aa r X i v : . [ m a t h . S G ] S e p Removing parametrized rays symplectically
B. Stratmann
Abstract
Let (
M, ω ) be a symplectic manifold. Let [0 , ∞ ) × Q ⊂ R × Q beconsidered as parametrized rays [0 , ∞ ) and let ϕ : [ − , ∞ ) × Q → M be an injective, proper, continuous map immersive on ( − , ∞ ) × Q .If for the standard vector field ∂∂t on R and any further vector field ν tangent to ( − , ∞ ) × Q the equation ϕ ∗ ω ( ∂∂t , ν ) = 0 holds then M and M \ ϕ ([0 , ∞ ) × Q ) are symplectomorphic. The question which subsets N of a symplectic manifold M can be chosensuch that M and M \ N are symplectomorphic has been treated in particularfor M = R n a time ago already, see e.g. [Gro85, McD87, MT93, Tra93]. Morerecently, X. Tang showed that for a general manifold M the subset N canbe chosen to be a ray if the ray possesses a “wide neighborhood” ([Tan18]).Roughly speaking a ray is a 2-ended connected non-compact 1-dimensionallocal submanifold whose one end closes up inside M while at the other end theembedding is proper. In this paper an extension to higher dimensional setsregarded as parametrized rays is provided. While for those higher dimensionalsets a condition is needed, this condition is trivially fulfilled for an isolatedray as treated in [Tan18].In order to state the theorem precisely let ∂∂t denote the standard vector fieldon R , i.e. whose flow consists of translations. Theorem.
Let ( M, ω ) be a symplectic manifold, Q some manifold and themap ϕ : [ − , ∞ ) × Q → M be injective, proper and continuous such that ϕ | ( − , ∞ ) × Q is immersive. If the equation ı ∂∂t ϕ ∗ ω = 0 (1) holds on ( − , ∞ ) × Q then M and M = M \ ϕ ([0 , ∞ ) × Q ) are symplecto-morphic.Additionally this symplectomorphism can be chosen to be the identity outsidesome selected neighborhood of ϕ ([ − , ∞ ) × Q ) . ψ s : M → M arising from a time-dependent vector field ξ s such that( ξ
1) the set [ s ′ ∈ [ s − ,s +1] { m ∈ M | ξ s ′ | m = 0 } is relatively compact for all s ∈ R ,( ξ
2) for all compact subsets K ⊂ M there is s K ∈ R such that ψ s | K = ψ s ′ | K for all s, s ′ ≥ s K and( ξ
3) for each compact set L ⊂ M there exists σ L ∈ R such that ψ σ | ψ − σL ( L ) = ψ σ ′ | ψ − σL ( L ) for all σ, σ ′ ≥ σ L .This curve of symplectomorphisms will arise from a time-dependent Hamil-tonian vector field ξ s which in turn will be the extension of a time-dependentvector field ζ s on ϕ (( − , ∞ ) × Q ) arising from a diffeotopy. The condition ı ∂∂t ϕ ∗ ω = 0 in the Theorem (Condition (1)) imposes few constraints on therank of ϕ ∗ ω such that there is little hope to extend ζ s to ξ s using localneighborhood models as classically known to exist around isotropic subman-ifolds. Therefore integrabilty of the Hamiltonian flow is requested explicitlyby Property ( ξ ξ
2) will ensure that the curve of sym-plectomorphism ψ s becomes locally stable on M for s → ∞ and thereforeconverges to a limit ψ : M → M still satisfying ψ ∗ ω = ω while Property ( ξ ψ . Proof.
