Classification of coadjoint orbits for symplectomorphism groups of surfaces
aa r X i v : . [ m a t h . S G ] N ov Classification of coadjoint orbits forsymplectomorphism groups of surfaces
Ilia Kirillov ∗ Abstract
We classify generic coadjoint orbits for symplectomorphism groups of com-pact symplectic surfaces with or without boundary. We also classify simpleMorse functions on such surfaces.
Contents ∗ Department of Mathematics, University of Toronto, Toronto, ON M5S 2E4, Canada; e-mail: [email protected] Introduction
The classification problem for coadjoint orbits for the action of symplectic (or area-preserving) diffeomorphisms in two dimensions was known to specialists in view of itsapplication in fluid dynamics since 1960s, and it was explicitly formulated in [3, seeSection I.5] in 1998. The same classification problem also arises in Poisson geometrysince coadjoint orbits are symplectic leaves of the Lie-Poisson bracket, and also inrepresentation theory in connection with the orbit method of A.Kirillov [13]. In therecent work [18] the orbit method was applied to the symplectomorphism group ofthe two-sphere. The classification of generic coadjoint orbits was obtained in [9, 10]for the case of closed surfaces. In [9] there is a list of open questions for the case ofsurfaces with boundary. We answer all those questions in Sections 3 and 4.The classification problem for coadjoint orbits for symplectomorphisms of a sur-face is closely related to a certain classification problem for functions. Hence we aregoing to address the following two problems:1. Classify generic coadjoint orbits of symplectomorphism groups of surfaces.2. Classify generic smooth functions on symplectic surfaces up to symplectomor-phisms.There is a number of papers devoted to classification of functions with non-degenerate critical points on surfaces. In [7] simple Morse functions on surfaceswith boundary were classified with respect to smooth left-right equivalence. In [10]simple Morse functions on closed compact symplectic surfaces were classified up toa symplectomorphism. We generalize the results of [10] to the case of surfaces withboundary. The classification of functions is given in Theorem 3.13. It is worthnoting that the classification in [10] is based on the classification of so-called simpleMorse fibrations obtained in [6]. The proof in the present paper uses a differentmethod, which also gives an alternative proof for Theorem 3.11 from [10] in the caseof closed surfaces.This paper is organised as follows: in Section 2 we give a local classificationof Morse functions up to symplectomorphisms. In Section 3 we solve the globalclassification problem of simple Morse functions. In Section 4 we use the resultsof Section 3 in order to classify generic coadjoint orbits of the symplectomorphismgroup and illustrate these results with examples. In Section 5 we discuss some openproblems in this area, and in Appendix A we present a hydrodynamical motivationfor the main classification theorem.
Acknowledgements
The author is grateful to A.Izosimov, B.Khesin, E.Kudryavtseva, V.Matveev, andA.Oshemkov for discussions and valuable comments, which significantly improvedthis paper. This research is supported in part by the Russian Science Foundation(grant No. 17-11-01303) and the Simons Foundation.2
Local classification of functions up to a symplec-tomorphism
Let p M, ω q be a connected compact two-dimensional symplectic surface with bound-ary B M. Definition 2.1 ([11]) . A function F : M Ñ R is called a simple Morse function ifthe following conditions hold:(i) All critical points of F are non-degenerate;(ii) F does not have critical points on the boundary B M ;(iii) The restriction of F to the boundary B M is a Morse function;(iv) All critical values of F and of its restriction F | B M are distinct. Proposition 2.2 ([11]) . Simple Morse functions form an open and dense subset inthe space of smooth functions in the C -topology. For a regular point O P M zB M of the function F there exists a coordinate chart p p, q q centered at O such that F “ p and ω “ dp ^ dq. For a regular point O P B M ofthe restriction F | B M there exists a coordinate chart p p, q q in M centered at O suchthat F “ p, ω “ dp ^ dq and the boundary B M is given by the equation t q “ u . Next, we present the normal forms for the pair p F, ω q in a neighbourhood of a criticalpoint for the function F and of its restriction F | B M . Theorem 2.3 ([5, 20]) . Let p M, ω q be a symplectic surface, and F : M Ñ R bea simple Morse function. Let O P M zB M be a critical point for the function F. Then there exists a coordinate chart p p, q q centered at O such that ω “ dp ^ dq and F “ λ ˝ S where S “ pq or S “ p ` q . Here λ is a smooth function of one variabledefined in some neighborhood of the origin P R and λ p q ‰ . Moreover:(i) In the case S “ p ` q , if p ˜ p, ˜ q q is another chart as above then ˜ p ` ˜ q “ p ` q . (ii) In the case S “ pq, if p ˜ p, ˜ q q is another chart as above then ˜ p ˜ q “ pq ` ψ p pq q , where ψ is a function of one variable flat at the origin. Furthermore, everyfunction of one variable that is flat at the origin can be obtained in this way.Proof. All statements of this theorem but the last one are proved in [5, 20]. Thelast statement is proved in [20, see Lemma . a ]. Theorem 2.4.
Let p M, ω q be a symplectic surface, and F : M Ñ R be a simpleMorse function. Let O P B M be a regular point of F and a non-degenerate criticalpoint of its restriction F | B M . Then there exists a coordinate chart p p, q q centered at O such that ω “ dp ^ dq and F “ λ ˝ S where S “ q ` p or S “ q ´ p . Here λ isa smooth function of one variable defined in some neighborhood of the origin P R and λ p q ‰ . In this chart M is defined by q ě and the boundary B M is given bythe equation t q “ u , see Figure 1. Moreover: i) In the case S “ q ` p , if p ˜ p, ˜ q q is another chart as above then ˜ p ` ˜ q “ p ` q . (ii) In the case S “ q ´ p , if p ˜ p, ˜ q q is another chart as above then ˜ q ´ ˜ p “ q ´ p ` ψ p q ´ p q , where ψ is a function of one variable flat at the origin.Furthermore, every function of one variable that is flat at the origin can beobtained in this way. pq (a) Case S “ q ` p . pq (b) Case S “ q ´ p . Figure 1: Level sets of the function S . The horizontal axis corresponds to theboundary B M. Before we proceed with the proof of this theorem let us formulate and prove twolemmas.
Lemma 2.5.
Let h and h be two smooth non-negative functions R ` Ñ R ` suchthat h i p q “ and h i p q ą for i “ , . Then the following statements are pairwiseequivalent:(i) The difference h ´ h is a function flat at the origin, i.e. the Taylor series J h and J h are equal to each other.(ii) The difference ? h ´ ? h is a smooth function R ` Ñ R . (iii) The difference ? h ´ ? h is a smooth function R ` Ñ R flat at the origin.Proof. The implication p iii q ùñ p ii q is evident so it enough to show that p i q ùñp iii q and p ii q ùñ p i q . Let us start with implication p i q ùñ p iii q . It followsfrom Hadamard’s lemma that there exist smooth functions ˜ h and ˜ h such that h i “ x ˜ h i and ˜ h i p q ‰ for i “ , . We have the following formula for the difference ? h ´ ? h : a h p x q ´ a h p x q “ ? x h p x q ´ h p x q b ˜ h p x q ` b ˜ h p x q . (1)It follows from the formula (1) that the difference ? h ´ ? h is smooth and flat atthe origin whenever the difference h ´ h is flat the origin.It remains to show that p ii q implies (i). Denote by g the smooth function ? h ´? h . Assume that the difference h ´ h is not flat at the origin. Then there existsa number n P N and a smooth non-zero function f : R ` Ñ R z such that h p x q ´ h p x q “ x n f p x q . It is useful to rewrite formula (1) in the following form: g p x q? x ˆb ˜ h p x q ` b ˜ h p x q ˙ “ x n f p x q . (2)4ormula (2) implies that the function f is flat at the origin whenever the function g is flat at the origin. The function f is a non-zero function so we conclude that thefunction g is not flat at the origin. Therefore there exists a number m P N and asmooth non-zero function ˜ g : R ` Ñ R z such that g p x q “ x m ˜ g p x q . Now we take thesquare of both sides of (2) and obtain the following formula: x ` m ˜ g p x q ˆb ˜ h p x q ` b ˜ h p x q ˙ “ x n f p x q . That gives us a contradiction since the Taylor series of the left hand side starts withan odd power of x and the Taylor series of the right hand side starts with an evenpower of x. We conclude that the function h ´ h is flat at the origin. Lemma 2.6.
