标题： 曲线点，周长和凸环面域的复杂性 显示英文标题

标题： Curvy points, the perimeter, and the complexity of convex toric domains

摘要： 我们研究了环面边界的相关曲率和周长概念及其对辛填充问题的影响；一个自然的背景是广义凸环面区域，我们也对此进行了研究，其中除了凸性外，对矩形多面体没有任何条件。 我们证明了ECH和基本ECH容量的次主导渐进行为在下极限中恢复了这些区域的周长，而无需任何通用性要求，因此周长是完全填充的障碍。 作为应用，我们给出了通过紧致流形的开子集或无边界的紧致流形的填充稳定性失败的第一个例子；这对长期超级递归有影响。 我们还表明，环面边界上的一个正曲率的光滑点会阻碍无限楼梯的存在，并在此基础上完全分类了具有无限楼梯的光滑（广义）凸环面区域。 我们还将一些定理扩展到了广义凸环面区域，特别是“凹到凸”嵌入定理和“累积点定理”。 一个弯曲点导致“无限复杂性”；我们提出了一个问题，即无限复杂性区域是否可能具有无限楼梯，并给出了具有无限楼梯和任意高有限复杂性的例子。 显示英文摘要

摘要： We study the related notions of curvature and perimeter for toric boundaries and their implications for symplectic packing problems; a natural setting for this is a generalized version of convex toric domain which we also study, where there are no conditions on the moment polytope at all aside from convexity. We show that the subleading asymptotics of the ECH and elementary ECH capacities recover the perimeter of such domains in their liminf, without any genericity required, and hence the perimeter is an obstruction to a full filling. As an application, we give the first examples of the failure of packing stability by open subsets of compact manifolds with smooth boundary or with no boundary at all; this has implications for long-term super-recurrence. We also show that a single smooth point of positive curvature on the toric boundary obstructs the existence of an infinite staircase, and we build on this to completely classify smooth (generalized) convex toric domains which have an infinite staircase. We also extend a number of theorems to generalized convex toric domains, in particular the "concave to convex", embedding theorem and the "accumulation point theorem". A curvy point forces "infinite complexity"; we raise the question of whether an infinitely complex domain can ever have an infinite staircase and we give examples with infinite staircases and arbitrarily high finite complexity.