In the first part of the proof a suitable diffeotopy θ s : [ − , ∞ ) × Q → [ − , ∞ ) × Q is constructed which in turn is constructed from a diffeotopy τ s of [ − , ∞ ).Fix b ∈ ( − ,
0) and a diffeomorphism τ : [ − , → [ − , ∞ ) with τ | [ − ,b ] =id [ − ,b ] . There is a diffeotopy τ s , i.e. a smooth curve s τ s of diffeomorphisms τ s : [ − , ∞ ) → [ − , ∞ ), such that( τ τ s = id for all s ≤ τ τ s | [ − ,b ] = id [ − ,b ] for all s ∈ R , 2 τ
3) for all s ∈ R the set [ s ′ ∈ [ s − ,s +1] { t ∈ ( − , ∞ ) | τ s ′ ( t ) = t } is relatively compact in ( − , ∞ ) and( τ
4) for each compact subset A ⊂ ( − ,
0) there is s A ∈ R such that τ s | A = τ | A for all s ≥ s A .In order to define a suitable diffeotopy θ ◦ s : [ − , ∞ ) × Q → [ − , ∞ ) × Q choose a function ρ : Q → (0 , ∞ ) such that the set { q ∈ Q | ρ ( q ) ≤ c } iscompact for each c ∈ R . Define θ ◦ s by θ ◦ s ( t, q ) = ( τ s − ρ ( q ) ( t ) , q ) as well as θ ( t, q ) = ( τ ( t ) , q ) . Denoting π Q : R × Q → Q the projection the maps θ ◦ s satisfy π Q ◦ θ ◦ s = π Q while for fixed q and s the map t τ s − ρ ( q ) ( t ) is a diffeomorphism of [ − , ∞ ).Thus θ ◦ s is a diffeomorphism. By construction it satisfies( θ θ ◦ s = id for all s ≤ θ θ ◦ s | [ − ,b ] × Q = id [ − ,b ] × Q for all s ∈ R ,( θ
3) for all s ∈ R the set [ s ′ ∈ [ s − ,s +1] { ( t, q ) ∈ ( − , ∞ ) × Q | θ ◦ s ′ ( t, q ) = ( t, q ) } is relatively compact in ( − , ∞ ) × Q and( θ
4) for each compact set A ⊂ [ − , × Q there is s A ∈ R such that θ ◦ s | A = θ | A for all s ≥ s A .Next, the time s of the diffeotopy will be deformed which will be helpful laterto cut off time-dependent functions (see ( χ κ : R → R and δ > κ | [ n − δ,n + δ ] = n for all n ∈ Z and definethe diffeotopy θ s = θ ◦ κ ( s ) satisfying likewise all the above properties ( θ θ ζ s on [ − , ∞ ) × Q . The property π Q ◦ θ s = π Q shows that the time-dependentvector field ζ s points in the direction of the rays, i.e. there is a time-dependentfunction λ s : [ − , ∞ ) × Q → R such that ζ s = λ s · ∂∂t . (2)Furthermore ζ s satisfies the following properties.3 ζ ζ s = 0 for all s ≤ s ∈ [ n − δ, n + δ ] for each n ∈ Z ,( ζ ζ s | [ − ,b ] × Q = 0 for all s ∈ R ,( ζ
3) for each s ∈ R the set [ s ′ ∈ [ s − ,s +1] { ( t, q ) ∈ ( − , ∞ ) × Q | ζ s ′ | ( t,q ) = 0 } is relatively compact in ( − , ∞ ) × Q and( ζ
4) for each compact subset A ⊂ [ − , × Q there is an s A ∈ R defining B = θ s A ( A ) such that ζ s | B = 0 for all s ≥ s A .Property ( θ
4) implies in particular that the sets C s = { ( t, q ) ∈ [ − , ∞ ) × Q | ζ s ′ | ( t,q ) = 0 for all s ′ ≥ s } exhaust [ − , ∞ ) × Q , i.e. S s ∈ R ˚ C s = [ − , ∞ ) × Q . This can be seen as follows.Let B be an open subset of [ − , ∞ ) × Q such that ¯ B is compact and ˚¯ B = B .Property ( θ
4) states that for A = θ − ( ¯ B ) there is s A ∈ R such that θ s | A = θ | A for all s ≥ s A hence ζ s | B = 0 for all s ≥ s A , i.e. B ⊂ ˚ C s A . Exhausting[ − , ∞ ) × Q by such sets B yields the statement. On the other hand, the sets θ − s ( ˚ C s ) ∩ ([ − , × Q ) exhaust [ − , × Q . Observe that θ | θ − s ( ˚ C s ) = θ s | θ − s ( ˚ C s ) by definition of C s , in particular θ − ( ˚ C s ) = θ − s ( ˚ C s ). As the sets ˚ C s exhaust[ − , ∞ ) × Q , the sets θ − ( ˚ C s ) exhaust [ − , × Q . In summary [ s ∈ R ˚ C s = [ − , ∞ ) × Q and [ s ∈ R θ − s ( ˚ C s ) ⊃ [ − , × Q (3)In the second part of the proof as the map ϕ given in the theorem is assumedinjective, proper and continuous the set [ − , ∞ ) × Q will be seen as a subsetof M for simplicity. Using Equation (2), Condition (1) requested to hold inthe Theorem reads ω ( ζ s , ν ) = 0 for all ν ∈ T (( − , ∞ ) × Q ) ⊂ T M
