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.

%B Mathematische Annalen %V 359 %P 379–425 %8 Jun %G eng %U https://doi.org/10.1007/s00208-013-1005-3 %R 10.1007/s00208-013-1005-3 %0 Journal Article %J Annales de l'Institut Henri Poincare (C) Non Linear Analysis %D 2014 %T Existence of immersed spheres minimizing curvature functionals in non-compact 3-manifolds %A Andrea Mondino %A Johannes Schygulla %K Direct methods in the calculus of variations %K General Relativity %K Geometric measure theory %K second fundamental form %K Willmore functional %XWe 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.

%B Annales de l'Institut Henri Poincare (C) Non Linear Analysis %V 31 %P 707 - 724 %G eng %U http://www.sciencedirect.com/science/article/pii/S0294144913000851 %R https://doi.org/10.1016/j.anihpc.2013.07.002 %0 Journal Article %J Calculus of Variations and Partial Differential Equations %D 2014 %T Existence of integral m-varifolds minimizing $\int |A|^p $ and $\int |H|^p$ , p>m, in Riemannian manifolds %A Andrea Mondino %XWe 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.

%B Calculus of Variations and Partial Differential Equations %V 49 %P 431–470 %8 Jan %G eng %U https://doi.org/10.1007/s00526-012-0588-y %R 10.1007/s00526-012-0588-y %0 Journal Article %J Journal of Geometric Analysis %D 2013 %T The Conformal Willmore Functional: A Perturbative Approach %A Andrea Mondino %XThe 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.

%B Journal of Geometric Analysis %V 23 %P 764–811 %8 Apr %G eng %U https://doi.org/10.1007/s12220-011-9263-3 %R 10.1007/s12220-011-9263-3