The general relativistic constraint equations

Abate M, Tovena F (2012) Curves and surfaces. Unitext, vol 55. Springer, Milan. https://doi.org/10.1007/978-88-470-1941-6 Aceña AE (2009) Convergent null data expansions at space-like infinity of stationary vacuum solutions. Ann Henri Poincaré 10:275–337. https://doi.org/10.1007/s00023-009-0406-z Albanese G, Rigoli M (2016) Lichnerowicz-type equations on complete manifolds. Adv Nonlinear Anal 5:223–250. https://doi.org/10.1515/anona-2015-0106 Albanese G, Rigoli M (2017) Lichnerowicz-type equations with sign-changing nonlinearities on complete manifolds with boundary. J Differ Equations 263:7475–7495. https://doi.org/10.1016/j.jde.2017.08.010 Allen PT, Clausen A, Isenberg J (2008) Near-constant mean curvature solutions of the Einstein constraint equations with non-negative Yamabe metrics. Class Quantum Grav 25:075009. https://doi.org/10.1088/0264-9381/25/7/075009 Ambrosio L (2004) Transport equation and Cauchy problem for \(BV\) vector fields. Invent Math 158:227–260. https://doi.org/10.1007/s00222-004-0367-2 Ambrosio L, Trevisan D (2017) Lecture notes on the DiPerna–Lions theory in abstract measure spaces. Ann Fac Sci Toulouse Math 26:729–766. https://doi.org/10.5802/afst.1551 Ambrosio L, Carlotto A, Massaccesi A (2018) Lectures on elliptic partial differential equations, Appunti. Scuola Normale Superiore di Pisa (Nuova Serie) [Lecture Notes. Scuola Normale Superiore di Pisa (New Series)], vol 18. Edizioni della Normale, Pisa. https://doi.org/10.1007/978-88-7642-651-3 Ambrozio L (2015) On perturbations of the Schwarzschild anti-de Sitter spaces of positive mass. Commun Math Phys 337:767–783. https://doi.org/10.1007/s00220-015-2360-6 Ambrozio L (2017) On static three-manifolds with positive scalar curvature. J Differ Geom 107:1–45. https://doi.org/10.4310/jdg/1505268028 An Z (2020) On mass-minimizing extension of Bartnik boundary data. arXiv e-prints arXiv:2007.05452 Anderson M (2018) On the conformal method for the Einstein constraint equations. arXiv e-prints arXiv:1812.06320 Anderson MT, Chruściel PT (2005) Asymptotically simple solutions of the vacuum Einstein equations in even dimensions. Commun Math Phys 260:557–577. https://doi.org/10.1007/s00220-005-1424-4 Anderson MT, Jauregui JL (2019) Embeddings, immersions and the Bartnik quasi-local mass conjectures. Ann Henri Poincaré 20:1651–1698. https://doi.org/10.1007/s00023-019-00786-3 Andersson L, Chruściel PT (1996) Solutions of the constraint equations in general relativity satisfying “hyperboloidal boundary conditions”. Diss Math (Rozprawy Mat) 355:100 Andersson L, Dahl M (1998) Scalar curvature rigidity for asymptotically locally hyperbolic manifolds. Ann Glob Anal Geom 16:1–27. https://doi.org/10.1023/A:1006547905892 Andersson L, Chruściel PT, Friedrich H (1992) On the regularity of solutions to the Yamabe equation and the existence of smooth hyperboloidal initial data for Einstein’s field equations. Commun Math Phys 149:587–612. https://doi.org/10.1007/bf02096944 Andersson L, Cai M, Galloway GJ (2008) Rigidity and positivity of mass for asymptotically hyperbolic manifolds. Ann Henri Poincaré 9:1–33. https://doi.org/10.1007/s00023-007-0348-2 Andersson L, Bäckdahl T, Joudioux J (2014) Hertz potentials and asymptotic properties of massless fields. Commun Math Phys 331:755–803. https://doi.org/10.1007/s00220-014-2078-x Arms JM, Marsden JE (1979) The absence of Killing fields is necessary for linearization stability of Einstein’s equations. Indiana Univ Math J 28:119–125 Arms JM, Marsden JE, Moncrief V (1982) The structure of the space of solutions of Einstein’s equations. II. Several Killing fields and the Einstein–Yang–Mills equations. Ann Phys 144:81–106. https://doi.org/10.1016/0003-4916(82)90105-1 Arnowitt R, Deser S, Misner CW (1959) Dynamical structure and definition of energy in general relativity. Phys Rev 116:1322–1330. https://doi.org/10.1103/physrev.116.1322 Arnowitt R, Deser S, Misner CW (1960) Canonical variables for general relativity. Phys Rev 117:1595–1602. https://doi.org/10.1103/physrev.117.1595 Arnowitt R, Deser S, Misner CW (1961) Coordinate invariance and energy expressions in general relativity. Phys Rev 122:997–1006. https://doi.org/10.1103/physrev.122.997 Arnowitt R, Deser S, Misner CW (1962) The dynamics of general relativity. In: Witten L (ed) Gravitation: an introduction to current research. Wiley, New York, pp 227–265 Aronszajn N (1957) A unique continuation theorem for solutions of elliptic partial differential equations or inequalities of second order. J Math Pures Appl 36:235–249 Ashtekar A, Horowitz GT (1982) Energy–momentum of isolated systems cannot be null. Phys Lett A 89:181–184. https://doi.org/10.1016/0375-9601(82)90203-1 Aubin T (1970) Métriques riemanniennes et courbure. J Differ Geom 4:383–424. https://doi.org/10.4310/jdg/1214429638 Aubin T (1976) Équations différentielles non linéaires et problème de Yamabe concernant la courbure scalaire. J Math Pures Appl 55:269–296 Aubin T (1998) Some nonlinear problems in Riemannian geometry. Springer monographs in mathematics. Springer, Berlin. https://doi.org/10.1007/978-3-662-13006-3 Bahouri H, Chemin JY (1999) Équations d’ondes quasilinéaires et estimations de Strichartz. Am J Math 121:1337–1377. https://doi.org/10.1353/ajm.1999.0038 Bamler R, Kleiner B (2017) Uniqueness and stability of Ricci flow through singularities. arXiv e-prints arXiv:1709.04122 Bamler R, Kleiner B (2019) Ricci flow and contractibility of spaces of metrics. arXiv e-prints arXiv:1909.08710 Bartnik R (1984) Existence of maximal surfaces in asymptotically flat spacetimes. Commun Math Phys 94:155–175. https://doi.org/10.1007/bf01209300 Bartnik R (1986) The mass of an asymptotically flat manifold. Commun Pure Appl Math 39:661–693. https://doi.org/10.1002/cpa.3160390505 Bartnik R (1989) New definition of quasilocal mass. Phys Rev Lett 62:2346–2348. https://doi.org/10.1103/physrevlett.62.2346 Bartnik R (1993) Quasi-spherical metrics and prescribed scalar curvature. J Differ Geom 37:31–71. https://doi.org/10.4310/jdg/1214453422 Bartnik R (1997) Energy in general relativity. Tsing Hua lectures on geometry & analysis (Hsinchu, 1990–1991). International Press, Cambridge, pp 5–27 Bartnik R (2002) Mass and 3-metrics of non-negative scalar curvature. In: Proceedings of the international congress of mathematicians, vol. II (Beijing, 2002). Higher Education Press, Beijing, pp 231–240 Bartnik R (2005) Phase space for the Einstein equations. Commun Anal Geom 13:845–885. https://doi.org/10.4310/cag.2005.v13.n5.a1 Bartnik R, Fodor G (1993) On the restricted validity of the thin sandwich conjecture. Phys Rev D 48:3596–3599. https://doi.org/10.1103/physrevd.48.3596 Bartnik R, Isenberg J (2004) The constraint equations. In: The Einstein equations and the large scale behavior of gravitational fields. Birkhäuser, Basel, pp 1–38. https://doi.org/10.1007/978-3-0348-7953-8_1 Baumgarte TW, Shapiro SL (2010) Numerical relativity. Solving Einstein’s equations on the computer. Cambridge University Press, Cambridge Baumgarte TW, Ó Murchadha N, Pfeiffer HP (2007) Einstein constraints: uniqueness and nonuniqueness in the conformal thin sandwich approach. Phys Rev D 75:044009. https://doi.org/10.1103/physrevd.75.044009 Beig R (1997) TT-tensors and conformally flat structures on 3-manifolds. In: Mathematics of gravitation, Part I (Warsaw, 1996). Banach Center Publ., vol 41. Polish Acad. Sci. Inst. Math., Warsaw, pp 109–118. https://doi.org/10.4064/-41-1-109-118 Beig R, Chruściel PT (1996) Killing vectors in asymptotically flat space-times. I. Asymptotically translational Killing vectors and the rigid positive energy theorem. J Math Phys 37:1939–1961. https://doi.org/10.1063/1.531497 