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Symplectic rigidity and weak commutativity 1 Simone Vazzoler Franco Cardin 1 0 2 Dipartimento di Matematica Pura ed Applicata n a Via Trieste, 63 - 35121 Padova, Italy J 4 January 25, 2011 2 ] G Abstract S . h An alternative proof of Eliashberg-Gromov’s C0-rigidity theorem is t a presented andanewnotion of weak Liebrackets forHamiltonian vector m fields is proposed and compared. [ A notion of Hamiltonian C0-commutativity, together with a related theo- 1 v rem linking it to the standard Poisson brackets, has been introduced in [1] 1 in connection with the problem of finding variational solutions of multi-time 0 5 Hamilton-Jacobi equations. The main theorem on this new environment is 4 based essentially on the use of Viterbo’s capacities. In that paper it has been . 1 foreseen that from that C0-commutativity framework Eliashberg-Gromov’s 0 1 theorem on symplectic rigidity could follow. An interesting proof of this fact 1 has been recently worked out by V. Humilière (see [7]) using the concept of : v pseudo-representations. i X The question on the C0-closure of the group of symplectomorphisms is widely r considered as a starting point for the study of symplectic topology, thus is a important to enrich this particular area with new proofs, eventually trying to give further elucidations to the subject. In this note, moving from the above C0-commutativity, an alternative proof of Eliashberg-Gromov’s theorem is presented which is based on simple algebraic arguments. Ageneralnotionofweakcommutativity ofvector fields(notnecessarily Hamil- tonian) is presented in the work by Rampazzo and Sussmann (see [9]), where they extend the usual Lie brackets even in the case of Lipschitz vector fields. In [1] has been suggested the existence of a possible relation between C0- commutativity and the notion presented in [9]. Here, we enter adding some 1 details on this matter. Firstly, on the line of thought of [9], we introduce the notion of weak Hamiltonian vector field and afterwards the notion of weak Lie brackets; this construction makes our definition coinciding with the one given in [9]. Lastly, we see that this setting is right suitable to provide the equiva- lence between C0-commutativity and weak commutativity of vector fields for Hamiltonians of class C1,1. 1 Generating Functions Quadratic at Infinity (GFQI) Let M be a paracompact n dimensional manifold and S C2(M;R). In what ∈ follows we suppose that the Palais-Smale condition holds: (PS) The pair (M,S) satisfies the (PS) condition if every sequence xk k N { } ∈ such that lim DS(x ) = 0 and S(x ) is bounded, admits a convergent k k n k k subsequenc→e.∞ We notice that paracompactness allows us to choose a Riemannian metric g on M such that, for example, DS(x) = g(x)DS(x),DS(x) . We will denote q k k h i Sλ the sublevel set relative to λ, i.e. Sλ = q M S(q) λ , and for every { ∈ | ≤ } α H (Sb,Sa) 0 we define ∗ ∈ \{ } c(α,S) = inf λ [a,b] : i α = 0 { ∈ ∗λ 6 } where i : H (Sb,Sa) H (Sλ,Sa) is the map induced by the natural inclu- ∗λ ∗ → ∗ sion i : Sλ ֒ Sb (if M is not compact we will work with compactly supported λ → differential forms). We recall some important properties of c(α,S): (i) c(α,S)isacritical valueofS: itis commonly saidthe“min-max” critical value; (ii) c(u v,S +S ) c(u,S )+c(v,S ); 1 2 1 2 · ≥ (iii) if α H (M) is the Poincaré dual of µ Hq(M), then c(µ, S) = n q ∈ − ∈ − c(α,S); in particular: − (iv) if 1 H0(M) and µ Hn(M), then c(1, S) = c(µ,S). ∈ ∈ − − Forthe proofssee [11]or [12]. Now we consider a closed differentiable manifold M and denote with L a Lagrangian submanifold of T M