Intersection probabilities for flats in hyperbolic space

• Published in: arXiv.org
• Main Authors: Sönmez, Ercan, Spanos, Panagiotis, Thäle, Christoph
• Format: Paper
• Language: English
• Published: Ithaca Cornell University Library, arXiv.org 15.07.2024
• Subjects: Euclidean geometry; Hyperbolic coordinates; Intersections; Manifolds (mathematics); Phase transitions
• ISSN: 2331-8422

Consider the \(d\)-dimensional hyperbolic space \(\mathbb{M}_K^d\) of constant curvature \(K<0\) and fix a point \(o\) playing the role of an origin. Let \(\mathbf{L}\) be a uniform random \(q\)-dimensional totally geodesic submanifold (called \(q\)-flat) in \(\mathbb{M}_K^d\) passing through \(o\) and, independently of \(\mathbf{L}\), let \(\mathbf{E}\) be a random \((d-q+\gamma)\)-flat in \(\mathbb{M}_K^d\) which is uniformly distributed in the set of all \((d-q+\gamma)\)-flats intersecting a hyperbolic ball of radius \(u>0\) around \(o\). We are interested in the distribution of the random \(\gamma\)-flat arising as the intersection of \(\mathbf{E}\) with \(\mathbf{L}\). In contrast to the Euclidean case, the intersection \(\mathbf{E}\cap \mathbf{L}\) can be empty with strictly positive probability. We determine this probability and the full distribution of \(\mathbf{E}\cap \mathbf{L}\). Thereby, we elucidate crucial differences to the Euclidean case. Moreover, we study the limiting behaviour as \(d\uparrow\infty\) and also \(K\uparrow 0\). Thereby we obtain a phase transition with three different phases which we completely characterize, including a critical phase with distinctive behavior and a phase recovering the Euclidean results. In the background are methods from hyperbolic integral geometry.