. a Does ρm = imply that M is homotopy equivalent to a P-manifold? b Does sg = or š fibre g = imply that M, g is a P-manifold? Question 2a is partially answered and further discussed in Section 3. Acknowledgements. We wish to thank the referee of our paper [30] for suggesting the present approach to Corollary. Another important stimulus was Gabriel Paternain s beautiful lecture on topological entropy at Max Planck Institute Leipzig in November We warmly thank Gabriel for corrections and many remarks, which considerably improved a first draft of the paper. Most of this paper was written at the turn of the year 2004/2005 at Hokkaido University at Sapporo. We cordially thank Kaoru Ono for valuable discussions and for his generous hospitality, and Theo Bühler, Steve Halperin, Otto van Koert, Leonardo Macarini and Matthias Schwarz for their help. 2. Proofs 2.. Proof of Theorem. Our proof is along the following lines. Using an idea from [30] we first show that fiberwise volume growth is a consequence of the growth of the dimension of certain Floer homology groups. Applying the isotopy invariance of Floer homology and a recent result of Abbondandolo and Schwarz, these homology groups are then seen to be isomorphic to the homology of the space of based loops in M not exceeding a certain energy. Step. Volume growth via Floer homology

12 2 Fiberwise volume growth via Lagrangian intersections Geometric set-up. Consider a closed Riemannian manifold M, g and a Hamiltonian H : [0, ] T M Ê as in Theorem. We abbreviate β = f r H, and we assume without loss of generality that β > 0. Fix ǫ > 0 and let K : [0, ] T M Ê be a C -smooth function such that 9 Kt, q, p = β p 2 if p r H + ǫ. For m =, 2,... we recursively define Kt m = K t + Kt m ϕ t K. Then ϕ m K = ϕ Km. For q 0, q M let Ω T M, q 0, q be the space of all paths x: [0, ] T M of Sobolev class W,2 such that x0 Tq 0 M and x Tq M. This space has a canonical Hilbert manifold structure, [44]. The action functional of classical mechanics A K m : Ω T M, q 0, q Ê associated with K m is defined as A K mx = λ ẋ K m t, x dt, 0 where λ = d j= p j dq j is the canonical -form on T M. This functional is C -smooth, and its critical points are precisely the elements of the space P q 0, q, K m of C -smooth paths x: [0, ] T M solving ẋt = X K m xt, t [0, ], xj Tq j M, j = 0,. Notice that the elements of P q 0, q, K m correspond to the intersection points of ϕ m K T q0 M and Tq M via the evaluation map x x. Fix now q 0 M, and let V q 0, K m be the set of those q M for which ϕ m K D q 0 r H + 2ǫ and D q r H + 2ǫ intersect transversely. Lemma 2.. The set V q 0, K m is open and of full measure in M. Proof. Since D q0 r H + 2ǫ is compact, V q 0, K m is open. Applying Sard s Theorem to the projection ϕ m K D q 0 r H + 2ǫ M one sees that V q 0, K m has full measure in M. Consider the annuli-bundle Aǫ = {q, p T M r H + ǫ p r H + 2ǫ}, set A q0 ǫ = Aǫ T q 0 M, and let Wq 0, K m be the set of those q V q 0, K m for which ϕ m K A q 0 ǫ A q ǫ is empty. For q Wq 0, K m the set P q 0, q, K m, r H + 2ǫ := {x P q 0, q, K m x D r H + 2ǫ} is finite and contained in D r H + ǫ. In view of the Lagrangian foliation given by the fibers of T M, the Maslov index µx of each x P q 0, q, K m, r H + 2ǫ is a well-defined integer, which in case of a geodesic Hamiltonian agrees with the Morse index of the corresponding geodesic path, see [, Section.2]. Lagrangian Floer homology. Floer homology for Lagrangian intersections was invented by Floer in a series of seminal papers, [23, 24, 25, 26]. We shall use a version of Lagrangian Floer homology described in [4, 30], with the only add on that we work over arbitrary coefficient fields p instead of 2 only. In the above situation, we define the k th Floer

