Existence of immersed spheres minimizing curvature functionals in non-compact 3-manifolds

Mondino, Andrea ; Schygulla, Johannes

Annales de l'I.H.P. Analyse non linéaire, Tome 31 (2014), p. 707-724 / Harvested from Numdam

Résumé

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 $ \overline{x}\in M$ with scalar curvature $ {R}^{M}(\overline{x})>0$ then there exists a smooth embedding $ f:{\mathbb{S}}^{2}\hookrightarrow M$ minimizing the Willmore functional $ \frac{1}{4}\int {|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 $ \overline{x}\in M$ with scalar curvature $ {R}^{M}(\overline{x})>6$ then there exists a smooth immersion $ f:{\mathbb{S}}^{2}\hookrightarrow M$ minimizing the functional $ \int (\frac{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:{\mathbb{S}}^{2}\hookrightarrow M$ for the functional $ \int (\frac{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.

Détail

Publié le : 2014-01-01

MR 3249810 | Zbl 1300.53042

DOI : https://doi.org/10.1016/j.anihpc.2013.07.002
Classification:  53C21,  53C42,  58E99,  35J60

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