Lev Buhovsky
In this paper we present an explicit construction of a relative symplectic packing. This confirms the sharpness of the upper bound for the relative packing of a ball into the pair (CP^2, T^2) of the standard complex projective plane and the Clifford torus, obtained by Biran and Cornea.
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In the present article we study basic aspects of the symplectic version of Clifford analysis associated to the symplectic Dirac operator. Focusing mostly on the symplectic vector space of real dimension $2$, this involves the analysis of first order symmetry operators, symplectic Clifford-Fourier transform, reproducing kernel for the symplectic Fischer product and the construction of bases of symplectic monogenics for the symplectic Dirac operator.
We construct explicit maximal symplectic packings of minimal rational and ruled symplectic 4-manifolds by few balls in a very simple way.
Hamiltonian linking and Symplectic packing
本文通过一个观察建立了标准辛球中两集合辛填装与哈密顿环绕之间的联系。
We establish a version of the Arnold conjecture, both the degenerate and non-degenerate case, for target manifolds equipped with Clifford pencils of symplectic structures and the domains (time-manifolds) equipped with frames of divergence-free vector fields. This result generalizes the original work on the hyperkahler Arnold conjecture by Hohloch, Noetzel and Salamon for three-dimensional time and also the previous work by the authors.
The main goal of this paper is to give constructive proofs of several existence results for symplectic embeddings. The strong relation between symplectic packings and singular symplectic curves, which can be derived from McDuff's inflations on the blow-ups, is revisited through a new inflation technique that lives at the level of the manifold. As an application, we explain constructions of maximal symplectic packings of $P^2$ by 6, 7 or 8 balls.
In this paper we describe the intersection between the balls of maximal symplectic packings of $¶^2$. This analysis shows the existence of singular points for maximal packings of $¶^2$ by more than three equal balls. It also yields a construction of a class of very regular examples of maximal packings by five balls.
We show that all closed symplectic 4-manifolds have the packing stability property: there are no obstructions beyond volume to embedding a collection of sufficiently small balls. This generalizes a theorem of Biran which gives the same result under the assumption that the symplectic form lies in a rational cohomology class.
We prove packing stability for any closed symplectic manifold with rational cohomology class. This will rely on a general symplectic embedding result for ellipsoids which assumes only that there is no volume obstruction and that the domain is sufficiently thin relative to the target. We also obtain easily computable bounds for the Embedded Contact Homology capacities which are sufficient to imply the existence of some volume preserving embeddings in dimension 4.
In this paper we prove the connectedness of symplectic ball packings in the complement of a spherical Lagrangian, S^2 or RP^2, in symplectic manifolds that are rational or ruled. Via a symplectic cutting construction this is a natural extension of McDuff's connectedness of ball packings in other settings and this result has applications to several different questions: smooth knotting and unknottedness results for spherical Lagrangians, the transitivity of the action of the symplectic Torelli group, classifying Lagrangian isotopy classes in the presence of knotting, and detecting Floer-theoreti…
Packing stability and the subleading asymptotics of symplectic Weyl laws
本文证明所有具光滑边界的紧致连通辛4-流形满足辛球填装稳定性，并由此得到辛Weyl律误差项的精确指数估计，同时构造了填装稳定性失效的近单位球星形域反例。
Given a symplectic manifold (M, ω) and a Lagrangian submanifold L, we construct versions of the symplectic blow-up and blow-down which are defined relative to L. Furthermore, we show that if M admits an anti-symplectic involution ϕ and we blow-up an appropriately symmetric embedding of symplectic balls, then there exists an anti-symplectic involution on the blow-up as well. We derive a homological condition which determines when the topology of a real Lagrangian surface L = Fix(ϕ) changes after a blow down, and we use these constructions to study the real packing numbers and packing stability …
We define and solve the toric version of the symplectic ball packing problem, in the sense of listing all 2n-dimensional symplectic-toric manifolds which admit a perfect packing by balls embedded in a symplectic and torus equivariant fashion. In order to do this we first describe a problem in geometric-combinatorics which is equivalent to the toric symplectic ball packing problem. Then we solve this problem using arguments from Convex Geometry and Delzant theory. Applications to symplectic blowing-up are also presented, and some further questions are raised in the last section.
