Geometric Relativity

Lee, Dan

doi:10.1090/gsm/201

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Geometric Relativity

About this Title

Dan A. Lee, CUNY Graduate Center and Queens College, New York, NY

Publication: Graduate Studies in Mathematics
Publication Year: 2019; Volume 201
ISBNs: 978-1-4704-5081-6 (print); 978-1-4704-5405-0 (online)
DOI: https://doi.org/10.1090/gsm/201
MathSciNet review: MR3970261
MSC: Primary 83C05; Secondary 83C57

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Ian Agol, The virtual Haken conjecture, Doc. Math. 18 (2013), 1045–1087. With an appendix by Agol, Daniel Groves, and Jason Manning. MR 3104553, DOI Incorrect DOI link
S. Alexakis, A. D. Ionescu, and S. Klainerman, Uniqueness of smooth stationary black holes in vacuum: small perturbations of the Kerr spaces, Comm. Math. Phys. 299 (2010), no. 1, 89–127. MR 2672799, DOI 10.1007/s00220-010-1072-1
William K. Allard, On the first variation of a varifold, Ann. of Math. (2) 95 (1972), 417–491. MR 307015, DOI 10.2307/1970868
F. J. Almgren Jr., Some interior regularity theorems for minimal surfaces and an extension of Bernstein’s theorem, Ann. of Math. (2) 84 (1966), 277–292. MR 200816, DOI 10.2307/1970520
Frederick J. Almgren Jr., Almgren’s big regularity paper, World Scientific Monograph Series in Mathematics, vol. 1, World Scientific Publishing Co., Inc., River Edge, NJ, 2000. $Q$-valued functions minimizing Dirichlet’s integral and the regularity of area-minimizing rectifiable currents up to codimension 2; With a preface by Jean E. Taylor and Vladimir Scheffer. MR 1777737
Lucas C. Ambrozio, On perturbations of the Schwarzschild anti-de Sitter spaces of positive mass, Comm. Math. Phys. 337 (2015), no. 2, 767–783. MR 3339162, DOI 10.1007/s00220-015-2360-6
Lars Andersson, Mingliang Cai, and Gregory J. Galloway, Rigidity and positivity of mass for asymptotically hyperbolic manifolds, Ann. Henri Poincaré 9 (2008), no. 1, 1–33. MR 2389888, DOI 10.1007/s00023-007-0348-2
Lars Andersson and Mattias Dahl, Scalar curvature rigidity for asymptotically locally hyperbolic manifolds, Ann. Global Anal. Geom. 16 (1998), no. 1, 1–27. MR 1616570, DOI 10.1023/A:1006547905892
Lars Andersson, Michael Eichmair, and Jan Metzger, Jang’s equation and its applications to marginally trapped surfaces, Complex analysis and dynamical systems IV. Part 2, Contemp. Math., vol. 554, Amer. Math. Soc., Providence, RI, 2011, pp. 13–45. MR 2884392, DOI 10.1090/conm/554/10958
Lars Andersson, Gregory J. Galloway, and Ralph Howard, A strong maximum principle for weak solutions of quasi-linear elliptic equations with applications to Lorentzian and Riemannian geometry, Comm. Pure Appl. Math. 51 (1998), no. 6, 581–624. MR 1611140, DOI 10.1002/(SICI)1097-0312(199806)51:6<581::AID-CPA2>3.3.CO;2-E
Lars Andersson, Marc Mars, and Walter Simon, Stability of marginally outer trapped surfaces and existence of marginally outer trapped tubes, Adv. Theor. Math. Phys. 12 (2008), no. 4, 853–888. MR 2420905
Lars Andersson and Jan Metzger, The area of horizons and the trapped region, Comm. Math. Phys. 290 (2009), no. 3, 941–972. MR 2525646, DOI 10.1007/s00220-008-0723-y
R. Arnowitt, S. Deser, and C. W. Misner, Energy and the criteria for radiation in general relativity, Phys. Rev. (2) 118 (1960), 1100–1104. MR 127945
R. Arnowitt, S. Deser, and C. W. Misner, Coordinate invariance and energy expressions in general relativity, Phys. Rev. (2) 122 (1961), 997–1006. MR 127946
R. Arnowitt, S. Deser, and C. W. Misner, The dynamics of general relativity, Gravitation: An introduction to current research, Wiley, New York, 1962, pp. 227–265. MR 0143629
Abhay Ashtekar and R. O. Hansen, A unified treatment of null and spatial infinity in general relativity. I. Universal structure, asymptotic symmetries, and conserved quantities at spatial infinity, J. Math. Phys. 19 (1978), no. 7, 1542–1566. MR 503432, DOI 10.1063/1.523863
Abhay Ashtekar and Gregory J. Galloway, Some uniqueness results for dynamical horizons, Adv. Theor. Math. Phys. 9 (2005), no. 1, 1–30. MR 2193368
Thierry Aubin, Métriques riemanniennes et courbure, J. Differential Geometry 4 (1970), 383–424 (French). MR 279731
