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

### Andrea Mondino

Scuola Normale Superiore, Piazza dei Cavalieri 7, 56126 Pisa, Italy### Johannes Schygulla

Mathematisches Institut, Universität Freiburg, Eckerstraße 1, 79104 Freiburg, Germany

## 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 x\limits^{¯} \in M with scalar curvature R^{M}(x\limits^{¯}) > 0 then there exists a smooth embedding $f:\mathbb{S}^{2}↪M$ 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 x\limits^{¯} \in M with scalar curvature R^{M}(x\limits^{¯}) > 6 then there exists a smooth immersion $f:\mathbb{S}^{2}↪M$ minimizing the functional $\int (\frac{1}{2}|A|^{2} + 1)$, where *A* is the second fundamental form. Finally, adding the bound $K^{M}⩽2$ to the last assumptions, we obtain a smooth minimizer $f:\mathbb{S}^{2}↪M$ 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.

## Cite this article

Andrea Mondino, Johannes Schygulla, Existence of immersed spheres minimizing curvature functionals in non-compact 3-manifolds. Ann. Inst. H. Poincaré Anal. Non Linéaire 31 (2014), no. 4, pp. 707–724

DOI 10.1016/J.ANIHPC.2013.07.002