Beig R, Chruściel PT (2017) Shielding linearized gravity. Phys Rev D 95:064063. https://doi.org/10.1103/physrevd.95.064063 Beig R, Chruściel PT (2020) On linearised vacuum constraint equations on Einstein manifolds. Class Quantum Grav. https://doi.org/10.1088/1361-6382/ab81cc Beig R, Chruściel PT, Schoen R (2005) KIDs are non-generic. Ann Henri Poincaré 6:155–194. https://doi.org/10.1007/s00023-005-0202-3 Bérard-Bergery L (1983) Scalar curvature and isometry group. Spectra of Riemannian manifolds. Kaigai Publications, Tokyo, pp 9–28 Besse AL (2008) Einstein manifolds. Classics in mathematics, reprint of the 1987 edition. Springer, Berlin. https://doi.org/10.1007/978-3-540-74311-8 Bessières L, Besson G, Maillot S (2011) Ricci flow on open 3-manifolds and positive scalar curvature. Geom Topol 15:927–975. https://doi.org/10.2140/gt.2011.15.927 Bieri L (2010) An extension of the stability theorem of the Minkowski space in general relativity. J Differ Geom 86:17–70. https://doi.org/10.4310/jdg/1299766683 Bizoń P, Pletka S, Simon W (2015) Initial data for rotating cosmologies. Class Quantum Grav 32:175015. https://doi.org/10.1088/0264-9381/32/17/175015 Bland J, Kalka M (1989) Negative scalar curvature metrics on noncompact manifolds. Trans Amer Math Soc 316:433–446. https://doi.org/10.1090/s0002-9947-1989-0987159-2 Borghini S, Mazzieri L (2018) On the mass of static metrics with positive cosmological constant: I. Class Quantum Grav 35:125001. https://doi.org/10.1088/1361-6382/aac081 Botvinnik B, Gilkey PB (1996) Metrics of positive scalar curvature on spherical space forms. Can J Math 48:64–80. https://doi.org/10.4153/cjm-1996-003-0 Botvinnik B, Hanke B, Schick T, Walsh M (2010) Homotopy groups of the moduli space of metrics of positive scalar curvature. Geom Topol 14:2047–2076. https://doi.org/10.2140/gt.2010.14.2047 Boucher W, Gibbons GW, Horowitz GT (1984) Uniqueness theorem for anti-de Sitter spacetime. Phys Rev D 30:2447–2451. https://doi.org/10.1103/physrevd.30.2447 Bourguignon JP, Ebin DG, Marsden JE (1976) Sur le noyau des opérateurs pseudo-différentiels à symbole surjectif et non injectif. C R Acad Sci Paris Sér A-B 282:A867–A870 Bray HL (2001) Proof of the Riemannian Penrose inequality using the positive mass theorem. J Differ Geom 59:177–267. https://doi.org/10.4310/jdg/1090349428 Brendle S, Chen SYS (2014) An existence theorem for the Yamabe problem on manifolds with boundary. J Eur Math Soc 16:991–1016. https://doi.org/10.4171/jems/453 Brendle S, Marques FC (2011) Scalar curvature rigidity of geodesic balls in \(S^n\). J Differ Geom 88:379–394. https://doi.org/10.4310/jdg/1321366355 Brendle S, Marques FC, Neves A (2011) Deformations of the hemisphere that increase scalar curvature. Invent Math 185:175–197. https://doi.org/10.1007/s00222-010-0305-4 Brill D, Cantor M (1981) The Laplacian on asymptotically flat manifolds and the specification of scalar curvature. Compos Math 43:317–330 Bunting GL, Masood-ul Alam AKM (1987) Nonexistence of multiple black holes in asymptotically Euclidean static vacuum space-time. Gen Relativ Gravit 19:147–154. https://doi.org/10.1007/bf00770326 Cabrera Pacheco AJ, Miao P (2018) Higher dimensional black hole initial data with prescribed boundary metric. Math Res Lett 25:937–956. https://doi.org/10.4310/mrl.2018.v25.n3.a10 Cabrera Pacheco AJ, Cederbaum C, McCormick S, Miao P (2017) Asymptotically flat extensions of CMC Bartnik data. Class Quantum Grav 34:105001. https://doi.org/10.1088/1361-6382/aa6921 Cabrera Pacheco AJ, Cederbaum C, McCormick S (2018) Asymptotically hyperbolic extensions and an analogue of the Bartnik mass. J Geom Phys 132:338–357. https://doi.org/10.1016/j.geomphys.2018.06.010 Caffarelli LA, Gidas B, Spruck J (1989) Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth. Commun Pure Appl Math 42:271–297. https://doi.org/10.1002/cpa.3160420304 Cantor M (1977) The existence of non-trivial asymptotically flat initial data for vacuum spacetimes. Commun Math Phys 57:83–96. https://doi.org/10.1007/bf01651695 Cantor M (1979) A necessary and sufficient condition for York data to specify an asymptotically flat spacetime. J Math Phys 20:1741–1744. https://doi.org/10.1063/1.524259 Carlotto A (2021) A survey on positive scalar curvature metrics. Boll Unione Mat Ital. https://doi.org/10.1007/s40574-020-00228-7 Carlotto A, Li C (2019) Constrained deformations of positive scalar curvature metrics. arXiv e-prints arXiv:1903.11772 Carlotto A, Schoen R (2016) Localizing solutions of the Einstein constraint equations. Invent Math 205:559–615. https://doi.org/10.1007/s00222-015-0642-4 Carlotto A, Chodosh O, Rubinstein YA (2015) Slowly converging Yamabe flows. Geom Topol 19:1523–1568. https://doi.org/10.2140/gt.2015.19.1523 Carlotto A, Chodosh O, Eichmair M (2016) Effective versions of the positive mass theorem. Invent Math 206:975–1016. https://doi.org/10.1007/s00222-016-0667-3 Carr R (1988) Construction of manifolds of positive scalar curvature. Trans Amer Math Soc 307:63–74. https://doi.org/10.1090/s0002-9947-1988-0936805-7 Cerf J (1968) Sur les difféomorphismes de la sphère de dimension trois \((\Gamma _{4}=0)\). Lecture notes in mathematics, vol 53. Springer, Berlin. https://doi.org/10.1007/bfb0060395 Chau A, Martens A (2020) Exterior Schwarzschild initial data for degenerate apparent horizons. arXiv e-prints arXiv:2004.09060 Chen PN, Wang MT (2015) Rigidity and minimizing properties of quasi-local mass. In: Cao HD, Schoen R, Yau ST (eds) Surveys in differential geometry 2014. Regularity and evolution of nonlinear equations. Surveys in differential geometry, vol 19. International Press, Somerville, pp 49–61 Chodosh O (2016) Large isoperimetric regions in asymptotically hyperbolic manifolds. Commun Math Phys 343:393–443. https://doi.org/10.1007/s00220-015-2457-y Chodosh O, Li C (2019) Generalized soap bubbles and the topology of manifolds with positive scalar curvature. arXiv e-prints arXiv:2008.11888 Chodosh O, Eichmair M, Shi Y, Yu H (2016) Isoperimetry, scalar curvature and mass in asymptotically flat Riemannian 3-manifolds. arXiv e-prints arXiv:1606.04626 Choquet-Bruhat Y (1952) Théorème d’existence pour certains systèmes d’équations aux dérivées partielles non linéaires. Acta Math 88:141–225. https://doi.org/10.1007/bf02392131 Choquet-Bruhat Y (1993) Solution des contraintes pour les équations d’Einstein sur une variété asymptotiquement euclidienne non maximale. C R Acad Sci Paris Sér I Math 317:109–114 Choquet-Bruhat Y (2004) Einstein constraints on compact \(n\)-dimensional manifolds. Class Quantum Grav 21:S127–S151. https://doi.org/10.1088/0264-9381/21/3/009 Choquet-Bruhat Y (2009) General relativity and the Einstein equations. Oxford mathematical monographs. Oxford University Press, Oxford Choquet-Bruhat Y (2015) Beginnings of the Cauchy problem for Einstein’s field equations. In: Bieri L, Yau ST (eds) Surveys in differential geometry 2015. One hundred years of general relativity. Surv. Differ. Geom., vol 20. International Press, Boston, pp 1–16. https://doi.org/10.4310/sdg.2015.v20.n1.a1 Choquet-Bruhat Y, Christodoulou D (1981) Elliptic systems in \(H_{s,\delta }\) spaces on manifolds which are Euclidean at infinity. Acta Math 146:129–150. https://doi.org/10.1007/bf02392460 Choquet-Bruhat Y, Deser S (1972) Stabilité initiale de l’espace temps de Minkowski. C R Acad Sci Paris Sér A-B 275:A1019–A1021 Choquet-Bruhat Y, Deser S (1973) On the stability of flat space. Ann Phys 81:165–178. https://doi.org/10.1016/0003-4916(73)90484-3 Choquet-Bruhat Y, Geroch R (1969) Global aspects of the Cauchy problem in general relativity. Commun Math Phys 14:329–335. https://doi.org/10.1007/bf01645389 Choquet-Bruhat Y, Moncrief V (2003) Nonlinear stability of an expanding