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 C1 norm and of compact support, we prove that if there is some point x¯∈M with scalar curvature RM(x¯)>0 then there exists a smooth embedding f:S2↪M minimizing the Willmore functional 14∫|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¯∈M with scalar curvature RM(x¯)>6 then there exists a smooth immersion f:S2↪M minimizing the functional ∫(12|A|2+1), where A is the second fundamental form. Finally, adding the bound KM⩽2 to the last assumptions, we obtain a smooth minimizer f:S2↪M for the functional ∫(14|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.