Let ψ : R ˆ r , s Ñ R be a smooth function such that the restriction ψ p¨ , ¨ , t q “ : ψ t is a function flat at the origin for each t P r , s . Consider the followingpartial differential equation pH tq ` H tp “ ψ t p q ´ p q . (3) We claim is that there exists a smooth solution H for equation (3) such that H | q “ “ and H t p¨ , ¨q is a function flat at the origin for each t P r , s . Proof.
We cannot directly apply the method of characteristics in this situation sincethe vector field BB p ` p BB q is tangent to t q “ u at the point p , q . However, one candefine a solution with the desired properties by the following explicit formula: H t p p, q q : “ $’’&’’% p p ` a p ´ q q ψ t p q ´ p q if p ă and q ă p p p ´ a p ´ q q ψ t p q ´ p q if p ě and q ă p pψ t p q ´ p q otherwiseWe want to prove that H t is a smooth function flat along the parabola t q “ p u for each t P r , s , and it is a solution to equation (3) for each t P r , s . Note thatthe function G defined by G p p, q q : “ pψ t p q ´ p q is a smooth function flat along theparabola t q “ p u , and is a solution to equation (3). Indeed, pG tq ` G tp “ p r pψ t p q ´ p qs q ` r pψ t p q ´ p qs p “ p d ψ t d x p q ´ p q ´ p d ψ t d x p q ´ p q ` ψ t p q ´ p q “ ψ t p q ´ p q , as required. To complete the proof it is enough to show that the function ˜ H : “ H ´ G is a smooth function flat along the parabola t q “ p u , and p ˜ H tq ` ˜ H tp “ . First of all, the function ˜ H is smooth outside the parabola t q “ p u , and continuouseverywhere. We can write the following formula for a partial derivative of ˜ H : B k ` l ˜ H B p k B q l “ ÿ i,j P ij p p q ψ p i q p q ´ p qp p ´ q q j { P ij P R r p s , and only a finite number of P ij are non-zero. Note that thefunction ψ p i q p q ´ p qp p ´ q q j { is smooth and equal to zero whenever q “ p . Now, the equality p ˜ H tq ` ˜ H tp “ directly follows from the fact that the function ˜ H is constant along the level sets ofthe function p p, q q ÞÑ q ´ p . Finally, the fact that the function φ t is flat at the originfor each t P r , s implies that the function H t is flat at the origin for each t P r , s . Now we proceed with the proof of Theorem 2.4.
Proof of Theorem 2.4.
Without loss of generality we can assume that F p O q “ . The main part of this theorem on the existence of a coordinate chart was provedin [14, 15].Let us prove statement p i q of this theorem. We need to prove the equality ˜ p ` ˜ q “ p ` q . In this case (see Figure 1, (a)) the region t F ď ε u is diffeomorphicto a closed ball provided that ε ą is sufficiently small. Therefore, the area of thisregion A F,ω p ε q : “ ż F ď ε ω is well-defined. Let p p, q q be a coordinate chart centered at O such that F “ λ p p ` q q and ω “ d p ^ d q. Then we can write an explicit formula for the function A F,ω p ε q : A F,ω p ε q “ ż ? λ ´ p ε q´ ? λ ´ p ε q p λ ´ p ε q ´ p q d p “ λ ´ p ε q { . So we conclude that the function λ (and thus the function q ` p ) is uniquelydetermined by the pair p F, ω q . Now consider the second case where in some local chart centered at O we have F “ λ p q ´ p q and ω “ d p ^ d q. Consider a smooth curve ℓ Ă M such that it istransversal to the right half of the parabola t q “ p u and these two curves intersecteach other at exactly one point. In coordinates p p, q q the curve ℓ can be describedas a graph some function g : ℓ “ tp p, q q : q “ g p p qu . We fix a number ε ă , and consider the region R ε (see Figure 2) bounded by theboundary curve B M “ t q “ u , the right half of the parabola t q “ p u X t p ě u , the right half of the parabola t q “ p ` λ ´ p ε qu X t p ě u , and the curve ℓ. F “ u “ t q “ p ut F “ ε u “ t q “ p ` λ ´ p ε qu ℓ “ t q “ g p p qu F “ λ p q ´ p q p R ε p p ℓ pq Figure 2: The area function A F,ω,ℓ “ ş R ε ω. Then the area of this region A F,ω,ℓ p ε q : “ ż R ε ω is well-defined. Denote by p the p -coordinate of the intersection t F “ ε u X B M, by p the p -coordinate of the intersection t F “ u X ℓ, and by p the p -coordinate ofthe intersection t F “ ε u X ℓ. The coordinates p , p , and p depend on ℓ, and alsothe coordinates p and p depend on ε. It follows from the implicit function theoremthat the coordinate p is a smooth function of ε. As for p , it is explicitly given by a ´ λ ´ p ε q . Note that p p ε q ă p ă p p ε q provided that ε is sufficiently small. Wehave the following formula for the function A F,ω,ℓ p ε q : A F,ω,ℓ p ε q “ ż p p ε q p d p ´ ż p p p ε q λ ´ p ε q d p ` ż p p ε q p p g p p q ´ p ´ λ ´ p ε qq d p “ p p ε q{ ´ p p ε q λ ´ p ε q ` smooth function of ε “ p´ λ ´ p ε qq { ` smooth function of ε. Now consider some other chart p ˜ p, ˜ q q such that F “ ˜ λ p q ´ p q , ω “ d˜ p ^ d˜ q, and theboundary B M is given by t ˜ q “ u . Then it is follows from above that p´ λ ´ p ε qq { ´ p´ ˜ λ ´ p ε qq { “ f p ε q where f is a smooth function of one variable. We want to prove that the Taylorseries J λ is equal to the Taylor series of J ˜ λ. It follows from Lemma 2.5 thatthe Taylor series J λ ´ p ε q is equal to the Taylor series J ˜ λ ´ p ε q . From here weconclude that J λ “ J ˜ λ. Finally, let us prove statement p ii q of this theorem. Let ψ : R Ñ R be a func-tion of one variable flat at the origin. The goal is to find a symplectomorphism Φ O is a fixed point for Φ , thesymplectomorphism Φ preserves the boundary B M, and Φ ˚ r q ´ p ` ψ p q ´ p qs “ q ´ p . Apply Moser’s path method and consider the family of functions f t : “ q ´ p ` tψ p q ´ p q for each t P r , s . Instead of looking for one symplectomorphism Φ , we will belooking for a family of symplectomorphisms Φ t such that Φ t ˚ f t “ q ´ p , (4) Φ t pB M q Ă B M for each t P r , s , and Φ t p O q “ O for each t P r , s . Let v t be thevector field corresponding to the flow Φ t : ddt Φ t “ v t ˝ Φ t . Differentiating (4) with respect to t, we obtain the following differential equation Φ t ˚ L v t f t ` Φ t ˚ df t dt “ , which we rewrite as Φ t ˚ ˆ L v t f t ` df t dt ˙ “ . Since Φ t is a diffeomorphism, it is equivalent to L v t f t ` ψ p q ´ p q “ . (5)Since v t has to preserve the symplectic structure ω, we will be looking for the field v t in the Hamiltonian form v t “ H tq BB p ´ H tp BB q (6)where H tq : “ B H t B q and H tp : “ B H t B p . Substitute the right-hand side of (6) into (5) toobtain the following partial differential equation ψ p q ´ p q ´ pH tq p ` tψ p q ´ p qq ´ H tp p ` tψ p q ´ p qq “ . Rewrite it as pH tq ` H tp “ ´ ψ p q ´ p q ` tψ p q ´ p q . Consider the family of functions ψ t p x q : “ ψ p x q ` tψ p x q for each t P r , s . Then our equation assumes the form pH tq ` H tp “ ψ t p q ´ p q .
