On stability of equivariant minimal tori in the 3-sphere. (English) Zbl 1297.53042
Summary: We prove that amongst the equivariant constant mean curvature tori in the 3-sphere, the Clifford torus is the only local minimum of the Willmore energy. All other equivariant minimal tori in the 3-sphere are local maxima of the Willmore energy.
MSC:
| 53C42 | Differential geometry of immersions (minimal, prescribed curvature, tight, etc.) |
References:
| [1] | Pinkall, U.; Sterling, I., On the classification of constant mean curvature tori, Ann. of Math., 130, 407-451 (1989) · Zbl 0683.53053 |
| [2] | Bobenko, A. I., All constant mean curvature tori in \(R^3, S^3, H^3\) in terms of theta-functions, Math. Ann., 290, 209-245 (1991) · Zbl 0711.53007 |
| [3] | Ercolani, N. M.; Knörrer, H.; Trubowitz, E., Hyperelliptic curves that generate constant mean curvature tori in \(R^3\), Integrable Systems (Luminy, 1991), (Progr. Math., vol. 115 (1993), Birkhäuser Boston: Birkhäuser Boston Boston, MA), 81-114 · Zbl 0867.35082 |
| [4] | Carberry, E., Minimal tori in \(S^3\), Pacific J. Math., 233, 1, 41-69 (2007) · Zbl 1158.53047 |
| [5] | Burstall, F. E.; Kilian, M., Equivariant harmonic cylinders, Q. J. Math., 57, 449-468 (2006) · Zbl 1158.53050 |
| [6] | Kilian, M.; Schmidt, M. U.; Schmitt, N., Flows of constant mean curvature tori in the 3-sphere: the equivariant case, J. Reine Angew. Math. (2013), in press |
| [7] | Grinevich, P. G.; Schmidt, M. U., Period preserving nonisospectral flows and the moduli space of periodic solutions of soliton equations, Physica D, 87, 1-4, 73-98 (1995) · Zbl 1194.35345 |
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