The next goal is to extend ζ s to a time-dependent vector field ξ s such that ı ξ s ω is a closed time-dependent 1-form on M . In fact, it is even possible toconstruct a time-dependent function g s : M → R with g s | [ − , ∞ ) × Q = 0 suchthat ı ζ s ω | ( − , ∞ ) × Q = d g s | ( − , ∞ ) × Q g s can be givenexplicitly. Using a partition of unity the result holds globally as the localfunctions g s vanish on the local submanifolds.Once a neighborhood U of [0 , ∞ ) × Q is selected as stated in the Theorem,deforming ϕ slightly on [ − , × Q , it can be assumed that [ − , ∞ ) × Q ⊂ U .Now the time-dependent function g s can be cut off to vanish outside of U .Furthermore in view of ( ζ g s can be required to satisfy additionally thatthe sets [ s ′ ∈ [ s − ,s +1] { m ∈ M | d g s ′ | m = 0 } are relatively compact for all s ∈ R .Starting from g s a time-dependent function f s will be defined as the limitof a sequence of time-dependent functions f n,s . Each such time-dependentfunction f n,s and f s define a time-dependent Hamitonian vector field by ı ξ n,s ω = d f n,s and ı ξ s ω = d f s with flows ψ n,s and ψ s respectively.Initialize f ,s = g s and set f n +1 ,s = χ n,s · f n,s for a sequence of smooth time-dependent functions χ n,s : M → [0 ,
1] whose properties will be specifiedbelow. Since for all time-dependent functions h s ∈ { f n,s , f s } the set [ s ′ ∈ [ s − ,s +1] { m ∈ M | d h s ′ | m = 0 } ⊂ [ s ′ ∈ [ s − ,s +1] { m ∈ M | d g s ′ | m = 0 } is relatively compact for all s ∈ R , the flows ψ n,s and ψ s are defined for all s ∈ R globally.In order to define χ n,s let L n be an exhausting sequence of compact subsetsof M , i.e. L = ∅ , L n ⊂ ˚ L n +1 and S n ∈ N L n = M . In view of (3), this choicecan be made such that L n ∩ [ − , ∞ ) × Q ⊂ ˚ C n for all n ∈ N . Analogouslyusing (3) again, an exhaustion K n of M \ ([0 , ∞ ) × Q ) is chosen such that K n ∩ [ − , × Q ⊂ θ − n ( ˚ C n ).The time-dependent functions χ n,s shall satisfy( χ χ n,s ( m ) = 1 if n ≥ s ,( χ χ n,s ( m ) = 1 if m ∈ [ − , ∞ ) × Q and f n,s ( m ) = 0.Property ( χ
1) implies that f n,s = f m,s and ξ n,s = ξ m,s and hence ψ n,s = ψ m,s for all n, m ≥ s . f n,s converges to f s for n → ∞ , and consequently ξ n,s converges to ξ s and so the sequence of corresponding flows ψ n,s converges to the flow ψ s of ξ s . The limits satisfy f n,s = f s and ξ n,s = ξ s and ψ n,s = ψ s for all n ≥ s . As the choice of χ n,s does not change ψ s ′ for all s ′ ≤ n , in particular ψ n , thefollowing third condition on χ n,s can be required, since the sequence χ n,s canbe defined inductively.( χ χ n,s ( m ) = 0 if s ∈ [ n + δ, ∞ ) and m ∈ L n ∪ ψ n ( K n )By making use of Property ( ζ
1) on the side, this condition ( χ
3) implies ψ s | K n = ψ s ′ | K n for all s, s ′ ≥ n . Thus, since K n has been chosen to exhaust M = M \ ([0 , ∞ ) × Q ), thediffeomorphisms ψ s converge for s → ∞ on M to a diffeomorphism ψ : M → ψ ( M ) ⊂ M . Recall that each diffeomorphism ψ n,s as well as eachdiffeomorphism ψ s is in fact a symplectomorphism, so the limit ψ satisfies ψ ∗ ω = ω .Furthermore all time-dependent functions g s , f n,s and f s and therefore alltime-dependent vector fields ξ n,s and ξ s are constructed to vanish outside U .Thus all flows ψ n,s and ψ s and a forteriori ψ equal the identity outside U .Finally ψ will be shown to be surjective to M . By construction ψ | [ − , × Q = θ the image of θ , namely [ − , ∞ ) × Q , is contained in the image of ψ . For acompact set L disjoint to [ − , ∞ ) × Q there is n ∈ N such that L ⊂ L n .Furthermore ψ n is a bijection of M and the restriction ψ n | M \ ([ − , ∞ ) × Q ) abijection of M \ ([ − , ∞ ) × Q ), i.e. L ⊂ ψ n ( M \ ([ − , ∞ ) × Q )). For each l ∈ L there is an m ∈ ψ − n ( L ) such that l = ψ s ( m ) = ψ s ′ ( m ) for all s, s ′ ≥ n with m [ − , ∞ ) × Q and hence as ψ ( m ) = ψ s ( m ) for all s ≥ n , theequation ψ ( m ) = l holds which finishes the proof showing surjectivity of ψ : M → M . Acknowledgement
The author expresses his gratitude to A. Weinstein fordrawing his interest to the question and to the result of X. Tang ([Tan18]) inthe discussion process of [Str20]. Furthermore he likes to thank A. Abbon-dandolo.
The research for this work was partially supported by the SFB/Transregio 191 “Symplek-tische Strukturen in Geometrie, Algebra und Dynamik”. eferences [Gro85] M. Gromov. Pseudo holomorphic curves in symplectic manifolds. Invent. Math. , 82(2):307–347, 1985.[McD87] D. McDuff. Symplectic structures on R n . In Aspects dynamiques ettopologiques des groupes infinis de transformation de la m´ecanique(Lyon, 1986) , volume 25 of
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Department of MathematicsRuhr-Universit¨at Bochum44780 BochumGermany
E-mail address [email protected]@rub.de