双曲空间中的移动圆盘与共形容量的最小化
Mobile disks in hyperbolic space and minimization of conformal capacity
AI总结：
本文研究双曲空间中可移动不相交圆盘星座的共形容量下界，结合边界积分方程法和hp-FEM进行数值模拟与最小化，发现极值情形下圆盘会彼此聚集，类似北极动物群落减少热量散失的行为。
摘要：
我们的研究重点是双曲空间中不相交圆盘的星座，即配备双曲度量的单位圆盘。每个星座对应一个集合 $E$，它是 $m>2$ 个圆盘的并集，这些圆盘具有双曲半径 $r_j>0, j=1,...,m$。圆盘的中心不是固定的，因此星座中的各个圆盘可以在不重叠且双曲半径保持不变的约束下移动。我们的主要目标是为给定星座的共形容量寻找计算下界。即使在 $m=3$ 或 $m=4$ 的最简单情形下，容量对中心和半径的依赖方式也非常复杂。在缺乏解析方法的情况下，我们的工作基于数值模拟，分别使用了两种不同的数值方法：边界积分方程法和 $hp$-FEM 方法。我们的模拟将容量计算与最小化方法相结合，产生了圆盘彼此紧挨着分组的极值情形。这类似于北极地区动物群落最小化热量流动的行为。
英文摘要：
Our focus is to study constellations of disjoint disks in the hyperbolic space, the unit disk equipped with the hyperbolic metric. Each constellation corresponds to a set $E$ which is the union of $m>2$ disks with hyperbolic radii $r_j>0, j=1,...,m$. The centers of the disks are not fixed and hence individual disks of the constellation are allowed to move under the constraints that they do not overlap and their hyperbolic radii remain invariant. Our main objective is to find computational lower bounds for the conformal capacity of a given constellation. The capacity depends on the centers and radii in a very complicated way even in the simplest cases when $m=3$ or $m=4$. In the absence of analytic methods our work is based on numerical simulations using two different numerical methods, the boundary integral equation method and the $hp$-FEM method, resp. Our simulations combine capacity computation with minimization methods and produce extremal cases where the disks of the constellation are grouped next to each other. This resembles the behavior of animal colonies minimizing heat flow in arctic areas.
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