8t follows from Lemma 2.6 that there exists a smooth solution H for equation 2.2such that H | q “ “ . The family of symplectomorphisms Φ t can be recovered as theflow of the corresponding field v t “ H tq BB p ´ H tp BB q . The condition H t | B M “ impliesthat the field v t has a zero restriction on the boundary t q “ u , and we concludethat the corresponding family Φ t preserves the boundary t q “ u , and Φ t p O q “ O for each t P r , s . Now applying the theorem on the smooth dependence of the flowon initial data one can conclude that the flow Φ t is well-defined for t P r , s . Hence,the diffeomorphism Φ has the desired properties.In the next section we are going to use these local results to obtain a globalclassification of simple Morse functions with respect to the action of the group SDiff p M q of symplectomorphisms of M. Let M be a compact connected oriented surface with boundary B M, and let F : M Ñ R be a simple Morse function. In what follows, by levels we mean con-nected component of level sets of F. Non-critical levels are diffeomorphic to a circleor a line segment. The surface M can be considered as a union of levels, and we geta foliation with singularities. The base space of this foliation is a finite connectedgraph Γ F (see Figure 3) whose vertices correspond to critical values of F or F | B M . This graph Γ F is called the Reeb graph of the function F. We denote the edges ofthe Reeb graph by solid lines if they correspond to circle components and by dashed lines if they correspond to segment components. We denote the union of solid (re-spectively, dashed) edges in Γ F by Γ sF and Γ dF , respectively. We denote the preimages π ´ p Γ sF q and π ´ p Γ dF q by M sF and M dF . Thus Γ F “ Γ sF Y Γ dF , and M “ M sF Y M dF . There are 7 possible types of vertices in the graph Γ F (see Table 1). The function F on M descends to a function f on the Reeb graph Γ F . It is also convenient toassume that Γ F is oriented: edges are oriented in the direction of increasing f.F Γ F M Figure 3: A torus with one hole with the height function on it and the correspondingReeb graph This graph is also called the Kronrod graph of a function, see [19, 1] v be a vertex of the Reeb graph Γ F . Let us fix a number ε ą such that f ´ pr f p v q ´ ε, f p v q ` ε sq X e is a proper subset of e for each edge edge e incident to v. Consider the preimage P εv : “ π ´ p f ´ pr f p v q ´ ε, f p v q ` ε sqq Ă M. The boundary B π ´ p P εv q is piecewisesmooth closed oriented curve. This curve is connected in the case where the vertex v is incident only to dashed edges, and its image π rB π ´ p P εv qs is a closed orientedcurve that passes edges incident to the vertex v in a certain cyclic order. Thisconstruction is nontrivial only in the case when there are at least three dashededges incident to the vertex v (otherwise, there is only one cyclic order at the setof edges incident to v ). Thus for an arbitrary II-vertex or IV-vertex (see Table 1)of the graph Γ F we have a natural cyclic order for the edges incident to this vertex.The above properties of the graph Γ F make it natural to introduce the followingdefinition of an abstract Reeb graph. Definition 3.1.
An (abstract) Reeb graph p Γ , f q is an oriented connected graph Γ with solid or dashed edges, cyclic order on the set of edges incident to II- orIV-vertices (see Table 1), and a continuous function f : Γ Ñ R , with the followingproperties:(i) Each vertex of Γ is of one of the 7 types from Table 1.(ii) The function f is strictly monotonic on each edge of Γ , and the edges of Γ areoriented towards the direction of increasing f . Definition 3.2.
Abstract Reeb graphs p Γ , f q and p Γ , g q are said to be equivalent by means of the isomorphism φ : Γ Ñ Γ if the map φ : (i) maps solid (respectively, dashed) edges to solid (respectively, dashed) edges;(ii) preserves the cyclic order on the set of edges incident to each II - or IV -vertex,i.e. if e follows e in the cyclic order, then φ p e q follows φ p e q ; (iii) takes the function g to the function f (i.e. f “ g ˝ φ ). In this subsection we follow [7]. Let M be a compact connected oriented surfacewith the boundary B M, and let F : M Ñ R be a simple Morse function. The imageof each boundary component of M is a closed curve in the graph Γ F . Informallyspeaking, the following definition describes those closed curves for an abstract Reebgraph.
Definition 3.3.
Let p Γ , f q be an abstract Reeb graph. A non-empty sequence ofedges p e , e , . . . , e n q together with a sequence p v , v , . . . , v n , v n ` “ v q of verticesis called a boundary cycle if the following three conditions hold:(i) All edges in the sequence are dashed.(ii) Each edge e i is incident to the vertices v i and v i ` for every i P t , . . . , n u . B C D E F (a) A B C D E F (b) Figure 4: An illustration to Definition 3.3: dashed Reeb graph with dim H p Γ q “ corresponding to both a disk with two holes p a q and torus with one hole p b q . Cuttingthe disk drawn here along the three dashed levels and then restoring the gluingswith opposite orientations, one obtains a torus with one hole. This figure is basedon Figure 5 from [9].(iii) If the vertex v i has three or more adjacent dashed edges, then the pair ( e i ´ , e i q of consecutive edges is also a consecutive pair of edges with respect to the cyclicorder on the set of edges incident to the vertex v i for every i P t , . . . , n u . We call two boundary cycles equivalent if they differ by the action of a cyclic group,i.e. the sets of vertices v . . . v n v and v i . . . v n v . . . v i ´ v i define the same topologicalcycle for each i P t , . . . , n u . In addition, in the case when a boundary cycle consistsonly of or -valent vertices (i.e. of vertices of type III and IV) we also call twoboundary cycles v v . . . v n ´ v n v and v n v n ´ . . . v v v n equivalent. We denote by σ p Γ q the number of (equivalence classes of) boundary cycles in Γ . Example 3.4.
