Geometry of generating function and Lagrangian spectral invariants

Transcription

1 Geometry of generating function and Lagrangian spectral invariants Yong-Geun Oh IBS Center for Geometry and Physics & POSTECH & University of Wisconsin-Madsion Paris, Alanfest, July 18, 2012 Yong-Geun Oh (IBS Center for Geometry and Physics & POSTECH Generating & function University of Wisconsin-Madsion) Alanfest 1 / 27

2 Outline 1 Hamilton-Jacobi equation and method of characteristics 2 Floer homology and basic phase function 3 Topological Hamiltonian flows 4 Piecewise smooth Hamiltonian geometry 5 Summary and future directions Yong-Geun Oh (IBS Center for Geometry and Physics & POSTECH Generating & function University of Wisconsin-Madsion) Alanfest 2 / 27

3 Hamilton-Jacobi equation Hamilton-Jacobi equation is a evolutionary first-order nonlinear PDE { ( ) S t + H t, q, S q = 0 S(0, q) = S 0 (q), where S : R R n R is a C 1 -function. When one takes optical-mechanical analogy as Hamilton originally did, S q0 (q) corresponds to optical length of the path from q 0 to q, the least time of propagation of light from q 0 to q. For a general smooth manifold N, the equation takes the form { S t + H (t, ds) = 0 (1) S(0, q) = S 0 (q) for a C 1 -function S : R N R. From this expression, we see that the natural space where the equation is defined involves the cotangent bundle T N: The map (t, q) (t, ds(t, q)) defines a section of the extended phase space R T N R N. ong-geun Oh (IBS Center for Geometry and Physics & POSTECH Generating & function University of Wisconsin-Madsion) Alanfest 3 / 27

4 On the extended phase space R T N, the equation can be written as that of differential forms n ds = Θ H dt, Θ = p dq = p i dq i where Θ is the Liouville one-form and H = H(t, q, p) is called the Hamiltonian of the mechanical system, and S is called the (phase space) generating function. These three are the major players in Hamiltonian mechanics and also in symplectic topology of cotangent bundles. Here is the classical action funcgtional i=1 Definition (Hamilton s action functional) A H (γ) = p dq H(t, γ(t)) dt where γ is a path in the phase space T N. γ Hamilton s action principle states that the trajectories of motion are extremals of the action functional. Yong-Geun Oh (IBS Center for Geometry and Physics & POSTECH Generating & function University of Wisconsin-Madsion) Alanfest 4 / 27

5 More precisely, we have Proposition (First variation formula) δa H (γ)(ξ) = dθ(ξ(0), γ X H (t, γ)) dt Θ(γ(0)), ξ(0) + Θ(γ(1), ξ(1). In particular, an extremal trajectory with natural boundary condition satisfies Hamilton s equation q i = H p i, ṗ i = H q i. The Hamiltonian vector field is defined by X H = n H H p i q i q i p i i=1 in canonical coordinates in which ω = dθ = i=1 dq i dp i is the symplectic form. Yong-Geun Oh (IBS Center for Geometry and Physics & POSTECH Generating & function University of Wisconsin-Madsion) Alanfest 5 / 27

6 Phase space generating function A boundary condition for the phase trajectories is called a natural boundary condition, if the boundary terms satisfy Θ(γ(0)), ξ(0) + Θ(γ(1), ξ(1) = 0. For example, the boundary condition p(γ(0)) o N, q(γ(1)) = Tq 0 N. is one such boundary condition. Definition (Phase space generating function) Consider the function h H : R T N R by h H (t, x) = Θ zx H [0,t] t 0 H(t, z H x (t)) dt where zx H is the Hamiltonian trajectory with final point or at t = 1 given by x T N. More explicitly it is given by the formula z H x (t) = φ t H (φ1 H ) 1 (x). Yong-Geun Oh (IBS Center for Geometry and Physics & POSTECH Generating & function University of Wisconsin-Madsion) Alanfest 6 / 27

