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1.
Felix Schlenk 《Israel Journal of Mathematics》2003,138(1):215-252
We study the rigidity and flexibility of symplectic embeddings in the model case in which the domain is a symplectic ellipsoid.
It is first proved that under the conditionr
n
2
≤2r
1
2
the symplectic ellipsoidE(r
1,…,r
n)with radiir
1≤…≤r
ndoes not symplectically embed into a ball of radius strictly smaller thanr
n.We then use symplectic folding to see that this condition is sharp. We finally sketch a proof of the fact that any connected
symplectic 4-manifold of finite volume can be asymptotically filled with skinny ellipoids. 相似文献
2.
Dan Mangoubi 《Mathematische Annalen》2008,341(1):1-13
We consider Riemannian metrics compatible with the natural symplectic structure on T
2 × M, where T
2 is a symplectic 2-torus and M is a closed symplectic manifold. To each such metric we attach the corresponding Laplacian and consider its first positive
eigenvalue λ1. We show that λ1 can be made arbitrarily large by deforming the metric structure, keeping the symplectic structure fixed. The conjecture is
that the same is true for any symplectic manifold of dimension ≥ 4. We reduce the general conjecture to a purely symplectic
question. 相似文献
3.
Paul Biran 《Inventiones Mathematicae》1999,136(1):123-155
We prove that for any closed symplectic 4-manifold (M,Ω) with [Ω]∈H
2(M, Q) there exists a number N
0 such that for every N≥N
0, (M,Ω) admits full symplectic packing by N equal balls. We also indicate how to compute this N
0. Our approach is based on Donaldson's symplectic submanifold theorem and on tools from the framework of Taubes theory of
Gromov invariants.
Oblatum 9-I-1998 & 1-VII-1998 / Published online: 14 January 1999 相似文献
4.
Yu. V. Eliseeva 《Differential Equations》2010,46(9):1339-1352
In the present paper, we prove comparison theorems for symplectic systems of difference equations, which generalize difference
analogs of canonical systems of differential equations. We obtain general relations between the number of focal points of
conjoined bases of two symplectic systems with matrices W
i
and $
\hat W_i
$
\hat W_i
as well as their corollaries, which generalize well-known comparison theorems for Hamiltonian difference systems. We consider
applications of comparison theorems to spectral theory and in the theory of transformations. We obtain a formula for the number
of eigenvalues λ of a symplectic boundary value problem on the interval (λ
1, λ
2]. For an arbitrary symplectic transformation, we prove a relationship between the numbers of focal points of the conjoined
bases of the original and transformed systems. In the case of a constant transformation, we prove a theorem that generalizes
the well-known reciprocity principle for discrete Hamiltonian systems. 相似文献
5.
Giovanni Mongardi 《Central European Journal of Mathematics》2012,10(4):1472-1485
Let X be a hyperk?hler manifold deformation equivalent to the Hilbert square of a K3 surface and let φ be an involution preserving the symplectic form. We prove that the fixed locus of φ consists of 28 isolated points and one K3 surface, and moreover that the anti-invariant lattice of the induced involution on H 2(X, ℤ) is isomorphic to E 8(−2). Finally we show that any couple consisting of one such manifold and a symplectic involution on it can be deformed into a couple consisting of the Hilbert square of a K3 surface and the involution induced by a symplectic involution on the K3 surface. 相似文献
6.
Symplectic instanton vector bundles on the projective space ℙ3 constitute a natural generalization of mathematical instantons of rank-2. We study the moduli space I n;r of rank-2r symplectic instanton vector bundles on ℙ3 with r ≥ 2 and second Chern class n ≥ r, n ≡ r (mod 2). We introduce the notion of tame symplectic instantons by excluding a kind of pathological monads and show that the locus I n;r * of tame symplectic instantons is irreducible and has the expected dimension, equal to 4n(r + 1) −r(2r + 1). 相似文献
7.
Roberto Paoletti 《manuscripta mathematica》2002,107(2):145-150
We study the stability of a compact Lagrangian submanifold of a symplectic manifold under perturbation of the symplectic
structure. If X is a compact manifold and the ω
t
are cohomologous symplectic forms on X, then by a well-known theorem of Moser there exists a family Φ
t
of diffeomorphisms of X such that ω
t
=Φ
t
*(ω0). If L⊂X is a Lagrangian submanifold for (X,ω0), L
t
=Φ
t
-1(L) is thus a Lagrangian submanifold for (X,ω
t
). Here we show that if we simply assume that L is compact and ω
t
|
L
is exact for every t, a family L
t
as above still exists, for
sufficiently small t. Similar results are proved concerning the stability of special Lagrangian and Bohr–Sommerfeld special Lagrangian submanifolds,
under perturbation of the ambient Calabi–Yau structure.
