Notes on the Schwarzian tensor and measured foliations at infinity of quasifuchsian manifolds

2017/08/06 by Jean‐Marc Schlenker, Schlenker, Jean-Marc
Mathematics · #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Geometry and complex manifolds

paper · pdf · doi:10.48550/arxiv.1708.01852

openalex publication_date 2017/08/06 · openalex created_date 2017/08/17 · openalex updated_date 2026/07/28

Abstract

The boundary at infinity of a quasifuchsian hyperbolic manifold is equiped with a holomorphic quadratic differential. Its horizontal measured foliation f can be interpreted as the natural analog of the measured bending lamination on the boundary of the convex core. This analogy leads to a number of questions. We provide a variation formula for the renormalized volume in terms of the extremal length \ext(f) of f, and an upper bound on \ext(f). We then describe two extensions of the holomorphic quadratic differential at infinity, both valid in higher dimensions. One is in terms of Poincaré-Einstein metrics, the other (specifically for conformally flat structures) of the second fundamental form of a hypersurface in a "constant curvature" space with a degenerate metric, interpreted as the space of horospheres in hyperbolic space. This clarifies a relation between linear Weingarten surfaces in hyperbolic manifolds and Monge-Ampère equations.

Citations

• On the Renormalized Volume of Hyperbolic 3-Manifolds
• Volume and Area Renormalizations for Conformally Compact Einstein Metrics

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