13 chain group CF k q 0, q, K m, r H + 2ǫ; p as the finite-dimensional p -vector space freely generated by the elements of P q 0, q, K m, r H + 2ǫ of Maslov index k, and the full Floer chain group as CF q 0, q, K m, r H + 2ǫ; p = k CF k q 0, q, K m, r H + 2ǫ; p. In order to define the Floer boundary operator, we follow [4, 72, 9] and consider the nonempty and connected set J of t-dependent smooth families J = {J t }, t [0, ], of ω-compatible almost complex structures on D r H + 2ǫ such that J t is convex and independent of t on Aǫ. This in particular means that J is invariant under the local flow of the Liouville vector field Y = p j p j on Aǫ. For J J, for smooth maps u from the strip S = Ê [0, ] to D r H + 2ǫ, and for x ± P q 0, q, K m, r H + 2ǫ consider Floer s equation 0 s u + J t u t u X K m t u = 0, us, j T q j M, j = 0,, lim s ± us, t = x± t uniformly in t. Lemma 2.2. Solutions of 0 are contained in Dr H + ǫ. Sketch of proof. Since q Wq 0, K m, lim us, t = s ± x± t Dr H + ǫ. In view of the strong maximum principle, the lemma follows from the convexity of J on Aǫ and from together with the fact that the special form 9 of K on Aǫ implies ω Y, JX K m = 0, cf. [4, 30]. We denote the set of solutions of 0 by M x, x +, K m, r H + 2ǫ;J. Note that the group Ê freely acts on M x, x +, K m, r H + 2ǫ;J by time-shift. Lemma 2.2 is an important ingredient to establish the compactness of the quotients M x, x +, K m, r H + 2ǫ;J/Ê. The other ingredient is that there is no bubbling-off of J-holomorphic spheres or discs. Indeed, [ω] vanishes on π 2 T M because ω = dλ is exact, and [ω] vanishes on π 2 T M, T q j M because λ vanishes on T q j M, j = 0,. See for instance [24] or [62] for more details. There exists a generic subset J reg of J such that for each J J reg the moduli space M x, x +, K m, r H + 2ǫ;J is a smooth manifold of dimension µx µx + for all x ± P q 0, q, K m, r H + 2ǫ, see [27]. Fix J J reg. It is shown in [, Section.4] that the manifolds M x, x +, K m, r H + 2ǫ;J can be oriented in a way which is coherent with gluing, so that with the usual definition of the Floer boundary operators k J : CF k q 0, q, K m, r H + 2ǫ; p CF k q 0, q, K m, r H + 2ǫ; p one finds k J k J = 0 for each k. The proof makes use of the compactness of the 0- and -dimensional components of M x, x +, K m, r H + 2ǫ;J/Ê, see [23, 64, ]. As our notation suggests, the Floer homology groups HF k q 0, q, K m, r H + 2ǫ; p := ker k J/ im k+ J 3

14 4 Fiberwise volume growth via Lagrangian intersections do not depend on the choices involved in their construction: They neither depend on coherent orientations up to canonical isomorphisms, [, Section.7], nor do they depend on J J reg up to natural isomorphisms, as a continuation argument shows, [23, 64]. The same continuation argument also shows that the Floer homology groups do not alter if we add to K m a function supported in [0, ] Dr H + ǫ. The function G m β := mβ p 2 is such a function, so that 2 HF q 0, q, K m, r H + 2ǫ; p = HF q0, q, G m β, r H + 2ǫ; p provided that q also belongs to V q 0, G m β. Remark 2.3. For most applications of Floer homology found so far, it suffices to work over the coefficient field 2. In view of Example c it is important that we can work over arbitrary coefficient fields p. Proposition 2.4. Assume that q Wq 0, G m β. i If M, g is energy hyperbolic, there exists p È such that dim HF q0, q, G m β, r H + 2ǫ; p e 2βr H CM,gm for all large enough m Æ. ii In general, for every δ > 0 there exists p È such that dim HF q0, q, G m β, r H + 2ǫ; p m cm δ for all large enough m Æ. Proposition 2.4 will be proved in the next step. In the remainder of this step we show Proposition 2.4 = Theorem. We show that Proposition 2.4i implies Theorem i. Assume that M, g is energy hyperbolic with exponent CM, g. Let K be as before, and pick q Wq 0, K m Wq 0, G m β. Since the generators of CF q 0, q, K m, r H + 2ǫ; p correspond to ϕ m K D q 0 r H + ǫ D q r H + ǫ, we find together with 2 and Proposition 2.4i that 3 # ϕ m K D q0 r H + ǫ D q r H + ǫ = dim CF q 0, q, K m, r H + 2ǫ; p dim HF q 0, q, K m, r H + 2ǫ; p = dim HF q0, q, G m β, r H + 2ǫ; p e 2βr HCM,gm for some p È and for m large enough. Recall now that β = f r H. We thus find a sequence ǫ i 0 and a sequence K i : [0, ] T M Ê of C -smooth functions such that K i t, q, p = β p 2 if p r H + ǫ i and such that K K i DrH +ǫ i is uniformly bounded in the C 2 -topology, K2 K i DrH H DrH in the C 2 -topology.