universe with the \(S^1\) isometry group. In: Partial differential equations and mathematical physics (Tokyo, 2001). Progr. Nonlinear Differential Equations Appl., vol 52. Birkhäuser, Boston, Boston, MA, pp 57–71. https://doi.org/10.1007/978-1-4612-0011-6_5 Choquet-Bruhat Y, York JW Jr (1980) The Cauchy problem. General relativity and gravitation, vol 1. Plenum, New York, pp 99–172 Choquet-Bruhat Y, Fisher A, Marsden J (1977) Équations des contraintes sur une variété non compacte. C R Acad Sci Paris Sér A-B 284:A975–A978 Choquet-Bruhat Y, Isenberg J, York JW Jr (2000) Einstein constraints on asymptotically Euclidean manifolds. Phys Rev D 61:084034. https://doi.org/10.1103/physrevd.61.084034 Choquet-Bruhat Y, Isenberg J, Pollack D (2006) The Einstein-scalar field constraints on asymptotically Euclidean manifolds. Chin Ann Math Ser B 27:31–52. https://doi.org/10.1007/s11401-005-0280-z Choquet-Bruhat Y, Isenberg J, Pollack D (2007a) Applications of theorems of Jean Leray to the Einstein-scalar field equations. J Fixed Point Theory Appl 1:31–46 Choquet-Bruhat Y, Isenberg J, Pollack D (2007b) The constraint equations for the Einstein-scalar field system on compact manifolds. Class Quantum Grav 24:809–828. https://doi.org/10.1088/0264-9381/24/4/004 Christodoulou D, Ó Murchadha (1981) The boost problem in general relativity. Commun Math Phys 80:271–300. https://doi.org/10.1007/bf01213014 Chruściel P (1986a) Boundary conditions at spatial infinity from a Hamiltonian point of view. In: Topological properties and global structure of space-time (Erice, 1985). NATO ASI Series B, vol 138. Plenum, New York, pp 49–59 Chruściel PT (1986b) A remark on the positive-energy theorem. Class Quantum Grav 3:L115–L121. https://doi.org/10.1088/0264-9381/3/6/002 Chruściel PT (1990) On space-times with \(U(1)\times U(1)\) symmetric compact Cauchy surfaces. Ann Phys 202:100–150. https://doi.org/10.1016/0003-4916(90)90341-k Chruściel PT (2005) Recent results in mathematical relativity. In: General relativity and gravitation. World Scientific, Hackensack, pp 36–55. https://doi.org/10.1142/9789812701688_0005 Chruściel PT (2019) Anti-gravity à la Carlotto-Schoen [after Carlotto and Schoen]. Astérisque 407:Exposé No. 1120, 1–25. https://doi.org/10.24033/ast.1058, s’eminaire Bourbaki. Vol. 2016/2017 Chruściel PT, Delay E (2002) Existence of non-trivial, vacuum, asymptotically simple spacetimes. Class Quantum Grav 19:L71–L79. https://doi.org/10.1088/0264-9381/19/9/101 Chruściel PT, Delay E (2003) On mapping properties of the general relativistic constraints operator in weighted function spaces, with applications, Mém. Soc. Math. France, vol 94. SMF. https://doi.org/10.24033/msmf.407 Chruściel PT, Delay E (2004) Manifold structures for sets of solutions of the general relativistic constraint equations. J Geom Phys 51:442–472. https://doi.org/10.1016/j.geomphys.2003.12.002 Chruściel PT, Delay E (2009) Gluing constructions for asymptotically hyperbolic manifolds with constant scalar curvature. Commun Anal Geom 17:343–381. https://doi.org/10.4310/cag.2009.v17.n2.a8 Chruściel PT, Delay E (2018) Exotic hyperbolic gluings. J Differ Geom 108:243–293. https://doi.org/10.4310/jdg/1518490818 Chruściel PT, Delay E (2019) The hyperbolic positive energy theorem. arXiv e-prints arXiv:1901.05263 Chruściel PT, Gicquaud R (2017) Bifurcating solutions of the Lichnerowicz equation. Ann Henri Poincaré 18:643–679 Chruściel PT, Herzlich M (2003) The mass of asymptotically hyperbolic Riemannian manifolds. Pac J Math 212:231–264. https://doi.org/10.2140/pjm.2003.212.231 Chruściel PT, Maerten D (2006) Killing vectors in asymptotically flat space-times. II. Asymptotically translational Killing vectors and the rigid positive energy theorem in higher dimensions. J Math Phys 47:022502. https://doi.org/10.1063/1.2167809 Chruściel PT, Mazzeo R (2015) Initial data sets with ends of cylindrical type: I. The Lichnerowicz equation. Ann Henri Poincaré 16:1231–1266 Chruściel PT, Jezierski J, Łȩski S (2004) The Trautman–Bondi mass of hyperboloidal initial data sets. Adv Theor Math Phys 8:83–139. https://doi.org/10.4310/atmp.2004.v8.n1.a2 Chruściel PT, Isenberg J, Pollack D (2005) Initial data engineering. Commun Math Phys 257:29–42. https://doi.org/10.1007/s00220-005-1345-2 Chruściel PT, Corvino J, Isenberg J (2010a) Initial data for the relativistic gravitational \(N\)-body problem. Class Quantum Grav 27:222002 Chruściel PT, Galloway GJ, Pollack D (2010b) Mathematical general relativity: a sampler. Bull Am Math Soc 47:567–638. https://doi.org/10.1090/s0273-0979-2010-01304-5 Chruściel PT, Corvino J, Isenberg J (2011) Construction of \(N\)-body initial data sets in general relativity. Commun Math Phys 304:637–647. https://doi.org/10.1007/s00220-011-1244-7 Chruściel PT, Mazzeo R, Pocchiola S (2013) Initial data sets with ends of cylindrical type: II. The vector constraint equation. Adv Theor Math Phys 17:829–865. https://doi.org/10.4310/atmp.2013.v17.n4.a4 Cook GB (2000) Initial data for numerical relativity. Living Rev Relativ 3:5. https://doi.org/10.12942/lrr-2000-5 Corvino J (2000) Scalar curvature deformation and a gluing construction for the Einstein constraint equations. Commun Math Phys 214:137–189. https://doi.org/10.1007/pl00005533 Corvino J, Huang LH (2016) Localized deformation for initial data sets with the dominant energy condition. arXiv e-prints 10.1007/s00526-019-1679-9. arXiv:1606.03078 Corvino J, Pollack D (2011) Scalar curvature and the Einstein constraint equations. In: Surveys in geometric analysis and relativity. Advanced Lectures in Mathematics, vol 20. International Press, Somerville, MA, pp 145–188 Corvino J, Schoen RM (2006) On the asymptotics for the vacuum Einstein constraint equations. J Differ Geom 73:185–217. https://doi.org/10.4310/jdg/1146169910 Corvino J, Eichmair M, Miao P (2013) Deformation of scalar curvature and volume. Math Ann 357:551–584. https://doi.org/10.1007/s00208-013-0903-8 Crandall MG, Rabinowitz PH (1971) Bifurcation from simple eigenvalues. J Funct Anal 8:321–340. https://doi.org/10.1016/0022-1236(71)90015-2 Cvetič M, Gibbons GW, Pope CN (2011) More about Birkhoff’s invariant and Thorne’s hoop conjecture for horizons. Class Quantum Grav 28:195001. https://doi.org/10.1088/0264-9381/28/19/195001 Czimek S (2018) An extension procedure for the constraint equations. Ann PDE 4:2. https://doi.org/10.1007/s40818-017-0039-3 Czimek S (2019) The localised bounded \(L^2\)-curvature theorem. Commun Math Phys 372:71–90. https://doi.org/10.1007/s00220-019-03458-9 Dahl M, Sakovich A (2015) A density theorem for asymptotically hyperbolic initial data satisfying the dominant energy condition. arXiv e-prints arXiv:1502.07487 Dahl M, Gicquaud R, Humbert E (2012) A limit equation associated to the solvability of the vacuum Einstein constraint equations by using the conformal method. Duke Math J 161:2669–2697. https://doi.org/10.1215/00127094-1813182 Dain S (2004) Trapped surfaces as boundaries for the constraint equations. Class Quantum Grav 21:555–573. https://doi.org/10.1088/0264-9381/22/4/c01 Dain S (2008) Proof of the angular momentum-mass inequality for axisymmetric black holes. J Differ Geom 79:33–67. https://doi.org/10.4310/jdg/1207834657 Dain S, Gabach Clément ME (2009) Extreme Bowen–York initial data. Class Quantum Grav 26:035020. https://doi.org/10.1088/0264-9381/26/3/035020 Dain S, Gabach Clément ME (2011) Small deformations of extreme Kerr black hole initial data. Class Quantum Grav 28:075003. https://doi.org/10.1088/0264-9381/28/7/075003 Dain S, Gabach-Clement ME (2018) Geometrical inequalities bounding angular momentum and charges in general relativity. Living Rev Relativ 21:5. https://doi.org/10.1007/s41114-018-0014-7 Daszuta