Constant Mean Curvature 1/2 Surfaces in H2 × R

dc.contributor.advisorRatzkin, Jesse
dc.contributor.authorChristian, Murray
dc.date.accessioned2020-02-25T11:37:29Z
dc.date.available2020-02-25T11:37:29Z
dc.date.issued2019
dc.date.updated2020-02-25T06:33:54Z
dc.description.abstractThis thesis lies in the field of constant mean curvature (cmc) hypersurfaces and specifically cmc 1/2 surfaces in the three-manifold H 2 × R. The value 1/2 is the critical mean curvature for H 2 × R, in that there do no exist closed cmc surfaces with mean curvature 1/2 or less. Daniel and Hauswirth have constructed a one-parameter family of complete, cmc 1/2 annuli that are symmetric about a reflection in the horizontal place H 2 × {0}, the horizontal catenoids. In this thesis we prove that these catenoids converge to a singular limit of two tangent horocylinders as the neck size tends to zero. We discuss the analytic gluing construction that this fact suggests, which would create a multitude of cmc 1/2 surfaces with positive genus. The main result of the thesis concerns a key step in such an analytic gluing construction. We construct families of cmc 1/2 annuli with boundary, whose single end is asymptotic to an end of a horizontal catenoid. We produce these families by solving the mean curvature equation for normal graphs off the end of a horizontal catenoid. This is a non-linear boundary value problem, which we solve by perturbative methods. To do so we analyse the linearised mean curvature operator, known as the Jacobi operator. We show that on carefully chosen weighted H¨older spaces the Jacobi operator can be inverted, modulo a finite-dimensional subspace, and provided the neck size of the horizontal catenoid is sufficiently small. Using these linear results we solve the boundary value problem for the mean curvature equation by a contraction mapping argument.
dc.identifier.apacitationChristian, M. (2019). <i>Constant Mean Curvature 1/2 Surfaces in H2 × R</i>. (). ,Faculty of Science ,Department of Maths and Applied Maths. Retrieved from http://hdl.handle.net/11427/31318en_ZA
dc.identifier.chicagocitationChristian, Murray. <i>"Constant Mean Curvature 1/2 Surfaces in H2 × R."</i> ., ,Faculty of Science ,Department of Maths and Applied Maths, 2019. http://hdl.handle.net/11427/31318en_ZA
dc.identifier.citationChristian, M. 2019. Constant Mean Curvature 1/2 Surfaces in H2 × R.en_ZA
dc.identifier.ris TY - Thesis / Dissertation AU - Christian, Murray AB - This thesis lies in the field of constant mean curvature (cmc) hypersurfaces and specifically cmc 1/2 surfaces in the three-manifold H 2 × R. The value 1/2 is the critical mean curvature for H 2 × R, in that there do no exist closed cmc surfaces with mean curvature 1/2 or less. Daniel and Hauswirth have constructed a one-parameter family of complete, cmc 1/2 annuli that are symmetric about a reflection in the horizontal place H 2 × {0}, the horizontal catenoids. In this thesis we prove that these catenoids converge to a singular limit of two tangent horocylinders as the neck size tends to zero. We discuss the analytic gluing construction that this fact suggests, which would create a multitude of cmc 1/2 surfaces with positive genus. The main result of the thesis concerns a key step in such an analytic gluing construction. We construct families of cmc 1/2 annuli with boundary, whose single end is asymptotic to an end of a horizontal catenoid. We produce these families by solving the mean curvature equation for normal graphs off the end of a horizontal catenoid. This is a non-linear boundary value problem, which we solve by perturbative methods. To do so we analyse the linearised mean curvature operator, known as the Jacobi operator. We show that on carefully chosen weighted H¨older spaces the Jacobi operator can be inverted, modulo a finite-dimensional subspace, and provided the neck size of the horizontal catenoid is sufficiently small. Using these linear results we solve the boundary value problem for the mean curvature equation by a contraction mapping argument. DA - 2019 DB - OpenUCT DP - University of Cape Town KW - applied maths LK - https://open.uct.ac.za PY - 2019 T1 - Constant Mean Curvature 1/2 Surfaces in H2 × R TI - Constant Mean Curvature 1/2 Surfaces in H2 × R UR - http://hdl.handle.net/11427/31318 ER - en_ZA
dc.identifier.urihttp://hdl.handle.net/11427/31318
dc.identifier.vancouvercitationChristian M. Constant Mean Curvature 1/2 Surfaces in H2 × R. []. ,Faculty of Science ,Department of Maths and Applied Maths, 2019 [cited yyyy month dd]. Available from: http://hdl.handle.net/11427/31318en_ZA
dc.language.rfc3066eng
dc.publisher.departmentDepartment of Maths and Applied Maths
dc.publisher.facultyFaculty of Science
dc.subjectapplied maths
dc.titleConstant Mean Curvature 1/2 Surfaces in H2 × R
dc.typeDoctoral Thesis
dc.type.qualificationlevelDoctoral
dc.type.qualificationnamePhD
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