Digital Object Identiﬁer (DOI) https://doi.org/10.1007/s00220-017-3004-9
Commun. Math. Phys. 357, 467–517 (2018)
Non-CMC Solutions of the Einstein Constraint Equations
on Compact Manifolds with Apparent Horizon Boundaries
Michael Holst , Caleb Meier, G. Tsogtgerel
Department of Mathematics, University of California San Diego, La Jolla, CA 92093, USA.
E-mail: email@example.com; firstname.lastname@example.org; email@example.com
Received: 22 October 2013 / Accepted: 5 August 2017
Published online: 2 November 2017 – © Springer-Verlag GmbH Germany 2017
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 com-
pact manifolds with boundary. By building in a natural way on our recent work in Holst
and Tsogtgerel (Class Quantum Gravity 30:205011, 2013),andHolstetal.(PhysRev
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 Dain and Maxwell for compact
manifolds with boundary, and subsequently proved existence of solutions to the Lich-
nerowicz 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 condi-
tions. 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.
MH was supported in part by NSF Awards 1217175, 1262982, and 1620366.
CM was supported by NSF Award 1065972.
GT was supported by an NSERC Discovery Grant and by an FQRNT Nouveaux Chercheurs Grant.