Bipolar Lawson Tau-Surfaces and Generalized Lawson Tau-Surfaces
Recently Penskoi [J. Geom. Anal. 25 (2015), 2645-2666, arXiv:1308.1628] generalized the well known two-parametric family of Lawson tau-surfaces τr,m minimally immersed in spheres to a three-parametric family Ta,b,c of tori and Klein bottles minimally immersed in spheres. It was remarked that this fa...
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| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Date: | 2016 |
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| Format: | Article |
| Language: | English |
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Інститут математики НАН України
2016
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/147428 |
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| Cite this: | Bipolar Lawson Tau-Surfaces and Generalized Lawson Tau-Surfaces / B. Causley // Symmetry, Integrability and Geometry: Methods and Applications. — 2016. — Т. 12. — Бібліогр.: 29 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862684121436782592 |
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| author | Causley, B. |
| author_facet | Causley, B. |
| citation_txt | Bipolar Lawson Tau-Surfaces and Generalized Lawson Tau-Surfaces / B. Causley // Symmetry, Integrability and Geometry: Methods and Applications. — 2016. — Т. 12. — Бібліогр.: 29 назв. — англ. |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | Recently Penskoi [J. Geom. Anal. 25 (2015), 2645-2666, arXiv:1308.1628] generalized the well known two-parametric family of Lawson tau-surfaces τr,m minimally immersed in spheres to a three-parametric family Ta,b,c of tori and Klein bottles minimally immersed in spheres. It was remarked that this family includes surfaces carrying all extremal metrics for the first non-trivial eigenvalue of the Laplace-Beltrami operator on the torus and on the Klein bottle: the Clifford torus, the equilateral torus and surprisingly the bipolar Lawson Klein bottle τ¯₃,₁. In the present paper we show in Theorem 1 that this three-parametric family Ta,b,c includes in fact all bipolar Lawson tau-surfaces τ¯r,m. In Theorem 3 we show that no metric on generalized Lawson surfaces is maximal except for τ¯₃,₁ and the equilateral torus.
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| first_indexed | 2025-12-07T15:58:02Z |
| format | Article |
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| id | nasplib_isofts_kiev_ua-123456789-147428 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-07T15:58:02Z |
| publishDate | 2016 |
| publisher | Інститут математики НАН України |
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| spelling | Causley, B. 2019-02-14T18:30:28Z 2019-02-14T18:30:28Z 2016 Bipolar Lawson Tau-Surfaces and Generalized Lawson Tau-Surfaces / B. Causley // Symmetry, Integrability and Geometry: Methods and Applications. — 2016. — Т. 12. — Бібліогр.: 29 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 58E11; 58J50; 49Q05; 35P15 DOI:10.3842/SIGMA.2016.009 https://nasplib.isofts.kiev.ua/handle/123456789/147428 Recently Penskoi [J. Geom. Anal. 25 (2015), 2645-2666, arXiv:1308.1628] generalized the well known two-parametric family of Lawson tau-surfaces τr,m minimally immersed in spheres to a three-parametric family Ta,b,c of tori and Klein bottles minimally immersed in spheres. It was remarked that this family includes surfaces carrying all extremal metrics for the first non-trivial eigenvalue of the Laplace-Beltrami operator on the torus and on the Klein bottle: the Clifford torus, the equilateral torus and surprisingly the bipolar Lawson Klein bottle τ¯₃,₁. In the present paper we show in Theorem 1 that this three-parametric family Ta,b,c includes in fact all bipolar Lawson tau-surfaces τ¯r,m. In Theorem 3 we show that no metric on generalized Lawson surfaces is maximal except for τ¯₃,₁ and the equilateral torus. This result was obtained during studies of the author at the National Research University –
 Higher School of Economics, Moscow and the author is very grateful for its hospitality. The
 author also thanks A.V. Penskoi for the statement of this problem, many useful discussions and
 invaluable help in preparing this manuscript. The research of the author was partially supported
 by an NSERC Postgraduate Fellowship and by AG Laboratory NRU-HSE, Russian Federation
 government grant, ag. 11.G34.31.0023. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Bipolar Lawson Tau-Surfaces and Generalized Lawson Tau-Surfaces Article published earlier |
| spellingShingle | Bipolar Lawson Tau-Surfaces and Generalized Lawson Tau-Surfaces Causley, B. |
| title | Bipolar Lawson Tau-Surfaces and Generalized Lawson Tau-Surfaces |
| title_full | Bipolar Lawson Tau-Surfaces and Generalized Lawson Tau-Surfaces |
| title_fullStr | Bipolar Lawson Tau-Surfaces and Generalized Lawson Tau-Surfaces |
| title_full_unstemmed | Bipolar Lawson Tau-Surfaces and Generalized Lawson Tau-Surfaces |
| title_short | Bipolar Lawson Tau-Surfaces and Generalized Lawson Tau-Surfaces |
| title_sort | bipolar lawson tau-surfaces and generalized lawson tau-surfaces |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/147428 |
| work_keys_str_mv | AT causleyb bipolarlawsontausurfacesandgeneralizedlawsontausurfaces |