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.

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 x ¯M with scalar curvature R M (x ¯)>0 then there exists a smooth embedding f:𝕊 2 M minimizing the Willmore functional 1 4|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 R M (x ¯)>6 then there exists a smooth immersion f:𝕊 2 M minimizing the functional (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:𝕊 2 M for the functional (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.

DOI: 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
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     title = {Existence of immersed spheres minimizing curvature functionals in non-compact 3-manifolds},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {707--724},
     publisher = {Elsevier},
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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/articles/10.1016/j.anihpc.2013.07.002/

[1] A. Ambrosetti, A. Malchiodi, Nonlinear Analysis and Semilinear Elliptic Problems, Camb. Stud. Adv. Math. , Cambridge Univ. Press (2007) | MR | Zbl

[2] D. Gilbarg, N.S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer (2001) | MR | Zbl

[3] E. Kuwert, A. Mondino, J. Schygulla, Existence of immersed spheres minimizing curvature functionals in compact 3-manifolds, arXiv:1111.4893 (2011) | MR | Zbl

[4] E. Kuwert, R. Schätzle, Removability of isolated singularities of Willmore surfaces, Ann. Math. 160 no. 1 (2004), 315 -357 | MR | Zbl

[5] T. Lamm, J. Metzger, Small surfaces of Willmore type in Riemannian manifolds, Int. Math. Res. Not. IMRN 19 (2010), 3786 -3813 | MR | Zbl

[6] T. Lamm, J. Metzger, F. Schulze, Foliations of asymptotically flat manifolds by surfaces of Willmore type, Math. Ann. 350 (2011), 1 -78 | MR | Zbl

[7] J.M. Lee, T.H. Parker, The Yamabe problem, Bull., New Ser., Am. Math. Soc. 17 no. 1 (July 1987), 37 -91 | MR

[8] P. Li, S.T. Yau, A new conformal invariant and its applications to the Willmore conjecture and the first eigenvalue on compact surfaces, Invent. Math. 69 (1982), 269 -291 | EuDML | MR | Zbl

[9] F. Link, PhD thesis, in preparation.

[10] A. Mondino, Some results about the existence of critical points for the Willmore functional, Math. Zeit. 266 no. 3 (2010), 583 -622 | MR | Zbl

[11] A. Mondino, The conformal Willmore Functional: a perturbative approach, J. Geom. Anal. (24 September 2011), 1 -48 | MR

[12] A. Neves, G. Tian, Existence and uniqueness of constant mean curvature foliation of asymptotically hyperbolic 3-manifolds, Geom. Funct. Anal. 19 (2009), 910 -942 | MR | Zbl

[13] T. Rivière, Analysis aspects of Willmore surfaces, Invent. Math. 174 no. 1 (2008), 1 -45 | MR | Zbl

[14] R. Schoen, S.T. Yau, On the proof of the positive mass conjecture in general relativity, Commun. Math. Phys. 65 (1979), 45 -76 | MR | Zbl

[15] J. Schygulla, Willmore minimizers with prescribed isoperimetric ratio, Arch. Ration. Mech. Anal. 203 no. 3 (2012), 901 -941 | MR | Zbl

[16] L. Simon, Existence of surfaces minimizing the Willmore functional, Commun. Anal. Geom. 1 no. 2 (1993), 281 -325 | MR | Zbl

[17] L. Simon, Lectures on Geometric Measure Theory, Proc. Centre Math. Anal. Austral. Nat. Univ. vol. 3 , Austral. Nat. Univ., Canberra, Australia (1983) | MR | Zbl

[18] T.J. Willmore, Riemannian Geometry, Oxford Sci. Publ. , Oxford University Press (1993) | MR | Zbl

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