JournalsaihpcVol. 31, No. 4pp. 707–724

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

We study curvature functionals for immersed 2-spheres in non-compact, three-dimensional Riemannian manifold (M,h)(M,h) without boundary. First, under the assumption that (M,h)(M,h) is the euclidean 3-space endowed with a semi-perturbed metric with perturbation small in C1C^{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:S2Mf:\mathbb{S}^{2}↪M minimizing the Willmore functional 14H2\frac{1}{4}\int |H|^{2}, where H is the mean curvature. Second, assuming that (M,h)(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:S2Mf:\mathbb{S}^{2}↪M minimizing the functional (12A2+1)\int (\frac{1}{2}|A|^{2} + 1), where A is the second fundamental form. Finally, adding the bound KM2K^{M}⩽2 to the last assumptions, we obtain a smooth minimizer f:S2Mf:\mathbb{S}^{2}↪M for the functional (14H2+1)\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