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Foliations of asymptotically flat manifolds by surfaces of Willmore type. (English) Zbl 1222.53028
In the frame of the study of the foliations of asymptotically flat manifolds, the authors are interested in constructing embedded spheres $$\Sigma$$ in a Riemannian manifold $$(M,g)$$ which satisfy the equation: $$- \Delta H -H |\AA|^{2} - ^{M}$$Rc$$(\nu, \nu)H = \lambda H$$, where $$H$$ is the mean curvature of $$\Sigma$$, $$\AA = A- \frac{1}{2}H \gamma$$ ($$\gamma$$ is the induced metric on $$\Sigma$$), $$^{M}$$Rc is the Ricci curvature of $$M$$ and $$\Delta$$ is the Laplace-Beltrami operator on $$\Sigma$$. The authors argue that the above equation is the most natural equation to consider when defining a geometric center of the Hawking mass. The preliminaries necessary for the understanding of their interesting results, discussions and other points of view on the subject are also presented.

##### MSC:
 53C12 Foliations (differential geometric aspects) 53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.) 53C80 Applications of global differential geometry to the sciences 83C99 General relativity
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##### References:
 [1] Bauer M., Kuwert E.: Existence of minimizing Willmore surfaces of prescribed genus. Int. Math. Res. Not. 10, 553–576 (2003) · Zbl 1029.53073 · doi:10.1155/S1073792803208072 [2] Christodoulou, D., Yau, S.-T.: Some remarks on the quasi-local mass. In: Mathematics and General Relativity (Santa Cruz, CA, 1986), vol. 71 of Contemp. Math., pp. 9–14. Amer. Math. Soc., Providence (1988) [3] De Lellis C., Müller S.: Optimal rigidity estimates for nearly umbilical surfaces. J. Differ. Geom. 69(1), 75–110 (2005) · Zbl 1087.53004 [4] De Lellis C., Müller S.: A C 0 estimate for nearly umbilical surfaces. Calc. Var. Partial Differ. Equ. 26(3), 283–296 (2006) · Zbl 1100.53005 · doi:10.1007/s00526-006-0005-5 [5] Gallot S., Hulin D., Lafontaine J.: Riemannian Geometry, 2nd edn. Springer, Berlin (1993) [6] Huang, L.-H.: Foliations by stable spheres with constant mean curvature for isolated systems with general asymptotics. Comm. Math. Phys. (to appear) · Zbl 1206.53028 [7] Huisken G.: An isoperimetric concept for the mass in general relativity. Oberwolfach Rep. 3(3), 1898–1899 (2006) [8] Huisken, G., Polden, A.: Geometric evolution equations for hypersurfaces. In: Calculus of variations and geometric evolution problems (Cetraro, 1996), vol. 1713 of Lecture Notes in Math., pp. 45–84. Springer, Berlin (1999) · Zbl 0942.35047 [9] Huisken G., Yau S.-T.: Definition of center of mass for isolated physical systems and unique foliations by stable spheres with constant mean curvature. Invent. Math. 124(1–3), 281–311 (1996) · Zbl 0858.53071 · doi:10.1007/s002220050054 [10] Kuwert E., Schätzle R.: The Willmore flow with small initial energy. J. Differ. Geom. 57(3), 409–441 (2001) · Zbl 1035.53092 [11] Kuwert E., Schätzle R.: Gradient flow for the Willmore functional. Comm. Anal. Geom. 10(2), 307–339 (2002) · Zbl 1029.53082 [12] MapleSoft. Maple. http://www.maplesoft.com [13] Mazzeo, R., Pacard, F.: Constant curvature foliations on asymptotically hyperbolic spaces. Rev. Mat. Iberoam (to appear) · Zbl 1214.53024 [14] Metzger J.: Foliations of asymptotically flat 3-manifolds by 2-surfaces of prescribed mean curvature. J. Differ. Geom. 77, 201–236 (2007) · Zbl 1140.53013 [15] Michael J.H., Simon L.M.: Sobolev and mean-value inequalities on generalized submanifolds of R n . Comm. Pure Appl. Math. 26, 361–379 (1973) · Zbl 0256.53006 · doi:10.1002/cpa.3160260305 [16] Neves A., Tian G.: Existence and uniqueness of constant mean curvature foliation of asymptotically hyperbolic 3-manifolds. Geom. Funct. Anal. 19(3), 910–942 (2009) · Zbl 1187.53027 · doi:10.1007/s00039-009-0019-1 [17] Neves A., Tian G.: Existence and uniqueness of constant mean curvature foliation of asymptotically hyperbolic 3-manifolds II. J. Reine Angew. Math. 641, 69–93 (2010) · Zbl 1194.53026 [18] Qing J., Tian G.: On the uniqueness of the foliation of spheres of constant mean curvature in asymptotically flat 3-manifolds. J. Am. Math. Soc. 20(4), 1091–1110 (2007) · Zbl 1142.53024 · doi:10.1090/S0894-0347-07-00560-7 [19] Schoen R.: The existence of weak solutions with prescribed singular behavior for a conformally invariant scalar equation. Comm. Pure Appl. Math. 41(3), 317–392 (1988) · Zbl 0674.35027 · doi:10.1002/cpa.3160410305 [20] Schoen, R., Yau, S.-T.: Lectures on Differential Geometry. In: Conference Proceedings and Lecture Notes in Geometry and Topology. International Press, Boston (1994) · Zbl 0830.53001 [21] Simon L.: Existence of surfaces minimizing the Willmore functional. Comm. Anal. Geom. 1(2), 281–326 (1993) · Zbl 0848.58012 [22] Simons J.: Minimal varieties in riemannian manifolds. Ann. Math. (2) 88, 62–105 (1968) · Zbl 0181.49702 · doi:10.2307/1970556 [23] Weiner J.L.: On a problem of Chen, Willmore, et al. Indiana Univ. Math. J. 27(1), 19–35 (1978) · Zbl 0368.53043 · doi:10.1512/iumj.1978.27.27003 [24] Ye, R.: Foliation by constant mean curvature spheres on asymptotically flat manifolds. In: Geometric Analysis and the Calculus of Variations, pp. 369–383. International Press, Cambridge (1996) · Zbl 0932.53026
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