isotopic to the zero ∗ section O by means of a compactly supported time one Hamiltonian flow ϕ1. M 2 Definition 1.1. A smooth function S : M Rk R is a generating function × → quadratic at infinity (GFQI) for a Lagrangian submanifold L if: (i) the map ∂S (q;ξ) (q;ξ) → ∂ξ has zero as a regular value; (ii) the map i : Σ M Rk T M s s ∗ ⊂ × → ∂S (q;ξ) (q; (q;ξ)) 7→ ∂q has image i (Σ ) = L, where Σ = (q;ξ) : ∂S(q;ξ) = 0 . s s s { ∂ξ } (iii) for ξ > C | | S(q;ξ) = ξTQξ where ξTQξ it is a non degenerate quadratic form; It is well known (see again [11]) that a generating function is unique up to three fundamental operations: diffeomorphisms on the fibers and addition of quadratic forms and constants. In more details, let S ,S be two GFQI. We 1 2 easy see that S and S are equivalent, i.e. they draw the same L, if there 1 2 exists a diffeomorphism Φ : M Rk M Rk × → × (q;ξ) (q;φ(q;ξ)) 7→ such that S (q;φ(q;ξ)) = S + c with c R. Moreover, we see also that a 1 2 ∈ GFQI S is still equivalent to S if 2 1 S (q;ξ,η) = S (q;ξ)+ηTBη 2 1 where ηTBη it is a non degenerate quadratic form on the fibers; S is said 2 stabilization of S . The following theorems ensure the existence and unicity of 1 a GFQI for L = ϕ1(O ), precisely up to the three operations above. M Theorem 1.2. Let O be the zero section of T M and (ϕt) a Hamiltonian M ∗ flow. Then the Lagrangian submanifold ϕ1(O ) admits a GFQI. M Proof. [10] 3 Theorem 1.3. Let S and S be two GFQI for L = ϕ1(O ). Then, up to the 1 2 M above three operations, S and S are equivalent. 1 2 Proof. [11, page 688] Remark 1.4. In literature sometimes local generating functions S(q,ξ) sat- isfying (i) and (ii), but not (iii), are called Morse families. It was known, be- fore [11], that the above three operations are locally characterizing any Morse family, see [13] and [8]. If S is a GFQI and c R is large enough, one has ∈ H (Sc,S c) H (M) H (D ,∂D ) ∗ − ∗ ∗ − − ≃ ⊗ whereD istheunitarydiscinthenegativeeigenspaceofQ. So,afterchoosing − α H (M) 0 , we can associate to it α T H (Sc,S c), where T is a ∗ ∗ − ∈ \ { } ⊗ ∈ generator of H (D ,∂D ) R. ∗ − − ≃ 2 The γ and γ metrics and C0-commuting Ha- b miltonians All the proofs in this section can be found in [1]. Definition 2.1. Let S be a GFQI for L = ϕ1(O ). We define M γ(L) = c(µ,S) c(1,S), 1 H0(M), µ Hn(M) − ∈ ∈ It is important to remark that this definition is well posed: even if c(µ,S) and c(1,S) depends on S, the difference does not depend on the function. On the set of the Lagrangiansubmanifolds of T M isotopic to the zero section, named ∗ L, we can define a metric. Definition 2.2. Given L ,L L we define 1 2 ∈ γ(L ,L ) = c(µ,S S ) c(1,S S ) 1 2 1 2 1 2 ⊖ − ⊖ where (S S )(q;ξ ,ξ ) = S (q;ξ ) S (q;ξ ). 1 2 1 2 1 1 2 2 ⊖ − In [11] it is shown that γ is a metric on the set L. If we define H (T M) = c ∗ C ([0,1] T M,R) (i.e. the set of the time dependent Hamiltonians with c∞ × ∗ compact support) and set HD (T M) to be the group of the time one maps c ∗ of H (T M), we can extend the γ metric to HD (T M) in this way c ∗ c ∗ γ(ϕ) = sup γ(ϕ(L),L) : L L { ∈ } b γ(ϕ,ψ) = γ(ϕψ 1) − b b 4 Proposition 2.3. γ defines a bi-invariant metric on HD (T M). c ∗ b c We will say that ϕ c-converges to ϕ (and we will write ϕ ϕ) if n n → lim γ(ϕ ,ϕ) = 0 n n →∞b We can extend γ to H (T M): if we choose H with flow ψt, we can define c ∗ γ(H) = sup bγ(ψt). The following inequality (see [1]) t [0,1] ∈ b b γ(ϕ) H C0 (2.1) ≤ k k b holds, where ϕ is the flow at time one of the Hamiltonian H(t,q,p) and H C0 = sup H(t,q,p) inf H(t,q,p) k k (t,q,p) −(t,q,p) If ϕ and ϕ are the time one flows of H and H respectively and we have that n n H H in the C0 topology, then