Hyperbolic Plateau problems

2005/06/13 by Graham Smith, Smith, Graham
Mathematics · #32Q65 #51M10 #53C42 #53C45 #53D10 #58D10 #58J05 #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Mathematical Dynamics and Fractals #Symplectic Geometry (math.SG) #math.DG #math.SG #msc:32Q65 #msc:51M10 #msc:53C42 #msc:53C45 #msc:53D10 #msc:58D10 #msc:58J05

paper · pdf · doi:10.48550/arxiv.math/0506231

Complete redraft of the preceding version. Much shorter. Ideas essentially identical though slightly refined. Hopefully clearer

openalex publication_date 2005/06/13 · arxiv created 2011/05/23 · arxiv updated 2011/05/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider surfaces of constant Gaussian curvature immersed in 3-dimensional manifolds, and we strengthen the compactness result of Labourie in the case where the ambient manifold is 3-dimensional hyperbolic space. This allows us to prove results of existence of solutions to the asymptotic Plateau problem, as defined by Labourie, and the continuous dependence of these solutions on the data.

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