15 Note that π: T M M is a Riemannian submersion with respect to the Riemannian metrics g and g. Applying 3 to K i we therefore find 4 µ g ϕ m Ki D q0 r H + ǫ i e 2βrHCM,gm µ g Wq0, Ki m Wq 0, G m β. Since ǫ i 0, we have lim µ g i ϕ m Ki Aǫ i = lim µ g ϕ m Gβ Aǫ i = 0, i so that, together with Lemma 2., 5 lim i µ g Wq0, K m i Wq 0, G m β = lim i µ g V q0, K m i V q 0, G m β = µ g M. Moreover, ǫ i 0 and K imply 6 lim µ g ϕ m K Dq0 i i r H + ǫ i \ D q0 r H = 0, and K2 implies 7 lim µ g ϕ m Ki D q0 r H = µ g ϕ m H D q0 r H. i Using 7, 6, 4 and 5 we find that µ g ϕ m H D q0 r H e 2βr HCM,gm µ g M for m large enough. Since q 0 M was arbitrary, this yields ˇv fibre ϕ H ; r H 2βr H CM, g, as claimed. In a similar way, Proposition 2.4ii implies that in general š fibre ϕ H ; r H cm δ for every δ > 0 and hence š fibre ϕ H ; r H cm. The estimate cm is guaranteed by Propositions and 2. 5 Step 2. From Floer homology to the homology of the path space In this step we prove Proposition 2.4. Viterbo [7, 72] was the first to notice that the Floer homology for periodic orbits of T M is isomorphic to the singular homology of the free loop space of M. Different proofs were found by Salamon Weber [63] and Abbondandolo Schwarz []. The work [] also establishes a relative version of this result: The Floer homology for Lagrangian intersections of T M is isomorphic to the singular homology of the based loop space of M. It is this version that we shall take advantage of. We abbreviate ρ = r H +2ǫ. In order to estimate the dimension of HF q0, q, G m β, ρ; p from below, we first describe these p -vector spaces in a somewhat different way. Set { P mβρ2 q0, q, G m β := x P } q 0, q, G m β AG m x mβρ2. β Lemma 2.5. P mβρ2 q 0, q, G m β = Pq 0, q, G m β, ρ.

