Reference : Notes on the Schwarzian tensor and measured foliations at infinity of quasifuchsian m...
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Notes on the Schwarzian tensor and measured foliations at infinity of quasifuchsian manifolds.
Schlenker, Jean-Marc mailto [University of Luxembourg > Faculty of Science, Technology and Communication (FSTC) > Mathematics Research Unit >]
[en] 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 $\ext(f)$. \par We then describe two extensions of the holomorphic quadratic differential at infinity, both valid in higher dimensions. One is in terms of Poincar\'e-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-Amp\`ere equations.
Notes aiming at clarifying the relations between different points of view and introducing one new notion, no real result. Not intended to be submitted at this point

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