An extension procedure for the constraint equations. (English) Zbl 1403.35287
Summary: Let \((\bar{g}, \bar{k})\) be a solution to the maximal constraint equations of general relativity on the unit ball \(B_1\) of \({\mathbb R}^3\). We prove that if \((\bar{g},\bar{k})\) is sufficiently close to the initial data for Minkowski space, then there exists an asymptotically flat solution (\(g, k\)) on \({\mathbb R}^3\) that extends \((\bar{g}, \bar{k})\). Moreover, (\(g, k\)) is bounded by \((\bar{g}, \bar{k})\) and has the same regularity. Our proof uses a new method of solving the prescribed divergence equation for a tracefree symmetric 2-tensor, and a geometric variant of the conformal method to solve the prescribed scalar curvature equation for a metric. Both methods are based on the implicit function theorem and an expansion of tensors based on spherical harmonics. They are combined to define an iterative scheme that is shown to converge to a global solution (\(g, k\)) of the maximal constraint equations which extends \((\bar{g},\bar{k})\).
MSC:
| 35Q75 | PDEs in connection with relativity and gravitational theory |
| 35B65 | Smoothness and regularity of solutions to PDEs |
| 35B40 | Asymptotic behavior of solutions to PDEs |
References:
| [1] | Adams, R., Fournier, J.: Sobolev Spaces, 2nd edn. Pure and Applied Mathematics, vol. 140. Elsevier/Academic Press, Amsterdam (2003) · Zbl 1098.46001 |
| [2] | Anderson, MT; Khuri, M, On the bartnik extension problem for the static vacuum Einstein equations, Class. Quantum Gravity, 30, 125005, (2013) · Zbl 1271.83013 · doi:10.1088/0264-9381/30/12/125005 |
| [3] | Anderson, MT, On the bartnik conjecture for the static vacuum Einstein equations, Class. Quantum Gravity, 33, 015001, (2016) · Zbl 1331.83020 · doi:10.1088/0264-9381/33/1/015001 |
| [4] | Bartnik, R, Existence of maximal surfaces in asymptotically flat spacetimes, Commun. Math. Phys., 94, 155-175, (1984) · Zbl 0548.53054 · doi:10.1007/BF01209300 |
| [5] | Bartnik, R, The mass of an asymptotically flat manifold, Commun. Pure Appl. Math., 39, 661-693, (1986) · Zbl 0598.53045 · doi:10.1002/cpa.3160390505 |
| [6] | Bartnik, R, Quasi-spherical metrics and prescribed scalar curvature, J. Differ. Geom., 37, 31-71, (1993) · Zbl 0786.53019 · doi:10.4310/jdg/1214453422 |
| [7] | Bartnik, R.: Energy in General Relativity. Tsing Hua Lectures on Geometry & Analysis (Hsinchu, 1990-1991), pp. 5-27. International Press, Cambridge, MA (1997) · Zbl 0884.53065 |
| [8] | Bartnik, R.: Mass and 3-metrics of non-negative scalar curvature. In: Proceedings of the International Congress of Mathematicians, vol. II, pp. 231-240. Higher Education Press, Beijing (2002) · Zbl 1009.53055 |
| [9] | Choquet-Bruhat, Y; Christodoulou, D, Elliptic systems in \(H_{s,δ }\)-spaces on manifolds which are Euclidean at infinity, Acta Math., 146, 129-150, (1981) · Zbl 0484.58028 · doi:10.1007/BF02392460 |
| [10] | Christodoulou, D., Klainerman, S.: The Global Nonlinear Stability of the Minkowski Space. Princeton Mathematical Series, vol. 41. Princeton University Press, Princeton (1993) · Zbl 0827.53055 |
| [11] | Chruściel, PT; Delay, E, On mapping properties of the general relativistic constraints operator in weighted function spaces, with applications, Mém. Soc. Math. Fr., 94, 1-103, (2003) · Zbl 1058.83007 |
| [12] | Chruściel, PT; Isenberg, J; Pollack, D, Initial data engineering, Commun. Math. Phys., 257, 29-42, (2005) · Zbl 1080.83002 · doi:10.1007/s00220-005-1345-2 |
| [13] | Corvino, J, Scalar curvature deformation and a gluing construction for the Einstein constraint equations, Commun. Math. Phys., 214, 137-189, (2000) · Zbl 1031.53064 · doi:10.1007/PL00005533 |
| [14] | Corvino, J; Schoen, R, On the asymptotics for the vacuum Einstein constraint equations, J. Differ. Geom., 73, 185-217, (2006) · Zbl 1122.58016 · doi:10.4310/jdg/1146169910 |
| [15] | Courant, R., Hilbert, D.: Methods of Mathematical Physics, vol. I. Interscience Publishers, Inc., London (1953) · Zbl 0051.28802 |
| [16] | Czimek, S.: Boundary harmonic coordinates and the localised bounded \(L^2\) curvature theorem. arXiv:1708.01667 · Zbl 1403.35287 |