Consider a disk with with holes and a torus with one hole, andconsider the height function on them (as shown in Figure 4). The correspondingReeb graphs are identical except for the cyclic orders at the vertices C and C . In case p a q of a disk with with holes there are three boundary cycles: B C D B ,C E D C , and A B D E F E C B A . In case p b q of a torus with one hole thereis only one boundary cycle: A B D E F E C D B C E D C D A . Proposition 3.5 ([7]) . Let M be a compact connected oriented surface with theboundary B M, and let F : M Ñ R be a simple Morse function. Then the number ofboundary cycles σ p Γ F q is equal to the number of boundary components dim H pB M q of the surface M. Theorem 3.6 ([7]) . The genus g p M q of a surface M is given by the followingformula: g p Γ q : “ ´ χ p Γ s q ` ´ χ p Γ d q ` p Γ s X Γ d q ´ σ p Γ F q ´ dim H p Γ s q ´ dim H p Γ d q ` , where χ p Γ s q is the Euler characteristic and σ p Γ F q is the number of boundary cycles. Theorem 3.6 motivates us to give the following definition.
Definition 3.7.
Let p Γ , f q be an abstract Reeb graph. Define the genus g p Γ q asthe number from the right-hand side of the formula in Theorem 3.6.11ype Level Sets ReebGraph AsymptoticsI µ pr v, x sq “ ψ p f p x qq a | f p x q| , where ψ p q “ , and ψ p q ‰ .II e e e µ pr v, x sq “ ε i ψ p f p x qq a | f p x q| ` η i p f p x qq , where ε “ ε “ ´ , ε “ , ψ p q “ ,ψ p q ‰ , and η ` η ` η “ .III e e µ pr v, x sq “ ε i ψ p f p x qq a | f p x q| ` η i p f p x qq , where ε “ ´ , ε “ , ψ p q “ , ψ p q ‰ ,and η ` η “ .IV e e e e µ pr v, x sq “ ε i ψ p f p x qq ln | f p x q| ` η i p f p x qq , where ε “ ε “ ´ , ε “ ε “ , ψ p q “ , ψ p q ‰ , and η ` η ` η ` η “ .V e e e µ pr v, x sq “ ε i ψ p f p x qq ln | f p x q| ` η i p f p x qq , where ε “ ε “ ´ , ε “ , ψ p q “ , ψ p q ‰ , and η ` η ` η “ .VI e e e µ pr v, x sq “ ε i ψ p f p x qq ln | f p x q| ` η i p f p x qq , where ε “ ε “ ´ , ε “ , ψ p q “ , ψ p q ‰ , and η ` η ` η “ .VII µ pr v, x sq “ ψ p f p x qq , where ψ p q “ , and ψ p q ‰ . Table 1: 7 types of neighborhoods of singular points with corresponding Reebgraphs and asymptotics for the measure on a Reeb graph (figures are partially takenfrom [9]). In order to simplify notation we assume that f p v q “ . If not, we replace f by ˜ f p x q : “ f p x q ´ f p v q . .3 Measured Reeb graphs Now, fix an area form ω on the surface M. Then the natural projection map π : M Ñ Γ F induces a measure µ : “ π ˚ ω on the graph Γ F . Definition 3.8.
A measure µ on a Reeb graph p Γ , f q is called quasi-smooth if thefollowing conditions hold.1. The measure µ has a C -smooth non-zero density d µ { d f in the complement Γ z V p Γ q .2. In a neighbourhood of each vertex the measure µ can be expressed by thecorresponding formula from Table 1. Proposition 3.9.
Let p M, ω q be a symplectic surface. Let also F : M Ñ R be asimple Morse function and π : M Ñ Γ F a natural projection. Then the measure µ : “ π ˚ ω is quasi-smooth.Proof. For vertices of types VI and VII this was proved in [10, Lemma . ]. Theproof is based on Theorem 2.3. The proof for other types follows the same lines,with the only difference that it uses both Theorems 2.3 and 2.4.The above properties of the measure µ make it natural to introduce the followingdefinition of an abstract measured Reeb graph. Definition 3.10. A measured Reeb graph p Γ , f, µ q is a Reeb graph p Γ , f q equippedwith a quasi-smooth measure µ. Definition 3.11.
Two measured Reeb graphs p Γ , f, µ q and p Γ , g, ν q are said to be equivalent by means of the isomorphism φ : Γ Ñ Γ if the map φ : (i) is an isomorphism between the Reeb graphs p Γ , f q and p Γ , g q ; (ii) pushes the measure µ to the measure ν. Definition 3.12.
A measured Reeb graph p Γ , f, µ q is compatible with p M, ω q if thefollowing conditions hold:(i) The genus g p Γ q of the graph Γ is equal to the genus g p M q of the surface M. (ii) The number σ p Γ q of boundary cycles is equal to the number dim H pB M q ofboundary components of the surface M. (iii) The volume of Γ with respect to the measure µ is equal to the area of thesurface M : ş Γ d µ “ ş M ω. Theorem 3.13. i) Let
F, G : M Ñ R be two simple Morse functions. Then thefollowing conditions are equivalent:a) There exists a symplectomorphism Φ : M Ñ M such that Φ ˚ F “ G. b) Measured Reeb graphs of F and G are isomorphic. oreover, every isomorphism φ : p Γ F , f, µ F q Ñ p Γ G , g, µ G q can be lifted to asymplectomorphism Φ : M Ñ M such that Φ ˚ F “ G. ii) For each measured Reeb graph p Γ , f, µ q compatible with p M, ω q there exists asimple Morse function F : M Ñ R such that the corresponding measured Reebgraph Γ F is isomorphic to p Γ , f, µ q . Proof.
Let us prove the first statement. The implication p a q ùñ p b q is evident, soit suffices to prove the implication p b q ùñ p a q . Let φ : Γ F Ñ Γ G be an isomorphismof measured Reeb graphs. We need to construct a symplectomorphism Φ : M Ñ M such that Φ ˚ F “ G and π G ˝ Φ “ φ ˝ π F . Let ℓ Ă M be a smooth oriented curve which is transversal to the level sets of thefunction F, it does not intersect the singular levels of the function F, and such thatthe function F is strictly increasing along the curve ℓ. Consider the Hamiltonian flow P tF corresponding to the function F. We denote by T F p p F , q F q the time necessary togo from the curve ℓ to the point p p F , q F q under the action of P t ; see Figure 5(a).The pair of functions p F, T F q forms a coordinate system in some neighborhood of ℓ such that ω “ d F ^ d T F (see proof in [2]). In particular, this construction worksfor the boundary curve B M ; see Figure 5(b). The range of the function T F alongthe non-critical level of F is a segment r , Π p F qs in the case when the F -level is asegment, and it is a half-interval r , Π p F qq in the case when the F -level is a circle.The function Π is called a period. It follows from Stokes’ theorem that Π p F q isequal to the derivative d µ d f . p F, qp F, T F q ℓF T F (a) p F, T F q p F, qB M B MF T F (b) Figure 5: An illustration to the definition of the function T F . Let e Ă Γ dF be a dashed edge. The formula p F, T F q ÞÑ p G, T G q defines a symplec-tomorphism from the interior of π ´ F p e q to the interior of π ´ G p φ p e qq . The condition φ ˚ µ F “ µ G guarantees that the periods of the functions T F and T G coincide andhence the symplectomorphism is well-defined. Now let e Ă Γ sF be a solid edge. Letus pick a smooth oriented curve ℓ Ă M which is transversal to the level sets of thefunction F, connects two singular levels of the function F, and such that the function F is strictly increasing along the curve ℓ. Then, as above, we obtain a symplecto-morphism from the interior of π ´ F p e q to the interior of π ´ G p φ p e qq . By applying thesame procedure to all edges of the graph Γ F we obtain a symplectomorphism Φ : π ´ r Γ F z V p Γ F qs Ñ π ´ r Γ G z V p Γ G qs Φ ˚ F “ G and π G ˝ Φ “ φ ˝ π F . Now let O be a singular point for the function F or its restriction F | B M . Thenthere is only one way to define the image of O : Φ p O q : “ π ´ G p φ p π F p O qqq . Let p p F , q F q (respectively, p p G , q G q ) be a chart centered at the point O (respectively, Φ p O q ) as in Theorem 2.3 or 2.4. Then the condition φ ˚ µ F “ µ G guarantees thatthe corresponding functions λ F and λ G are the same or they differ by a functionflat at the origin. In the latter case it follows from Theorem 2.3 or 2.4 that we canreplace the chart p p F , q F q with a chart p ˜ p F , ˜ q F q such that ˜ λ F “ λ G . So without lossof generality we may assume that λ F “ λ G . Therefore, one can define Φ in someneighbourhood U O of O by the formula Φ : p p F , q F q ÞÑ p p G , q G q . This local symplectomorphism Φ extends uniquely to a semi-local symplectomor-phism Φ : π ´ F p π F p U qq Ñ π ´ G pr φ ˝ π F sp U qq . Without loss of generality we may assume that π ´ F p π F p U O qq is a “standard” neigh-bourhood of the singular level π ´ F p π F p O qq (see Table 1), i.e. it is a connectedcomponent of the set t P P M : | F p P q ´ F p O q | ă ε u containing the point O andthe number ε ą is sufficiently small so that these “standard” neighbourhoods fordistinct O are pairwise disjoint. Denote by U F,ε the union of all these neighbour-hoods. By applying the same procedure to all singular points of the function F orits restriction F | B M we obtain a symplectomorphism Φ : U F,ε Ñ U G,ε . such that Φ ˚ F “ G and π G ˝ Φ “ φ ˝ π F . So the isomorphism φ : Γ F Ñ Γ G is lifted to a symplectomorphism Φ : π ´ r Γ F z V p Γ F qs Ñ π ´ r Γ G z V p Γ G qs and to a symplectomorphism Φ : U F,ε Ñ U G,ε . However, these two symplectomorphisms not necessarily define a global symplecto-morphism of the surface M. Let e Ă Γ dF be a dashed edge. Then the intersection U F,ε X π ´ p e q is a disjoint union of two rectangles and the ratio Φ ´ ˝ Φ is a symplec-tic automorphism of this union preserving each component. The only symplecticautomorphism of a fibered rectangle is the identity. So the symplectomorphisms Φ ˝ Θ and Φ agree with each other on the preimage π ´ F p e q of the edge e. Nowlet e Ă Γ sF be a solid edge. Then the intersection U F,ε X π ´ p e q is a disjoint unionof two open cylinders and the ratio Φ ´ ˝ Φ is a symplectic automorphism of thisunion preserving each component. Any symplectic automorphism of a cylinder isa Hamiltonian automorphism. The same holds for their union. The correspondingHamiltonian H t extends (using a bump function) to a smooth function on all of M in such a way that its support is in the preimage of the edge e for all t P r , s . Θ . Now we have Φ | π ´ F p e q “ p Φ ˝ Θ q| π ´ F p e q , i.e. the symplectomorphisms Φ ˝ Θ and Φ do agree witheach other on the preimage π ´ F p e q of the edge e. By applying the same procedure toall solid edges of Γ F we obtain a globally defined symplectomorphism Φ : M Ñ M such that Φ ˚ F “ G and π G ˝ Φ “ φ ˝ π F . This completes the proof of part i q .Now let us prove the second statement of the theorem. Given a triple p Γ , f, µ q we need to construct a quadruple p ˜ M , ˜ π, ˜ F , ˜ ω q such that ˜ F “ f ˝ ˜ π and ˜ π ˚ ˜ ω “ µ. Ifthis is done then ż ˜ M ˜ ω “ ż Γ d µ “ ż M ω and it follows from Moser’s theorem [17] that there is a diffeomorphism Φ : ˜ M Ñ M such that Φ ˚ ω “ ˜ ω so that one can take F “ ˜ F ˝ Φ ´ . It follows from [7] that thereexists a surface ˜ M with a simple Morse function ˜ F and a projection ˜ π : ˜ M Ñ Γ such that ˜ F “ f ˝ π. It remains to construct a symplectic form ˜ ω such that ˜ π ˚ ˜ ω “ µ. Let O be a singular point of the function ˜ F or its restriction ˜ F | B M . It follows fromthe proofs of Theorem 2.3 and Theorem 2.4 that there exists a symplectic form ω O P Ω p M q such that π ˚ p ω O q| U “ µ | U for some neighborhood U of the vertex ˜ π p O q . Using an appropriate partition of unity we construct a symplectic form ˜ ω asa combination of forms ω O , such that ˜ π ˚ ˜ ω “ µ . Let p M, ω q be a compact symplectic surface. The Lie group SDiff p M q of all sym-plectomorphisms of M has the Lie algebra svect p M q of divergence-free vector fieldson M tangent to the boundary B M. The regular dual space svect ˚ p M q can be identi-fied with the space of cosets Ω p M q{ d Ω p M q (see Appendix). Moreover, the naturalaction of the group SDiff p M q on the space of cosets Ω p M q{ dΩ p M q by means ofpull-backs coincides with the coadjoint action of the group of symplectomorphisms SDiff p M q : Ad ˚ Φ r α s “ r Φ ˚ α s , where Φ P SDiff p M q is a symplectomorphism and α P Ω p M q is a 1-form.Define the exterior derivative operator d on the space of cosets t α ` d f | f P C p M qu by the formula d r α s : “ d α. (This operator is well-defined on the cosetssince d p α ` d f q “ d α. q Consider the following mapping: curl : Ω p M q{ dΩ p M q Ñ C p M q , defined by taking a vorticity function d α { ω “ : curl r α s . It is easy to see that if theboundary B M of the surface M is not empty then the mapping curl is a surjection.In the case of a closed surface M there is a relation: ż M curl r α s ω “ and the mapping curl is surjective onto the space of zero-mean functions.16uppose that cosets r α s and r β s belong to the same coadjoint orbit of SDiff p M q .Then by definition there is a symplectomorphism Φ such that r Φ ˚ β s “ r α s and thefollowing diagram is commutative: r β s Φ ˚ / / curl (cid:15) (cid:15) r α s curl (cid:15) (cid:15) curl r β s φ ˚ / / curl r α s Definition 4.1.
