ON THE HOFER GEOMETRY INJECTIVITY RADIUS CONJECTURE 6 1 YASHASAVELYEV 0 2 Abstract. We verifyheresomevariants oftopological anddynamical flavor b oftheinjectivityradiusconjectureinHofergeometry,Lalonde-Savelyev [4]in e the caseofHam(S2,ω)andHam(Σ,ω), forΣaclosedpositivegenus surface. F In particular we show that any loop in Ham(S2,ω), respectively Ham(Σ,ω) 7 withL+Hoferlengthlessthanarea(S2)/2,respectivelyanyL+ lengthiscon- tractible through (L+) Hofer shorter loops, in the C∞ topology. We also ] prove some stronger variants of this statement on the loop space level. One G dynamical type corollary is that there are no smooth, positive Morse index (Ustilovsky) geodesics, in Ham(S2,ω), respectively in Ham(Σ,ω) with L+ S Hofer length lessthan area(S2)/2, respectively anylength. Theabove condi- . h tiononthegeodesicscanbeexpandedasanexplicitandelementarydynamical t conditionontheassociatedHamiltonianflow. Wealsogivesomespeculations a onconnectionsofthislaterresultwithcurvaturepropertiesoftheHamiltonian m diffeomorphismgroupofsurfaces. [ 3 v 0 1. Introduction 4 7 One of the most fundamental objects associated to a Finsler manifold is its 2 injectivity radius function. The group Ham(M,ω), with its bi-invariant Finsler 0 Hofer metric, has no well defined exponential map, so that if we ask about its . 1 injectivity radius we need to decide how to interpret this. One simple way to 0 interpretistoaskforthesizeofthelargestmetricepsilon-ballwhichiscontractible, 5 or has null-homotopic inclusion map into the total space Ham(M,ω). 1 In this formulation we studied the question of injectivity radius for Ham(M,ω) : v in Lalonde-Savelyev [4], and formulated there the conjecture that the injectivity i X and (or) weak injectivity radius of Ham(M,ω) is positive. In this note we shall verify some variants of this conjecture in the case of sur- r a faces. The main new ingredient is a kind of “curvature” 1 flow for connection type symplectic forms on surface Hamiltonian fibrations over S2, which exists in the presence of certain foliations by holomorphic curves, using which we also obtain a curve shortening algorithm. It is interesting to speculate whether this is at all related to symplectic Ricci type flows, but to emphasize, our “flow” is of very elementarynature,atleastassumingthestate-of-the-artinGromov-Wittentheory. Using these developments we obtain some dynamical applications, which are formulatedin terms of certain non-existence results of Ustilovsky geodesics. These mayalsobeinterpretedasinjectivityradiusstatementsbutfromaFinslergeometric ratherthanpurelymetricpointofview. Thisalsousestheauthor’s(virtual)Morse Keywords andphrases. Hofermetric,Floertheory,Gromov-Wittentheory,curveshortening. 1Thisisaslightmisnomerascurvaturemaynotbemonotonicallydecreasingunderflow. But probablythebestwaytothinkaboutit. 1 2 YASHASAVELYEV theory for the Hofer length functional. We follow this with some speculations on the “curvature” of the Hamiltonian group of surfaces. 1.1. Statements. GivenasmoothfunctionH :M2n×(S1 =R/Z)→R,(timeisin S1forconvenienceandlateruse)thereisanassociatedtime-dependentHamiltonian vector field X , 0≤t≤1, defined by t (1.1) ω(X ,·)=−dH (·). t t The vector field X generates a path γ : [0,1] → Diff(M), starting at id. Given t sucha path γ, its end point γ(1) is calleda Hamiltonian symplectomorphism. The space of Hamiltonian symplectomorphisms forms a group, denoted by Ham(M,ω). In particular the path γ above lies in Ham(M,ω). Given a general smooth path γ, the positive Hofer length, L+(γ) is defined by 1 L+(γ):= maxHγdt, t Z0 M where Hγ is a generating function for the path t7→γ(0)−1γ(t), 0≤t≤1, normalized to have zero mean at each moment. Part of the reason for interest in the above functional as well as in the original Hofer length functional, which is obtained by integration of the L∞ norm of the generatingfunction, is due to existence of some deep connections of this with both the theory of dynamical systems and with Floer-Gromov-Wittentheory. Notation 1.1. Let ΩcHam(M,ω):={γ ∈ΩHam(M,ω)|L+(γ) 1, while Ham(S2,ω) g is a rough CAT(k) space for some k >0, see [1], with respect to the positive Hofer length functional. 