7 Configuration space generating function h H satisfies the equation h H t + H = 0, p = h H q where h H is defined on the phase space but not on the configuration space. If the Lagrangian submanifold φ 1 H (o N) has no caustic, then the function f H defined by f H (t, q) = h H (t, σ F (q)), σ H = π 1 L H = df H becomes a solution to Hamilton-Jacobi equation. When L H := φ 1 H (o N) has caustic, we need to select a single-valued branch out of h H and L H. Call the pair (f H, σ H ) a selection rule. Question What kind of selection rule will make f H continuous?, How do we make this selection rule canonical so that it varies continuously over the change of Hamiltonian H? Yong-Geun Oh (IBS Center for Geometry and Physics & POSTECH Generating & function University of Wisconsin-Madsion) Alanfest 7 / 27

8 Wave front and selection rule The extended phase space R t T N has its companion, called one-jet space J 1 (N) which itself has the form R z T N. The space carries a natural geometric structure, called a contact structure induced by the contact form dz p dq. The combined space R t R z T N can be formed into T (R N) with symplectic form dt dz dθ = dt dz + i dq i dp i. The pair (h H, L H ) R z T N = J 1 (N) forms a Legendrian submanifold at each time t, and its time propagation is called the wave front propagation. We call the image W H of the front projection R z T N R z N the front propagation. The time-wise selection rule (f H, σ H ) defines a single valued branch of the wave front propagation. Yong-Geun Oh (IBS Center for Geometry and Physics & POSTECH Generating & function University of Wisconsin-Madsion) Alanfest 8 / 27

9 Selection rule p + z q Yong-Geun Oh (IBS Center for Geometry and Physics & POSTECH Generating & function University of Wisconsin-Madsion) Alanfest 9 / 27

10 Floer trajectory equation We study the (negative) L 2 -gradient flow of the action functional u τ (τ) + grad g J A H (u(τ)) = 0 (2) where u : R Ω(o N, T q N) is a path on the path space Ω(o N, T q N) = {γ : [0, 1] T N γ(0) o N, γ(1) T q N}. Here J is an almost complex structure compatible to dθ =: ω 0, i.e., the pairing g J := ω 0 (, J ) becomes a Riemannian metric. We have the formula for the grad gj A H (γ) grad gj A H (γ)(t) = J γ t in general and so (2) is equivalent to the perturbed Cauchy-Riemann equation for the map u : R [0, 1] T N, { u τ + J ( u t X H (t, u) ) = 0 u(τ, 0) o N, u(τ, 1) Tq N. (3) Yong-Geun Oh (IBS Center for Geometry and Physics & POSTECH Generating & function University of Wisconsin-Madsion) Alanfest 10 / 27

11 Floer complex Form the free abelian group, called Floer chain group, { N } CF (H; o N, Tq N) = a i [z i ] z i Crit A H Ω(o N, Tq N). i=1 An element α in this group is called a Floer chain. For any Floer chain α, we define its level by λ H (α) := max A H(z) z supp α Define the boundary map (J,H) by the matrix element n(z, z + ; H, J) := #(M(z, z + ; H, J)) where M(z, z + ; J, H) is the moduli space of solutions of (3). We call the complex (CF(H, J), (H,J) ) the Floer complex of (H, J). Theorem ( ;1997) HF(H; o N, T q N) = Z. Denote by [q] its generator. Yong-Geun Oh (IBS Center for Geometry and Physics & POSTECH Generating & function University of Wisconsin-Madsion) Alanfest 11 / 27

12 Basic phase function Definition Define a function f H : N R by and call it the basic phase function. Theorem ( ;1997) f H (q) := inf α [q] λ H(α) 1 The map (q, F) f F (q) is continuous in q N and satisfies f F f F C 0 F F := 1 0 osc(f t F t ) dt 2 (q, df F (q)) φ 1 F (o N) whenever df F (q) is defined. 3 Define S F (v, q) := f F v (q) where F v is the time-reparameterization such that F t (t, x) = vf(vt, x) which generates the flow t φ vt F. Then S F satisfies Hamilton-Jacobi equation. Yong-Geun Oh (IBS Center for Geometry and Physics & POSTECH Generating & function University of Wisconsin-Madsion) Alanfest 12 / 27

