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

Publication Type | Journal Article |

Year of Publication | 2014 |

Authors | Mondino, A, Schygulla, J |

Journal | Annales de l'Institut Henri Poincare (C) Non Linear Analysis |

Volume | 31 |

Pagination | 707 - 724 |

ISSN | 0294-1449 |

Keywords | Direct methods in the calculus of variations; General Relativity; Geometric measure theory; second fundamental form; Willmore functional |

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 $\bar{x}\in M$ 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. |

URL | http://www.sciencedirect.com/science/article/pii/S0294144913000851 |

DOI | 10.1016/j.anihpc.2013.07.002 |

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

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