A coset r α s P Ω p M q{ dΩ p M q is called simple Morse if curl r α s isa simple Morse functions. A coadjoint orbit O is called simple Morse if some (andhence every) coset r α s P O is simple Morse.With every simple Morse coset r α s P Ω p M q{ dΩ p M q one can associate a mea-sured Reeb graph Γ F r α s . If two simple Morse cosets r α s and r β s belong to the samecoadjoint orbit then the corresponding Reeb graphs are isomorphic.Let F : M Ñ M be a simple Morse function. Suppose that cosets r α s and r β s have isomorphic Reeb graphs. Then it follows from Theorem 3.13 that thereexists a symplectomorphism Φ such that Φ ˚ curl r β s “ curl r α s . Therefore, the 1-form Φ ˚ r β s ´ r α s is closed. Since this 1-form is not necessarily exact, the cosets r α s and r β s do not necessarily belong to the same coadjoint orbit. Nevertheless, we concludethat the space of coadjoint orbits corresponding to the same measured Reeb graph isfinite-dimensional and its dimension is at most dim H p M q . Throughout this section,unless otherwise stated, all (co)homology groups will be with coefficients in R . In [10] the notion of a circulation function was introduced for the case of closedsurfaces. In the case of surfaces with boundary we need a modification of thatdefinition. Take a point x P Γ sF which is not a vertex. Then π ´ p x q is a circle C. Itis naturally oriented as the boundary of the set of smaller values of the function F .The integral of a coset r α s over C is well-defined. Thus we obtain a function C r α s : Γ sF z V p Γ sF q Ñ R , defined by C r α s p x q “ ş π ´ p x q α. Proposition 4.2 ([10]) . The function C r α s “ ş π ´ p x q α has the following properties.(i) Assume that x an y are two interior points of some edge e Ă Γ sF , and that e is pointing from x towards y . Then C r α s satisfies the Newton-Leibniz formula C r α s p y q ´ C r α s p x q “ ż yx f dµ (ii) for all vertices of Γ s which do not belong to Γ d the function C r α s satisfies theKirchhoff rule at v : ÿ e Ñ v lim x e ÝÑ v C r α s p x q “ ÿ e Ð v lim x e ÝÑ v C r α s p x q , where the notation e Ñ v stands for the set of edges pointing at the vertex v ,and e Ð v stands for the set of solid edges pointing away from v . f on the subgraph Γ sF can be recovered from the circula-tion function C by the formula: f “ d C { d µ . It follows from Proposition 4.2 that thedifference C r α s ´ C r β s is as an element of the relative homology group H p Γ F , Γ dF q . The above properties of the circulation function C r α s make it natural to introducethe following definition of an abstract circulation function. Definition 4.3.
Let p Γ , f, µ q be a measured Reeb graph. Any function C : Γ s z V p Γ s q Ñ R satisfying properties listed in Proposition 4.2 is called a cir-culation function (an antiderivative) . Proposition 4.4.
Let p Γ , f, µ q be a measured Reeb graph.i) If the subgraph Γ d is not empty, then the pair p f, µ q on Γ admits an antideriva-tive.ii) If the subgraph Γ d is empty, then the pair p f, µ q on Γ admits an antiderivativeif and only if ş Γ f d µ “ .iii) If the pair p f, µ q admits an antiderivative, then the set of antiderivatives of p f, µ q is an affine space whose underlying vector space is the relative homologygroup H p Γ , Γ d q . Proof.
To prove this result one applies Proposition . in [8] to the graph Γ s , withthe set of boundary vertices defined as those vertices that belong to Γ d . In this subsection we follow [8]. Let p M, ω q be a symplectic surface with boundary B M. Denote by CB p M q Ă C p M q the space of Morse functions on M constant onthe boundary B M, and without critical points on the boundary B M. Elements of CB p M q are called functions of CB ´ type. Definition 4.5.
A coset r α s P Ω p M q{ dΩ p M q is said to be of CB -type if curl r α s P CB p M q . A coadjoint orbit O called to be of CB -type if some (and hence every)coset r α s P O is of CB -type.All definitions from the present paper such as Reeb graph, compatibility condi-tions, circulation graph, etc. can be modified for the case of functions and cosets of CB ´ type, see details in [8]. The result we are interested in can be formulated asfollows. Theorem 4.6 ([8]) . Let M be a connected symplectic surface with or without bound-ary. Then coadjoint orbits of SDiff p M q of CB ´ type are in one-to-one correspon-dence with (isomorphism classes of ) circulation graphs p Γ , f, µ, C q compatible with M . In other words, the following statements hold:i) For a symplectic surface M and cosets of CB ´ type r α s , r β s P svect ˚ p M q thefollowing conditions are equivalent:a) r α s and r β s lie in the same orbit of the SDiff p M q coadjoint action;b) circulation graphs Γ r α s and Γ r β s corresponding to the cosets r α s and r β s are isomorphic.ii) For each circulation graph Γ which is compatible with M , there exists a generic r α s P svect ˚ p M q such that Γ r α s “ p Γ , f, µ, C q . F M ˜Γ F π i Figure 6: An illustration to the definition of the graph ˜Γ F . In the case of surfaces with boundary circulation functions do not form a completeset of invariants for coadjoint orbits, i.e. the equality C r α s “ C r β s does not in generalimply that cosets α and β belong to the same coadjoint orbit. Example 4.7.
Consider the disk with two holes from Figure 4(a). In this case thereare no circulation functions since there are no solid edges in the Reeb graph. Onthe other hand, in this case there are no nontrivial symplectomorphisms preservingthe function hence the dimension of the space of coadjoint orbits is equal to the firstBetti number of the surface, i.e. it is equal to two.It turns out that it is possible to define some additional invariants: integrals ofcosets over certain cycles associated with the pair p M, F q in invariant way.There is a unique way to lift each edge e Ă Γ F to a smooth oriented (anddiffeomorphic to a segment) curve ˜ e Ă B M such thati) π p ˜ e q “ e ; ii) for each x P e the regular F -level π ´ p x q is pointed in the direction of thecurve ˜ e. We define the subset ˜ E F Ă M to be the union ˜ E F : “ ď e P E p Γ dF q ˜ e. We also define the subset ˜ V F Ă M to be ˜ V F : “ π ´ p V p Γ dF qzB Γ dF q where B Γ dF is the set of boundary vertices (i.e. vertices of types I, III, or V) of thegraph Γ dF . And, finally, define the subset ˜Γ F to be the union of ˜ E F and ˜ V F (seeFigure 6). The set ˜Γ F is a topological graph embedded into the surface M. Wedenote by i the inclusion ˜Γ F ã ÝÑ M. Lemma 4.8.
The map i ˝ p : ˜Γ F Ñ Γ F is a homotopy equivalence.Proof. Consider the graph ˜Γ F { ˜ E F obtained from ˜Γ F by contracting each connectedcomponent of a singular F -level in ˜Γ F to a point. Denote by p the projection19 Γ F Ñ ˜Γ F { ˜ E F . The map p is a homotopy equivalence since each singular F -level in ˜Γ F is connected and simply connected. The map i ˝ π factors (in a unique way)through ˜Γ F { ˜ E F , i.e. there exists a map ˜ π : ˜Γ F { ˜ E F Ñ Γ F such that i ˝ π “ p ˝ ˜ π. The map ˜ π is a homeomorphism. We conclude that the map i ˝ p is a homotopyequivalence as a composition of the homotopy equivalence p and the homeomorphism ˜ π. Let r α s P Ω p M q{ dΩ p M q be a coset of a one-form. There is a natural way todefine the restriction i ˚ r α s P H p ˜Γ F q . First, we define the restriction i ˚ α as a one-cochain such that i ˚ α p e q : “ ş e α for each edge e Ă ˜Γ F . Now we take i ˚ r α s : “ r i ˚ α s . The cohomology class i ˚ r α s is well-defined since each exact one-form d f restrictsto the exact one-cochain i ˚ d f. It follows from Lemma 4.8 that i ˚ ˝ π ˚ : H p Γ F q Ñ H p ˜Γ F q is an isomorphism. Hence with each coset r α s P Ω p M q{ dΩ p M q we can alsoassociate an element ξ r α s P H p Γ F q defined by the formula ξ r α s : “ p i ˚ ˝ π ˚ q ´ p i ˚ r α sq . Next, we generalize the notion of a circulation graph from [10].
Definition 4.9.