13 HJE for continuous Hamiltonian H It is clear that even for continuous Hamiltonian H = H(t, q, p), the equation S (t, q) + H(t, ds(t, q)) = 0 (4) t makes sense. However the method of characteristic does not: It requires to solve Hamilton s equation q = H p, ṗ = H q whose existence, uniqueness and continuous dependence of the solutions require that H should be at least C 2 (or C 1,1.) The usual method to solve this kind of problems is via considering a smooth approximation of the equation, say, by taking a smooth approximation H i of H. Yong-Geun Oh (IBS Center for Geometry and Physics & POSTECH Generating & function University of Wisconsin-Madsion) Alanfest 13 / 27

14 Topological Hamiltonian Flows We ask Question Approximate H by smooth Hamiltonians H i. Can we construct a solution for (4) by taking a limit of solutions S i? The answer is No for an ad-hoc solution for (1) but Yes for the basic phase function solution S F. Motivated by the fact that the method of characteristic requires the existence of the flow, we introduce Definition (Topological Hamiltonian flows; Müller-Oh 2005) A continuous flow t λ(t) in Sympeo(M, ω) := Symp(M, ω) is called a topological Hamiltonian flow if it allows a smooth approximation 1 H i H in in L (1, ). 2 φ t H i λ(t) uniformly on compact subset of t R. ong-geun Oh (IBS Center for Geometry and Physics & POSTECH Generating & function University of Wisconsin-Madsion) Alanfest 14 / 27

15 Hamiltonian homeomorphisms (or hameomorphisms) We introduce Definition (Müller-Oh, 2005) Define the set of hameomorphisms to be the subset Hameo(M, ω) = {ψ Sympeo(M, ω) ψ = λ(1), λ P ham }. Theorem (Müller-Oh, 2005) Hameo(M, ω) is a normal subgroup of Sympeo(M, ω). In 2 dimension, this together with some smoothing result implies Corollary Hameo(D 2, D 2 ) is a normal subgroup of the area preserving homeomorphism group Homeo Ω (D 2, D 2 ) for the standard area form Ω or with respect to the Lebesque measure on D 2 C. Yong-Geun Oh (IBS Center for Geometry and Physics & POSTECH Generating & function University of Wisconsin-Madsion) Alanfest 15 / 27

16 The uniqueness of the flow a continuous Hamiltonian is proved by Müller-Oh using the energy-capacity inequality (Hofer-Zehender, Lalonde-McDuff), and the uniquenss of Hamiltonian for given topological Hamiltonian flow is proved by Viterbo (closed case for C 0 ), Oh (open case for C 0 ) and by Buhovsky-Seyfaddini both open and closed case for L (1, ). Here is a 30-year old open problem in dynamical systems Question (Mather,??) Is Homeo Ω (D 2, D 2 ) a simple group? Diff Ω (D 2, D 2 ) is shown to be not simple by Banyaga (1978) using the presence of non-trivial homomorphism Cal : Diff Ω (D 2, D 2 ) R, called Calabi homomorphism. For Homeo Ω (M) with dim M 3, Fathi (1980) proved it is not simple again using the presence of non-trivial mass flow homomorphism. Yong-Geun Oh (IBS Center for Geometry and Physics & POSTECH Generating & function University of Wisconsin-Madsion) Alanfest 16 / 27

17 Calabi homomorphism and its extension Start with the path space Definition Let λ = φ H P ham (Diff Ω (D 2, D 2 ), id). We define Cal path (φ H ) := 1 By a simple application of Stokes formula, Proposition 0 D 2 H(t, x) Ω dt (5) The value Cal path (φ H ) depends only on the time-one map. Therefore Cal path descends to a non-trivial homomorphism Question Cal : Diff Ω (D 2, D 2 ) R. Is Cal continuous in C 0 -topology of Diff Ω (D 2, D 2 )? Yong-Geun Oh (IBS Center for Geometry and Physics & POSTECH Generating & function University of Wisconsin-Madsion) Alanfest 17 / 27

