# Non-CMC Solutions of the Einstein Constraint Equations on Compact Manifolds with Apparent Horizon Boundaries

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## Abstract

In this article we continue our effort to do a systematic development of the solution theory for conformal formulations of the Einstein constraint equations on compact manifolds with boundary. By building in a natural way on our recent work in Holst and Tsogtgerel (Class Quantum Gravity 30:205011, 2013), and Holst et al. (Phys Rev Lett 100(16):161101, 2008, Commun Math Phys 288(2):547–613, 2009), and also on the work of Maxwell (J Hyperbolic Differ Eqs 2(2):521–546, 2005a, Commun Math Phys 253(3):561–583, 2005b, Math Res Lett 16(4):627–645, 2009) and Dain (Class Quantum Gravity 21(2):555–573, 2004), under reasonable assumptions on the data we prove existence of both near- and far-from-constant mean curvature (CMC) solutions for a class of Robin boundary conditions commonly used in the literature for modeling black holes, with a third existence result for CMC appearing as a special case. Dain and Maxwell addressed initial data engineering for space-times that evolve to contain black holes, determining solutions to the conformal formulation on an asymptotically Euclidean manifold in the CMC setting, with interior boundary conditions representing excised interior black hole regions. Holst and Tsogtgerel compiled the interior boundary results covered by Dain and Maxwell, and then developed general interior conditions to model the apparent horizon boundary conditions of Dainand Maxwell for compact manifolds with boundary, and subsequently proved existence of solutions to the Lichnerowicz equation on compact manifolds with such boundary conditions. This paper picks up where Holst and Tsogtgerel left off, addressing the general non-CMC case for compact manifolds with boundary. As in our previous articles, our focus here is again on low regularity data and on the interaction between different types of boundary conditions. While our work here serves primarily to extend the solution theory for the compact with boundary case, we also develop several technical tools that have potential for use for other cases.

