# Existence of immersed spheres minimizing curvature functionals in non-compact 3-manifolds

**accepted**
**year:** 2012

**journal:** Annales IHP-Anal. Non Lin.

**abstract:** 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 $R^M(\overline{x})>0$ then there exists a smooth embedding $f:S^2 \hookrightarrow M$ 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 $\overline{x} \in M$ with scalar curvature $R^M(\overline{x})>6$ then there exists a smooth immersion $f:S^2 \hookrightarrow M$ 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 $f:S^2 \hookrightarrow M$ 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.

The paper is available on the
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