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Constant Mean Curvature 1/2 Surfaces in H2 × R
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...
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| Format: | Thesis |
| Language: | English |
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Department of Maths and Applied Maths
2020
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| _version_ | 1867613142821896192 |
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| access_status_str | Open Access |
| author | Christian, Murray |
| author2 | Ratzkin, Jesse |
| author_browse | Christian, Murray Ratzkin, Jesse |
| author_facet | Ratzkin, Jesse Christian, Murray |
| author_sort | Christian, Murray |
| collection | Thesis |
| description | 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. |
| format | Thesis |
| id | oai:open.uct.ac.za:11427/31318 |
| institution | University of Cape Town (South Africa) |
| language | eng |
| last_indexed | 2026-06-10T12:31:26.417Z |
| license_str | Not specified — see source repository |
| provenance_str_mv | Harvested via OAI-PMH from UCTD — University of Cape Town Open Access Repository |
| publishDate | 2020 |
| publishDateRange | 2020 |
| publishDateSort | 2020 |
| publisher | Department of Maths and Applied Maths |
| publisherStr | Department of Maths and Applied Maths |
| record_format | dspace |
| source_str | UCTD — University of Cape Town Open Access Repository |
| spelling | oai:open.uct.ac.za:11427/31318 Constant Mean Curvature 1/2 Surfaces in H2 × R Christian, Murray Ratzkin, Jesse applied maths 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. 2020-02-25T11:37:29Z 2020-02-25T11:37:29Z 2019 2020-02-25T06:33:54Z Doctoral Thesis Doctoral PhD http://hdl.handle.net/11427/31318 eng application/pdf Department of Maths and Applied Maths Faculty of Science |
| spellingShingle | applied maths Christian, Murray Constant Mean Curvature 1/2 Surfaces in H2 × R |
| thesis_degree_str | Doctoral |
| title | Constant Mean Curvature 1/2 Surfaces in H2 × R |
| title_full | Constant Mean Curvature 1/2 Surfaces in H2 × R |
| title_fullStr | Constant Mean Curvature 1/2 Surfaces in H2 × R |
| title_full_unstemmed | Constant Mean Curvature 1/2 Surfaces in H2 × R |
| title_short | Constant Mean Curvature 1/2 Surfaces in H2 × R |
| title_sort | constant mean curvature 1 2 surfaces in h2 r |
| topic | applied maths |
| url | http://hdl.handle.net/11427/31318 |
| work_keys_str_mv | AT christianmurray constantmeancurvature12surfacesinh2r |