18 Wild homeomorphism The answer is No: There exists a sequence φ i Diff Ω (D 2, D 2 ) such that φ i id in C 0 but Cal(φ i ) = 1. (6) Example Consider the set of dyadic numbers 1 for k = 0,. Let (r, θ) be polar 2 k coordinates on D 2 with the standard area form ω = r dr dθ. Consider maps φ k : D 2 D 2 of the form given by φ k = φ ρk : (r, θ) (r, θ + ρ k (r)) : ρ k : (0, 1] [0, ) supported in (0, 1). Then φ ρk autonomous Hamiltonian given by is generated by an r F φk (r, θ) = sρ k (s) ds. 1 Yong-Geun Oh (IBS Center for Geometry and Physics & POSTECH Generating & function University of Wisconsin-Madsion) Alanfest 18 / 27

19 Example continued The Calabi invariant of φ k becomes ( r ) Cal(φ k ) = sρ k (s) ds r dr dθ = π D 2 We now choose ρ k inductively in the following way: 1 Cal(φ 1 ) = 1. 2 ρ k has support in 1 2 k < r < 1 2 k 1 3 For each k = 1,, we have 1 ρ k (r) = 2 4 ρ k 1 (2r) 1 0 r 3 ρ k (r) dt. for r ( 1 2 k, 1 2 k 1 ). Since φ k s have disjoint supports by construction, we can take the infinite product φ := Π k=0 φ k. We conjecture that φ is an area preserving homeomorphism group, but not a hameomorphism, which we call a wild homeomorphism. Yong-Geun Oh (IBS Center for Geometry and Physics & POSTECH Generating & function University of Wisconsin-Madsion) Alanfest 19 / 27

20 The main slogan then in this regard is the phenomenon (6) cannot happen if φ i s are anchored by a topological Hamiltonian. Conjecture Cal path descends to a homomorphism Cal : Hameo(D 2, D 2 ) R. Or equivalently, lim i Cal path (φ Fi ) = 0 whenever F i satisfies 1 φ t F i uniformly converges over t [0, 1] and φ 1 F i id. 2 F i converges in L (1, ). Corollary Homeo Ω (D 2, D 2 ) is not simple. Conjecture The above infinite product is not contained in Hameo(D 2, D 2 ) and so Hameo(D 2, D 2 ) is a proper normal subgroup of Homeo Ω (D 2, D 2 ). ong-geun Oh (IBS Center for Geometry and Physics & POSTECH Generating & function University of Wisconsin-Madsion) Alanfest 20 / 27

21 Everything is a Lagrangian submanifold! We can geometrize the Hamiltonian flow by considering its graphs using the following standard result, Theorem (Darboux-Weinstein) There exists a symplectic diffeomorphism U (M M, ω ω) V (T, dθ). Consider the family Graph φ t F = {(φt F (y), y) y M} (M M, ω ω) which defines a deformation of the diagonal under the Hamiltonian flow φ t F on M M where F is the time-dependent Hamiltonian function given by F(t, x) := F(t, π 1 (x)) = F(t, x), x = (x, y). We call this kind of processes Lagrangianization in general. Yong-Geun Oh (IBS Center for Geometry and Physics & POSTECH Generating & function University of Wisconsin-Madsion) Alanfest 21 / 27

22 Basic phase function versus Calabi invariant Denote by F the mean-normalized Hamiltonian for a given F. Theorem ( ;2011) Let λ = φ F be any topological Hamiltonian path supp F U in P ham (Sympeo U (M, ω), id) and with U = M \ B where B is a closed subset of nonempty interior. Suppose φ 1 F i id. Then ( ) lim f F i (x) = Cal U (F) = lim Cal U (F i ) i i (7) uniformly, for any choice of an approximating sequence F i of F with supp F i U. Applying this theorem to D 2 S 2, the extension problem of Cal to Hameo(D 2, D 2 ) is reduced to the vanishing lim i f Fi = 0. After a long excursion into Floer homology theory, we reduce this vanishing to that of the average of f F with respect to the sequence of measures on as chosen above. Yong-Geun Oh (IBS Center for Geometry and Physics & POSTECH Generating & function University of Wisconsin-Madsion) Alanfest 22 / 27

