Minimizers of the Willmore functional with a small area constraint
Lamm, Tobias ; Metzger, Jan
Annales de l'I.H.P. Analyse non linéaire, Tome 30 (2013), p. 497-518 / Harvested from Numdam
Access to full text
Full (PDF)
Full (DJVU)

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.

@article{AIHPC_2013__30_3_497_0,
     author = {Lamm, Tobias and Metzger, Jan},
     title = {Minimizers of the Willmore functional with a small area constraint},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     volume = {30},
     year = {2013},
     pages = {497-518},
     doi = {10.1016/j.anihpc.2012.10.003},
     zbl = {1290.49090},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIHPC_2013__30_3_497_0}
}

Lamm, Tobias; Metzger, Jan. Minimizers of the Willmore functional with a small area constraint. Annales de l'I.H.P. Analyse non linéaire, Tome 30 (2013) pp. 497-518. doi : 10.1016/j.anihpc.2012.10.003. http://gdmltest.u-ga.fr/item/AIHPC_2013__30_3_497_0/

[1] M. Bauer, E. Kuwert, Existence of minimizing Willmore surfaces of prescribed genus, Int. Math. Res. Not. 10 (2003), 553-576 | 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] C. De Lellis, S. Müller, Optimal rigidity estimates for nearly umbilical surfaces, J. Differential Geom. 69 (2005), 75-110 | MR 2169583 | Zbl 1087.53004

[4] C. De Lellis, S. Müller, A C 0 estimate for nearly umbilical surfaces, Calc. Var. Partial Differential Equations 26 (2006), 283-296 | MR 2232206 | Zbl 1100.53005

[5] D. Fischer-Colbrie, R. Schoen, The structure of complete stable minimal surfaces in 3-manifolds of nonnegative scalar curvature, Comm. Pure Appl. Math. 33 (1980), 199-211 | MR 562550 | Zbl 0439.53060

[6] E. Kuwert, Y. Li, W 2,2 -conformal immersions of a closed Riemann surface into ℝ n , Comm. Anal. Geom. 20 (2012), 313-340 | MR 2928715 | Zbl 1271.53010

[7] E. Kuwert, A. Mondino, J. Schygulla, Existence of immersed spheres minimizing curvature functionals in compact 3-manifolds, preprint, 2011. | MR 3201902

[8] E. Kuwert, R. Schätzle, Minimizers of the Willmore functional under fixed conformal class, J. Differential Geom., in press. | MR 3024303

[9] T. Lamm, J. Metzger, Small surfaces of Willmore type in Riemannian manifolds, Int. Math. Res. Not. 19 (2010), 3786-3813 | MR 2725514 | Zbl 1202.53056

[10] T. Lamm, J. Metzger, F. Schulze, Foliations of asymptotically flat manifolds by surfaces of Willmore type, Math. Ann. 350 (2011), 1-78 | MR 2785762 | Zbl 1222.53028

[11] A. Mondino, Some results about the existence of critical points for the Willmore functional, Math. Z. 266 (2010), 583-622 | MR 2719422 | 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. | MR 3023857

[13] A. Mondino, T. Rivière, Willmore spheres in compact Riemannian manifolds, preprint, 2012. | MR 2989995

[14] T. Rivière, Variational principles for immersed surfaces with L 2 -bounded second fundamental form, preprint, 2010. | MR 3276154

[15] J. Schygulla, Willmore minimizers with prescribed isoperimetric ratio, Arch. Ration. Mech. Anal. 203 (2012), 901-941 | MR 2928137 | Zbl 1288.74027

[16] L. Simon, Existence of surfaces minimizing the Willmore functional, Comm. Anal. Geom. 1 (1993), 281-326 | MR 1243525 | Zbl 0848.58012

[17] M. Struwe, Variational Methods, Ergeb. Math. Grenzgeb. vol. 34, Springer Verlag, Berlin (2008) | MR 2431434