We provide a complete study of the existence and uniqueness of solutions to the Lichnerowicz equation in general relativity with arbitrary mean curvature.
REFERENCES
1.
Adams
, D. R.
and Hedberg
, L. I.
, “Function spaces and potential theory
,” in Grundlehren der Mathematischen Wissenschaften
, Fundamental Principles of Mathematical Sciences Vol. 314 (Springer-Verlag
, Berlin
, 1996
).2.
Aubin
, T.
, Some Nonlinear Problems in Riemannian Geometry
, Springer Monographs in Mathematics (Springer-Verlag
, Berlin
, 1998
).3.
Bartnik
, R.
and Isenberg
, J.
, “The constraint equations
,” in The Einstein Equations and the Large Scale Behavior of Gravitational Fields
(Birkhäuser
, Basel
, 2004
), pp. 1
–38
.4.
Brézis
, H.
and Oswald
, L.
, “Remarks on sublinear elliptic equations
,” Nonlinear Anal.
10
, 55
–64
(1986
).5.
Choquet-Bruhat
, Y.
, General Relativity and the Einstein Equations
, Oxford Mathematical Monographs (Oxford University Press
, Oxford
, 2009
).6.
Dahl
, M.
, Gicquaud
, R.
, and Humbert
, E.
, “A limit equation associated to the solvability of the vacuum Einstein constraint equations by using the conformal method
,” Duke Math. J.
161
(14
), 2669
–2697
(2012
).7.
Dilts
, J.
and Maxwell
, D.
, “Yamabe classification and prescribed scalar curvature in the asymptotically Euclidean setting
,” Commun. Anal. Geom.
26
(5
), 1127
–1168
(2018
).8.
Gicquaud
, R.
, “Prescribed non positive scalar curvature on asymptotically hyperbolic manifolds with application to the Lichnerowicz equation
,” Communications in Analysis and Geometry
,arXiv:1909.05343.9.
Gicquaud
, R.
, “Solutions to the Einstein constraint equations with a small TT-tensor and vanishing Yamabe invariant
,” Ann. Henri Poincarè
22
, 2407
–2435
(2021
).10.
Gicquaud
, R.
and Anh Ngô
, Q.
, “A new point of view on the solutions to the Einstein constraint equations with arbitrary mean curvature and small TT-tensor
,” Classical Quantum Gravity
31
(19
), 195014
(2014
).11.
Gilbarg
, D.
and Trudinger
, N. S.
, Elliptic Partial Differential Equations of Second Order
, Classics in Mathematics (Springer-Verlag
, Berlin
, 2001
), reprint of the 1998 edition.12.
Holst
, M.
, Nagy
, G.
, and Tsogtgerel
, G.
, “Far-from-constant mean curvature solutions of Einstein’s constraint equations with positive Yamabe metrics
,” Phys. Rev. Lett.
100
(16
), 161101
(2008
).13.
Holst
, M.
, Nagy
, G.
, and Tsogtgerel
, G.
, “Rough solutions of the Einstein constraints on closed manifolds without near-CMC conditions
,” Commun. Math. Phys.
288
(2
), 547
–613
(2009
).14.
Isenberg
, J.
, “Constant mean curvature solutions of the Einstein constraint equations on closed manifolds
,” Classical Quantum Gravity
12
(9
), 2249
–2274
(1995
).15.
Lee
, J. M.
and Parker
, T. H.
, “The Yamabe problem
,” Bull. Am. Math. Soc.
17
(1
), 37
–91
(1987
).16.
Maxwell
, D.
, “Rough solutions of the Einstein constraint equations on compact manifolds
,” J. Hyperbolic Differ. Equations
2
(2
), 521
–546
(2005
).17.
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
); available at https://www.intlpress.com/site/pub/pages/journals/items/mrl/content/vols/0016/0004/a006/.18.
Ouyang
, T.
, “On the positive solutions of semilinear equations Δu + λu + hup = 0 on compact manifolds. II
,” Indiana Univ. Math. J.
40
(3
), 1083
–1141
(1991
).19.
Ouyang
, T.
, “On the positive solutions of semilinear equations Δu + λu − hup = 0 on the compact manifolds
,” Trans. Am. Math. Soc.
331
(2
), 503
–527
(1992
).20.
Rauzy
, A.
, “Courbures scalaires des variétés d’invariant conforme négatif
,” Trans. Am. Math. Soc.
347
(12
), 4729
–4745
(1995
).21.
Tang
, J. J.
, “Solvability of the equation on manifolds
,” Proc. Am. Math. Soc.
121
(1
), 83
–92
(1994
).22.
Taylor
, M. E.
, Partial Differential Equations III: Nonlinear Equations
, 2nd ed., Applied Mathematical Sciences Vol. 117 (Springer
, New York
, 2011
).© 2022 Author(s). Published under an exclusive license by AIP Publishing.
2022
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