Received: 29 May 2001/ Revised version: 17 October 2001 相似文献
8.
In 1995 Dusa McDuff and Dietmar Salamon conjectured the existence of symplectic 4–manifolds (X,ω) which satisfy b+=1, K2=0, K·ω>0, and which fail to be of Lefschetz type. This is equivalent to finding a symplectic, homology T2×S2 manifold with nontorsion canonical class and a cohomology ring which is not isomorphic to the cohomology ring of T2×S2. They needed such examples to complete a list of possible symplectic 4–manifolds with b+=1. In that same year Tian-Jun Li and Ai-ko Liu, working from a different point of view, questioned whether there existed
symplectic 4–manifolds with b+=1 with Seiberg- Witten invariants that did not depend on the chamber structure of the moduli space. The purpose of this paper
is to construct an infinite number of examples which satisfy both requirements.
The author was partially supported by NSF grant DMS-0406021. 相似文献
9.
N. A. Vavilov 《Journal of Mathematical Sciences》2008,151(3):2937-2948
Let Γ = GSp(2l, R) be the general symplectic group of rank l over a commutative ring R such that 2 ∈ R*; and let ν be a symmetric equivalence
relation on the index set {1,…, l, −l,…, 1} all of whose classes contain at least 3 elements. In the present paper, we prove
that if a subgroup H of Γ contains the group EΓ(ν) of elementary block diagonal matrices of type ν, then H normalizes the subgroup generated by all elementary symplectic
transvections Tij(ξ) ∈ H. Combined with the previous results, this completely describes the overgroups of subsystem subgroups in this case.
Similar results for subgroups of GL(n, R) were established by Z. I. Borewicz and the author in the early 1980s, while for GSp(2l, R) and GO(n, R) they have been announced by the author in the late 1980s, but a complete proof for the symplectic case has not been
published before. Bibliography: 74 titles.
__________
Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 349, 2007, pp. 5–29. 相似文献
10.
11.
Asymplectic integration of a Poisson manifold (M, Λ) is a symplectic groupoid (Γ,η) whichrealizes the given Poisson manifold, i.e. such that the space of units Γ0 with the induced Poisson structure Λ0 is isomorphic to (M, Λ). This notion was introduced by A. Weinstein in [28] in order to quantize Poisson manifolds by quantizing their symplectic
integration. Any Poisson manifold can be integrated by alocal symplectic groupoid ([4], [13]) but already for regular Poisson manifolds there are obstructions to global integrability
([2], [6], [11], [17], [28]).
The aim of this paper is to summarize all the known obstructions and present a sufficient topological condition for integrability
of regular Poisson manifolds; we will indeed describe a concrete procedure for this integration. Further our criterion will
provide necessary and sufficient if we require Γ to be Hausdorff, which is a suitable condition to proceed to Weinstein’s
program of quantization. These integrability results may be interpreted as a generalization of the Cartan-Smith proof of Lie’s
third theorem in the infinite dimensional case.
Recherche supportée par D.G.I.C.Y.T. Espagne (Proyecto PB90-0765) et Xunta de Galicia (Proxecto XUGA20704B90) 相似文献
Recherche supportée par D.G.I.C.Y.T. Espagne (Proyecto PB90-0765) et Xunta de Galicia (Proxecto XUGA20704B90) 相似文献
12.
Lower estimates for the mazimal weight multiplicities in irreducible representations of algebraic groups of type D
n
in characteristic 2 are found. lf n ≥ 8, then either such a multiplicity is at least n—4— [n]4, where [n]4 is the residue of n modulo 4, or all of its weight multiplicities arc equal to 1. For groups of types B
n
and D
n
in odd characteristic and of type C
n
, in characteristics greater than 7, similar results were obtained earlier. Bibliography: 11 titles. 相似文献
13.
Karl Friedrich Siburg 《manuscripta mathematica》1993,78(1):149-163
Some years ago a class of new symplectic invariants was discovered, the so-called symplectic capacities. In this note we compute
the Hofer-Zehnder-capacityc
HZ
, which is defined via periodic solutions of Hamiltonian differential equations, for two-dimensional, connected manifoldsM with an area element ω. It turns out thatc
HZ
(M,ω) is just the area |∫
M
ω|. Moreover, some examples illustrate the dynamics standing behind the definition ofc
HZ
. In the last part we treat the special case of the real plane where also another type of capacities exists, an example of
which is Hofer's displacement energy. 相似文献
14.
