The Cauchy problem for Liouville equation and Bryant surfaces

2003/10/07 by José A. Gálvez, Jose A. Galvez, Pablo Mira +2
Mathematics · #53A10 #53J15 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #math.AP #math.DG #msc:53A10 #msc:53J15

paper · pdf · doi:10.48550/arxiv.math/0310092

34 pages, 4 figures

arxiv created 2003/10/07 · openalex publication_date 2003/10/07 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We give a construction that connects the Cauchy problem for Liouville elliptic equation with a certain initial value problem for mean curvature one surfaces in hyperbolic 3-space H3, and solve both of them. We construct the only mean curvature one surface in H3 that passes through a given curve with given unit normal along it, and provide diverse applications. In particular, topics like period problems, symmetries, finite total curvature, planar geodesics, rigidity, etc. of surfaces are treated.

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