Existence of immersed spheres minimizing curvature functionals in non-compact 3-manifolds
Annales de l'I.H.P. Analyse non linéaire, Volume 31 (2014) no. 4, p. 707-724
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We study curvature functionals for immersed 2-spheres in non-compact, three-dimensional Riemannian manifold (M,h) without boundary. First, under the assumption that (M,h) is the euclidean 3-space endowed with a semi-perturbed metric with perturbation small in C 1 norm and of compact support, we prove that if there is some point x ¯M with scalar curvature R M (x ¯)>0 then there exists a smooth embedding f:𝕊 2 M minimizing the Willmore functional 1 4|H| 2 , where H is the mean curvature. Second, assuming that (M,h) is of bounded geometry (i.e. bounded sectional curvature and strictly positive injectivity radius) and asymptotically euclidean or hyperbolic we prove that if there is some point x ¯M with scalar curvature R M (x ¯)>6 then there exists a smooth immersion f:𝕊 2 M minimizing the functional (1 2|A| 2 +1), where A is the second fundamental form. Finally, adding the bound K M 2 to the last assumptions, we obtain a smooth minimizer f:𝕊 2 M for the functional (1 4|H| 2 +1). The assumptions of the last two theorems are satisfied in a large class of 3-manifolds arising as spacelike timeslices solutions of the Einstein vacuum equation in case of null or negative cosmological constant.

DOI : https://doi.org/10.1016/j.anihpc.2013.07.002
Classification:  53C21,  53C42,  58E99,  35J60
Keywords: L 2 second fundamental form, Willmore functional, Direct methods in the calculus of variations, Geometric measure theory, General Relativity
@article{AIHPC_2014__31_4_707_0,
     author = {Mondino, Andrea and Schygulla, Johannes},
     title = {Existence of immersed spheres minimizing curvature functionals in non-compact 3-manifolds},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     publisher = {Elsevier},
     volume = {31},
     number = {4},
     year = {2014},
     pages = {707-724},
     doi = {10.1016/j.anihpc.2013.07.002},
     zbl = {1300.53042},
     mrnumber = {3249810},
     language = {en},
     url = {http://www.numdam.org/item/AIHPC_2014__31_4_707_0}
}
Mondino, Andrea; Schygulla, Johannes. Existence of immersed spheres minimizing curvature functionals in non-compact 3-manifolds. Annales de l'I.H.P. Analyse non linéaire, Volume 31 (2014) no. 4, pp. 707-724. doi : 10.1016/j.anihpc.2013.07.002. http://www.numdam.org/item/AIHPC_2014__31_4_707_0/

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