Existence of immersed spheres minimizing curvature functionals in non-compact 3-manifolds
Annales de l'I.H.P. Analyse non linéaire, Volume 31 (2014) no. 4, p. 707-724

We study curvature functionals for immersed 2-spheres in non-compact, three-dimensional Riemannian manifold $\left(M,h\right)$ without boundary. First, under the assumption that $\left(M,h\right)$ 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 ${R}^{M}\left(\overline{x}\right)>0$ then there exists a smooth embedding $f:{𝕊}^{2}↪M$ minimizing the Willmore functional $\frac{1}{4}\int {|H|}^{2}$, where H is the mean curvature. Second, assuming that $\left(M,h\right)$ 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 ${R}^{M}\left(\overline{x}\right)>6$ then there exists a smooth immersion $f:{𝕊}^{2}↪M$ minimizing the functional $\int \left(\frac{1}{2}{|A|}^{2}+1\right)$, where A is the second fundamental form. Finally, adding the bound ${K}^{M}⩽2$ to the last assumptions, we obtain a smooth minimizer $f:{𝕊}^{2}↪M$ for the functional $\int \left(\frac{1}{4}{|H|}^{2}+1\right)$. 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.

DOI : https://doi.org/10.1016/j.anihpc.2013.07.002
Classification:  53C21,  53C42,  58E99,  35J60
Keywords: ${L}^{2}$ second fundamental form, Willmore functional, Direct methods in the calculus of variations, Geometric measure theory, General Relativity
@article{AIHPC_2014__31_4_707_0,
author = {Mondino, Andrea and Schygulla, Johannes},
title = {Existence of immersed spheres minimizing curvature functionals in non-compact 3-manifolds},
journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
publisher = {Elsevier},
volume = {31},
number = {4},
year = {2014},
pages = {707-724},
doi = {10.1016/j.anihpc.2013.07.002},
zbl = {1300.53042},
mrnumber = {3249810},
language = {en},
url = {http://www.numdam.org/item/AIHPC_2014__31_4_707_0}
}

Mondino, Andrea; Schygulla, Johannes. Existence of immersed spheres minimizing curvature functionals in non-compact 3-manifolds. Annales de l'I.H.P. Analyse non linéaire, Volume 31 (2014) no. 4, pp. 707-724. doi : 10.1016/j.anihpc.2013.07.002. http://www.numdam.org/item/AIHPC_2014__31_4_707_0/

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