A measured Reeb graph p Γ , f, µ q endowed with a circulationfunction C and an element ξ P H p Γ d q is called a augmented circulation graph p Γ , f, µ, C , ξ q . Two augmented circulation graphs are isomorphic if they are isomorphic as mea-sured Reeb graphs, and the isomorphism between them preserves all additional data.An augmented circulation graph p Γ , f, µ, C , ξ q is compatible with a symplectic sur-face p M, ω q if the corresponding measured Reeb graph p Γ , f, µ q is compatible with p M, ω q (see Definition 3.12). Theorem 4.10.
Let p M, ω q be a connected symplectic surface with or without bound-ary. Then generic coadjoint orbits of SDiff p M q are in one-to-one correspondencewith (isomorphism classes of ) augmented circulation graphs p Γ , f, µ, C q compatiblewith M . In other words, the following statements hold:i) For a symplectic surface M and generic cosets r α s , r β s P svect ˚ p M q the follow-ing conditions are equivalent:a) r α s and r β s lie in the same orbit of the SDiff p M q coadjoint action;b) augmented circulation graphs Γ r α s and Γ r β s corresponding to the cosets r α s and r β s are isomorphic.ii) For each augmented circulation graph Γ which is compatible with M , thereexists a generic r α s P svect ˚ p M q such that Γ r α s “ p Γ , f, µ, C , ξ q . Corollary 4.11.
The space of coadjoint orbits of the group
SDiff p M q correspondingto the same measured Reeb graph p Γ , f, µ q is a finite-dimensional affine space andits dimension is dim H p Γ , Γ d q ` dim H p Γ d q . Remark 4.12.
It follows from the long exact sequence for the pair p Γ , Γ d q that dim H p Γ , Γ d q ` dim H p Γ d q “ dim H p Γ q ´ dim H p Γ d q ` . Therefore, the space of coadjoint orbits of the group
SDiff p M q corresponding to thesame measured Reeb graph p Γ , f, µ q has dimension dim H p Γ q in the case when thesubgraph Γ d is connected. 20 xample 4.13. Consider the torus with one boundary component from Figure 3with the height function F on it, and the corresponding Reeb graph Γ F . In this case H p Γ dF q “ and H p Γ F , Γ dF q “ . Therefore, the corresponding space of coadjointorbits is one-dimensional.Before we proceed with the proof of Theorem 4.10 let us formulate and provetwo lemmas.
Lemma 4.14.
Let M be a connected oriented surface with non-empty boundary,and let F be a simple Morse function on M . Then dim H p M dF q “ dim H p Γ dF q ` dim H p Γ sF X Γ dF q . Proof.
Let ˜ M be the smooth surface obtained from the surface M dF by contractingeach circle in M dF X M sF to a point. It is clear that dim H p M dF q “ dim H p ˜ M q ` dim H p Γ sF X Γ dF q . Let p be the canonical projection M Ñ ˜ M .
The function F descends to a simpleMorse function ˜ F : ˜ M Ñ R such that F “ ˜ F ˝ p. The Reeb graph Γ ˜ F consistsonly of dashed edges, and it is coincides with Γ dF . Then the surface ˜ M is homotopyequivalent to graph Γ ˜ F . Therefore, dim H p M dF q “ dim H p ˜ M q ` dim H p Γ sF X Γ dF q “ dim H p Γ ˜ F q ` dim H p Γ sF X Γ dF q“ dim H p Γ dF q ` dim H p Γ sF X Γ dF q . Lemma 4.15.
Let M be a connected oriented surface possibly with boundary, andlet F be a simple Morse function on M . Assume that r γ s P H p M q is such thatthe integral of γ over any F -level vanishes, and ξ r γ s is a zero element in H p Γ dF q . Then there exists a C function H : M Ñ R (with zero restriction on the surface M dF ) such that the one-form H d F is closed, and its cohomology class is equal to r γ s .Moreover, H can be chosen in such a way that the ratio H { F is a smooth function.Proof. Denote by i d the inclusion M dF X M sF ã ÝÑ M dF , and denote by π d the restrictionof the projection π : M Ñ Γ F on the surface M dF . Note that the homomorphism p π d q ˚ : H p M dF q Ñ H p Γ dF q is a surjection, and Im p i d q ˚ Ă Ker p π d q ˚ . It follows fromLemma 4.14 that dim H p M dF q “ dim H p Γ dF q ` dim H p Γ sF X Γ dF q . Hence the image of the homomorphism p π d q ˚ coincides with the kernel of the homo-morphism p π d q ˚ . From above we conclude that the homomorphism π ˚ : H p Γ dF q Ñ H p M dF q is an injection, and Im p π d q ˚ “ Ker p i d q ˚ . Denote by i the inclusion M dF ã ÝÑ M F . Since the integral of r γ s over any connectedcomponent of any closed F -level vanishes and ξ r γ s is a zero element in H p Γ dF q , the cohomology class r i ˚ γ s is a zero element in H p M dF q . Consider the long exactcohomology sequence for the pair p M F , M dF q :0 Ñ H p M dF qÑ H p M F q Ñ H p M F , M dF q Ñ H p M F q Ñ H p M dF q Ñ . r γ s on M belongs to the kernel of the homomorphism i ˚ : H p M F q Ñ H p M dF q . Hence it belongs to image of the homomorphism H p M F , M dF q Ñ H p M F q , i.e. there exists a one-form ˜ γ such that r ˜ γ s “ r γ s and ˜ γ | M dF “ . Denote by π s the restriction of the projection π : M Ñ Γ F on the surface M sF . Thehomomorphism p π s q ˚ : H p M sF q Ñ H p Γ sF q is a surjection, and its kernel is generatedby those homology classes which are homologous to regular F -levels. From abovewe conclude that the homomorphism p π s q ˚ : H p Γ sF q Ñ H p M sF q is an injection, and Im p π s q ˚ “ Ann Ker p π s q . Therefore, there exists a one-cochain α P H p Γ sF q of theform α “ ř e P E p Γ sF q α e e ˚ such that r ˜ γ s “ p π s q ˚ r α s . Recall that the function f is the pushforward of the function F to the graph Γ F . Consider a continuous function h : Γ F Ñ R such thati) it is a smooth function of f in a neighborhood of each point x P Γ F ;ii) it vanishes whenever x is sufficiently close to a vertex;iii) h | Γ dF “ iv) for each edge e , we have α p e q “ ż e h d f. Obviously, such a function does exist. Now, lifting h to M , we obtain a smoothfunction H with the desired properties. Proof of Theorem 4.10.
