Apollonian地毯与超Kähler流形的Kähler锥边界
Apollonian carpets and the boundary of the Kahler cone of a hyperkahler manifold
AI总结：
该研究探讨超Kähler流形丰沛锥的迷向边界与全纯自同构群动力学的关系，证明迷向边界上无理点均属极限集，实解析曲线含于测地球面，还研究了Apollonian地毯的分形几何及其与自同构群的关联。
摘要：
紧Kähler n维流形M的丰沛锥是其Kähler锥与由整(1,1)-类生成的实子空间的交。它的迷向边界是其边界上所有满足∫_M ηⁿ=0的点η的集合。我们关注超Kähler流形丰沛锥的迷向边界形状与其全纯自同构群G的动力学之间的关系。在此情形下，丰沛锥的射影化可实现为双曲空间H中的一个开的、局部多面体的子集。迷向边界S可实现为H的双曲边界（绝对形）A的一个子集，而A自然等同于一个欧氏球面。显然，迷向边界S包含G作用在其丰沛锥上的极限集。我们证明反过来，S上的所有无理点都属于该极限集。利用N. Shah关于双曲流形上测地流作用下曲线的极限分布的一个结果，我们证明S中的每条实解析曲线都包含在S的一个测地球面中；而当存在这类曲线时，极限集就是这些测地球面之并的闭包。我们研究了这类被称为Apollonian地毯的分形集的几何性质，并建立了Apollonian地毯与自同构群结构之间的联系。
英文摘要：
The ample cone of a compact Kahler $n$-manifold $M$ is the intersection of its Kahler cone and the real subspace generated by integer (1,1)-classes. Its isotropic boundary is the set of all points $η$ on its boundary such that $\int_M η^n=0$. We are interested in the relation between the shape of the isotropic boundary of the ample cone of a hyperkahler manifold and the dynamics of its holomorphic automorphism group $G$. In this case, the projectivization of the ample cone is realized as an open, locally polyhedral subset in a hyperbolic space $H$. The isotropic boundary $S$ is realized as a subset of the hyperbolic boundary (the absolute) $A$ of $H$, which is naturally identified with a Euclidean sphere. It is clear that the isotropic boundary $S$ contains the limit set of $G$ acting on its ample cone. We prove that, conversely, all irrational points on $S$ belong to the limit set. Using a result of N. Shah about limiting distributions of curves under geodesic flow on hyperbolic manifolds, we prove that every real analytic curve in $S$ is contained in a geodesic sphere in $S$,and in presence of such curves the limit set is the closure of the union of these geodesic spheres. We study the geometry of such fractal sets, called Apollonian carpets, and establish the link between the Apollonian carpet and the structure of the automorphism group.
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