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) no. 4, pp. 707-724.

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 $R^M\left(\overline{x}\right)>0$ then there exists a smooth embedding $f:{𝕊}^2\hookrightarrow M$ minimizing the Willmore functional $\frac{1}{4}\int {\left|H\right|}^2$, 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 $R^M\left(\overline{x}\right)>6$ then there exists a smooth immersion $f:{𝕊}^2\hookrightarrow M$ minimizing the functional $\int (\frac{1}{2}{\left|A\right|}^2+1)$, where A is the second fundamental form. Finally, adding the bound $K^M⩽2$ to the last assumptions, we obtain a smooth minimizer $f:{𝕊}^2\hookrightarrow M$ for the functional $\int (\frac{1}{4}{\left|H\right|}^2+1)$. 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.

Classification : 53C21, 53C42, 58E99, 35J60
Mots clés : $ {L}^{2}$ second fundamental form, Willmore functional, Direct methods in the calculus of variations, Geometric measure theory, General Relativity

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