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},
pages = {497--518},
publisher = {Elsevier},
volume = {30},
number = {3},
year = {2013},
doi = {10.1016/j.anihpc.2012.10.003},
zbl = {1290.49090},
language = {en},
url = {http://www.numdam.org/articles/10.1016/j.anihpc.2012.10.003/}
}
TY - JOUR AU - Lamm, Tobias AU - Metzger, Jan TI - Minimizers of the Willmore functional with a small area constraint JO - Annales de l'I.H.P. Analyse non linéaire PY - 2013 SP - 497 EP - 518 VL - 30 IS - 3 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.anihpc.2012.10.003/ DO - 10.1016/j.anihpc.2012.10.003 LA - en ID - AIHPC_2013__30_3_497_0 ER -
%0 Journal Article %A Lamm, Tobias %A Metzger, Jan %T Minimizers of the Willmore functional with a small area constraint %J Annales de l'I.H.P. Analyse non linéaire %D 2013 %P 497-518 %V 30 %N 3 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.anihpc.2012.10.003/ %R 10.1016/j.anihpc.2012.10.003 %G en %F 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) no. 3, pp. 497-518. doi : 10.1016/j.anihpc.2012.10.003. http://www.numdam.org/articles/10.1016/j.anihpc.2012.10.003/
[1] , , Existence of minimizing Willmore surfaces of prescribed genus, Int. Math. Res. Not. 10 (2003), 553-576 | Zbl
[2] J. Chen, Y. Li, Bubble tree of a class of conformal mappings and applications to the Willmore functional, preprint, 2011.
[3] , , Optimal rigidity estimates for nearly umbilical surfaces, J. Differential Geom. 69 (2005), 75-110 | MR | Zbl
[4] , , A estimate for nearly umbilical surfaces, Calc. Var. Partial Differential Equations 26 (2006), 283-296 | MR | Zbl
[5] , , The structure of complete stable minimal surfaces in 3-manifolds of nonnegative scalar curvature, Comm. Pure Appl. Math. 33 (1980), 199-211 | MR | Zbl
[6] , , -conformal immersions of a closed Riemann surface into , Comm. Anal. Geom. 20 (2012), 313-340 | MR | Zbl
[7] E. Kuwert, A. Mondino, J. Schygulla, Existence of immersed spheres minimizing curvature functionals in compact 3-manifolds, preprint, 2011. | MR
[8] E. Kuwert, R. Schätzle, Minimizers of the Willmore functional under fixed conformal class, J. Differential Geom., in press. | MR
[9] , , Small surfaces of Willmore type in Riemannian manifolds, Int. Math. Res. Not. 19 (2010), 3786-3813 | MR | Zbl
[10] , , , Foliations of asymptotically flat manifolds by surfaces of Willmore type, Math. Ann. 350 (2011), 1-78 | MR | Zbl
[11] , Some results about the existence of critical points for the Willmore functional, Math. Z. 266 (2010), 583-622 | MR | Zbl
[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
[13] A. Mondino, T. Rivière, Willmore spheres in compact Riemannian manifolds, preprint, 2012. | MR
[14] T. Rivière, Variational principles for immersed surfaces with -bounded second fundamental form, preprint, 2010. | MR
[15] , Willmore minimizers with prescribed isoperimetric ratio, Arch. Ration. Mech. Anal. 203 (2012), 901-941 | MR | Zbl
[16] , Existence of surfaces minimizing the Willmore functional, Comm. Anal. Geom. 1 (1993), 281-326 | MR | Zbl
[17] , Variational Methods, Ergeb. Math. Grenzgeb. vol. 34, Springer Verlag, Berlin (2008) | MR
Cité par Sources :
