Bounds for several-disk packings of hyperbolic surfaces. (English) Zbl 1445.52014
The paper under review considers the problem of packing disjoint open metric disks in hyperbolic surfaces. The general motivating question the author states is the following: for a fixed \(k \in \mathbb N\) and topological manifold \(M\) that admits a complete constant-curvature metric of finite volume, what is the supremal density of packings of \(M\) by \(k\) balls of equal radius, taken over all such metrics with fixed curvature?
Briefly mentioning the Euclidean case of this problem, the author then focuses on the \(2\)-dimensional hyperbolic situation. Here he proves an upper bound on the radius of a packing of a complete, finite-area, \(n\)-cusped hyperbolic surface by \(k\) disks of equal radius in terms of \(k, n\) and the Euler characteristic of this surface. The author gives topological conditions for the surface to have such a packing, and discusses specific situations for which his bound is attained, thus showing its optimality. Such optimal surfaces can be orientable or non-orientable, but there are only finitely many of them for each fixed \(k, n\) and Euler characteristic.
Briefly mentioning the Euclidean case of this problem, the author then focuses on the \(2\)-dimensional hyperbolic situation. Here he proves an upper bound on the radius of a packing of a complete, finite-area, \(n\)-cusped hyperbolic surface by \(k\) disks of equal radius in terms of \(k, n\) and the Euler characteristic of this surface. The author gives topological conditions for the surface to have such a packing, and discusses specific situations for which his bound is attained, thus showing its optimality. Such optimal surfaces can be orientable or non-orientable, but there are only finitely many of them for each fixed \(k, n\) and Euler characteristic.
Reviewer: Lenny Fukshansky (Claremont)
MSC:
| 52C15 | Packing and covering in \(2\) dimensions (aspects of discrete geometry) |
| 05B45 | Combinatorial aspects of tessellation and tiling problems |
| 57M50 | General geometric structures on low-dimensional manifolds |
Software:
kepler98References:
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