Maximal surfaces and the universal Teichmüller space

Details Maximal surfaces and the universal Teichmüller space Maximal surfaces and the universal Teichmüller space

2009-11-20

arxiv.org/abs/0911.4124v1

Inventiones Mathematicae 182(2010):279-333

Mathematics

Differential Geometry

31 pages, 3 figures

Scientific paper

We show that any element of the universal Teichm\"uller space is realized by a unique minimal Lagrangian diffeomorphism from the hyperbolic plane to itself. The proof uses maximal surfaces in the 3-dimensional anti-de Sitter space. We show that, in $AdS^{n+1}$, any subset $E$ of the boundary at infinity which is the boundary at infinity of a space-like hypersurface bounds a maximal space-like hypersurface. In $AdS^3$, if $E$ is the graph of a quasi-symmetric homeomorphism, then this maximal surface is unique, and it has negative sectional curvature. As a by-product, we find a simple characterization of quasi-symmetric homeomorphisms of the circle in terms of 3-dimensional projective geometry.

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