Anisotropic Einstein data with isotropic non negative prescribed scalar curvature
Annales de l'I.H.P. Analyse non linéaire, Volume 32 (2015) no. 2, pp. 401-428.

We construct time-symmetric black hole initial data for the Einstein equations with prescribed scalar curvature, or more precisely a piece of such initial data contained inside the black hole. In this case, the Einstein constraint equations translate into a parabolic equation, with radius as ‘time’ variable, for a metric component u that undergoes blow up. The metric itself is regular up to and including the surface at the blow up radius, which is a minimal surface.We show the existence of Einstein constrained data with blow up profiles that are anisotropic (i.e. not O(3) symmetric) although the scalar curvature was isotropically prescribed.Our results are based on center manifold theory for quasilinear parabolic equations and on equivariant bifurcation theory for not necessarily self-similar solutions of a self-similarly rescaled equation.

Nous construisons des données initiales de trou noir à symétrie temporelle pour les équations d'Einstein dont la courbure scalaire est prescrite ; plus précisément une partie de telles données initiales contenue à l'intérieur du trou noir. Dans ce cas, les contraintes d'Einstein peuvent être exprimées à l'aide d'une équation parabolique dont la variable « temps » est le rayon, vérifiée par une composante u de la métrique qui subit une explosion en « temps » fini. La métrique elle-même est régulière jusqu'à la surface au rayon de l'explosion (inclue) ; cette surface est une surface minimale.Nous montrons l'existence de données vérifiant les contraintes d'Einstein, dont le profil d'explosion est anisotrope (i.e. elles ne sont pas O(3)-symétriques) alors que la courbure scalaire a été prescrite de façon isotrope.Nos résultats sont basés sur la théorie des variétés centrales pour les équations paraboliques quasi-linéaires et sur la théorie équivariante des bifurcations pour des solutions de l'équation en variables auto-similaires, dont l'évolution n'est pas nécessairement auto-similaire.

@article{AIHPC_2015__32_2_401_0,
     author = {Fiedler, Bernold and Hell, Juliette and Smith, Brian},
     title = {Anisotropic {Einstein} data with isotropic non negative prescribed scalar curvature},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {401--428},
     publisher = {Elsevier},
     volume = {32},
     number = {2},
     year = {2015},
     doi = {10.1016/j.anihpc.2014.01.002},
     zbl = {1332.35358},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1016/j.anihpc.2014.01.002/}
}
TY  - JOUR
AU  - Fiedler, Bernold
AU  - Hell, Juliette
AU  - Smith, Brian
TI  - Anisotropic Einstein data with isotropic non negative prescribed scalar curvature
JO  - Annales de l'I.H.P. Analyse non linéaire
PY  - 2015
DA  - 2015///
SP  - 401
EP  - 428
VL  - 32
IS  - 2
PB  - Elsevier
UR  - http://archive.numdam.org/articles/10.1016/j.anihpc.2014.01.002/
UR  - https://zbmath.org/?q=an%3A1332.35358
UR  - https://doi.org/10.1016/j.anihpc.2014.01.002
DO  - 10.1016/j.anihpc.2014.01.002
LA  - en
ID  - AIHPC_2015__32_2_401_0
ER  - 
%0 Journal Article
%A Fiedler, Bernold
%A Hell, Juliette
%A Smith, Brian
%T Anisotropic Einstein data with isotropic non negative prescribed scalar curvature
%J Annales de l'I.H.P. Analyse non linéaire
%D 2015
%P 401-428
%V 32
%N 2
%I Elsevier
%U https://doi.org/10.1016/j.anihpc.2014.01.002
%R 10.1016/j.anihpc.2014.01.002
%G en
%F AIHPC_2015__32_2_401_0
Fiedler, Bernold; Hell, Juliette; Smith, Brian. Anisotropic Einstein data with isotropic non negative prescribed scalar curvature. Annales de l'I.H.P. Analyse non linéaire, Volume 32 (2015) no. 2, pp. 401-428. doi : 10.1016/j.anihpc.2014.01.002. http://archive.numdam.org/articles/10.1016/j.anihpc.2014.01.002/

[1] M. Abramowitz, I.A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, (1970) | Zbl

[2] R. Bartnik, Existence of maximal surfaces in asymptotically flat spacetimes, Commun. Math. Phys. 94 (1984), 155 -175 | Zbl

[3] R. Bartnik, Quasi-spherical metrics and prescribed scalar curvature, J. Differ. Geom. 37 (1993), 31 -71 | Zbl

[4] R. Bartnik, J. Isenberg, The constraint equations, P.T. Chruściel, H. Friedrich (ed.), The Einstein Equations and the Large Scale Behavior of Gravitational Fields: 50 Years of the Cauchy Problem in General Relativity, Birkhäuser, Basel, Switzerland (2004), 1 -34

[5] P. Brunovskỳ, B. Fiedler, Connecting orbits in scalar reaction diffusion equations. II: The complete solution, J. Differ. Equ. 81 (1989), 106 -135 | Zbl

[6] M. Cantor, The existence of non-trivial asymptotically flat initial data for vacuum spacetimes, Commun. Math. Phys. 57 (1977), 83 -86 | Zbl

[7] M. Cantor, D. Brill, The Laplacian on asymptotically flat manifolds and the specification of scalar curvature, J. Differ. Geom. 43 (1981), 317 -330 | EuDML | Numdam | Zbl

[8] M. Cantor, A. Fischer, J. Marsden, N. Ō Murchadha, The existence of maximal slicings in asymptotically flat spacetimes, Commun. Math. Phys. 49 (1976), 187 -190 | Zbl

[9] J. Carr, Applications of Center Manifold Theory, Appl. Math. Sci. vol. 35 , Springer (1981)

