On the stability of the CMC Clifford tori as constrained Willmore surfaces. (English) Zbl 1273.53005
The tori \(T_r =rS^1\times sS^1\subset S^3\), where \(r^2 +s^2 =1\) are constrained Willmore surfaces, i.e., critical points of the Willmore functional among tori of the same conformal type. The authors give conditions for the \(T_r\) to be stable critical points.
Reviewer: A. Arvanitoyeorgos (Patras)
MSC:
| 53A05 | Surfaces in Euclidean and related spaces |
| 53A30 | Conformal differential geometry (MSC2010) |
| 58E30 | Variational principles in infinite-dimensional spaces |
References:
| [1] | Alías L.J., Piccione P.: Bifurcation of constant mean curvature tori in Euclidean spheres. J. Geom. Anal. 18, 687-703 (2011) · Zbl 1151.53053 · doi:10.1007/s12220-008-9029-8 |
| [2] | Bohle C., Peters P., Pinkall U.: Constrained Willmore surfaces. Calc. Var. Partial Differential Equations 38, 45-74 (2008) · Zbl 1153.53037 |
| [3] | Fischer A., Tromba A.: On a purely “Riemannian ” proof of the structure and dimension of the unramified moduli space of a compact Riemann surface. Math. Ann. 267, 311-345 (1984) · Zbl 0518.32015 · doi:10.1007/BF01456093 |
| [4] | Guo Z., Li H., Wang C.: The second variational formula for Willmore submanifolds in Sn. Results Math. 40, 205-225 (2001) · Zbl 1163.53312 · doi:10.1007/BF03322706 |
| [5] | Kilian, M., Schmidt, M.U., Schmitt, N.: Flows of constant mean curvature tori in the 3-sphere: the equivariant case (2010). http://arxiv.org/abs/1011.2875 · Zbl 1329.53087 |
| [6] | Kuwert, E., Schätzle, R.: Minimizers of the Willmore functional under fixed conformal class (2010). http://arxiv.org/abs/1009.6168 · Zbl 1276.53010 |
| [7] | Lamm T., Metzger J., Schulze F.: Foliations of asymptotically flat manifolds by surfaces of Willmore type. Math. Ann. 350, 1-78 (2011) · Zbl 1222.53028 · doi:10.1007/s00208-010-0550-2 |
| [8] | Lorenz, J.: Die Zweite Variation des Willmore-Funktionals und Constrained Willmore-Tori in S3. Diplom Thesis, Universität Freiburg (2012) · Zbl 1153.53037 |
| [9] | Ndiaye, C.B., Schätzle, R.: New Explicit examples of constrained Willmore minizers. Oberwolfach Report, No. 38, pp. 17-19 (2011) |
| [10] | Schätzle, R.: Conformally constrained Willmore immersions (2012). http://sfbtr71.de/index.php?page=publications&action=show&id=71 · Zbl 1282.53007 |
| [11] | Tromba A.J.: Teichmüller Theory in Riemannian Geometry. Birkhäuser, Basel (1992) · Zbl 0785.53001 · doi:10.1007/978-3-0348-8613-0 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
