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Hyperbolic Plateau problems | | Graham Smith
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13 Jun 2005 | | Subject: | Differential Geometry; Symplectic Geometry MSC-class: 32Q65; 51M10; 53C42; 53C45; 53D10; 58D10; 58J05 | math.DG math.SG | | Abstract: | Using the notion of Plateau problems defined by Labourie in ``Un lemme de Morse pour les surfaces convexes’’, we show that, for every $kin(0,1)$, and for every locally conformal mapping from the Poincaré disc $Bbb{D}$ into the ideal boundary $partial_inftyBbb{H}^3$ of three dimensional hyperbolic space $Bbb{H}^3$, viewed as a Plateau problem, there is a unique convex immersed surface $Sigma$ in $Bbb{H}^3$ of constant Gaussian curvature equal to $k$ which is a solution to this problem. We thus obtain a homeomorphism between the space of locally conformal mappings from $Bbb{D}$ into $hat{Bbb{C}}congpartial_inftyBbb{H}^3$ and a family of holomorphic immersions from $Bbb{D}$. | | Source: | arXiv, math.DG/0506231 | | Services: | Forum | Review | PDF | Favorites | |
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