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## References

- 1.Allen P.T., Clausen A., Isenberg J.: Near-constant mean curvature solutions of the Einstein constraint equations with non-negative Yamabe metrics.. Class. Quantum Gravity
**25**(7), 075009, 15 (2008)MathSciNetCrossRefMATHGoogle Scholar - 2.Aubin T.: Nonlinear Analysis on Manifolds. Monge-Ampére Equations. Springer, New York (1982)CrossRefMATHGoogle Scholar
- 3.Behzadan, A., Holst, M.: Multiplication in Sobolev–Slobodeckij spaces, revisited. Submitted for publication. Available as arXiv:1512.07379 [math.AP]
- 4.Behzadan, A., Holst, M.: On certain geometric operators between Sobolev spaces of sections of tensor bundles on compact manifolds equipped with rough metrics. Submitted for publication. Available as arXiv:1704.07930 [math.AP]
- 5.Choquet-Bruhat Y.: Einstein constraints on compact n-dimensional manifolds. Class. Quantum Gravity
**21**, S127– (2004)ADSMathSciNetCrossRefMATHGoogle Scholar - 6.Choquet-Bruhat Y., Isenberg J., York J.: Einstein constraint on asymptotically Euclidean manifolds. Phys. Rev. D
**61**, 084034 (2000)ADSMathSciNetCrossRefGoogle Scholar - 7.Dahl M., Gicquaud R., Humbert E.: A limit equation associated to the solvability of the vacuum Einstein constraint equations using the conformal method. Duke Math. J.
**161**(14), 2669–2697 (2012)MathSciNetCrossRefMATHGoogle Scholar - 8.Dain S.: Trapped surfaces as boundaries for the constraint equations. Class. Quantum Gravity
**21**(2), 555–573 (2004)ADSMathSciNetCrossRefMATHGoogle Scholar - 9.Dain S.: Generalized Korn’s inequality and conformal Killing vectors. Calc. Var.
**25**(4), 535–540 (2006) Available as arXiv:gr-qc/0505022.MathSciNetCrossRefMATHGoogle Scholar - 10.Dilts J.: The Einstein constraint equations on compact manifolds with boundary. Class. Quantum Gravity
**31**, 125009 (2014)ADSMathSciNetCrossRefMATHGoogle Scholar - 11.Escobar J.F.: The Yamabe problem on manifolds with boundary. J. Differ. Geom.
**35**(1), 21–84 (1992)MathSciNetCrossRefMATHGoogle Scholar - 12.Escobar J.F.: Conformal deformation of a Riemannian metric to a constant scalar curvature metric with constant mean curvature on the boundary. Indiana Univ. Math. J.
**45**(4), 917–943 (1996)MathSciNetCrossRefMATHGoogle Scholar - 13.Hebey, E.: Sobolev spaces on Riemannian manifolds, volume 1635 of Lecture notes in mathematics. Springer, Berlin (1996)Google Scholar
- 14.Holst M., Nagy G., Tsogtgerel G.: Far-from-constant mean curvature solutions of Einstein’s constraint equations with positive Yamabe metrics. Phys. Rev. Lett.
**100**(16), 161101 (2008)ADSMathSciNetCrossRefMATHGoogle Scholar - 15.Holst M., Nagy G., Tsogtgerel G.: Rough solutions of the Einstein constraints on closed manifolds without near-CMC conditions. Commun. Math. Phys.
**288**(2), 547–613 (2009)ADSMathSciNetCrossRefMATHGoogle Scholar - 16.Holst M., Tsogtgerel G.: The Lichnerowicz equation on compact manifolds with boundary. Class. Quantum Gravity
**30**, 205011 (2013)ADSMathSciNetCrossRefMATHGoogle Scholar - 17.Isenberg J.: Constant mean curvature solution of the Einstein constraint equations on closed manifold. Class. Quantum Gravity
**12**, 2249–2274 (1995)ADSMathSciNetCrossRefMATHGoogle Scholar - 18.Isenberg J., Moncrief V.: A set of nonconstant mean curvature solution of the Einstein constraint equations on closed manifolds. Class. Quantum Gravity
**13**, 1819–1847 (1996)ADSMathSciNetCrossRefMATHGoogle Scholar - 19.Lee J., Parker T.: The Yamabe problem. Bull. Am. Math. Soc.
**17**(1), 37–91 (1987)MathSciNetCrossRefMATHGoogle Scholar - 20.Maxwell D.: Rough solutions of the Einstein constraint equations on compact manifolds. J. Hyp. Differ. Eqs.
**2**(2), 521–546 (2005)MathSciNetCrossRefMATHGoogle Scholar - 21.Maxwell D.: Solutions of the Einstein constraint equations with apparent horizon boundaries. Commun. Math. Phys.
**253**(3), 561–583 (2005)ADSMathSciNetCrossRefMATHGoogle Scholar - 22.Maxwell D.: Rough solutions of the Einstein constraint equations. J. Reine Angew. Math.
**590**, 1–29 (2006)MathSciNetCrossRefMATHGoogle Scholar - 23.Maxwell D.: A class of solutions of the vacuum Einstein constraint equations with freely specified mean curvature.. Math. Res. Lett.
**16**(4), 627–645 (2009)MathSciNetCrossRefMATHGoogle Scholar - 24.Palais R.: Seminar on the Atiyah–Singer index theorem. Princeton University Press, Princeton (1965)MATHGoogle Scholar
- 25.Wald R.: General Relativity. The University of Chicago Press, Chicago (1984)CrossRefMATHGoogle Scholar
- 26.York, J., Piran, T.: The initial value problem and beyond. In R. A. Matzner and L. C. Shepley, editors, Spacetime and Geometry: The Alfred Schild Lectures, pp. 147–176. University of Texas Press, Austin (Texas) (1982)Google Scholar
- 27.Zeidler, E.: Nonlinear Functional Analysis and its Applications II/A, Linear Monotone Operators. Springer, New York (1990)Google Scholar