B, Frauendiener J (2019) Numerical initial data deformation exploiting a gluing construction: I. Exterior asymptotic Schwarzschild. Class Quantum Grav 36:185008. https://doi.org/10.1088/1361-6382/ab34d8 De Lellis C (2008) Ordinary differential equations with rough coefficients and the renormalization theorem of Ambrosio [after Ambrosio, DiPerna, Lions]. Astérisque 317:Exposé No. 972, 175–203. Séminaire Bourbaki. Vol. 2006/2007 Delay E (2020) Hilbert manifold structure on fibers of the scalar curvature and the constraint operator. arXiv e-prints arXiv:2003.02129 Delay E, Fougeirol J (2016) Hilbert manifold structure for asymptotically hyperbolic relativistic initial data. arXiv e-prints arXiv:1607.05616 Delay E, Mazzieri L (2014) Refined gluing for vacuum Einstein constraint equations. Geom Dedicata 173:393–415. https://doi.org/10.1007/s10711-013-9948-9 Dilts J (2014) The Einstein constraint equations on compact manifolds with boundary. Class Quantum Grav 31:125009 arXiv:1310.2303 Dilts J (2015) The Einstein constraint equations on asymptotically Euclidean manifolds. PhD thesis, University of Oregon. arXiv:1507.01913 [gr-qc] Dilts J, Isenberg J (2016) Existence and blowup results for asymptotically Euclidean initial data sets generated by the conformal method. Phys Rev D 94:104046. https://doi.org/10.1103/physrevd.94.104046 Dilts J, Leach J (2015) A limit equation criterion for solving the Einstein constraint equations on manifolds with ends of cylindrical type. Ann Henri Poincaré 16:1583–1607. https://doi.org/10.1007/s00023-014-0357-x Dilts J, Maxwell D (2018) Yamabe classification and prescribed scalar curvature in the asymptotically Euclidean setting. Commun Anal Geom 26:1127–1168. https://doi.org/10.4310/cag.2018.v26.n5.a5 Dilts J, Isenberg J, Mazzeo R, Meier C (2014) Non-CMC solutions of the Einstein constraint equations on asymptotically Euclidean manifolds. Class Quantum Grav 31:065001. https://doi.org/10.1088/0264-9381/31/6/065001 Dinkelbach J, Leeb B (2009) Equivariant Ricci flow with surgery and applications to finite group actions on geometric 3-manifolds. Geom Topol 13:1129–1173. https://doi.org/10.2140/gt.2009.13.1129 DiPerna RJ, Lions PL (1989) Ordinary differential equations, transport theory and Sobolev spaces. Invent Math 98:511–547. https://doi.org/10.1007/bf01393835 Douglis A, Nirenberg L (1955) Interior estimates for elliptic systems of partial differential equations. Commun Pure Appl Math 8:503–538. https://doi.org/10.1002/cpa.3160080406 Ebin DG (1970) The manifold of Riemannian metrics. In: Global Analysis (Proc. Sympos. Pure Math., Vol. XV, Berkeley, Calif., 1968). American Mathematical Society, Providence, R.I., pp 11–40. https://doi.org/10.1090/pspum/015/0267604 Eichmair M (2013) The Jang equation reduction of the spacetime positive energy theorem in dimensions less than eight. Commun Math Phys 319:575–593. https://doi.org/10.1007/s00220-013-1700-7 Eichmair M, Huang LH, Lee DA, Schoen R (2016) The spacetime positive mass theorem in dimensions less than eight. J Eur Math Soc 18:83–121. https://doi.org/10.4171/jems/584 Escobar JF (1992a) Conformal deformation of a Riemannian metric to a scalar flat metric with constant mean curvature on the boundary. Ann Math 136:1–50. https://doi.org/10.2307/2946545 Escobar JF (1992b) The Yamabe problem on manifolds with boundary. J Differ Geom 35:21–84. https://doi.org/10.4310/jdg/1214447805 Fischer AE, Marsden JE (1972) The Einstein evolution equations as a first-order quasi-linear symmetric hyperbolic system. I. Commun Math Phys 28:1–38. https://doi.org/10.1007/bf02099369 Fischer AE, Marsden JE (1973) Linearization stability of the Einstein equations. Bull Am Math Soc 79:997–1003. https://doi.org/10.1090/s0002-9904-1973-13299-9 Fischer AE, Marsden JE (1975a) Deformations of the scalar curvature. Duke Math J 42:519–547. https://doi.org/10.1215/s0012-7094-75-04249-0 Fischer AE, Marsden JE (1975b) Linearization stability of nonlinear partial differential equations. In: Differential geometry (Proc. Sympos. Pure Math., Vol. XXVII, Part 2, Stanford Univ., Stanford, Calif., 1973). pp 219–263. https://doi.org/10.1090/pspum/027.2/0383456 Fischer AE, Marsden JE, Moncrief V (1980) The structure of the space of solutions of Einstein’s equations. I. One Killing field. Ann Inst Henri Poincare 33:147–194 Friedrich H (1985) On the hyperbolicity of Einstein’s and other gauge field equations. Commun Math Phys 100:525–543. https://doi.org/10.1007/bf01217728 Friedrich H (1986) On the existence of \(n\)-geodesically complete or future complete solutions of Einstein’s field equations with smooth asymptotic structure. Commun Math Phys 107:587–609. https://doi.org/10.1007/bf01205488 Friedrich H (1995) Einstein equations and conformal structure: existence of anti-de Sitter-type space-times. J Geom Phys 17:125–184. https://doi.org/10.1016/0393-0440(94)00042-3 Friedrich H (2007) Static vacuum solutions from convergent null data expansions at space-like infinity. Ann Henri Poincaré 8:817–884. https://doi.org/10.1007/s00023-006-0323-3 Friedrich H (2011) Yamabe numbers and the Brill–Cantor criterion. Ann Henri Poincare 12:1019–1025. https://doi.org/10.1007/s00023-011-0102-7 Friedrich H (2018) Peeling or not peeling—is that the question? Class Quantum Grav 35:083001. https://doi.org/10.1088/1361-6382/aaafdb Friedrich H, Rendall A (2000) The Cauchy problem for the Einstein equations. In: Einstein’s field equations and their physical implications. Lecture notes in physics, vol 540. Springer, Berlin, pp 127–223. https://doi.org/10.1007/3-540-46580-4_2 Friedrich H, Schmidt BG (1987) Conformal geodesics in general relativity. Proc R Soc London Ser A 414:171–195 Gabach Clément ME (2010) Conformally flat black hole initial data with one cylindrical end. Class Quantum Grav 27:125010. https://doi.org/10.1088/0264-9381/27/12/125010 Gallot S, Hulin D, Lafontaine J (2004) Riemannian geometry, 3rd edn. Universitext. Springer, Berlin. https://doi.org/10.1007/978-3-642-18855-8 Galloway GJ, Miao P, Schoen R (2015) Initial data and the Einstein constraint equations. General relativity and gravitation. Cambridge University Press, Cambridge, pp 412–448 Gibbons G (2009) Birkhoff’s invariant and Thorne’s hoop conjecture. arXiv e-prints arXiv:0903.1580 Gicquaud R (2010) De l’équation de prescription de courbure scalaire aux équations de contrainte en relativité générale sur une variété asymptotiquement hyperbolique. J Math Pures Appl 94:200–227. https://doi.org/10.1016/j.matpur.2010.03.011 Gicquaud R (2018) Solutions to the Einstein constraint equations with a small TT-tensor and vanishing yamabe invariant. arXiv e-prints arXiv:1802.05080 Gicquaud R (2019) Prescribed non positive scalar curvature on asymptotically hyperbolic manifolds with application to the Lichnerowicz equation. arXiv e-prints arXiv:1909.05343 Gicquaud R, Ngô QA (2014) A new point of view on the solutions to the Einstein constraint equations with arbitrary mean curvature and small TT-tensor. Class Quantum Grav 31:190514. https://doi.org/10.1088/0264-9381/31/19/195014 Gicquaud R, Nguyen C (2016) Solutions to the Einstein-scalar field constraint equations with a small TT-tensor. Calc Var 55:29. https://doi.org/10.1007/s00526-016-0963-1 Gicquaud R, Sakovich A (2012) A large class of non-constant mean curvature solutions of the Einstein constraint equations on an asymptotically hyperbolic manifold. Commun Math Phys 310:705–763. https://doi.org/10.1007/s00220-012-1420-4 Gidas B, Ni WM, Nirenberg L (1979) Symmetry and related properties via the maximum principle. Commun Math Phys 68:209–243. https://doi.org/10.1007/bf01221125 Gidas B, Ni WM, Nirenberg L (1981) Symmetry of positive solutions of nonlinear elliptic equations in \({\mathbb{R}}^{n}\). In: Mathematical analysis and applications, Part A. Adv. in Math. Suppl. Stud., vol 7. Academic Press, New York-London, pp 369–402 Gilbarg D, Trudinger NS (2001) Elliptic partial differential equations of second order. Classics in Mathematics, Springer, Berlin. https://doi.org/10.1007/978-3-642-61798-0, reprint of the 1998 edition Gromov M (1969) Stable mappings of foliations into manifolds. Izv Akad Nauk SSSR Ser Mat 33:707–734. https://doi.org/10.1070/im1969v003n04abeh000796 Gromov M (2018) A dozen problems, questions and conjectures about positive scalar curvature. In: Foundations of mathematics and physics one century after Hilbert. Springer. https://doi.org/10.1007/978-3-319-64813-2_6 Gromov M (2019) No metrics with positive scalar curvatures on aspherical 5-manifolds. arXiv e-prints arXiv:2009.05332 Gromov M, Lawson HB Jr (1980a) The classification of simply connected manifolds of positive scalar curvature. Ann Math 111:423–434. https://doi.org/10.2307/1971103 Gromov M, Lawson HB Jr (1980b) Spin and scalar curvature in the presence of a fundamental group. I. Ann Math 111:209–230. https://doi.org/10.2307/1971198 Gromov M, Lawson HB Jr (1983) Positive scalar curvature and the Dirac operator on complete Riemannian manifolds. Inst Hautes Études Sci Publ Math 58:83–196. https://doi.org/10.1007/bf02953774 Guillemin V, Pollack A (1974) Differential topology. Prentice-Hall, Englewood Cliffs. https://doi.org/10.1090/chel/370 Müller zum Hagen H, Robinson DC, Seifert HJ (1973) Black holes in static vacuum space-times. Gen Relativ Gravit 4:53–78. https://doi.org/10.1007/bf00769760 Hamilton RS (1982) Three-manifolds with positive Ricci curvature. J Differ Geom 17:255–306. https://doi.org/10.4310/jdg/1214436922 Hanke B (2019) Positive scalar curvature on manifolds with odd order abelian fundamental groups. arXiv e-prints arXiv:1908.00944 Hanke B, Schick T, Steimle W (2014) The space of metrics of positive scalar curvature. Publ Math Inst Hautes Études Sci 120:335–367. https://doi.org/10.1007/s10240-014-0062-9 Hebey E, Veronelli G (2014) The Lichnerowicz equation in the closed case of the Einstein-Maxwell theory. Trans Amer Math Soc 366:1179–1193. https://doi.org/10.1090/S0002-9947-2013-05790-X Hebey E, Pacard F, Pollack D (2008) A variational analysis of Einstein-scalar field Lichnerowicz equations on compact Riemannian manifolds. Commun Math Phys 278:117–132. https://doi.org/10.1007/s00220-007-0377-1 Herzlich M (2005) Mass formulae for asymptotically hyperbolic manifolds. In: AdS/CFT correspondence: Einstein metrics and their conformal boundaries. IRMA Lect. Math. Theor. Phys., vol 8. Eur. Math. Soc., Zürich, pp 103–121 Hirsch S, Lesourd M (2019) On the moduli space of asymptotically flat manifolds with boundary and the constraint equations. arXiv e-prints arXiv:1911.02687 Hitchin N (1974) Harmonic spinors. Adv Math 14:1–55. https://doi.org/10.1016/0001-8708(74)90021-8 Holst M, Meier C (2015) Non-CMC solutions to the Einstein constraint equations on asymptotically Euclidean manifolds with apparent horizon boundaries. Class Quantum Grav 32:025006. https://doi.org/10.1088/0264-9381/32/2/025006 Holst M, Tsogtgerel G (2013) The Lichnerowicz equation on compact manifolds with boundary. Class Quantum Grav 30:205011. https://doi.org/10.1088/0264-9381/30/20/205011 Holst M, Nagy G, Tsogtgerel G (2008) Far-from-constant mean curvature solutions of Einstein’s constraint equations with positive Yamabe metrics. Phys Rev Lett 100:161101. https://doi.org/10.1103/physrevlett.100.161101 Holst M, Nagy G, Tsogtgerel G (2009) Rough solutions of the Einstein constraints on closed manifolds without near-CMC conditions. Commun Math Phys 288:547–613. https://doi.org/10.1007/s00220-009-0743-2 Holst M, Sarbach O, Tiglio M, Vallisneri M (2016) The emergence of gravitational wave science: 100 years of development of mathematical theory, detectors, numerical algorithms, and data analysis tools. Bull Am Math Soc 53:513–554. https://doi.org/10.1090/bull/1544 Holst M, Meier C, Tsogtgerel G (2018) Non-CMC solutions of the Einstein constraint equations on compact manifolds with apparent horizon boundaries. Commun Math Phys 357:467–517. https://doi.org/10.1007/s00220-017-3004-9 Huang LH, Lee DA (2020a) Bartnik minimers and improvability of the dominant energy scalar. arXiv e-prints arXiv:2007.00593 Huang LH, Lee DA (2020b) Equality in the spacetime positive mass theorem. Commun Math Phys 376:2379–2407. https://doi.org/10.1007/s00220-019-03619-w Huang LH, Schoen R, Wang MT (2011) Specifying angular momentum and center of mass for vacuum initial data sets. Commun Math Phys 306:785–803. https://doi.org/10.1007/s00220-011-1295-9 Huang LH, Jang HC, Martin D (2020) Mass rigidity for hyperbolic manifolds. Commun Math Phys 376:2329–2349. https://doi.org/10.1007/s00220-019-03623-0 Hughes TJR, Kato T, Marsden JE (1977) Well-posed quasi-linear second-order hyperbolic systems with applications to nonlinear elastodynamics and general relativity. Arch Ration Mech Anal 63:273–294. https://doi.org/10.1007/bf00251584 Huisken G, Ilmanen T (2001) The inverse mean curvature flow and the Riemannian Penrose inequality. J Differ Geom 59:353–437. https://doi.org/10.4310/jdg/1090349447 Huneau C (2015) Constraint equations for \(3+1\) vacuum Einstein equations with a translational space-like Killing field in the asymptotically flat case II. Asymptot Anal 96:51–89. https://doi.org/10.3233/asy-151333 Huneau C (2016) Constraint equations for \(3+1\) vacuum Einstein equations with a translational space-like Killing field in the asymptotically flat case. Ann Henri Poincare 17:271–299. https://doi.org/10.1007/s00023-014-0392-7 Isenberg J (1995) Constant mean curvature solutions of the Einstein constraint equations on closed manifolds. Class Quantum Grav 12:2249–2274. https://doi.org/10.1088/0264-9381/12/9/013 Isenberg J (2014) The initial value problem in general relativity. In: Springer handbook of spacetime. Springer, Dordrecht, pp 303–321. https://doi.org/10.1007/978-3-642-41992-8_16 Isenberg J, Moncrief V (1996) A set of nonconstant mean curvature solutions of the Einstein constraint equations on closed manifolds. Class Quantum Grav 13:1819–1847. https://doi.org/10.1088/0264-9381/13/7/015 Isenberg J, Ó Murchadha N (2004) Non-CMC conformal data sets which do not produce solutions of the Einstein constraint equations. Class Quantum Grav 21:S233–S241. https://doi.org/10.1088/0264-9381/21/3/013 Isenberg J, Park J (1997) Asymptotically hyperbolic non-constant mean curvature solutions of the Einstein constraint equations. Class Quantum Grav 14:A189–A201. https://doi.org/10.1088/0264-9381/14/1A/016 Isenberg J, Mazzeo R, Pollack D (2002) Gluing and wormholes for the Einstein constraint equations. Commun Math Phys 231:529–568. https://doi.org/10.1007/s00220-002-0722-3 Isenberg J, Maxwell D, Pollack D (2005) A gluing construction for non-vacuum solutions of the Einstein-constraint equations. Adv Theor Math Phys 9:129–172. https://doi.org/10.4310/atmp.2005.v9.n1.a3 Isenberg J, Lee JM, Stavrov Allen I (2010) Asymptotic gluing of asymptotically hyperbolic solutions to the Einstein constraint equations. Ann Henri Poincare 11:881–927. https://doi.org/10.1007/s00023-010-0049-0 Israel W (1967) Event horizons in static vacuum space-times. Phys Rev 164:1776–1779. https://doi.org/10.1103/physrev.164.1776 Jaco W (1980) Lectures on three-manifold topology, CBMS regional conference series in mathematics, vol 43. American Mathematical Society, Providence, RI. https://doi.org/10.1090/cbms/043 Jin Q, Li Y, Xu H (2008) Symmetry and asymmetry: the method of moving spheres. Adv Differ Equations 13:601–640. https://projecteuclid.org/euclid.ade/1355867331 Jin ZR (1988) A counterexample