Tubular neighborhood of totally geodesic hypersurfaces in hyperbolic manifolds
This paper contains several results on hyperbolic manifolds (constant curvature -1). The first result is the following higher-dimensional generalization of the well-known collar lemma on surfaces: Let \(M^ n\) be a hyperbolic manifold of dimension \(n\) and \(\Sigma\) an embedded totally geodesic hypersurface of area \(A\). Then \(\Sigma\) has a tubular neighborhood of width \(c_ n(A)\), where \(c_ n\) is a function for which the author gives an explicit formula in terms of the volume of the hyperbolic ball of radius \(n\). The author proves also the following related results: There exists a constant \(a_ n\), depending only on \(n\), such that if \(M^ n\) contains \(K\) disjoint closed embedded totally geodesic hypersurfaces, then \(\text{Vol}(M) > Ka_ n\). In dimensions \(n \geq 4\), if \(M^ n_ i\) is a sequence of (not necessarily distinct) hyperbolic \(n\)-manifolds each containing an embedded totally geodesic closed hypersurface of area \(A_ i\), where \(A_ i \to \infty\), then the volumes of the tubular neighborhoods of width \(c_ n(A_ i)\) tend to infinity. In contrast, for \(n = 3\), there exist examples of totally geodesic closed surfaces embedded in hyperbolic 3-manifolds whose areas get arbitrarily large, but whose best possible tubular neighborhoods have bounded volume. There are several other corollaries and applications, in particular a hyperbolic quantitative version of the Klein-Maskit combination theorem (in all dimensions) for free products of Fuchsian groups, as well as upper bounds for the lowest Dirichlet eigenvalue for domains in \(M^ n\). The reviewer considers that the paper is very well written. The proofs are self- contained and involve basic hyperbolic geometry.
scientific article; zbMATH DE number 617412
scientific article