On the Convergence of Non-Integer Linear Hopf Flow
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2022
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| _version_ | 1866915496169832448 |
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| author | Guilfoyle, Brendan Robson, Morgan |
| author_facet | Guilfoyle, Brendan Robson, Morgan |
| contents | The evolution of a rotationally symmetric surface by a linear combination of its radii of curvature equation is considered. It is known that if the coefficients form certain integer ratios the flow is smooth and can be integrated explicitly. In this paper the non-integer case is considered for certain values of the coefficients and with mild analytic restrictions on the initial surface.
We prove that if the focal points at the north and south poles on the initial surface coincide, the flow converges to a round sphere. Otherwise the flow converges to a non-round Hopf sphere. Conditions on the fall-off of the astigmatism at the poles of the initial surface are also given that ensure the convergence of the flow.
The proof uses the spectral theory of singular Sturm-Liouville operators to construct an eigenbasis for an appropriate space in which the evolution is shown to converge. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2205_15978 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | On the Convergence of Non-Integer Linear Hopf Flow Guilfoyle, Brendan Robson, Morgan Differential Geometry Analysis of PDEs Classical Analysis and ODEs Spectral Theory Primary 35K10, Secondary 53A05 The evolution of a rotationally symmetric surface by a linear combination of its radii of curvature equation is considered. It is known that if the coefficients form certain integer ratios the flow is smooth and can be integrated explicitly. In this paper the non-integer case is considered for certain values of the coefficients and with mild analytic restrictions on the initial surface. We prove that if the focal points at the north and south poles on the initial surface coincide, the flow converges to a round sphere. Otherwise the flow converges to a non-round Hopf sphere. Conditions on the fall-off of the astigmatism at the poles of the initial surface are also given that ensure the convergence of the flow. The proof uses the spectral theory of singular Sturm-Liouville operators to construct an eigenbasis for an appropriate space in which the evolution is shown to converge. |
| title | On the Convergence of Non-Integer Linear Hopf Flow |
| topic | Differential Geometry Analysis of PDEs Classical Analysis and ODEs Spectral Theory Primary 35K10, Secondary 53A05 |
| url | https://arxiv.org/abs/2205.15978 |