to the Yamabe problem for complete noncompact manifolds. In: Chern SS (ed) Partial differential equations (Tianjin, 1986). Lecture Notes in Mathematics, vol 1306. Springer, Berlin, pp 93–101. https://doi.org/10.1007/bfb0082927 Joudioux J (2017) Gluing for the constraints for higher spin fields. J Math Phys 58:111513. https://doi.org/10.1063/1.5001004 Joyce D (2003) Constant scalar curvature metrics on connected sums. Int J Math Math Sci 2003:405–450. https://doi.org/10.1155/s016117120310806x Kánnár J (1996) Hyperboloidal initial data for the vacuum Einstein equations with cosmological constant. Class Quantum Grav 13:3075–3084. https://doi.org/10.1088/0264-9381/13/11/021 Kazdan JL (1985) Prescribing the curvature of a Riemannian manifold, CBMS regional conference series in mathematics, vol 57. American Mathematical Society, Providence, RI. https://doi.org/10.1090/cbms/057 Kazdan JL, Warner FW (1975a) Existence and conformal deformation of metrics with prescribed Gaussian and scalar curvatures. Ann Math 101:317–331. https://doi.org/10.2307/1970993 Kazdan JL, Warner FW (1975b) Prescribing curvatures. In: Chern SS, Osserman R (eds) Differential geometry (Proceedings of the Symposium Pure Mathematics, Vol. XXVII, Stanford University, Stanford, Calif., (1973) Part 2. American Mathematical Society, Providence, RI, pp 309–319 Kazdan JL, Warner FW (1975c) Scalar curvature and conformal deformation of Riemannian structure. J Differ Geom 10:113–134. https://doi.org/10.4310/jdg/1214432678 Khavkine I (2015) Topology, rigid cosymmetries and linearization instability in higher gauge theories. Ann Henri Poincare 16:255–288. https://doi.org/10.1007/s00023-014-0321-9 Khuri M, Weinstein G, Yamada S (2017) Proof of the Riemannian Penrose inequality with charge for multiple black holes. J Differ Geom 106:451–498. https://doi.org/10.4310/jdg/1500084023 Klainerman S (2010) PDE as a unified subject. In: Alon N, Bourgain J, Connes A, Gromov M, Milman V (eds) Visions in mathematics: GAFA 2000 special volume, part I. Birkhäuser, Basel, pp 279–315. https://doi.org/10.1007/978-3-0346-0422-2_10, originally published in GAFA Geometr Funct Anal (2000) Klainerman S, Rodnianski I (2005a) The causal structure of microlocalized rough Einstein metrics. Ann Math 161:1195–1243. https://doi.org/10.4007/annals.2005.161.1195 Klainerman S, Rodnianski I (2005b) Rough solutions of the Einstein-vacuum equations. Ann Math 161:1143–1193. https://doi.org/10.4007/annals.2005.161.1143 Klainerman S, Rodnianski I, Szeftel J (2015) The bounded \(L^2\) curvature conjecture. Invent Math 202:91–216. https://doi.org/10.1007/s00222-014-0567-3 Kleiner B, Lott J (2008) Notes on Perelman’s papers. Geom Topol 12:2587–2855. https://doi.org/10.2140/gt.2008.12.2587 Kleiner B, Lott J (2017) Singular Ricci flows I. Acta Math 219:65–134. https://doi.org/10.4310/acta.2017.v219.n1.a4 Kosinski AA (1993) Differential manifolds. Pure and applied mathematics, vol 138. Academic Press Inc, Boston Kreck M, Stolz S (1993) Nonconnected moduli spaces of positive sectional curvature metrics. J Am Math Soc 6:825–850. https://doi.org/10.1090/s0894-0347-1993-1205446-4 Lang S (2002) Algebra. Revised third edition, Graduate texts in mathematics, vol 211. Springer, New York Lawson HB Jr (1970) Complete minimal surfaces in \(S^{3}\). Ann Math 92:335–374. https://doi.org/10.2307/1970625 Lawson HB Jr, Michelsohn ML (1989) Spin geometry. Princeton mathematical series, vol 38. Princeton University Press, Princeton Leach J (2014) A far-from-CMC existence result for the constraint equations on manifolds with ends of cylindrical type. Class Quantum Grav 31:035003. https://doi.org/10.1088/0264-9381/31/3/035003 Leach J (2016) Non-constant mean curvature trumpet solutions for the Einstein constraint equations. Class Quantum Grav 33:145001. https://doi.org/10.1088/0264-9381/33/14/145001 LeBrun C (1995) On the scalar curvature of complex surfaces. Geom Funct Anal 5:619–628. https://doi.org/10.1007/bf01895835 LeBrun C (1999) Kodaira dimension and the Yamabe problem. Commun Anal Geom 7:133–156. https://doi.org/10.4310/cag.1999.v7.n1.a5 Lee DA (2019) Geometric relativity. Graduate studies in mathematics, vol 201. American Mathematical Society, Providence. https://doi.org/10.1090/gsm/201 Lee JM, Parker TH (1987) The Yamabe problem. Bull Am Math Soc 17:37–91. https://doi.org/10.1090/s0273-0979-1987-15514-5 LeFloch PG, Nguyen TC (2019) The seed-to-solution for the Einstein equations and the asymptotic localization problem. arXiv e-prints arXiv:1903.00243 Lesourd M, Unger R, Yau ST (2020) Positive scalar curvature on noncompact manifolds and the Liouville theorem. arXiv e-prints arXiv:2009.12618 Li J, Yu P (2015) Construction of Cauchy data of vacuum Einstein field equations evolving to black holes. Ann Math 181:699–768. https://doi.org/10.4007/annals.2015.181.2.6 Lichnerowicz A (1944) L’intégration des équations de la gravitation relativiste et le problème des \(n\) corps. J Math Pures Appl 23:37–63 Lichnerowicz A (1963) Spineurs harmoniques. C R Acad Sci Paris 257:7–9 Lockhart RB, McOwen RC (1983) On elliptic systems in \({\mathbb{R}}^{n}\). Acta Math 150:125–135 Lockhart RB, McOwen RC (1985) Elliptic differential operators on noncompact manifolds. Ann Scuola Norm Sup Pisa Cl Sci 12:409–447 Lohkamp J (1999) Scalar curvature and hammocks. Math Ann 313:385–407. https://doi.org/10.1007/s002080050266 Lohkamp J (2015) Skin structures on minimal hypersurfaces. arXiv e-prints arXiv:1512.08249 Lohkamp J (2016) The higher dimensional positive mass theorem II. arXiv e-prints arXiv:1612.07505 Ma L, Wei J (2013) Stability and multiple solutions to Einstein-scalar field Lichnerowicz equation on manifolds. J Math Pures Appl 99:174–186. https://doi.org/10.1016/j.matpur.2012.06.009 Maerten D (2006) Positive energy-momentum theorem for AdS-asymptotically hyperbolic manifolds. Ann Henri Poincare 7:975–1011. https://doi.org/10.1007/s00023-006-0273-9 Mantoulidis C, Miao P (2017a) Mean curvature deficit and quasi-local mass. In: Bieri L, Chruściel PT, Yau ST (eds) Nonlinear analysis in geometry and applied mathematics. Harvard CMSA series in mathematics, vol 1. International Press, Somerville, pp 99–107 Mantoulidis C, Miao P (2017b) Total mean curvature, scalar curvature, and a variational analog of Brown–York mass. Commun Math Phys 352:703–718. https://doi.org/10.1007/s00220-016-2767-8 Mantoulidis C, Schoen R (2015) On the Bartnik mass of apparent horizons. Class Quantum Grav 32:205002. https://doi.org/10.1088/0264-9381/32/20/205002 Marques FC (2005) Existence results for the Yamabe problem on manifolds with boundary. Indiana Univ Math J 54:1599–1620. https://doi.org/10.1512/iumj.2005.54.2590 Marques F (2012) Deforming three-manifolds with positive scalar curvature. Ann Math 176:815–863. https://doi.org/10.4007/annals.2012.176.2.3 Mars M (2013) Constraint equations for general hypersurfaces and applications to shells. Gen Relativ Gravit 45:2175–2221. https://doi.org/10.1007/s10714-013-1579-9 Marsden JE, Tipler FJ (1980) Maximal hypersurfaces and foliations of constant mean curvature in general relativity. Phys Rep 66:109–139. https://doi.org/10.1016/0370-1573(80)90154-4 Maxwell D (2005a) Rough solutions of the Einstein constraint equations on compact manifolds. J Hyperbol Differ Equations 2:521–546. https://doi.org/10.1142/s021989160500049x Maxwell D (2005b) Solutions of the Einstein constraint equations with apparent horizon boundaries. Commun Math Phys 253:561–583. https://doi.org/10.1007/s00220-004-1237-x Maxwell D (2006) Rough solutions of the Einstein constraint equations. J Reine Angew Math 590:1–29. https://doi.org/10.1515/crelle.2006.001 Maxwell D (2009) A class of solutions of the vacuum Einstein constraint equations