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.

10aDirect methods in the calculus of variations10aGeneral Relativity10aGeometric measure theory10asecond fundamental form10aWillmore functional1 aMondino, Andrea1 aSchygulla, Johannes uhttp://www.sciencedirect.com/science/article/pii/S029414491300085101126nas a2200169 4500008004100000022001400041245009000055210006900145260000800214300001400222490000800236520060400244100001800848700002000866700002400886856004600910 2014 eng d a1432-180700aExistence of immersed spheres minimizing curvature functionals in compact 3-manifolds0 aExistence of immersed spheres minimizing curvature functionals i cJun a379–4250 v3593 aWe 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.

1 aKuwert, Ernst1 aMondino, Andrea1 aSchygulla, Johannes uhttps://doi.org/10.1007/s00208-013-1005-301095nas a2200145 4500008004100000022001400041245011100055210006900166260000800235300001400243490000700257520061900264100002000883856004600903 2014 eng d a1432-083500aExistence of integral m-varifolds minimizing $\int |A|^p $ and $\int |H|^p$ , p>m, in Riemannian manifolds0 aExistence of integral mvarifolds minimizing int Ap and int Hp pm cJan a431–4700 v493 aWe 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.

1 aMondino, Andrea uhttps://doi.org/10.1007/s00526-012-0588-y00887nas a2200145 4500008004100000022001400041245006300055210005800118260000800176300001400184490000700198520047000205100002000675856004600695 2013 eng d a1559-002X00aThe Conformal Willmore Functional: A Perturbative Approach0 aConformal Willmore Functional A Perturbative Approach cApr a764–8110 v233 aThe conformal Willmore functional (which is conformal invariant in general Riemannian manifolds $(M,g)$ is studied with a perturbative method: the Lyapunov–Schmidt reduction. Existence of critical points is shown in ambient manifolds $(\mathbb{R}^3,g_\epsilon)$ – 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.

1 aMondino, Andrea uhttps://doi.org/10.1007/s12220-011-9263-3