2. Proof of Theorem 1.5 We firstverifythe caseofM =S2. Thefollowingargumentworksthesameway for any 00, for v 6=0 . For every l ∈ Ω~,contHam(S2) we get a Hamiltonian S2 fibration X over CP1, by l using l as a clutching map: X =S2×D2 ⊔S2×D2/∼. l − + π Recall that a coupling form, see [2] for a Hamiltonian fibration M →P ֒→X is a closed 2-form α on the total space, which restricts to the symplectic form on the fibers, modeled by (M,ω), and which satisfies: e π α=0, ∗ whereπ istheintegrationoverthefibermap. By[2]suchformsαalwaysexistfora ∗ Hamiltonianfibration,andareuniquelyedeterminedby the associatedHamiltonian connections, defined by declaring the horizontal subspaces to beethe α-orthogonal e ON THE INJECTIVITY RADIUS CONJECTURE 5 subspacestotheverticaltangentspaces. WehavethecouplingformΩ onS2×D2 l,− − defined by Ω =ω−d(η(r)·Hldθ), e l,− where 0 ≤ r ≤ 1,0 ≤ θ ≤ 1, (which are our modified polar coordinates) Hl is the normalized generating functeion for l, and η : [0,1] → [0,1] is a smooth function satisfying: 0≤η′(r), and r2 if 0≤r ≤1−2κ (2.3) η(r)= (1 if 1−κ≤r ≤1, forasmallκ>0. Underthegluingrelation∼,Ω correspondstotheformω near l,− the boundary of S2 ×D2, so we may extend trivially over D2 to get a coupling + + form Ω on X . The coupling form Ω determinees a Hamiltonian connection A as l l l l described above. This in turn determines an almost complex structure J , by first l fixingea family {j(z)}, z ∈CP1 smoeoth in z: j(z) is an almost complex structure l l on the fiber π−1(z), where l π :X →CP1 l l is the projection,witheachj(z)compatible with the symplectic formonthe fiber, l and then defining J to coincide with {j (z)} on the vertical tangent bundle, to l l preservethehorizontaldistributionofA andtohaveaholomorphicprojectionmap l to (CP1,j), where j is the almost complex structure which preserves the standard orientation on CP1. As each l is contractible and so X trivializable, we may l consider the moduli spaces M(J ), l consisting of stable J holomorphic sections σ of X , in the class of the constant l l section [const], and in particular satisfying: h[Ω ],[σ]i=0. l By a stable holomorphic section we mean a stable J -holomorphic map σ into X , l l e in the classical sense, [6] with domain an umarked nodal Riemann sphere, one of those components is called principal. The restriction σ of σ, to the principal princ component is a J holomorphic section, i.e. we have a commutative diagram: l X ②②<< l σpr②in②②c②② πl (cid:15)(cid:15) CP1 id // CP. All the other components of σ are vertical, that is they are J -holomorphic maps l into the fibers of X . l Lemma 2.1. The moduli space M(J ) has no nodal curves as elements. l Proof. Suppose otherwise, then there is a stable J -holomorphic section σ of X , l l with total homology class [const], and consequently having a principal component σ which is a smooth J -holomorphic section of X with: princ l l h[Ω ],[σ ]i≤−~ l princ e 6 YASHASAVELYEV as~istheminimalenergyofanon-constantholomorphicsphereinS2,i.e. area(S2,ω). However in this case the classical energy inequality for holomorphic curves gives: ~≤area+(Ω ), l where area+ is the functional on the space of coupling forms e area+(Ω)=inf α Ω+π∗(α) is symplectic , α (cid:26)ZCP1 (cid:12) (cid:27) (cid:12) for α a 2-form on CP1e, with positive in(cid:12)teegral. On the other hand by direct calcu- (cid:12) lation we have area+(Ω )=L+(l)<~. l This gives a contradiction. (cid:3) e By automatic transversality see [6, Appendix C] M(J ) is regular i.e. the as- l sociated real linear Cauchy-Riemann operator is transverse for all σ ∈ M(J ). As l these are embedded we may use positivity of intersections, see [6, Section 2.6], to infer that M(J ) determines a smooth foliation of X by [const] class holomorphic l l sections. In particular ev :M(J )→S2, l obtainedbyevaluatingasectionat0∈CP1,isadiffeomorphism,anddeterminesa canonical smooth trivialization of X . Let Θ denote the corresponding horizontal l l distribution. Consequently for an appropriately smooth family: f :Sk →Ω~,contHam(S2,ω), we get natural (up to choices of almost complex structures) smooth trivialization of the bundle P →Sk, f with fiber over s ∈ Sk being X ≡ X . This is a trivialization of a bundle s f(s) with structure group Ω2Diff(S2). What is important for what follows is that this structure group is a subgroup of the group of smooth bundle maps of S2×CP1 → CP1. Let tr :P →(S2×CP1)×Sk, f denote this trivialization. Set {F = (tr )∗ω}, where tr denotes the restriction s s s of tr to the fiber X . By the above tr is a smooth bundle map and hence each s s F is a connection type closed form, which just means the restriction to each fiber s of π : X → CP1, is symplectic. This in turn means that the smooth bundle s s connection determined by declaring horizontal spaces to be F -orthogonal spaces s to the vertical tangent spaces, is Hamiltonian (with respect to the family of fiber symplectic forms determined by the restrictions.) Also by construction each F vanishes on the horizontal distribution Θ , and s f(s) so τ =F +ρπ∗τ, s s s is a symplectic form on X for any small ρ > 0, and where τ a fixed, area one s symplectic form on CP1. Likewise α =Ω +area+(Ω )π∗τ +ρπ∗τ, s f(s) f(s) s s e e ON THE INJECTIVITY RADIUS CONJECTURE 7 is symplectic, and α is positive on the horizontal distribution Θ . s f(s) Let α , t∈[0,1] denote the convex linear combination s,t α =(t)τ +(1−t)α . s,t s s Note that α is nondegenerate and hence symplectic for every s,t, as this is a s,t convex linear combination of symplectic forms tamed by the same almost complex structure J(A ), where A is the connection determined by Ω as before, and s s f(s) so α is also tamed by this complex structure. Moreover for a fixed fiber S2, s,t z αs,t is a convex linear combination of cohomologous symplecetic forms Fs|S2 and z Ωf(s)|Sz2. By construction (S02,Ωf(s)|S02) is naturally symplectomorphic to (S2,ω), consequently applying Moser’s argument we get that αs,t|S02 is naturally symplec- teomorphic to (S2,ω), for each te. We shall use this further on in the “radialtrivial- ization”construction. Notation 2.2. Denote by α the coupling form determined by α . s,t s,t Lemma2.3. area+(α )isboundedfromabovebyafunctionb(s,t)non-increasing s,t with t, with b(s,1)=ρ, andewith b(s,0)=L+(f(s))+ρ. e Proof. Define (2.4) area(α):=Vol(X ,α)/Vol(S2,ω). s Note that area+(α )≤area(α ), s,t s,t which follows by the fact that e α =α +π∗τ′, s,t s,t s for an area form τ′ by the construction. Set b(s,t) = area(α ), then by direct s,t calculation e b(s,0)=L+(f(s))+ρ, and b(s,1)=ρ. (cid:3) Each α determines a loop f in Ham(S2,ω), first by using the natural iden- s,t s,t tification of(S02,αs,t|S02) with (S2,ω)as describedabove,then identifying the fiber over 0∈eCP1 with the fiber over ∞∈CP1 by αs,t-parallel transport map over the θ =0 ray from 0eto ∞ (0=0∈D2, ∞=0∈D2 in our coordinates from before), − + andthenforeveryotherθ rayfrom0to∞getteinganelementfs,t(θ)∈Ham(S2,ω) by α parallel transport over this ray. Clearly f = f(s), and f = id and for s,t s,0 s,1 eacht,f(s,t)isaloopofHamiltoniansymplectomorphismsof(S2,ω)(bytheiden- tificeation above.) On the other hand by the proof of [10, Lemma 3.3A] (cf. proof of [5, Lemma 2.2]) we have: (2.5) L+(f )≤area+(α ). s,t s,t Consequently, by the Lemma 2.3 we get that L+(f ) as a function of t is s,t bounded from above by the non-increasing conteinuous function b(s,t) with maxi- mumL+(f(s))+ρ,andwithb(s,1)=ρ. TakingρsmallenoughsothatL+(f )+ρ< s ~ for each s, we obtain a null-homotopy of f in Ω~,contHam(S2,ω). 8 YASHASAVELYEV SowegetthatΩ~,contHam(S2,ω)isweaklycontractible. NextnotethatΩ~Ham(S2,ω) is an open subset of ΩHam(S2,ω), which by Milnor [9], has the homotopy type of a CW complex, as it is the loop space of a Frechet manifold, hence of an abso- lute neighborhood retract, and so Ω~Ham(S2,ω) has the homotopy type of a CW complex. So it follows that Ω~,contHam(S2,ω) is contractible. For a Σ as in the statement of the theorem, we may proceed with exactly the same argument upon noting that now there is no bubbling at all, so that we don’t need to restrict to short loops. (cid:3) 3. A virtually perfect Morse theory for the Hofer length functional and positive Morse index geodesics in Ham(S2,ω),Ham(Σ,ω) Let us review our notion of a local unstable manifold in the setting of loops. Definition 3.1. Let γ ∈ΩHam(M,ω) be an index k, L+ Ustilovsky geodesic and letBk denotethestandardk-ballinRk,centeredattheorigin0. LetΩHam(M,ω) E denote the E sub-level set of the loop space, with respect to L+, with 0

On the Hofer geometry injectivity radius conjecture

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