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

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 $K^M⩽2$ 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.