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On Otsuki tori and their Willmore energy. (English) Zbl 1250.53010

An Otsuki torus [T. Otsuki, Am. J. Math. 92, 145–173 (1970; Zbl 0196.25102)] is a compact, minimal and embedded hypersurface in \(\mathbb{S}^3 \subset \mathbb{R}^4\) that can be parametrized by \[ \begin{aligned} x &= \sqrt{1-h^2-(h')^2} \cos\alpha, \\ y &= \sqrt{1-h^2-(h')^2} \sin\alpha, \\ u &= h(\theta)\sin\theta + h'(\theta)\cos\theta, \\ v &= h'(\theta)\sin\theta - h(\theta)\cos\theta \end{aligned} \] where \(h(\theta)\) is a (necessarily periodic) solution to a certain differential equation with a rationality constraint on its minimal period. Alternative definitions are conceivable as well. Otsuki tori (with the single exception of the Clifford torus) are in a natural one-to-one correspondence with rational numbers in the interval \((1/2, \sqrt{2}/2)\). If \(p/q\) is a reduced rational number in this interval, the corresponding Otsuki torus is denoted by \(O_{p/q}\).
The authors’ main result is the estimate \(4 \pi q < W(O_{p/q}) < \sqrt{2} \pi^2 q\) for the Willmore energy \(W(O_{p/q})\) of Otsuki tori. If the torus is invariant under the antipodal map, the lower bound can be improved to \(W(O_{p/q}) > 32\pi\). These results show that the Willmore conjecture holds for Otsuki tori.

MSC:

53A30 Conformal differential geometry (MSC2010)
53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature

Citations:

Zbl 0196.25102
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References:

[1] Otsuki, T., Minimal hypersurfaces in a Riemannian manifold of constant curvature, Amer. J. Math., 92, 145-173 (1970) · Zbl 0196.25102
[2] A.V. Penskoi, Extremal spectral properties of Otsuki tori, 2011. arXiv:math.SP/1108.5160vl.; A.V. Penskoi, Extremal spectral properties of Otsuki tori, 2011. arXiv:math.SP/1108.5160vl. · Zbl 1271.58007
[3] Hsiang, W. Y.; Lawson, H. B., Minimal submanifold of low cohomogeneity, J. Differential Geom., 5, 1-38 (1971) · Zbl 0219.53045
[4] Willmore, T. J., Note on embedded surfaces, An. Şiinţ. Univ. AI. I. Cuza Iasi Ser. B, 11, 493-496 (1965) · Zbl 0171.20001
[5] Topping, P. M., Towards the Willmore conjecture, Calc. Var. Partial Differential Equations, 11, 361-393 (2000) · Zbl 1058.53060
[6] Langer, J.; Singer, D., On the total squared curvature on closed curves in space forms, J. Differential Geom., 20, 1-22 (1984) · Zbl 0554.53013
[7] Li, P.; Yau, S. T., A new conformal invariant and its applications to the Willmore conjecture and the first eigenvalue of compact surfaces, Invent. Math., 69, 269-291 (1982) · Zbl 0503.53042
[8] Ros, A., The Willmore conjecture in the real projective space, Math. Res. Lett., 6, 487-493 (1999) · Zbl 0951.53044
[9] Topping, P. M., An approach to the Willmore conjecture, Clay Math. Proc., 2, 769-772 (2005) · Zbl 1110.53004
[10] Simon, L., Existence of surfaces minimizing the Willmore functional, Comm. Anal. Geom., 2, 281-326 (1993) · Zbl 0848.58012
[11] Barros, M.; Ferrández, A., Willmore energy estimates in conformal Berger sphere, Chaos Solitons Fractals, 44, 515-521 (2011) · Zbl 1223.53027
[12] F.C. Marques, A. Neves, Min-Max theory and the Willmore conjecture, 2012. arXiv:math.DG/1202.6036vl.; F.C. Marques, A. Neves, Min-Max theory and the Willmore conjecture, 2012. arXiv:math.DG/1202.6036vl. · Zbl 1297.49079
[13] Bryant, R. L., A duality theorem for Willmore surfaces, J. Differential Geom., 20, 23-53 (1984) · Zbl 0555.53002
[14] Ejiri, N., Willmore surfaces with a duality in \(S^N(1)\), Proc. Lond. Math. Soc. (3), 57, 383-416 (1988) · Zbl 0671.53043
[15] Li, H.; Vrancken, L., New examples of Willmore surfaces in \(S^n\), Ann. Global Anal. Geom., 23, 205-225 (2003) · Zbl 1033.53049
[16] Mondino, A., Some results about the existence of critical points for the Willmore functional, Math. Z., 266, 583-622 (2010) · Zbl 1205.53046
[17] Musso, E., Willmore surfaces in the four-sphere, Ann. Global Anal. Geom., 8, 21-41 (1990) · Zbl 0705.53028
[18] Pinkall, U., Hopf tori in \(S^3\), Invent. Math., 81, 379-386 (1985) · Zbl 0585.53051
[19] Otsuki, T., On a differential equation related with differential geometry, Mem. Fac. Sci. Kyushu Univ. Ser. A, 47, 245-281 (1993) · Zbl 0802.34047
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