论文标题

在复曲面符号四个manifolds中的无限楼梯上

On infinite staircases in toric symplectic four-manifolds

论文作者

Cristofaro-Gardiner, Dan, Holm, Tara S., Mandini, Alessia, Pires, Ana Rita

论文摘要

McDuff和Schlenk的有影响力的结果断言,当四维符号椭球时编码的函数可以嵌入到一个四维球中具有出色的结构:该功能具有无限的角落,由奇数index fibonacci数字确定,由奇数属于fibonacci数字确定,以形成一个无限层面的楼梯。 最近,这项工作引起了人们对理解其他符号4个manifolds的椭圆形嵌入函数的兴趣,该功能由无限楼梯部分描述。我们提供了一个一般框架,用于分析这个问题的大型目标(称为有限型凸形旋转域),我们证明,这将概括为封闭的旋转曲面象征性4个manifolds。当目标是有限类型时,我们证明任何无限楼梯都必须具有唯一的累积点A_0,作为对显式二次方程的解决方案。此外,我们证明A_0处的嵌入功能必须等于经典卷的下限。特别是,我们的结果阻碍了我们显示的无限楼梯的存在。 在理性凸形的特殊情况下,我们可以说更多。我们猜想了无限楼梯存在问题的完整答案,这是六个家庭的区别,这些家族的瞬间是反射性的。然后,我们使用两种工具为我们的六个家庭提供了无限楼梯的存在的统一证明。首先,我们使用几乎要复曲面纤维的递归家族来查找符合性嵌入。对于第二个工具,我们发现了凸晶格路径的递归族,这些家族为嵌入提供了障碍物。最后,我们将猜想减少了,这些猜想是与Hardy和Littlewood的经典作品有关的数量理论中唯一的无限楼梯到一个问题。

An influential result of McDuff and Schlenk asserts that the function that encodes when a four-dimensional symplectic ellipsoid can be embedded into a four-dimensional ball has a remarkable structure: the function has infinitely many corners, determined by the odd-index Fibonacci numbers, that fit together to form an infinite staircase. This work has recently led to considerable interest in understanding when the ellipsoid embedding function for other symplectic 4-manifolds is partly described by an infinite staircase. We provide a general framework for analyzing this question for a large family of targets, called finite type convex toric domains, which we prove generalizes the class of closed toric symplectic 4-manifolds. When the target is of finite type, we prove that any infinite staircase must have a unique accumulation point a_0, given as the solution to an explicit quadratic equation. Moreover, we prove that the embedding function at a_0 must be equal to the classical volume lower bound. In particular, our result gives an obstruction to the existence of infinite staircases that we show is powerful. In the special case of rational convex toric domains, we can say more. We conjecture a complete answer to the question of existence of infinite staircases, in terms of six families that are distinguished by the fact that their moment polygon is reflexive. We then provide a uniform proof of the existence of infinite staircases for our six families, using two tools. For the first, we use recursive families of almost toric fibrations to find symplectic embeddings. For the second tool, we find recursive families of convex lattice paths that provide obstructions to embeddings. We conclude by reducing our conjecture that these are the only infinite staircases among rational convex toric domains to a question in number theory related to a classic work of Hardy and Littlewood.

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