with freely specified mean curvature. Math Res Lett 16:627–645. https://doi.org/10.4310/mrl.2009.v16.n4.a6 Maxwell D (2011) A model problem for conformal parameterizations of the Einstein constraint equations. Commun Math Phys 302:697–736. https://doi.org/10.1007/s00220-011-1187-z Maxwell D (2014) The conformal method and the conformal thin-sandwich method are the same. Class Quantum Grav 31:145006. https://doi.org/10.1088/0264-9381/31/14/145006 Mazzeo R, Pacard F (2006) Maskit combinations of Poincaré–Einstein metrics. Adv Math 204:379–412. https://doi.org/10.1016/j.aim.2005.06.001 Mazzeo R, Pollack D, Uhlenbeck K (1995) Connected sum constructions for constant scalar curvature metrics. Topol Methods Nonlinear Anal 6:207–233. https://doi.org/10.12775/tmna.1995.042 Mazzeo R, Pollack D, Uhlenbeck K (1996) Moduli spaces of singular Yamabe metrics. J Am Math Soc 9:303–344. https://doi.org/10.1090/s0894-0347-96-00208-1 Mazzieri L (2008) Generalized connected sum construction for nonzero constant scalar curvature metrics. Commun Partial Differ Equations 33:1–17. https://doi.org/10.1080/03605300600856741 Mazzieri L (2009a) Generalized connected sum construction for scalar flat metrics. Manuscr Math 129:137–168. https://doi.org/10.1007/s00229-009-0250-y Mazzieri L (2009b) Generalized gluing for Einstein constraint equations. Calc Var Partial Differ Equations 34:453–473. https://doi.org/10.1007/s00526-008-0191-4 McCormick S (2015) A note on mass-minimising extensions. Gen Relativ Gravit 47:145. https://doi.org/10.1007/s10714-015-1993-2 McFeron D, Székelyhidi G (2012) On the positive mass theorem for manifolds with corners. Commun Math Phys 313:425–443. https://doi.org/10.1007/s00220-012-1498-8 Miao P (2002) Positive mass theorem on manifolds admitting corners along a hypersurface. Adv Theor Math Phys 6:1163–1182. https://doi.org/10.4310/atmp.2002.v6.n6.a4 Miao P (2015) Quasi-local mass via isometric embeddings: a review from a geometric perspective. Class Quantum Grav 32:233001. https://doi.org/10.1088/0264-9381/32/23/233001 Miao P, Tam LF (2016) Evaluation of the ADM mass and center of mass via the Ricci tensor. Proc Amer Math Soc 144:753–761. https://doi.org/10.1090/proc12726 Miao P, Xie N (2018) On compact 3-manifolds with nonnegative scalar curvature with a CMC boundary component. Trans Amer Math Soc 370:5887–5906. https://doi.org/10.1090/tran/7500 Miao P, Xie N (2019) Bartnik mass via vacuum extensions. Int J Math 30:1940006. https://doi.org/10.1142/s0129167x19400068 Miao P, Wang Y, Xie N (2020) On Hawking mass and Bartnik mass of CMC surfaces. Math Res Lett 27:855–885. https://doi.org/10.4310/mrl.2020.v27.n3.a12 Michel B (2011) Geometric invariance of mass-like asymptotic invariants. J Math Phys 52:052504. https://doi.org/10.1063/1.3579137 Milnor JW (1962) A unique decomposition theorem for \(3\)-manifolds. Am J Math 84:1–7. https://doi.org/10.2307/2372800 Milnor JW (1965) Topology from the differentiable viewpoint. Weaver, The University Press of Virginia, Charlottesville, VA, Based on notes by David W. Weaver Min-Oo M (1989) Scalar curvature rigidity of asymptotically hyperbolic spin manifolds. Math Ann 285:527–539. https://doi.org/10.1007/bf01452046 Min-Oo M (1998) Scalar curvature rigidity of certain symmetric spaces. In: Geometry, topology, and dynamics (Montreal, PQ, 1995). CRM Proc. Lecture Notes, vol 15. American Mathematical Society, Providence, RI, pp 127–136 Minguzzi E (2019) Lorentzian causality theory. Living Rev Relativ 22:3. https://doi.org/10.1007/s41114-019-0019-x Misner CW, Thorne KS, Wheeler JA (1973) Gravitation. W. H. Freeman, San Francisco Moishezon BG, Robb A, Teicher M (1996) On Galois covers of Hirzebruch surfaces. Math Ann 305:493–539. https://doi.org/10.1007/bf01444235 Moncrief V (1975) Spacetime symmetries and linearization stability of the Einstein equations I. J Math Phys 16:493–498. https://doi.org/10.1063/1.522572 Moncrief V (1976) Space-time symmetries and linearization stability of the Einstein equations II. J Math Phys 17:1893–1902. https://doi.org/10.1063/1.522814 Moncrief V (2013) Reflections on the U(1) problem in general relativity. J Fixed Point Theory Appl 14:397–418. https://doi.org/10.1007/s11784-014-0159-2 Moore JD (1996) Lectures on Seiberg–Witten invariants. Lecture notes in mathematics, vol 1629. Springer, Berlin. https://doi.org/10.1007/bfb0092948 Morgan J, Tian G (2014) The geometrization conjecture. Clay mathematics monographs, vol 5. American Mathematical Society/Clay Mathematics Institute, Providence/Cambridge Mounoud P (2015) Metrics without isometries are generic. Monatsh Math 176:603–606. https://doi.org/10.1007/s00605-014-0614-6 Nardmann M (2010) A remark on the rigidity case of the positive energy theorem. arXiv e-prints arXiv:1004.5430 Ngô QA, Xu X (2012) Existence results for the Einstein-scalar field Lichnerowicz equations on compact Riemannian manifolds. Adv Math 230:2378–2415. https://doi.org/10.1016/j.aim.2012.04.007 Ngô QA, Xu X (2015) Existence results for the Einstein-scalar field Lichnerowicz equations on compact Riemannian manifolds in the null case. Commun Math Phys 334:193–222. https://doi.org/10.1007/s00220-014-2133-7 Nirenberg L (2001) Topics in nonlinear functional analysis. Courant Lecture Notes in Mathematics, vol 6. American Mathematical Society, Providence, RI, New York University, Courant Institute of Mathematical Sciences, New York. https://doi.org/10.1090/cln/006 Obata M (1962) Certain conditions for a Riemannian manifold to be isometric with a sphere. J Math Soc Japan. https://doi.org/10.2969/jmsj/01430333 Obata M (1971) The conjectures on conformal transformations of Riemannian manifolds. J Differ Geom 6:247–258. https://doi.org/10.4310/jdg/1214430407 Ó Murchadha N, York JW Jr (1973) Existence and uniqueness of solutions of the Hamiltonian constraint of general relativity on compact manifolds. J Math Phys 14:1551–1557. https://doi.org/10.1063/1.1666225 Ó Murchadha N, York JW Jr (1974) Initial-value problem of general relativity. I. General formulation and physical interpretation. Phys Rev D 10:428–436. https://doi.org/10.1103/physrevd.10.428 O’Neill B (1983) Semi-Riemannian geometry. Pure and applied mathematics, vol 103. Academic Press, New York Parker T, Taubes CH (1982) On Witten’s proof of the positive energy theorem. Commun Math Phys 84:223–238. https://doi.org/10.1007/bf01208569 Perelman G (2002) The entropy formula for the Ricci flow and its geometric applications. arXiv e-prints arXiv:math/0211159 Perelman G (2003a) Finite extinction time for the solutions to the Ricci flow on certain three-manifolds. arXiv e-prints. arXiv:math/0307245 Perelman G (2003b) Ricci flow with surgery on three-manifolds. arXiv e-prints arXiv:math/0303109 Petersen P (2006) Riemannian geometry. Graduate texts in mathematics, vol 171, 2nd edn. Springer, New York. https://doi.org/10.1007/978-3-319-26654-1 Pfeiffer HP, York JW Jr (2003) Extrinsic curvature and the Einstein constraints. Phys Rev D 67:044022. https://doi.org/10.1103/physrevd.67.044022 Pfeiffer HP, York JW Jr (2005) Uniqueness and nonuniqueness in the Einstein constraints. Phys Rev Lett 95:091101. https://doi.org/10.1103/physrevlett.95.091101 Premoselli B (2014) The Einstein-scalar field constraint system in the positive case. Commun Math Phys 326:543–557 Regge T, Teitelboim C (1974) Role of surface integrals in the Hamiltonian formulation of general relativity. Ann Phys 88:286–318. https://doi.org/10.1016/0003-4916(74)90404-7 Reiser P (2019) Moduli spaces of metrics of positive scalar curvature on topological spherical space forms. arXiv e-prints 10.4153/s0008439520000132. arXiv:1909.09512 Rendall AD (2004) Accelerated cosmological expansion due to a scalar field whose