Moebius structures, hyperbolic ends and $k$-surfaces in hyperbolic space

We present a basic introduction to the theories of Möbius structures and hyperbolic ends and we study their applications to the theory of k-surfaces in 3-dimensional hyperbolic space. Classification AMS: 57K32, 53C18, 53C42, 58D10

[1]  I. Chavel.  Riemannian Geometry: Subject Index , 2006 .

[2]  Franccois Fillastre,et al.  Spherical, hyperbolic, and other projective geometries: convexity, duality, transitions , 2016, Eighteen Essays in Non-Euclidean Geometry.

[3]  Hyperbolic Plateau problems , 2005, math/0506231.

[4]  O. Lehto,et al.  Univalent functions and Teichm?uller space , 1986 .

[5]  Kevin P. Scannell.  Flat conformal structures and the classification of de Sitter manifolds , 1999 .

[6]  D. Dumas.  Complex projective structures , 2009, 0902.1951.

[7]  M. Gromov.  Sign and geometric meaning of curvature , 1991 .

[8]  L. Andersson,et al.  Cosmological time versus CMC time in spacetimes of constant curvature , 2012 .

[9]  J. Espinar,et al.  Hypersurfaces in H n+1 and conformally invariant equations: the generalized Christoffel and Nirenberg problems , 2007, 0706.2314.

[10]  Hypersurfaces in Hn and the space of its horospheres , 2001, math/0101248.

[11]  Albert Marden,et al.  The monodromy groups of Schwarzian equations on closed Riemann surfaces , 1995 .

[12]  W. Thurston.  Three dimensional manifolds, Kleinian groups and hyperbolic geometry , 1982 .

[13]  Graham Smith,et al.  Group actions and scattering problems in Teichmüller theory , 2016, 1605.04563.

[14]  J. Milnor.  Hyperbolic geometry: The first 150 years , 1982 .

[15]  B. Deroin,et al.  Branched projective structures with Fuchsian holonomy , 2012, 1203.6038.

[16]  Graham Smith.  Moduli of flat conformal structures of hyperbolic type , 2008, 0804.0744.

[17]  W. Thurston.  The geometry and topology of three-manifolds , 1979 .

[18]  F. Labourie.  Problèmes de Monge-Ampère, courbes holomorphes et laminations , 1997 .

[19]  M. Gromov,et al.  Manifolds of Nonpositive Curvature , 1985 .

[20]  William M. Goldman,et al.  Projective structures with Fuchsian holonomy , 1987 .

[21]  Global singularity theory for the Gauss curvature equation , 2012, Ensaios Matemáticos.