16 6 Fiberwise volume growth via Lagrangian intersections Proof. Assume that x P q 0, q, G m β. For t0 [0, ] we choose geodesic normal coordinates q near πxt 0 in M. With respect to these coordinates the equation ẋ = X G m β x at t 0 reads ṗt 0 = 0, qt 0 = 2mβpt 0. Therefore, λ ẋt 0 G m β xt 0 = 2mβ p 2 mβ p 2 = mβ p 2 = G m β xt 0. Since G m β is autonomous, G m β xt 0 does not depend on t 0. Integrating over [0, ] we thus obtain A G m β x = G m β x. The lemma follows. We define the k th Floer chain group CF mβρ2 k q0, q, G m β ; p as the finite-dimensional p -vector space freely generated by the elements of P mβρ2 q0, q, G m β of Maslov index k. Lemma 2.5 yields Lemma 2.6. CF mβρ2 q0, q, G m β ; p = CF q0, q, G m β, ρ; p. Denote by Ĵ the set of families Ĵ = { } Ĵ t of almost complex structures on T M such that Ĵ Dρ J and Ĵ is invariant under the flow of Y on T M \ Dr H + ǫ. For Ĵ Ĵ, for smooth maps u: S T M, and for x ± P mβρ2 q0, q, G m β consider Floer s equation s u + Ĵtu t u X G m β u = 0, 8 us, j Tq j M, j = 0,, lim us, t = s ± x± t uniformly in t. We denote the set of solutions of 8 by M x mβρ2, x +, G m β ;Ĵ. Lemmata 2.5 and 2.2 imply Lemma 2.7. M x mβρ2, x +, G m β ;Ĵ = M x, x +, G m β, ρ;j. A standard argument shows that A G m β x A G m β x + for each u M x mβρ2, x +, G m β ;Ĵ. For generic Ĵ boundary operators Ĵ the usual definition of the Floer boundary operator therefore yields k Ĵ : CF mβρ 2 k q0, q, G m β ; p CF mβρ 2 k q0, q, G m β ; p. As before, their homology groups HF mβρ2 k q0, q, G m β ; p neither depend on coherent orientations of M x mβρ2, x +, G m β ;Ĵ nor on Ĵ. Lemmata 2.6 and 2.7 imply that kĵ = k J, whence Proposition 2.8. HF q0, q, G m β, ρ; p = HF mβρ 2 q0, q, G m β ; p.

17 For q 0, q M let Ω M, q 0, q be the space of all paths q: [0, ] M of Sobolev class W,2 such that q0 = q 0 and q = q. Again, this space has a canonical Hilbert manifold structure. The energy functional E : Ω M, q 0, q Ê is defined as Eq = 2 0 qt 2 dt. For a > 0 we consider the sublevel sets E a q 0, q := {q Ω M, q 0, q Eq a}. Proposition 2.9. HF mβρ2 q0, q, G m β ; p = H E 2βρm2 q 0, q ; p. Proof. Let L: TM Ê be the Legendre transform of G m β. Applying Theorem 3. of [] to G m β and L, we obtain { q0, q, G m β ; } p = H q Ω M, q 0, q L qt, qtdt mβρ 2 ; p. HF mβρ2 Notice now that Lq, v = 4mβ v 2. The set {... } on the right hand side therefore equals E 2βρm2 q 0, q, and so Proposition 2.9 follows. 0 7 Remark. Strictly speaking, Abbondandolo and Schwarz work in [] with almost complex structures which are close to the almost complex structure interchanging the horizontal and vertical tangent bundles of T M, g. They need to work with such almost complex structures in order to prove a subtle L -estimate for solutions of Floer s equation below a given action which is crucial for obtaining their Theorem 3. for a large class of Hamiltonians. For the special Hamiltonians G m β appearing in our situation, one can work with convex almost complex structures Ĵ Ĵ and does not need their L -estimate, since boundedness of solutions of Floer s equation below a given action then follows from Lemma 2.7. Propositions 2.8 and 2.9 yield Proposition 2.0. HF q0, q, G m β, ρ; p = H E 2βρm 2 q 0, q ; p. The dimension of this vector space is dim H k E 2 2βρm2 q 0, q ; p k 0 k 0 dim ι k H k E 2 2βρm2 q 0, q ; p, and the growth in m of the sequence on the right does not depend on q 0 and q. Since ρ = r H + 2ǫ > r H, Proposition 2.4i now follows in view of the definition of CM, g, and Proposition 2.4ii follows in view of the definition of cm Proof of Proposition. Consider a closed connected Riemannian manifold M, g with finite π M. We first verify that we can suppose M to be simply connected. Lemma 2.. Let M, g be the universal cover of M, g.