[10] I. Chavel, Riemannian Geometry – A Modern Introduction, Cambridge University Press (2006) | Zbl

[11] P. Chossat, R. Lauterbach, The instability of axisymmetric solutions in problems with spherical symmetry, SIAM J. Math. Anal. 1 (1989), 31 -38 | Zbl

[12] P. Chossat, R. Lauterbach, Methods in Equivariant Bifurcation and Dynamical Systems, Adv. Ser. Nonlinear Dyn. vol. 15 , World Scientific (2000) | Zbl

[13] P. Chossat, R. Lauterbach, I. Melbourne, Steady-state bifurcation with O(3)-symmetry, Arch. Ration. Mech. Anal. 113 (1991), 313 -376 | Zbl

[14] G. Cicogna, Symmetry breakdown from bifurcation, Lett. Nuovo Cimento 31 (1981), 600 -602

[15] M.G. Crandall, P.H. Rabinowitz, Bifurcation from simple eigenvalues, J. Funct. Anal. 8 (1971), 321 -340 | Zbl

[16] B. Fiedler, Global Bifurcation of Periodic Solutions with Symmetry, Lect. Notes Math. vol. 1309 , Springer (1988) | Zbl

[17] B. Fiedler, K. Mischaikow, Dynamics of bifurcations for variational problems with O(3)-equivariance: A Conley index approach, Arch. Ration. Mech. Anal. 119 (1992), 145 -196 | Zbl

[18] M. Golubitsky, D.G. Schaeffer, I. Stewart, Singularities and Groups in Bifurcation Theory, vol. II, Appl. Math. Sci. vol. 69 , Springer (1988)

[19] M. Haragus, G. Iooss, Local Bifurcations, Center Manifolds, and Normal Forms in Infinite-dimensional Dynamical Systems, Universitext , Springer (2010)

[20] J. Hell, Conley index at infinity, Topol. Methods Nonlinear Anal. 42 (2013), 137 -168 | Zbl

[21] D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lect. Notes Math. vol. 840 , Springer (1981) | Zbl

[22] E. Ihrig, M. Golubitsky, Pattern selection with O(3) symmetry, Physica D 13 (1984), 1 -33 | Zbl

[23] H. Koch, On center manifolds, Nonlinear Anal. 28 (1997), 1227 -1248 | Zbl

[24] R. Lauterbach, Bifurcation with O(3)-symmetry, University of Augsburg (1988)

[25] R. Lauterbach, Dynamics near steady state bifurcations in problems with spherical symmetry, M. Roberts, I. Stewart (ed.), Singularity Theory and Its Applications, Warwick, 1989, Part II, Lect. Notes Math. vol. 1463 , Springer (1991), 256 -265

[26] A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, Birkhäuser (1995) | Zbl

[27] A. Lunardi, Interpolation Theory, Scuola Normale Superiore, Pisa (2009) | Zbl

[28] G. Da Prato, A. Lunardi, Stability, instability and center manifold for fully nonlinear autonomous parabolic equations in Banach space, Arch. Ration. Mech. Anal. 58 (1988), 115 -141 | Zbl

[29] A. Mielke, Locally invariant manifolds for quasilinear parabolic equations, Rocky Mt. J. Math. 21 (1991), 707 -714 | Zbl

[30] H. Ringström, The Cauchy Problem in General Relativity, European Mathematical Society (2009) | Zbl

[31] D.H. Sattinger, Group Theoretic Methods in Bifurcation Theory, Lect. Notes Math. vol. 762 , Springer (1979) | Zbl

[32] Y. Shi, L. Tam, Positive mass theorem and the boundary behaviors of compact manifolds with nonnegative scalar curvature, J. Differ. Geom. 62 (2002), 79 -125 | Zbl

[33] L. Simon, Asymptotics for a class of non-linear evolution equations, with applications to geometric problems, Ann. Math. 118 (1983), 525 -571 | Zbl

[34] G. Simonett, Center manifolds for quasilinear reaction–diffusion systems, Differ. Integral Equ. 8 (1995), 753 -796 | Zbl

[35] B. Smith, Blow-up in the parabolic scalar curvature equation, arXiv:0705.3774 (2012)

[36] B. Smith, Black hole initial data with a horizon of prescribed geometry, Gen. Relativ. Gravit. 41 (2009), 1013 -1024 | Zbl

[37] B. Smith, Black hole initial data with a horizon of prescribed intrinsic and extrinsic geometry, Complex Analysis and Dynamical Systems IV. Part 2. General Relativity, Geometry, and PDE. Proceedings of the 4th Conference on Complex Analysis and Dynamical Systems, Nahariya, Israel, 2009, American Mathematical Society (2011) | Zbl

[38] B. Smith, G. Weinstein, On the connectedness of the space of initial data for the Einstein equations, Electron. Res. Announc. Am. Math. Soc. 6 (2000), 52 -63 | EuDML | Zbl

[39] B. Smith, G. Weinstein, Quasi-convex foliations and asymptotically flat metrics of non-negative scalar curvature, Commun. Anal. Geom. 12 (2004), 511 -551 | Zbl

[40] A. Vanderbauwhede, G. Iooss, Center Manifolds in Infinite Dimensions, Dyn. Rep. Expo. Dyn. Syst. New Ser. vol. 1 , Springer (1992), 125 -163 | Zbl

[41] A. Vanderbauwhede, Local Bifurcation and Symmetry, Res. Notes Math. vol. 75 , Pitman, London (1982) | Zbl

[42] A. Vanderbauwhede, Centre Manifolds, Normal Forms and Elementary Bifurcations, Dyn. Rep. vol. 2 , Springer (1989), 89 -169 | Zbl

Cited by Sources: