Cuschieri, Thomas (2009) Complete noncompact CMC surfaces in hyperbolic 3-space. PhD thesis, University of Warwick.

PDF WRAP_THESIS_Cuschieri_2009.pdf - Requires a PDF viewer such as GSview, Xpdf or Adobe Acrobat Reader Download (493Kb)

Abstract

In this thesis we study the asymptotic Plateau problem for surfaces with constant mean curvature (CMC) in hyperbolic 3-space H3. We give a new, geometrically transparent proof of the existence of a CMC surface spanning any given Jordan curve on the sphere at infinity of H3, for mean curvature lying in the range (-1,1). Our proof does not require methods from geometric measure theory, and yields an immersed disk as solution. We then study the dependence of the solution surface on the boundary data. We view the set of H-surfaces (CMC surfaces with mean curvature equal to H) as consisting of the conformal H-harmonic maps. We therefore begin by showing smooth dependence on boundary data for H-harmonic maps (with |H| < 1) which solve a Dirichlet problem at infinity. This is achieved by showing that the linearised H-harmonic map operator is invertible as a map between appropriate function spaces. Finally we show smooth dependence on boundary data for H-surfaces which lie in a neighbourhood of the totally umbilic spherical caps {H}. This is achieved by studying the mapping properties of the so-called conformality operator. We use methods from complex geometry to show that the linearisation of this operator at a cap H is an isomorphism for all H ∈ (−1, 1).

Item Type:	Thesis or Dissertation (PhD)
Subjects:	Q Science > QA Mathematics
Library of Congress Subject Headings (LCSH):	Surfaces of constant curvature, Boundary value problems -- Asymptotic theory, Spherical functions, Dirichlet problem -- Asymptotic theory, Asymptotic expansions
Date:	October 2009
Institution:	University of Warwick
Theses Department:	Mathematics Institute
Thesis Type:	PhD
Publication Status:	Unpublished
Supervisor(s)/Advisor:	Micallef, Mario
Format of File:	pdf
Extent:	77 leaves : charts
Language:	eng
URI:	http://wrap.warwick.ac.uk/id/eprint/3135

Request changes to a record

Actions (login required)

View Item