18 8 Fiberwise volume growth via Lagrangian intersections a The manifold M is elliptic resp. hyperbolic if and only if M is elliptic resp. hyperbolic, and ρm = ρ M. b The Riemannian manifold M, g is energy hyperbolic if and only if M, g is energy hyperbolic, in which case CM, g = C M, g. In general, cm = c M. Proof. We only give a proof of CM, g = C M, g, since it essentially contains a proof of the other assertions. Choose a point q 0 M over q 0 M. The map Ω π: Ω M, q 0 Ω M, q 0 induced by the projection π: M M maps Ω M, q 0 homeomorphically onto the component C of the constant loop γ q 0 ; since g is the lift of g, it in fact maps each E a q 0 homeomorphically onto E a q 0 C. Therefore, C M, g CM, g. In order to prove the reverse inequality, we choose in each other component C i of Ω M, q 0 a loop γ i, i = 2,...,l = #π M. Fix ǫ 0,. For γ, γ Ω M, q 0 define γ ǫ γ Ω M, q 0 by γ ǫ γ t = { γ t ǫ, 0 t ǫ, γ t ǫ ǫ, ǫ t. It is easy to see that the map C C i, γ γ i ǫ γ, is a homotopy equivalence with homotopy inverse Γ i : C i C, γ γ i ǫ γ. Notice that E γ ǫ γ = ǫ Eγ + ǫ Eγ for all γ, γ Ω M, q 0. Abbreviating Ei a q 0 = E a q 0 C i and E = max { Eγ i i = 2,..., l }, we therefore have Γ i E a i q 0 E a ǫ + E ǫ q 0. Since Γ i : C i C is a homotopy equivalence, it follows that Therefore, dim ι k H k Ei a q 0; p = dim ι k H k Γ i Ei a q 0; p dim ι k H k E a ǫ + E ǫ q 0 ; p dim ι k H k E a q 0 ; p = = dim ι k H k E a ǫ + E ǫ q0 ; p. l dim ι k H k Ei a q 0 ; p i= l dim ι k H k E a ǫ + E ǫ q0 ; p,

19 9 and so = lim inf m lim inf m m log dim ι k H k E 2 m2 q 0 ; p k 0 m log l dim ι k H k E m 2+ ǫ E 2 ǫ q 0 ; p k 0 ǫ lim inf m m log dim ι k H k E 2 m2 q 0 ; p. k 0 Since this is true for all p È, we obtain CM, g ǫ C M, g, and since ǫ 0, was arbitrary, we conclude that CM, g C M, g. In view of Lemma 2. we can assume that M, g is simply connected throughout the proof of Proposition. We next prove the implication M hyperbolic M, g energy hyperbolic and the inequalities cm ρm. The length functional L: Ω M, q 0 Ê is defined as Lq = 0 qt dt. For a > 0 we consider the sublevel sets L a q 0 := {q Ω M, q 0 Lq a}. Lemma 2.2. There exists a constant C G > 0 depending only on M, g such that each element of H k Ω M, q 0 ; p can be represented by a cycle in E 2 C G k 2 q 0. In particular, m dim ι k H k E 2 C G m 2 q 0 ; p dim H k Ω M, q 0 ; p k 0 k=0 for all m. Proof. According to a result of Gromov, there exists a constant C G > 0 depending only on M, g such that each element of H k Ω M, q 0 ; p can be represented by a cycle lying in L C G 2k q 0. Gromov s original proof of this result in [33] is very short. Detailed proofs can be found in [55] and [35, Chapter 7A]. Let k be the k-dimensional standard simplex, and let ψ = i n iψ i : k L C G 2k q 0 be an integral cycle. For notational convenience we pretend that ψ consists of only one simplex. By suitably reparametrizing each path ψs near t = 0 and t = and then smoothing each path with the same heat kernel we obtain a homotopic and hence homologous cycle ψ : k L C G k q 0 consisting of smooth paths. We identify ψ with the map k [0, ] M, s, t ψ s, t := ψ st. Endow the manifold M [0, ] with the product Riemannian metric, and set q 0 = q 0, 0. We lift ψ to the cycle ψ : k Ω M [0, ], q 0 defined by ψ s, t = ψ s, t, t. This cycle consists of smooth paths whose tangent vectors do not vanish. For each s let

20 20 Fiberwise volume growth via Lagrangian intersections ψ σs be the reparametrization of ψ s proportional to arc length. The homotopy Ψ: [0, ] k Ω M [0, ], q 0 defined by Ψτ, s t = ψ s, τt + τσs shows that ψ is homologous to the cycle ψ 2 s := Ψ, s. Its projection ψ 2 to Ω M, q 0 is homologous to ψ and lies in L CG k q 0. Since for each s the path ψ 2 s is parametrized proportional to arc length, we conclude that E ψ 2 s E ψ2 s = L ψ2 s 2 2 { L ψ2 s } 2 + C 2 G 2 k 2 + C 2 2 G k 2 = 2 for each s, so that indeed ψ 2 E 2 C G k 2 q 0. Recall that ΩM, q 0 is the space of continuous paths q: [0, ] M with q0 = q = q 0 endowed with the compact open topology. According to [5, Chapter 7] or [43, Theorem.2.0], the inclusion Ω M, q 0 ΩM, q 0 is a homotopy equivalence. The homotopy type of these spaces does not depend on q 0 and is denoted ΩM. Together with Lemma 2.2 we find m 9 dim ι k H k E 2 C G m 2 q 0 ; p dim H k ΩM; p = r m M; p k 0 for all m Æ and p È. It follows that M, g is energy hyperbolic if M is hyperbolic and that cm ρm. The estimate ρm follows from Proposition on page 483 of Serre s seminal work [67]: Applied to our closed simply connected manifold M of dimension d, it guarantees that for every integer i 0 there exists an integer j {,..., d } such that H i+j ΩM; p 0. We finally prove the implication M, g energy hyperbolic M hyperbolic and the inequality cm ρm. Choose convm, g > 0 so small that for each q M the ball of radius convm, g centered at q is geodesically convex. According to the proof of Theorem 3.4 in [5], H k E a q 0 ; p = 0 k=0 if k > d a convm, g + for all a > 0, q 0 M and p È. Here, r = max {n n r}. We therefore find a constant D such that Dm dim ι k H k E 2 m2 q 0 ; p = dim ι k H k E 2 m2 q 0 ; p k 0 for all m, from which the two claims follow. k=0 Dm k=0 dim H k ΩM; p

21 2.3. Proof of Proposition 2. Consider a closed connected Riemannian manifold M, g with infinite π M. The starting point of the proof is the crude estimate dim ι k H k E a q 0 ; p dim ι0 H 0 E a q 0 ; p. k 0 Denote by Π a E q 0 resp. Π a L q 0 the set of those homotopy classes of W,2 -paths q: [0, ] M with q0 = q = q 0 which can be represented by a path of energy resp. length at most a. The vector space ι 0 H 0 E a q 0 ; p is freely generated by the elements of Π a E q 0, i.e., dim ι 0 H 0 E a q 0 ; p = #Π a E q 0. Moreover, #Π 2 m2 E q 0 = #Π m L q 0 for all m. Let now S = {h,...,h #S } be a generating set of π M. In view of the definition of ls, g in Section.5 we can represent each h j by a smooth loop based at q 0 of length no more than ls, g. In view of the triangle inequality and the definition of the growth function γ S we obtain #Π m L q 0 γ S m/ls, g, where again r = max {n n r}. Summarizing, we have dim ι k H k E 2 m2 q 0 ; p γs m/ls, g k 0 for all m Æ and all p È, and so Proposition 2 follows in view of the definitions of νm, g and σ π M Proof of Proposition 3. Consider a closed connected Riemannian manifold M, g. Let K be a finite simply connected CW complex and f : K M a continuous map, and fix p È. According to the proof of Lemma 2. in [58], there exists a natural number C G depending only on K, f, M, g such that given any homology class [ψ] H k ΩK; p, the class Ωf k [ψ] = [f ψ] H k ΩM; p can be represented by a k-cycle in L CG 2k q 0 Ω M, q 0. As in the proof of Lemma 2.2 we then see that [f ψ] can be represented by a k-cycle ψ E 2 C G k 2 q 0. For h π M we denote by C h the component of Ω M, q 0 containing h, and abbreviate Eh aq 0 = E a q 0 C h. Since K is simply connected, ψ E 2 C G k 2 q 0. We conclude that m 20 r m M, K, f; p = rank H k Ωf; p = k=0 m dim H k Ωf; p H k ΩK; p k=0 m k=0 dim ι k H k E 2 C G m 2 q 0 ; p