Gregor Kemper 《manuscripta mathematica》1996,90(1):343-363
A general method is developed to attack Noether's Problem constructively by trying to find minimal bases consisting of rational
invariants which are quotients of polynomials of small degrees. This approach turns out to be successful for many small groups
and for most of the classical groups with their natural representations. The applications include affirmative answers to Noether's
Problem for the conformal symplectic groups CSp
2n
(q), for the simple subgroups Ω
n
(q) of the orthogonal groups forn andq odd, for some other subgroups of orthogonal groups and for the special unitary groups SU
n
(q
2).
The author was supported by the Graduate College “Modelling and Scientific Computing in Mathematics and Science” during this
work 相似文献
15.
16.
Let M be a quantizable symplectic manifold. If ψt is a loop in the group {Ham}(M) of Hamiltonian symplectomorphisms of M and A is a 2k-cycle in M, we define a symplectic action κA(ψ)∊ U(1) around ψt(A), which is invariant under deformations of ψ, and such that κA(ψ) depends only on the homology class of A. Using properties of κA( ) we determine a lower bound for ♯π1(Ham(O)), where O is a quantizable coadjoint orbit of a compact Lie group. In particular we prove that ♯π1(Ham(CPn)) ≥ n+1.
Mathematics Subject Classifications (2000): 53D05, 57S05, 57R17, 57T20. 相似文献
17.
François Laudenbach 《Commentarii Mathematici Helvetici》1995,70(1):558-614
Let }L
t{,t ∈ [0, 1], be a path of exact Lagrangian submanifolds in an exact symplectic manifold that is convex at infinity and of dimension
≥6. Under some homotopy conditions, an engulfing problem is solved: the given path }L
t{ is conjugate to a path of exact submanifolds inT
*Lo. This impliesL
t must intersectL
o at as many points as known by the generating function theory. Our Engulfing theorem depends deeply on a new flexibility property
of symplectic structures which is stated in the first part of this work.
相似文献
18.
Lower estimates for the maximal weight multiplicities in irreducible representations of the algebraic groups of type C
n
in characteristic p ≤ 7 are found. If n ≥ 8 and p ≠ 2 , then for an irreducible representation either such a multiplicity is at least n− 4 − [n]4,where [n]4 is the residue of n modulo 4, or all the weight multiplicities are equal to 1.For p = 2, the situation is more complicated, and for every n and l there exists a class of representations with the maximal weight multiplicity equal to 2
l
. For symplectic groups in characteristic p > 7 and spinor groups similar results were obtained earlier. Bibliography: 15 titles. 相似文献
19.
M.-C. Arnaud 《Publications Mathématiques de L'IHéS》2009,109(1):1-17
A theorem due to G. D. Birkhoff states that every essential curve which is invariant under a symplectic twist map of the annulus
is the graph of a Lipschitz map. We prove: if the graph of a Lipschitz map h:T→R is invariant under a symplectic twist map, then h is a little bit more regular than simply Lipschitz (Theorem 1); we deduce that there exists a Lipschitz map h:T→R whose graph is invariant under no symplectic twist map (Corollary 2).
Assuming that the dynamic of a twist map restricted to a Lipschitz graph is bi-Lipschitz conjugate to a rotation, we obtain
that the graph is even C
1 (Theorem 3).
Then we consider the case of the C
0 integrable symplectic twist maps and we prove that for such a map, there exists a dense G
δ
subset of the set of its invariant curves such that every curve of this G
δ
subset is C
1 (Theorem 4). 相似文献
20.
This paper studies the question of when a loop φ={φ
t
}0≤
t
≤1 in the group Symp(M,ω) of symplectomorphisms of a symplectic manifold (M,ω) is isotopic to a loop that is generated by a time-dependent Hamiltonian function. (Loops with this property are said to
be Hamiltonian.) Our main result is that Hamiltonian loops are rigid in the following sense: if φ is Hamiltonian with respect
to ω, and if φ′ is a small perturbation of φ that preserves another symplectic form ω′, then φ′ is Hamiltonian with respect
to ω′. This allows us to get some new information on the structure of the flux group, i.e. the image of π1(Symp(M,ω)) under the flux homomorphism. We give a complete proof of our result for some manifolds, and sketch the proof in general.
The argument uses methods developed by Seidel for studying properties of Hamiltonian loops via the quantum homology of M.
Oblatum 31-X-1997 & 20-III-1998 / Published online: 14 October 1998 相似文献