×

zbMATH — the first resource for mathematics

Minimizers of the Willmore functional with a small area constraint. (English) Zbl 1290.49090
Summary: We show the existence of a smooth spherical surface minimizing the Willmore functional subject to an area constraint in a compact Riemannian three-manifold, provided the area is small enough. Moreover, we partially classify complete surfaces of Willmore type with positive mean curvature in Riemannian three-manifolds.

MSC:
49Q20 Variational problems in a geometric measure-theoretic setting
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Bauer, M.; Kuwert, E., Existence of minimizing Willmore surfaces of prescribed genus, Int. Math. Res. Not., 10, 553-576, (2003) · Zbl 1029.53073
[2] J. Chen, Y. Li, Bubble tree of a class of conformal mappings and applications to the Willmore functional, preprint, 2011.
[3] De Lellis, C.; Müller, S., Optimal rigidity estimates for nearly umbilical surfaces, J. Differential Geom., 69, 75-110, (2005) · Zbl 1087.53004
[4] De Lellis, C.; Müller, S., A \(C^0\) estimate for nearly umbilical surfaces, Calc. Var. Partial Differential Equations, 26, 283-296, (2006) · Zbl 1100.53005
[5] Fischer-Colbrie, D.; Schoen, R., The structure of complete stable minimal surfaces in 3-manifolds of nonnegative scalar curvature, Comm. Pure Appl. Math., 33, 199-211, (1980) · Zbl 0439.53060
[6] Kuwert, E.; Li, Y., \(W^{2, 2}\)-conformal immersions of a closed Riemann surface into \(\mathbb{R}^n\), Comm. Anal. Geom., 20, 313-340, (2012) · Zbl 1271.53010
[7] E. Kuwert, A. Mondino, J. Schygulla, Existence of immersed spheres minimizing curvature functionals in compact 3-manifolds, preprint, 2011. · Zbl 1295.53028
[8] E. Kuwert, R. Schätzle, Minimizers of the Willmore functional under fixed conformal class, J. Differential Geom., in press.
[9] Lamm, T.; Metzger, J., Small surfaces of Willmore type in Riemannian manifolds, Int. Math. Res. Not., 19, 3786-3813, (2010) · Zbl 1202.53056
[10] 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
[11] Mondino, A., Some results about the existence of critical points for the Willmore functional, Math. Z., 266, 583-622, (2010) · Zbl 1205.53046
[12] A. Mondino, The conformal Willmore functional: a perturbative approach, J. Geom. Anal., http://dx.doi.org/10.1007/s12220-011-9263-3, in press. · Zbl 1276.53068
[13] A. Mondino, T. Rivière, Willmore spheres in compact Riemannian manifolds, preprint, 2012.
[14] T. Rivière, Variational principles for immersed surfaces with \(L^2\)-bounded second fundamental form, preprint, 2010.
[15] Schygulla, J., Willmore minimizers with prescribed isoperimetric ratio, Arch. Ration. Mech. Anal., 203, 901-941, (2012) · Zbl 1288.74027
[16] Simon, L., Existence of surfaces minimizing the Willmore functional, Comm. Anal. Geom., 1, 281-326, (1993) · Zbl 0848.58012
[17] Struwe, M., Variational methods, Ergeb. Math. Grenzgeb., vol. 34, (2008), Springer Verlag Berlin
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.