22 22 Fiberwise volume growth via Lagrangian intersections for all m Æ. For h π M let l h be the length of the shortest curve in Ω M, q 0 representing h, and let e h = 2 l2 h be its energy. Arguing as in the proof of Lemma 2. with ǫ = /2 we find that dim ι k H k E a q 0; p dim ι k H k E 2a+2e h h q 0 ; p for all a > 0. Using this estimate and the notation from the proof of Proposition 2 we can estimate dim ι k H k E 4a q 0 ; p = dim ι k H k E 4a h q 0; p Together with #Π 2 C G m 2 E that h π M h π M h Π a E q 0 dim ι k H k E 2a e h q 0 ; p dim ι k H k E a q 0 ; p. q 0 = #Π C G m L q 0 γ S C G m/ls, g and 20 we conclude dim ι k H k E 2 C G m 2 q 0 ; p dim ι k H k E 2CG m2 q 0 ; p γs C G m/ls, g k 0 k 0 γ S C G m/ls, g r m M, K, f; p, and so i and the first inequality in ii of Proposition 3 follow in view of definitions. The second inequality in ii has already been obtained in Propositions and Proof of Corollary. For M = S the claim follows from an elementary topological argument, see [30]. We can therefore assume that M, g is a P-manifold of dimension d 2. Let Ham c T M be the group of diffeomorphisms of T M generated by compactly supported Hamiltonians H : [0, ] T M Ê, and let Symp c 0 T M be the group of diffeomorphisms of T M which are isotopic to the identity through a family of symplectomorphisms supported in a compact subset of T M. Since d 2, Ham c T M = Symp c 0 T M see [30, Lemma 2.8]. Consider now a symplectomorphism ψ Symp c T M such that [ψ] = [ϑ m ] π 0 Symp c T M for a twist ϑ = ϑ f on T M and some m \ {0}. Then ψϑ m Symp c 0 T M = Ham c T M, so that we find a compactly supported Hamiltonian function H : [0, ] T M Ê with ϕ H = ϑ m ψ. Then ψ = ϑ m ϕ H = ϕ mf ϕ H = ϕ K, where Kt, q, p = mf p + H t, ϑ m f q, p. Choose r k so large that H is supported in [0, ] { p r K }. Since ϑ m f preserves the levels { p = const}, we then have Kt, q, p = mf p for p r k. Since f r K = and m 0, Theorem ii applies and yields š fibre ϕ K ; r K, as claimed.

23 3. More on P-manifolds Much information on P-manifolds can be found in the book [8] and in Section 0.0 of [7]. In this section we give a few additional results on the topology of P-manifolds. It is easy to see that P-manifolds of dimension at least 2 have finite fundamental group, [8, 7.38]. Proposition 3.. Let M, g be a P-manifold of dimension at least 2. Then M is elliptic, and cm = ρm =. Proof. Recall from Discussion 4a in Section.7 that an elementary computation shows š fibre ϑ f = for every twist on T M. The proposition thus follows from Discussion ii. The above proof of the inequality ρm relies on Floer homology and Gromov s theorem invoked earlier. We now prove this inequality by elementary means. A version of the argument proves cm, cf. [66, p. 29]. An elementary proof of ρm. Recall that we scaled g such that all unit-speed geodesics have minimal period. For such a geodesic γ : Ê M and t > 0 we let ind γt be the number of linearly independent Jacobi fields along γs, s [0, t], which vanish at γ0 and γt. If ind γt > 0, then γt is said to be conjugate to γ0 along γ. The index of γ [0,a] defined as ind γ [0,a] = ind γt t ]0,a[ is a finite number, and according to [8,.98 and 7.25] the number k = ind γ [0,] is the same for all unit-speed geodesics γ on M, g. Fix now q 0 M and choose q which is not conjugate to q 0. Then there are only finitely many geodesic segments γ,...,γ n from q 0 to q of length smaller than, see [8, 7.4]. Let k,...,k n be their indices. The energy functional on Ω M; q 0, q is Morse with indices k i +ld +k, where i n and l = 0,, 