Horospherical means and uniform distribution of curves of constant geodesic curvature
type: Item
instance of: scholarly article
title: Horospherical means and uniform distribution of curves of constant geodesic curvature
zbMATH Open document ID: 0939.53018
DOI: 10.1007/PL00004745
author: Leon Karp
author: Norbert Peyerimhoff
published in: Mathematische Zeitschrift
publication date: 1999-09-29T00:00:00Z
review text: Let \(\mathbb{R} H^n\) denote the \(n\)-dimensional real hyperbolic space of constant sectional curvature \(K\equiv 1\), and let \(S_p(r)\) denote the sphere of radius \(r\) at a fixed point \(p\) of \(\mathbb{R} H^n\). For a continuous function \(f:\mathbb{R} H^n\to\mathbb{R}\), let \(M_p(f)(r)= (1/\text{vol}(S_P(r))) \int_{S_p (r)}f dS\) for every \(r>0\), the spherical mean of \(f\) for radius \(r\). A result of P. Günther says that if \(M\) is any compact quotient manifold of \(\mathbb{R} H^n\) with covering projection \(\pi:\mathbb{R} H^n\to M\), then \((1/\text{vol} (M))\int_Mf dM= \lim_{r\to \infty}M_p(f \circ\pi)(r)\). A key feature of this result is that the spheres \(\{S_p(r): r>0\}\) are the leaves of a codimension 1 foliation \({\mathcal F}\) of \(\mathbb{R} H^n-\{p\}\) in which each leaf has constant mean curvature and such that the 1-dimensional orthogonal foliation \({\mathcal F}^\perp\) is integrable with leaves that are geodesics of \(\mathbb{R} H^n\). In the present article, the authors extend the result and method of proof of Günther to the case where \(\mathbb{R} H^n\) is replaced by an arbitrary complete rank 1 Riemannian symmetric space \(\widetilde M\) with \(K<0\), and the pair of foliations \({\mathcal F}\) and \({\mathcal F}^\perp\) has the same properties as above but the leaves of \({\mathcal F}\) are allowed to be noncompact hypersurfaces of \(\widetilde M\). In this case, each leaf of \({\mathcal F}\) must possess an admissible exhaustion \(\{K_j:j\geq 1\}\) by compact sets; that is, area \((\partial K_j)/ \text{volume} (K_j)\to 0\) as \(j \to\infty\). An admissible exhaustion \(\{K_j\}\) on one leaf \(F=F\times\{0\}\) of \({\mathcal F}\) induces an admissible exhaustion \(\{K_j(t)\}\) on the leaf \(F \times \{t\}\) by flowing along the unit speed geodesics orthogonal to \({\mathcal F}\) for a time \(t\). The main result says that \(\lim_{j\to\infty} (1/\text{vol} (K_j))\int_{K_j} (f\circ\pi) dK_j=(1/ \text{vol} (M)) \int_Mf dM\), where…
Mathematics Subject Classification ID: 53C12
Mathematics Subject Classification ID: 53C42
Mathematics Subject Classification ID: 53C35
zbMATH DE Number: 1336869
zbMATH Keywords: horospherical means
zbMATH Keywords: constant mean curvature
zbMATH Keywords: rank 1 symmetric space
zbMATH Keywords: hyperbolic space
zbMATH Keywords: admissible exhaustion
reviewed by: Patrick Eberlein
MaRDI profile type: Publication
Wikidata QID: Q124999890
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description: scientific article; zbMATH DE number 1336869