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} } @article {Mondino2014, title = {Existence of integral m-varifolds minimizing $\int |A|^p $ and $\int |H|^p$ , p>m, in Riemannian manifolds}, journal = {Calculus of Variations and Partial Differential Equations}, volume = {49}, number = {1}, year = {2014}, month = {Jan}, pages = {431{\textendash}470}, abstract = {We prove existence of integral rectifiable $m$-dimensional varifolds minimizing functionals of the type $\int |H|^p$ and $\int |A|^p$ in a given Riemannian $n$-dimensional manifold $(N,g)$, $2 \leq m\<n$ and $p\>m$ under suitable assumptions on $N$ (in the end of the paper we give many examples of such ambient manifolds). To this aim we introduce the following new tools: some monotonicity formulas for varifolds in ${\mathbb{R }^S}$ involving $\int\ |H|^p$to\ avoid degeneracy of the minimizer, and a sort of isoperimetric inequality to bound the mass in terms of the mentioned functionals.

}, issn = {1432-0835}, doi = {10.1007/s00526-012-0588-y}, url = {https://doi.org/10.1007/s00526-012-0588-y}, author = {Andrea Mondino} } @article {Mondino2013, title = {The Conformal Willmore Functional: A Perturbative Approach}, journal = {Journal of Geometric Analysis}, volume = {23}, number = {2}, year = {2013}, month = {Apr}, pages = {764{\textendash}811}, abstract = {The conformal Willmore functional (which is conformal invariant in general Riemannian manifolds $(M,g)$ is studied with a perturbative method: the Lyapunov{\textendash}Schmidt reduction. Existence of critical points is shown in ambient manifolds $(\mathbb{R}^3,g_\epsilon)$ {\textendash} where $g_\epsilon$ is a metric close and asymptotic to the Euclidean one. With the same technique a non-existence result is proved in general Riemannian manifolds $(M,g)$ of dimension three.

}, issn = {1559-002X}, doi = {10.1007/s12220-011-9263-3}, url = {https://doi.org/10.1007/s12220-011-9263-3}, author = {Andrea Mondino} }