23 Cliff-wall surgery p z 0 1 q Yong-Geun Oh (IBS Center for Geometry and Physics & POSTECH Generating & function University of Wisconsin-Madsion) Alanfest 23 / 27

24 Cliff-wall Lagrangian surgery Lemma We start with the following lemma where Ω 1 = π 1 ω f Fi (π ΣFi ; ) ρ Fi = h Fi Ω n 1 σ Fi Here we regard the basic Lagrangian selector σ Fi as a rectifiable integral Lagrangian current so that we can do integration by parts. is not a cycle. There is a natural cycle σ Fi σ add F i = σ Fi + σ Fi ;[ +] + σ Fi ; 2 where σ Fi ;[ +] is the cliff-wall chain and σ Fi ; 2 is the union of triangles over the 3-valent graphs of the Maxwell set. We call this cycle the basic Lagrangian cycle or the cliff-wall surgery of φ 1 F i (o N ). Yong-Geun Oh (IBS Center for Geometry and Physics & POSTECH Generating & function University of Wisconsin-Madsion) Alanfest 24 / 27

25 Piecewise smooth Hamiltonian geometry Definition Let ψ : L T N be a piecewise smooth continuous map. 1 We call ψ an isotropic map if it satisfies ψ ω = 0 at all smooth points of L. 2 We call ψ a Lagrangian immersion if its derivative dψ has rank n on a dense open subset of L in addition. 3 We call ψ an exact Lagrangian immersion if the equation ψ θ = df is satisfied whereever the form ψ θ is smooth and the function f : L R is piecewise smooth and continuous. The crucial point in the above definition is the requirement of continuity of the hamiltonian functions. With this definition, we obtain Theorem ( ;2013) There exists a continuous piecewise smooth exact Lagrangian homotoy from Σ add G \ Σ G to φ 1 G (o ) \ Σ G fixing their boundaries. Yong-Geun Oh (IBS Center for Geometry and Physics & POSTECH Generating & function University of Wisconsin-Madsion) Alanfest 25 / 27

26 Future directions In our discussion, we implicitly assumed the following things; 1 N is compact without boundary. For example, we do not deal with the classical case N = R n. This is an artifact of the current technology of Floer homology theory. 2 We assume the Hamiltonian is asymptotically constant. The case of most interest in relation to Aubry-Mather theory and Weak KAM theory is the case when the Hamiltonian is convex and superlinear. 3 We assume that N has no boundary. We believe that any of these restrictions can be overcome by suitably refining the current technology of Floer homology theory allowing either more general Hamiltonians or more general singular Lagrangian submanifolds. A fundamental problem in relation to C 0 -symplectic topology then will be to generalize Floer homology theory to Lagrangian currents. Such a need happens to also arise in recent developments related to the study of Fukaya category and homological mirror symmetry. ong-geun Oh (IBS Center for Geometry and Physics & POSTECH Generating & function University of Wisconsin-Madsion) Alanfest 26 / 27

27 There have appeared many instances of study of singular Lagrangian submanifolds most notably by the study of Arnold, Givental, Zakalyukin and others in the point of view of singularity theory and its classifications. These singular Lagrangian varieties are natural objects of study that appear when one takes a limit of various deformations of varieties. These singularities are not severe ones in the point of geometric measure theory. We suspect that when full power of the existing geometric measure theory such as presented in Federer s book is unleashed and combined with the method of pseudoholomorphic curve theory, it will generate a variety of new developments in hard symplectic topology (in the terminology of Gromov). Geometric measure theory has already been proven to be a powerful tool in Riemannian geometry and in other areas of analysis and PDE s. Thank you, Alan, for your teaching and support! Yong-Geun Oh (IBS Center for Geometry and Physics & POSTECH Generating & function University of Wisconsin-Madsion) Alanfest 27 / 27