Bounding the bending of a hyperbolic 3-manifold
The authors obtain bounds on the total bending of the boundary of the convex core of a hyperbolic 3-manifold \(N\). The convex core \(C(N)\) is obtained as follows: The hyperbolic manifold \(N\) is a quotient \(N = {\mathbf H}^3/\Gamma\), where hyperbolic 3-space \({\mathbf H}^3\) is the universal cover of \(N\) and \(\Gamma\approx\pi_1(N)\) is the group of covering translations. The group \(\Gamma\) has limit set \(\Lambda\subset{\mathbf S}^2= \partial{\mathbf H}^3\). The set \(\Lambda\) has convex hull \(\text{CH}(\Lambda)\subset{\mathbf H}^3\). The convex core \(C(N)\) is the quotient \(\text{CH}(\Lambda)/\Gamma\). The boundary \(\partial C(N)\) of \(C(N)\) is a hyperbolic surface in its intrinsic metric. It is totally geodesic except along a geodesic lamination called the bending lamination. The total bending of this lamination is, roughly, the integral of a product, namely the product of lamination length and bending angle. The authors obtain an upper bound on total bending that depends linearly on the Euler characteristic of \(\partial C(N)\) and takes into account the injectivity radius of \(\partial CH(\Lambda)\). If \(\partial C(N)\) has a nontrivial short curve compressible in \(C(N)\), the authors obtain a lower bound on the total bending. Calculations depend on an analysis of the intersections of geodesic curves with the bending lamination.
scientific article; zbMATH DE number 5117292