| [17] | Delay, E, Smooth compactly supported solutions of some underdetermined elliptic PDE, with gluing applications, Commun. Partial Differ. Equ., 37, 1689-1716, (2012) · Zbl 1268.58018 · doi:10.1080/03605302.2012.711794 |
| [18] | Fischer, A; Marsden, J, Deformations of the scalar curvature, Duke Math. J., 42, 519-547, (1975) · Zbl 0336.53032 · doi:10.1215/S0012-7094-75-04249-0 |
| [19] | Gilbarg, D., Trudinger, N.: Elliptic Partial Differential Equations of Second Order. Reprint of the 1998 edition. Classics in Mathematics. Springer (2001) · Zbl 1042.35002 |
| [20] | Hill, E, The theory of vector spherical harmonics, Am. J. Phys., 22, 211-214, (1954) · Zbl 0057.05303 · doi:10.1119/1.1933682 |
| [21] | Huisken, G; lmanen, T, The inverse mean curvature flow and the Riemannian Penrose inequality, J. Differ. Geom., 59, 353-437, (2001) · Zbl 1055.53052 · doi:10.4310/jdg/1090349447 |
| [22] | Isenberg, J; Mazzeo, R; Pollack, D, On the topology of vacuum spacetimes, Ann. Henri Poincaré, 4, 369-383, (2003) · Zbl 1026.83008 · doi:10.1007/s00023-003-0133-9 |
| [23] | Isett, P., Oh, S.-J.: On nonperiodic euler flows with Hölder regularity. arXiv:1402.2305v2 · Zbl 1338.35344 |
| [24] | Klainerman, S; Rodnianski, I; Szeftel, J, The bounded \(L^2\) curvature conjecture, Invent. Math., 202, 91-216, (2015) · Zbl 1330.53089 · doi:10.1007/s00222-014-0567-3 |
| [25] | Maxwell, D, Rough solutions of the Einstein constraint equations, J. Reine Angew. Math., 590, 1-29, (2006) · Zbl 1088.83004 · doi:10.1515/CRELLE.2006.001 |
| [26] | Maxwell, D.: Initial data for black holes and rough spacetimes. Dissertation, University of Washington (2004) |
| [27] | Miao, P, On existence of static metric extensions in general relativity, Commun. Math. Phys., 241, 27-46, (2003) · Zbl 1149.83311 · doi:10.1007/s00220-003-0925-2 |
| [28] | Oh, S.-J., Tataru, D.: Local well-posedness of the (4+1)-dimensional Maxwell-Klein-Gordon equation at energy regularity. Ann. PDE 2(1) (2016) · Zbl 1364.35198 |
| [29] | Ratiu, T., Abraham, R., Marsden, J.E.: Manifolds, Tensor Analysis, and Applications. Global Analysis. Addison-Wesley, Reading (1983) · Zbl 0508.58001 |
| [30] | Sandberg, V, Tensor spherical harmonics on \(S^2\) and \(S^3\) as eigenvalue problems, J. Math. Phys., 19, 2441-2446, (1978) · Zbl 0425.31010 · doi:10.1063/1.523649 |
| [31] | Sharples, J, Local existence of quasi-spherical space-time initial data, J. Math. Phys., 46, 052501, (2005) · Zbl 1110.83004 · doi:10.1063/1.1864250 |
| [32] | Shi, Y; Tam, L-F, Positive mass theorem and the boundary behaviors of compact manifolds with nonnegative scalar curvature, J. Differ. Geom., 62, 79-125, (2002) · Zbl 1071.53018 · doi:10.4310/jdg/1090425530 |
| [33] | Shi, Y; Tam, L-F, Quasi-local mass and the existence of horizons, Commun. Math. Phys., 274, 277-295, (2007) · Zbl 1206.83069 · doi:10.1007/s00220-007-0273-8 |
| [34] | Smith, B; Weinstein, G, Quasiconvex foliations and asymptotically flat metrics on non-negative scalar curvature, Commun. Anal. Geom., 12, 511-551, (2004) · Zbl 1073.53039 · doi:10.4310/CAG.2004.v12.n3.a2 |
| [35] | Smith, B; Weinstein, G, On the connectedness of the space of initial data for the Einstein equations, Electron. Res. Announc. Am. Math. Soc., 6, 52-63, (2000) · Zbl 0969.83006 · doi:10.1090/S1079-6762-00-00081-0 |
| [36] | Stein, E.: Singular Integrals and Differentiability Properties of Functions. Princeton Mathematical Series, vol. 30. Princeton University Press, Princeton (1971) · doi:10.1515/9781400883882 |
| [37] | Taylor, M.: Partial Differential Equations I: Basic Theory. Springer, New York (1999). (2nd printing) · Zbl 0924.34001 |
| [38] | Taylor, M.: Partial Differential Equations III: Nonlinear Equations. Springer, New York (2011). (2nd printing) · Zbl 1206.35004 · doi:10.1007/978-1-4419-7049-7 |
| [39] | Wald, R.: General Relativity. University of Chicago Press, Chicago, IL (1984) · Zbl 0549.53001 · doi:10.7208/chicago/9780226870373.001.0001 |
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