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 $\bar{x}\in M$ with scalar curvature $R^M(\bar{x})\>0$ 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 $\bar{x}\in M$ with scalar curvature $R^M(\bar{x})\>6$ 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 $K^M \leq 2$ 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.

}, keywords = {Direct methods in the calculus of variations, General Relativity, Geometric measure theory, second fundamental form, Willmore functional}, issn = {0294-1449}, doi = {https://doi.org/10.1016/j.anihpc.2013.07.002}, url = {http://www.sciencedirect.com/science/article/pii/S0294144913000851}, author = {Andrea Mondino and Johannes Schygulla} } @article {Kuwert2014, title = {Existence of immersed spheres minimizing curvature functionals in compact 3-manifolds}, journal = {Mathematische Annalen}, volume = {359}, number = {1}, year = {2014}, month = {Jun}, pages = {379{\textendash}425}, abstract = {We study curvature functionals for immersed 2-spheres in a compact, three-dimensional Riemannian manifold $M$. Under the assumption that the sectional curvature $K^M$ is strictly positive, we prove the existence of a smooth immersion $f:{\mathbb{S}}^2 \rightarrow M$ minimizing the $L^2$ integral of the second fundamental form. Assuming instead that $K^M \leq 2 $ and that there is some point $\bar{x}\in M$ with scalar curvature $R^M(\bar{x})\>6$, we obtain a smooth minimizer $f:{\mathbb{S}}^2 \rightarrow M$ for the functional $\int \frac{1}{4}|H|^2+1$, where $H$ is the mean curvature.

}, issn = {1432-1807}, doi = {10.1007/s00208-013-1005-3}, url = {https://doi.org/10.1007/s00208-013-1005-3}, author = {Kuwert, Ernst and Andrea Mondino and Johannes Schygulla} }