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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Опубліковано в: :Symmetry, Integrability and Geometry: Methods and Applications
Дата:2016
Автор: Causley, B.
Формат: Стаття
Мова:Англійська
Опубліковано: Інститут математики НАН України 2016
Онлайн доступ:https://nasplib.isofts.kiev.ua/handle/123456789/147428
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Назва журналу:Digital Library of Periodicals of National Academy of Sciences of Ukraine
Цитувати: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
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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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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