Let us prove the first statement. The implication (a) ñ (b)is immediate, so it suffices to prove the implication (b) ñ (a). Let φ : Γ r α s Ñ Γ r β s be an isomorphism of augmented circulation graphs. By Theorem 3.13, φ can belifted to a symplectomorphism Φ : M Ñ M that maps the function F “ curl r α s tothe function G “ curl r β s . Therefore, the -form γ defined by γ “ Φ ˚ β ´ α is closed.Assume that Ψ : M Ñ M is a symplectomorphism which maps the function F to itself and is isotopic to the identity. Then the composition r Φ “ Φ ˝ Ψ ´ maps F to G , and r r Φ ˚ β ´ α s “ r Φ ˚ β ´ Ψ ˚ α s “ r γ s ´ r Ψ ˚ α ´ α s . We claim that Ψ can be chosen in such a way that r Φ ˚ β ´ α is exact, i.e. one hasthe equality of the cohomology classes r Ψ ˚ α ´ α s “ r γ s . Moreover, let us show that there exists a time-independent symplectic vector field X that preserves F and satisfies r Ψ ˚ t α ´ α s “ t r γ s , (7)where Ψ t is the phase flow of X . Differentiating (7) with respect to t , we get in theleft-hand side r Ψ ˚ t L X α s “ r L X α s “ r i X d α s “ r F ¨ i X ω s , ue v uw Figure 7: The illustration to the construction of the graph Γ . The vertex w subdi-vides the edge e with endpoints t v, u u into the edges e v Ñ w and e w Ñ u . since L X α is closed and Ψ ˚ t does not change its cohomology class. Thus r F ¨ i X ω s “ r γ s . (8)Since Φ preserves the circulation function, the integrals of γ over all connectedcomponents of F -levels vanish. In addition, ξ r Φ ˚ α s “ ξ r α s . Therefore, by Lemma4.15, there exists a smooth function H such that r γ s “ r H d F s . Now we set X : “ HF ω ´ d F. It is easy to see that the vector field X is zero on M d , symplectic, preserves thelevels of F , and satisfies the equation (8). Therefore, its phase flow map Ψ “ Ψ has the required properties.Now let us prove the second statement. It follows from Theorem 3.13 that thereexist a symplectic surface p M, ω q and a simple Morse function F : M Ñ R suchthat Γ F “ Γ . Consider the surface M s and the restriction F | M s of the function F to the surface M s . The restriction F | M sF is a Morse function, and it is constanton the boundary B M sF since it is formed by some of closed F -levels. However, itis not necessarily a function of CB -type since it has hyperbolic critical points onthe boundary whenever the graph Γ F has vertices of type V. In order to apply theTheorem 4.6 we need to ‘cut out’ from M s these hyperbolic critical points. Let v P Γ F be a vertex of type V, let e Ă Γ F be the only solid edge incident to v, andalso let u P Γ sF be the only other vertex adjacent to e. The edge e , with endpoints t v, u u can be uniquely subdivided into two edges, say e v Ñ w and e w Ñ u , connectingto a new vertex w such that µ p e v Ñ w q “ µ p e w Ñ u q . After that we cut out the edge e v Ñ w . Denote by Γ the (abstract) measured Reeb graph obtained by applying theabove procedure to all vertices of type V in the graph Γ F (see Figure 7). Denoteby M Ă M sF the preimage π ´ p Γ q It is clear from the above that the restriction F | ˜ M is a function of CB -type. Therefore, it follows from Theorem 4.6 that thereexists a one-form α on M such that C r α s “ C | Γ . It is clear that the form α can beextended (along the cylinders M s z M on all of M s in such a way that C r α s “ C | Γ . On the other hand, there exists a one-form α on M such that rp i d q ˚ α s “ rp i s q ˚ α s and ξ r α s “ ξ since dim H p M dF q “ dim H p Γ dF q ` dim H p M dF X M sF q . Using an appropriate partition of unity we construct a one-form α (as a combinationof one-forms α and α ) such that C r α s “ C r α s “ C and ξ r α s “ ξ r α s “ ξ. Hence theaugmented circulation graph Γ r α s coincides with Γ . Conclusion
In this paper we have classified simple Morse functions on symplectic surfaces andgeneric coadjoint orbits of symplectomorphism groups of surfaces. This allowed usto completely resolve the questions posed in [9]. The answer to Problem . onreconstruction of a surface with boundary from its Reeb graph was given in thework [7]. As described in Section 3 the idea is to add a cyclic order for the dashededges incident to II- or IV-vertices to the structure of an abstract Reeb graph in orderto reconstruct the corresponding surface. Table 1 gives an answer to Problem . on measure asymptotics on the graph. The required by Problem . compatibilityconditions are described in Definition 3.12. And finally, Theorem 4.10 describes therequired by Problem . additional invariants for coadjoint orbits in case of surfaceswith boundary.One should mention two other relevant but not overlapping with us classificationresults for symplectic surfaces:a) Dufour, Molino, and Toulet classified in [6] simple Morse fibrations on closedsymplectic surfaces under the action of symplectic diffeomorphisms.b) Bolsinov [4] and Kruglikov [16] classified Hamiltonian vector fields on surfacesup to the action of arbitrary diffeomorphisms.It would be interesting to extend those classifications for surfaces with boundary.(Note that in [16] in the contrast with the present work Hamiltonian functions areassumed to be constant on the boundary.) It also would be very interesting toclassify Morse functions and Morse orbits for the action ofa) the group Ham p M q of Hamiltonian diffeomorphisms of a surface M ; b) the connected component SDiff p M q of the identity in the group SDiff p M q forthe case of surfaces M with boundary.This would generalise the corresponding results of [10] and the present work to theseimportant subgroups of the symplectomorphism groups. Appendix A Euler’s equation and coadjoint orbits
In this Appendix we describe following [10] the hydrodynamical motivation of theabove classification problems. Consider a symplectic surface p M, ω q with boundary B M. We denote by
SDiff p M q the Lie group of all symplectomorphisms of M, andby svect p M q the corresponding Lie algebra of divergence-free vector fields on M. Alinear functional f on svect p M q is called regular if there exists a smooth 1-form ξ f such that the value of f on a vector field v is just the pairing between ξ f and v : f p v q “ ż M ξ f p v q ω. The space svect ˚ p M q of regular functionals on svect p M q is a dense subset in the spaceof all continues linear functionals on svect p M q . It turns out that the space of regularfunctionals svect ˚ p M q can be identified with the space of cosets Ω p M q{ d Ω p M q , since exact -forms give zero functionals on divergence-free vector fields. Moreover,24he natural action of the group SDiff p M q on the space of cosets Ω p M q{ d Ω p M q bymeans of pull-backs coincides with the coadjoint action of the group of symplecto-morphisms SDiff p M q . The proof of this fact can be found in [3] (see Section I.8).More information about infinite-dimensional Lie groups can be found in [12].Now let us fix a Riemannian metric p¨ , ¨q on the surface M such that the corre-sponding area form coincides with the symplectic form ω. The motion of an inviscidincompressible fluid on M is governed by the classical Euler equation B t v ` ∇ v v “ ´ ∇ p (9)describing an evolution of a divergence-free velocity field v of a fluid flow in M , where div v “ and the field v is tangent to the boundary B M. The pressure function p entering the Euler equation is defined uniquely modulo an additive constant by thisequation along with the divergence-free constraint on the velocity v .The Riemannian metric p¨ , ¨q allows us to identify the (smooth parts of) the Liealgebra and its dual by means of the so-called inertia operator: given a vector field v on M one defines the 1-form α “ v as the pointwise inner product with vectorsof the velocity field v : v p W q : “ p v, W q for all W P T x M. The Euler equation (9)rewritten on 1-forms is B t α ` L v α “ ´ dP for the 1-form α “ v and an appropriate function P on M . In terms of the cosetsof 1-forms r α s “ t α ` df | f P C p M qu P Ω p M q{ dΩ p M q , the Euler equation looksas follows: B t r α s ` L v r α s “ (10)on the dual space g ˚ , where L v is the Lie derivative along the field v .The Euler equation (10) shows that the coset of 1-forms r α s evolves by an area-preserving change of coordinates, i.e. during the Euler evolution it remains in thesame coadjoint orbit in g ˚ . This is why invariants of coadjoint orbits of cosets r α s describe first integrals, called Casimirs, of the Euler equation, and their completeclassification is important in many areas of ideal fluid dynamics. References [1] G. Adelson-Welsky and A. Kronrode. Sur les lignes de niveau des fonctionscontinues poss´edant des d´eriv´ees partielles.
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