Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 12 (2016), 009, 11 pages      arXiv:1406.4652      https://doi.org/10.3842/SIGMA.2016.009

Bipolar Lawson Tau-Surfaces and Generalized Lawson Tau-Surfaces

Broderick Causley
Department of Mathematics and Statistics, McGill University, Burnside Hall, 805 Sherbrooke Street West, Montreal, QC H3A 0B9, Canada

Received November 27, 2015, in final form January 21, 2016; Published online January 25, 2016

Abstract
Recently Penskoi [J. Geom. Anal. 25 (2015), 2645-2666, arXiv:1308.1628] generalized the well known two-parametric family of Lawson tau-surfaces $\tau_{r,m}$ minimally immersed in spheres to a three-parametric family $T_{a,b,c}$ 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 $\tilde{\tau}_{3,1}$. In the present paper we show in Theorem 1 that this three-parametric family $T_{a,b,c}$ includes in fact all bipolar Lawson tau-surfaces $\tilde{\tau}_{r,m}$. In Theorem 3 we show that no metric on generalized Lawson surfaces is maximal except for $\tilde{\tau}_{3,1}$ and the equilateral torus.

Key words: bipolar surface; Lawson tau-surface; minimal surface; extremal metric.

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References

  1. Bando S., Urakawa H., Generic properties of the eigenvalue of the Laplacian for compact Riemannian manifolds, Tôhoku Math. J. 35 (1983), 155-172.
  2. Berger M., Sur les premières valeurs propres des variétés riemanniennes, Compositio Math. 26 (1973), 129-149.
  3. Burstall F.E., Harmonic tori in spheres and complex projective spaces, J. Reine Angew. Math. 469 (1995), 149-177.
  4. Byrd P.F., Friedman M.D., Handbook of elliptic integrals for engineers and scientists, Die Grundlehren der mathematischen Wissenschaften, Vol. 67, 2nd ed., Springer-Verlag, New York - Heidelberg, 1971.
  5. Carberry E., Harmonic maps and integrable systems, in Geometry and Topology Down Under, Contemp. Math., Vol. 597, Amer. Math. Soc., Providence, RI, 2013, 139-163, arXiv:1211.3101.
  6. Colbois B., El Soufi A., Extremal eigenvalues of the Laplacian in a conformal class of metrics: the 'conformal spectrum', Ann. Global Anal. Geom. 24 (2003), 337-349, math.DG/0409316.
  7. El Soufi A., Giacomini H., Jazar M., A unique extremal metric for the least eigenvalue of the Laplacian on the Klein bottle, Duke Math. J. 135 (2006), 181-202, math.MG/0701773.
  8. El Soufi A., Ilias S., Riemannian manifolds admitting isometric immersions by their first eigenfunctions, Pacific J. Math. 195 (2000), 91-99.
  9. El Soufi A., Ilias S., Laplacian eigenvalue functionals and metric deformations on compact manifolds, J. Geom. Phys. 58 (2008), 89-104, math.MG/0701777.
  10. Erdélyi A., Magnus W., Oberhettinger F., Tricomi F.G., Higher transcendental functions. Vol. III, McGraw-Hill Book Company, New York, 1955.
  11. Hitchin N.J., Harmonic maps from a $2$-torus to the $3$-sphere, J. Differential Geom. 31 (1990), 627-710.
  12. Hsiang W.-Y., Lawson Jr. H.B., Minimal submanifolds of low cohomogeneity, J. Differential Geom. 5 (1971), 1-38.
  13. Jakobson D., Nadirashvili N., Polterovich I., Extremal metric for the first eigenvalue on a Klein bottle, Canad. J. Math. 58 (2006), 381-400, math.SP/0311484.
  14. Kao C., Lai R., Osting B., Maximization of Laplace-Beltrami eigenvalues on closed Riemannian surfaces, arXiv:1405.4944.
  15. Karpukhin M.A., Nonmaximality of extremal metrics on a torus and the Klein bottle, Sb. Math. 204 (2013), 1728-1744, arXiv:1210.8122.
  16. Karpukhin M.A., Spectral properties of bipolar surfaces to Otsuki tori, J. Spectr. Theory 4 (2014), 87-111, arXiv:1205.6316.
  17. Karpukhin M.A., Spectral properties of a family of minimal tori of revolution in five-dimensional sphere, Canad. Math. Bull. 58 (2015), 285-296, arXiv:1301.2483.
  18. Korevaar N., Upper bounds for eigenvalues of conformal metrics, J. Differential Geom. 37 (1993), 73-93.
  19. Lapointe H., Spectral properties of bipolar minimal surfaces in ${\mathbb S}^4$, Differential Geom. Appl. 26 (2008), 9-22, math.DG/0511443.
  20. Lawson Jr. H.B., Complete minimal surfaces in $S^{3}$, Ann. of Math. 92 (1970), 335-374.
  21. Mironov A.E., New examples of Hamilton-minimal and minimal Lagrangian submanifolds in ${\mathbb C}^n$ and ${\mathbb C}{\rm P}^n$, Sb. Math. 195 (2004), 85-96, math.DG/0309128.
  22. Mironov A.E., Finite-gap minimal Lagrangian surfaces in ${\mathbb C}{\rm P}^2$, in Riemann Surfaces, Harmonic Maps and Visualization, OCAMI Stud., Vol. 3, Osaka Munic. Univ. Press, Osaka, 2010, 185-196, arXiv:1005.3402.
  23. Nadirashvili N., Berger's isoperimetric problem and minimal immersions of surfaces, Geom. Funct. Anal. 6 (1996), 877-897.
  24. Nadirashvili N., Sire Y., Isoperimetric inequality for the third eigenvalue of the Laplace-Beltrami operator on $\mathbb{S}^2$, arXiv:1506.07017.
  25. Penskoi A.V., Extremal spectral properties of Lawson tau-surfaces and the Lamé equation, Mosc. Math. J. 12 (2012), 173-192, arXiv:1009.0285.
  26. Penskoi A.V., Extremal metrics for the eigenvalues of the Laplace-Beltrami operator on surfaces, Russian Math. Surveys 68 (2013), 1073-1130.
  27. Penskoi A.V., Extremal spectral properties of Otsuki tori, Math. Nachr. 286 (2013), 379-391, arXiv:1108.5160.
  28. Penskoi A.V., Generalized Lawson tori and Klein bottles, J. Geom. Anal. 25 (2015), 2645-2666, arXiv:1308.1628.
  29. Yang P.C., Yau S.T., Eigenvalues of the Laplacian of compact Riemann surfaces and minimal submanifolds, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 7 (1980), 55-63.

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