ϕ c ϕ. n n → → 3 C0-commuting Hamiltonians and Eliashberg- Gromov’s theorem Definition 3.1. Let H,K be two autonomous Hamiltonians. We will say that H and K C0-commutes if there exist two sequences H ,K of C1 Hamiltonians n n C0-converging to H and K respectively such that, in the C0-topology: lim H ,K = 0 n n n { } →∞ Proposition 3.2. Let H,K be two autonomous Hamiltonians of class C1,1 and suppose that H,K is small in the C0 norm. If ϕt,ψs are the flows of { } H and K respectively, then the isotopy t ϕtψsϕ tψ s is generated by a C0 − − 7→ small Hamiltonian. Definition 3.1 is a good extension of the standard Poisson brackets commuta- tion since the following theorem does hold. Theorem 3.3 (Cardin, Viterbo [1]). Let H and K be two Hamiltonians of class C1,1. If they C0-commute then H,K = 0 in the usual sense. { } The last theorem can be extended also to the affine at infinity case (see [7] Lemma 10). Another important generalization of the previous theorem can be found in [5]. In what follows we will consider only sequences of symplectomorphisms n Φ(n) that arebounded deformations of the identity, more precisely 7→ such that supp(Φ(n)) Id is compact. − 5 Theorem 3.4 (Symplectic rigidity, [3], [4], [2], [6]). The group of compactly supported symplectomorphisms is C0-closed in the group of all diffeomorphisms of R2d. Proof. To fix the notations: (q,p) = (q ,...,q ,p ,...,p ) R2d and denote 1 d 1 d ∈ with (Q(n)(q,p),...,Q(n)(q,p),P(n)(q,p),...,P(n)(q,p)) 1 d 1 d a sequence of symplectic transformations C0-converging to (Q(q,p),P(q,p)). Note that we have to prove only Q ,P = 1. In fact, the other relations i i { } Q ,Q = 0 = P ,P ,and Q ,P = 0fori = j, areautomaticallysatisfied i j i j i j { } { } { } 6 using Theorem 3.3 and Lemma 10 in [7]. Now we define a new sequence (using the previous one) 1 d Q(n) = Q(n) + P(n) ei i √1d kX=d1 k P(n) = P(n) + Q(n) ei i √d kX=1 k Clearly Q(n),P(n) = 0, in fact i i { } e e 1 d Q(n),P(n) = Q(n),P(n) + ( P(n),P(n) + Q(n),Q(n) ) { i i } { i i } √d X { k i } { i k } e e k=1 1 d + P(n),Q(n) = 1 1 = 0 d X{ k k } − k=1 Using again Theorem 3.3 and Lemma 10 in [7], we get Q ,P = 0: passing i i { } to the limit, e e Q ,P = Q ,P + 1 d ( P ,P + Q ,Q )+ 1 d P ,Q , { i i} { i i} √d Pk=1 { k i} { i k} d Pk=1{ k k} e e = Q ,P + 1 d P ,Q = 0 { i i} d Pk=1{ k k} Define (just to semplify the notations) C (q,p) = Q ,P . For every fixed i i i { } (q,p) R2d the last homogeneous linear system reads ∈ d 1 1 ... 1 C 0 1  − − −     1 d 1 ... 1 C 0 2  −... −... ... −...  ...  = ...            1 1 ... d 1Cd 0 − − − that has C = C = ... = C as solution; in fact, the d d matrix has 1 2 d × determinant equal to zero: if we sum the last d 1 rows we get the opposite − 6 of the first row; in particular the rank of the matrix is d 1, so the subspace − of solutions has dimension 1, spanned by the above equal components vector. Recalling the Jacobi identity f, g,h + g, h,f + h, f,g = 0 { { }} { { }} { { }} we obtain, considering terms like Q , Q ,P for i = j, i j j { { }} 6 0 = Q , Q ,P + Q , P ,Q + P , Q ,Q i j j j j i j i j { { }} { { }} { { }} and since Q ,P = Q ,Q = 0, we get i j i j { } { } Q , Q ,P = 0 i j j { { }} Analogously, starting with P , Q ,P we obtain i j j { { }} P , Q ,P = 0 i j j { { }} Once we have posed C (q,p) = C (q,p) = ... = C (q,p) = C(q,p), using the 1 2 d previous relations, we have Q ,C = 0  1 { } {Q2...,C} = 0 P ,C = 0 {{Pdd−,C1 } =} 0 that is a homogeneous linear system of the type A DC = 0 · Q ... Q Q ... Q C 0 − 1,p1 − 1,pd 1,q1 1,qd ,q1   Q ... Q Q ... Q C 0 − ...2,p1 ... − ...2,pd 2...,q1 ... 2...,qd ...,q2 = ...           −Pd,p1 ... −Pd,pd Pd,q1 ... Pd,qdC,pd 0 From the fact that Φ : (q,p) (Q ,...,Q ,P ,...,P ) is a diffeomorphism 1 d 1 d 7→ we have detA = 0 (because A = DΦ E where E is the symplectic matrix) 6 · and so C(q,p) = C, a constant. It remains to show that C = 1. This comes from the fact that the limit is a deformation of the identity, i.e. outside a compact set of R2d we have (Q ,...,Q ,P ,...,P ) = (q ,...,q ,p ,...,p ) 1 d 1 d 1 d 1 d and so q ,q = p ,p = 0 and q ,p = δ outside this compact set. From i j i j i j ij { } { } { } the fact that the Poisson brackets are (at least) continuous then we must have C = 1. 7 4 Connection with Lie brackets ItispossibletoextendtheconnectionbetweenthePoissonbracketsandtheLie brackets. We proceed in this way: we define a “weak” Hamiltonian vector field (so that it is defined even if the Hamiltonian is not C1 but only Lipschitz) and then we extend the definition of Lie brackets using this vector field. First of all we recall here the definition of weak Lie brackets by Rampazzo and Sussmann (see [9] section 5) and then, following that line of thought, we propose a definition of weak Hamiltonian vector field. Definition 4.1 (Rampazzo, Sussmann [9]). Let f,g be two locally Lipschitz vector fields on Rn. We define the Lie bracket of f and g at x, and we will write [f,g](x), to be the convex hull of the set of all vectors v = lim(Df(x ) g(x ) Dg(x ) f(x )) j j j j j · − · →∞ for all sequences xj j N such that { } ∈ 1. x Diff(f) Diff(g) for all j; j ∈ ∩ 2. lim x = x; j j →∞ 3. the limit v exists. Inspired by Definition 4.1, we introduce Definition 4.2. LetH be a Lipschitz continuous compactlysupported Hamilto- nian on R2n. We define the weak Hamiltonian vector field X as the following H set valued vector field: EDH(q,p), if (q,p) Diff(H) X :=  ∈ H ch v : v = limEDH(q ,p ), (q ,p ) (q,p) , otherwise j j j j { ∀ → }  where E is the standard symplectic matrix, ch is the convex hull and the sequences (q ,p ) belong to Diff(H). j j The weak Hamiltonian vector field has two properties: (i) X (q,p) is a non empty, closed and convex set of R2n; H (ii) ifH C1 thentheweakvectorfieldcoincideswiththeusualHamiltonian ∈ vector field. 8 It is well known that if H,K are two C2 Hamiltonians then the following equality holds: [X ,X ] = X H K H,K { } We can extend this relation to the case when H and K are C1,1. Definition 4.3. Let H,K C1,1. We define the weak Lie brackets as ∈ JX ,X K := X H K H,K { } Note that X and X are well defined, but their Lie brackets are not (because H K X and X are only Lipschitz vector fields). Because of the definition of X H K H the following proposition holds: Proposition 4.4. The weak Lie brackets have two properties: (i) the definition coincides with the one given in [9] (i.e. these weak Lie brackets are compatible with Rampazzo-Sussmann’s ones); (ii) if H,K C2 then JX ,X K = [X ,X ]. H K H K ∈ In this constructed framework we are ready to state the following result in- volving the C0-commutativity: Proposition 4.5. If H,K are C0-commuting C1,1 Hamiltonians then weak commutativity of vector fields holds: JX ,X K = 0. Conversely, if we have H K JX ,X K = 0 then H,K = 0. In particular H,K C0-commute. H K { } Proof. If H,K C0-commute then we have H,K = 0 and so we get easily { } fromthedefinitionJX ,X K = 0. ConverselyifJX ,X K = 0thentheirflows H K H K commute (see [9]) and so H,K commute and in particular C0-commute. References [1] F.CardinandC.Viterbo. CommutingHamiltoniansandHamilton-Jacobi multi-time equations. Duke Math. J., 144(2):235–284, 2008. [2] I.Ekeland and H.Hofer. Symplectic topologyandHamiltonian dynamics. 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