potential has a positive lower bound. Class Quantum Grav 21:2445–2454. https://doi.org/10.1088/0264-9381/21/9/018 Rendall AD (2005) Intermediate inflation and the slow-roll approximation. Class Quantum Grav 22:1655–1666. https://doi.org/10.1088/0264-9381/22/9/013 Rendall AD (2006) Mathematical properties of cosmological models with accelerated expansion. In: Analytical and numerical approaches to mathematical relativity. Lecture notes in physics, vol 692. Springer, Berlin, pp 141–155. https://doi.org/10.1007/3-540-33484-X_7 Ringström H (2009) The Cauchy problem in general relativity. ESI lectures in mathematics and physics. European Mathematical Society, Zürich. https://doi.org/10.4171/053 Robinson DC (1977) A simple proof of the generalization of Israel’s theorem. Gen Relativ Gravit 8:695–698. https://doi.org/10.1007/bf00756322 Rosenberg J, Stolz S (2001) Metrics of positive scalar curvature and connections with surgery. In: Surveys on surgery theory, vol. 2. Annals of Mathematics Studies, vol 149. Princeton University Press, Princeton, NJ, pp 353–386. https://doi.org/10.1515/9781400865215-010 Ruberman D (1998) An obstruction to smooth isotopy in dimension 4. Math Res Lett 5:743–758 Sacks J, Uhlenbeck K (1981) The existence of minimal immersions of 2-spheres. Ann Math 113:1–24. https://doi.org/10.2307/1971131 Sahni V (2005) Dark matter and dark energy. In: Papantonopoulos E (ed) Physics of the early universe. Lecture notes in physics, vol 653. Springer, Berlin. https://doi.org/10.1007/978-3-540-31535-3_5 Sakovich A (2010) Constant mean curvature solutions of the Einstein-scalar field constraint equations on asymptotically hyperbolic manifolds. Class Quantum Grav 27:245019. https://doi.org/10.1088/0264-9381/27/24/245019 Sakovich A (2020) The Jang equation and the positive mass theorem in the asymptotically hyperbolic setting. arXiv e-prints arXiv:2003.07762 Sbierski J (2016) On the existence of a maximal Cauchy development for the Einstein equations: a dezornification. Ann Henri Poincare 17:301–329. https://doi.org/10.1007/s00023-015-0401-5 Schick T (1998) A counterexample to the (unstable) Gromov–Lawson–Rosenberg conjecture. Topology 37:1165–1168. https://doi.org/10.1016/s0040-9383(97)00082-7 Schick T (2014) The topology of positive scalar curvature. In: Proceedings of the International Congress of Mathematicians—Seoul 2014, vol II, 1285–1307, Kyung Moon Sa, Seoul. Kyung Moon Sa, Seoul Schleich K, Witt DM (2010) A simple proof of Birkhoff’s theorem for cosmological constant. J Math Phys 51:112502. https://doi.org/10.1063/1.3503447 Schoen R (1984) Conformal deformation of a Riemannian metric to constant scalar curvature. J Differ Geom 20:479–495. https://doi.org/10.4310/jdg/1214439291 Schoen R (1988) The existence of weak solutions with prescribed singular behavior for a conformally invariant scalar equation. Commun Pure Appl Math 41:317–392. https://doi.org/10.1002/cpa.3160410305 Schoen R (1989) Variational theory for the total scalar curvature functional for Riemannian metrics and related topics. In: Topics in calculus of variations (Montecatini Terme, 1987). Lecture notes in mathematics, vol 1365. Springer, Berlin, pp 120–154. https://doi.org/10.1007/bfb0089180 Schoen R, Yau ST (1979a) Existence of incompressible minimal surfaces and the topology of three-dimensional manifolds with nonnegative scalar curvature. Ann Math 110:127–142. https://doi.org/10.2307/1971247 Schoen R, Yau ST (1979b) On the proof of the positive mass conjecture in general relativity. Commun Math Phys 65:45–76. https://doi.org/10.1007/bf01940959 Schoen R, Yau ST (1979c) On the structure of manifolds with positive scalar curvature. Manuscr Math 28:159–183. http://eudml.org/doc/154634 Schoen R, Yau ST (1981) Proof of the positive mass theorem II. Commun Math Phys 79:231–260. https://doi.org/10.1007/bf01942062 Schoen R, Yau ST (1988) Conformally flat manifolds, Kleinian groups and scalar curvature. Invent Math 92:47–71. https://doi.org/10.1007/bf01393992 Schoen R, Yau ST (2017) Positive scalar curvature and minimal hypersurface singularities. arXiv e-prints arXiv:1704.05490 Smith B, Weinstein G (2004) Quasiconvex foliations and asymptotically flat metrics of non-negative scalar curvature. Commun Anal Geom 12:511–551. https://doi.org/10.4310/cag.2004.v12.n3.a2 Smith HF, Tataru D (2005) Sharp local well-posedness results for the nonlinear wave equation. Ann Math 162:291–366. https://doi.org/10.4007/annals.2005.162.291 Stolz S (1992) Simply connected manifolds of positive scalar curvature. Ann Math 136:511–540. https://doi.org/10.2307/2946598 Strauss WA (1989) Nonlinear wave equations, CBMS regional conference series in mathematics, vol 73. American Mathematical Society, Providence. https://doi.org/10.1090/cbms/073 Szabados L (2009) Quasi-local energy-momentum and angular momentum in general relativity. Living Rev Relativ 12:4. https://doi.org/10.12942/lrr-2009-4 Taubes CH (1994) The Seiberg–Witten invariants and symplectic forms. Math Res Lett 1:809–822. https://doi.org/10.4310/mrl.1994.v1.n6.a15 Teicher M (1999) Hirzebruch surfaces: degenerations, related braid monodromy, Galois covers. In: Algebraic geometry: Hirzebruch 70 (Warsaw, 1998). Contemp. Math., vol 241. American Mathematical Society, Providence, pp 305–325. https://doi.org/10.1090/conm/241/03642 Thornburg J (1987) Coordinates and boundary conditions for the general relativistic initial data problem. Class Quantum Grav 4:1119–1131. https://doi.org/10.1088/0264-9381/4/5/013 Wald R (1984) General relativity. University of Chicago Press, Chicago. https://doi.org/10.7208/chicago/9780226870373.001.0001 Walsh DM (2007) Non-uniqueness in conformal formulations of the Einstein constraints. Class Quantum Grav 24:1911–1925. https://doi.org/10.1088/0264-9381/24/8/002 Wang MT (2019) Quasi-local and total angular momentum in general relativity. In: Ji L, Yang L, Yau ST (eds) Proceedings of the seventh international congress of Chinese mathematicians, Vol. I. Advanced lectures in mathematics, vol 43. International Press, Somerville, MA, pp 457–472 Wang X (2001) The mass of asymptotically hyperbolic manifolds. J Differ Geom 57:273–299. https://doi.org/10.4310/jdg/1090348112 Waxenegger G, Beig R, Ó Murchadha N (2011) Existence and uniqueness of Bowen–York trumpets. Class Quantum Grav 28:245002. https://doi.org/10.1088/0264-9381/28/24/245002 Wiemeler M (2020) On moduli spaces of positive scalar curvature metrics on highly connected manifolds. Int Math Res Not rnz386. https://doi.org/10.1093/imrn/rnz386. arXiv:1610.09658 Witten E (1981) A new proof of the positive energy theorem. Commun Math Phys 80:381–402. https://doi.org/10.1007/bf01208277 Yamabe H (1960) On a deformation of Riemannian structures on compact manifolds. Osaka Math J 12:21–37. https://projecteuclid.org/euclid.ojm/1200689814 Yau ST (1982) Problem section. In: St Yau (ed) Seminar on differential geometry. Annals of mathematics studies, vol 102. Princeton University Press, Princeton, pp 669–706. https://doi.org/10.1515/9781400881918-035 Yip P (1987) A strictly-positive mass theorem. Commun Math Phys 108:653–665. https://doi.org/10.1007/bf01214423 York JW Jr (1973) Conformally invariant orthogonal decomposition of symmetric tensors on Riemannian manifolds and the initial-value problem of general relativity. J Math Phys 14:456–464. https://doi.org/10.1063/1.1666338 York JW Jr (1999) Conformal “thin-sandwich” data for the initial-value problem of general relativity. Phys Rev Lett 82:1350–1353. https://doi.org/10.1103/physrevlett.82.1350 Zhang X (2004) A definition of total energy–momenta and the positive mass theorem on asymptotically hyperbolic 3-manifolds I. Commun Math Phys 249:529–548. https://doi.org/10.1007/s00220-004-1056-0