2,.... Since d + k, we conclude that dim H j Ω M; q 0, q ; p n for all j Æ and p È. Recall now that the inclusion Ω M; q 0, q ΩM; q 0, q is a homotopy equivalence. Therefore, dim H j ΩM; p n for all j and p, and so ρm, as as claimed. The subsequent results on the topology of P-manifolds will rely only on the possibly weaker assumption ρm =. We shall therefore assume throughout that M is a closed connected manifold with finite π M and ρm =. We first consider the case that M is simply connected. All such known P-manifolds are diffeomorphic to a CROSS S d, P n, ÀP n or P 2 with its standard smooth structure. Proposition 3.2. Mc Cleary [49] Assume that M is simply connected. Then ρm = if and only if M has the integral cohomology ring of a CROSS. Corollary 3.3. A simply connected manifold with ρm = is either homeomorphic to S d or homotopy equivalent to P n or has the integral cohomology ring of ÀP n or P 2. 23

24 24 Fiberwise volume growth via Lagrangian intersections Proof. If H M; = H S d ;, then M is a simply connected homotopy sphere by Hurewicz s Theorem, and hence homeomorphic to S d by the proof of the Poincaré conjecture. If H M; = H P n ;, then M is homotopy equivalent to P n by a theorem of Klingenberg [42]. In small dimensions, we obtain Corollary 3.4. Assume that M is a simply connected d-manifold with ρm =. If d = 3, then M is diffeomorphic to S 3. If d = 4, then M is homeomorphic to S 4 or P 2. If d = 5, then M is diffeomorphic to S 5. If d = 6, then M is diffeomorphic to S 6 or homotopy equivalent to P 3. If d = 7, then M is homeomorphic to S 7. Remarks 3.5. a According to [73], the set of closed smooth 6-manifolds homotopy equivalent to P 3 splits into infinitely many diffeomorphism types M k, k ; they are determined by their first Ponrjagin class p M k = 48k + 4a 2 where a is a generator of H 2 M k ;. It would be interesting to see whether P 3 = M 0 is the only diffeomorphism type carrying a P-metric. b Corollaries 3.3 and 3.4 partially answer Question 2a in the simply connected case. There are, however, closed connected 8- and 6-manifolds M with the integral cohomology ring of ÀP 2 and P 2 and hence with ρm = which are not homotopy equivalent to ÀP 2 and P 2, see [6, 7], and it is unknown whether these homotopy types can be realized by P-manifolds. In order to study manifolds with ρm = and non-trivial π M, we shall need the following well-known result, whose proof is given for the reader s convenience. Proposition 3.6. Assume that X is a retract of a finite simplicial complex whose integral cohomology ring is the one of P 2n with n or of ÀP n with n 2 or of P 2. Then each continuous self mapping of X has a fixed point. Proof. We closely follow [37]. The cohomology ring H X; = α / α n+ is generated by an element α H k X;, where k {2, 4, 8}. If H X; is the cohomology ring of P 2n with n or of ÀP n with n even or of P 2, the claim thus follows at once from the Lefschetz fixed point theorem, see for instance [37, Theorem 2.C.3] Assume now that X has the integral cohomology ring of ÀP n, n 3. The image of α in H 4 X; 3, which is again denoted by α, generates H X; 3. Let f be a continuous self mapping of X, and define its degree d f = d 3 f 3 by f α = d f α. The following lemma shows that d f = 0 or d f =. If d f = 0, the Lefschetz number of f : H X; H X; is n j=0 3lj for some l, and if d f =, it is n j=0 + 3lj for some l. Since both numbers are not 0, the proposition again follows from the Lefschetz fixed point theorem. Lemma 3.7. The degree d f satisfies d f = d 2 f in 3.