Let M be a symplectic-toric manifold of dimension at least four. This paper investigates the so called symplectic ball packing problem in the toral equivariant setting. We show that the set of toric symplectic ball packings of M admits the structure of a convex polytope. Previous work of the first author shows that up to equivalence, only CP^1 x CP^1 and CP^2 admit density one packings when n=2 and only CP^n admits density one packings when n>2. In contrast, we show that for a fixed n>=2 and each r in (0, 1), there are uncountably many inequivalent 2n-dimensional symplectic-toric manifolds wit…
A note on the cardinality of Lagrangian packings
研究在给定哈密顿同伦类中能否打包不可数多个拉格朗日子流形，探讨C^∞和C^0版本的问题。
Boundaries of open symplectic manifolds and the failure of packing stability
本文通过构造辛同胚于圆盘型有界开区域的例子，给出堆积稳定性猜想的反例，并利用ECH谱次领头渐近与闵可夫斯基维数建立分形外尔定律，揭示开辛流形边界定义的障碍。
We exhibit monotone Lagrangian tori inside the standard symplectic four-dimensional unit ball that become Hamiltonian isotopic to the Clifford torus, i.e.~the standard product torus, only when considered inside a strictly larger ball (they are not even symplectomorphic to a standard torus inside the unit ball). These tori are then used to construct new examples of symplectic embeddings of toric domains into the unit ball which are symplectically knotted in the sense of J.~Gutt and M.~Usher. We also give a characterisation of the Clifford torus inside the ball as well as the projective plane in…
A Hermitian refinement of symplectic Clifford analysis
本文通过在标准辛流形上引入复结构，建立辛克利福德分析的埃尔米特精细化，构造两个辛狄拉克算子并研究其不变方程组的解空间，最终利用 transvector 代数给出多项式空间的显式 Fischer 分解。
Let M be a closed symplectic manifold of volume V. We say that M admits an unobstructed symplectic packing by balls if any collection of symplectic balls (of possibly different radii) of total volume less than V admits a symplectic embedding to M. In 1994 McDuff and Polterovich proved that symplectic packings of Kahler manifolds can be characterized in terms of the Kahler cones of their blow-ups. When M is a Kahler manifold which is not a union of its proper subvarieties (such a manifold is called Campana simple) these Kahler cones can be described explicitly using the Demailly and Paun struct…
Let M be a closed symplectic manifold of volume V. We say that the symplectic packings of M by ellipsoids are unobstructed if any collection of disjoint symplectic ellipsoids (possibly of different sizes) of total volume less than V admits a symplectic embedding to M. We show that the symplectic packings by ellipsoids are unobstructed for all even-dimensional tori equipped with Kahler symplectic forms and all closed hyperkahler manifolds of maximal holonomy, or, more generally, for closed Campana simple manifolds (that is, Kahler manifolds that are not unions of their complex subvarieties), as…
In this article we explore a symplectic packing problem where the targets and domains are $2n$-dimensional symplectic manifolds. We work in the context where the manifolds have first homology group equal to $\mathbb{Z}^n$, and we require the embeddings to induce isomorphisms between first homology groups. In this case, Maley, Mastrangeli and Traynor showed that the problem can be reduced to a combinatorial optimization problem, namely packing certain allowable simplices into a given standard simplex. They designed a computer program and presented computational results. In particular, they dete…
Packing Lagrangian tori
本文研究辛流形中积分拉格朗日环面的填充问题，证明 $S^2 \ imes S^2$ 中克利福德环面具有极大性，并证明当 $a,b>2$ 时辛多圆盘 $P(a,b)$ 的标准积分拉格朗日环面补集中仍存在其他积分拉格朗日环面。
In analogy to the relation between symplectic packings and symplectic blow ups we show that multiple point Seshadri constants on projective complex surfaces can be calculated as the supremum of radii of multiple Kähler ball embeddings.
We determine the Lagrangian monodromy group L(T) and the smooth monodromy group S(T) of a Clifford torus T in the symplectic 4-space. We show that L(T) is isomorphic to the infinite dihedral group, and S(T) is generated by three reflections. We give explicit formulas for both groups. We also show that if a Lagrangian torus is smoothly isotopic to a Clifford torus then the smooth isotopy can be chosen to be Lagrangian outside of a disc.
We completely solve the symplectic packing problem with equally sized balls for any rational, ruled, symplectic 4-manifolds. We give explicit formulae for the packing numbers, the generalized Gromov widths, the stability numbers, and the corresponding obstructing exceptional classes. As a corollary, we give explicit values for when an ellipsoid of type $E(a, b)$, with $\frac{b}{a} \in \N$, embeds in a polydisc $P(s,t)$. Under this integrality assumption, we also give an alternative proof of a recent result of M. Hutchings showing that the ECH capacities give sharp inequalities for embedding el…
On Symplectic Packing Problems in Higher Dimensions
本文证明将辛曲面因子加入辛球打包问题可降维至四维情形，并猜想高维球打包由Gromov双球定理和体积控制，同时建立稳定化版本及凹环面域嵌入的类似结论。
Packing Integral Tori in Del Pezzo Surfaces
本文将 Hind 和 Kerman 的积分拉格朗日环面堆积结果从二维球面的乘积推广到 Del Pezzo 曲面，证明总能找到仅由 Clifford 环面构成的最大积分堆积。
Seshadri constants on $\mathbb{P}^1\times\mathbb{P}^1$, and applications to the symplectic packing problem
本文研究了$\mathbb{P}^1\times\mathbb{P}^1$上的Seshadri常数，通过反射方法给出了辛打包问题的显式公式，并探讨了奇偶$r$对结果的影响。
We show that the union of a meridian and a longitude of the symplectic 2-torus is superheavy in the sense of Entov-Polterovich. By a result of Entov-Polterovich, this implies that the product of this union and the Clifford torus of $\mathbb{C}P^n$ with the Fubini-Study symplectic form cannot be displaced by symplectomorphisms.
We define new Hamiltonian isotopy invariants for a monotone Lagrangian torus embedded in a symplectic 4-manifold. We show that, in the standard symplectic 4-space, these invariants distinguish a monotone Clifford torus from a Chekanov torus.
We prove that the count of Maslov index 2 $J$-holomorphic discs passing through a generic point of a real Lagrangian submanifold in a closed spherically monotone symplectic manifold must be even. As a corollary, we exhibit a genuine real symplectic phenomenon in terms of involutions, namely that the Chekanov torus $\mathbb{T}_{\text{Chek}}$ in $S^2\times S^2$, which is a monotone Lagrangian torus not Hamiltonian isotopic to the Clifford torus $\mathbb{T}_{\text{Clif}}$, can be seen as the fixed point set of a smooth involution, but not of an antisymplectic involution.
We consider symplectic manifolds with Hamiltonian torus actions which are "almost but not quite completely integrable": the dimension of the torus is one less than half the dimension of the manifold. We provide a complete set of invariants for such spaces when they are "centered" and the moment map is proper. In particular, this classifies the moment map preimages of all sufficiently small open sets, which is an important step towards global classification. As an application, we construct a full packing of each of the Grassmanians Gr^+(2,R^5) and Gr^+(2,R^6) by two equal symplectic balls.
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