Thierry Aubin, Équations différentielles non linéaires et problème de Yamabe concernant la courbure scalaire, J. Math. Pures Appl. (9) 55 (1976), no. 3, 269–296. MR 431287
Sheldon Axler, Paul Bourdon, and Wade Ramey, Harmonic function theory, 2nd ed., Graduate Texts in Mathematics, vol. 137, Springer-Verlag, New York, 2001. MR 1805196, DOI 10.1007/978-1-4757-8137-3
Robert Bartnik, The mass of an asymptotically flat manifold, Comm. Pure Appl. Math. 39 (1986), no. 5, 661–693. MR 849427, DOI 10.1002/cpa.3160390505
Robert Bartnik, New definition of quasilocal mass, Phys. Rev. Lett. 62 (1989), no. 20, 2346–2348. MR 996396, DOI 10.1103/PhysRevLett.62.2346
Robert Bartnik, Quasi-spherical metrics and prescribed scalar curvature, J. Differential Geom. 37 (1993), no. 1, 31–71. MR 1198599
Robert Bartnik, Phase space for the Einstein equations, Comm. Anal. Geom. 13 (2005), no. 5, 845–885. MR 2216143
Robert Beig and Piotr T. Chruściel, Killing vectors in asymptotically flat space-times. I. Asymptotically translational Killing vectors and the rigid positive energy theorem, J. Math. Phys. 37 (1996), no. 4, 1939–1961. MR 1380882, DOI 10.1063/1.531497
Arthur L. Besse, Einstein manifolds, Classics in Mathematics, Springer-Verlag, Berlin, 2008. Reprint of the 1987 edition. MR 2371700
Lydia Bieri, Part I: Solutions of the Einstein vacuum equations, Extensions of the stability theorem of the Minkowski space in general relativity, AMS/IP Stud. Adv. Math., vol. 45, Amer. Math. Soc., Providence, RI, 2009, pp. 1–295. MR 2537047, DOI 10.1090/amsip/045
G. D. Birkhoff, Relativity and modern physics. With the cooperation of R. E. Langer, Harvard University Press, Cambridge, MA, 1923.
John Bland and Morris Kalka, Negative scalar curvature metrics on noncompact manifolds, Trans. Amer. Math. Soc. 316 (1989), no. 2, 433–446. MR 987159, DOI 10.1090/S0002-9947-1989-0987159-2
E. Bombieri, E. De Giorgi, and E. Giusti, Minimal cones and the Bernstein problem, Invent. Math. 7 (1969), 243–268. MR 250205, DOI 10.1007/BF01404309
H. Bondi, M. G. J. van der Burg, and A. W. K. Metzner, Gravitational waves in general relativity. VII. Waves from axi-symmetric isolated systems, Proc. Roy. Soc. London Ser. A 269 (1962), 21–52. MR 147276, DOI 10.1098/rspa.1962.0161
Hubert Lewis Bray, The Penrose inequality in general relativity and volume comparison theorems involving scalar curvature, ProQuest LLC, Ann Arbor, MI, 1997. Thesis (Ph.D.)–Stanford University. MR 2696584
Hubert L. Bray, Proof of the Riemannian Penrose inequality using the positive mass theorem, J. Differential Geom. 59 (2001), no. 2, 177–267. MR 1908823
H. Bray, S. Brendle, M. Eichmair, and A. Neves, Area-minimizing projective planes in 3-manifolds, Comm. Pure Appl. Math. 63 (2010), no. 9, 1237–1247. MR 2675487, DOI 10.1002/cpa.20319
Hubert Bray, Simon Brendle, and Andre Neves, Rigidity of area-minimizing two-spheres in three-manifolds, Comm. Anal. Geom. 18 (2010), no. 4, 821–830. MR 2765731, DOI 10.4310/CAG.2010.v18.n4.a6
Hubert Bray and Felix Finster, Curvature estimates and the positive mass theorem, Comm. Anal. Geom. 10 (2002), no. 2, 291–306. MR 1900753, DOI 10.4310/CAG.2002.v10.n2.a3
Hubert L. Bray and Marcus A. Khuri, A Jang equation approach to the Penrose inequality, Discrete Contin. Dyn. Syst. 27 (2010), no. 2, 741–766. MR 2600688, DOI 10.3934/dcds.2010.27.741
Hubert L. Bray and Dan A. Lee, On the Riemannian Penrose inequality in dimensions less than eight, Duke Math. J. 148 (2009), no. 1, 81–106. MR 2515101, DOI 10.1215/00127094-2009-020
Glen E. Bredon, Topology and geometry, Graduate Texts in Mathematics, vol. 139, Springer-Verlag, New York, 1997. Corrected third printing of the 1993 original. MR 1700700
Simon Brendle and Fernando C. Marques, Scalar curvature rigidity of geodesic balls in $S^n$, J. Differential Geom. 88 (2011), no. 3, 379–394. MR 2844438
Simon Brendle, Fernando C. Marques, and Andre Neves, Deformations of the hemisphere that increase scalar curvature, Invent. Math. 185 (2011), no. 1, 175–197. MR 2810799, DOI 10.1007/s00222-010-0305-4
Dieter R. Brill, On the positive definite mass of the Bondi-Weber-Wheeler time-symmetric gravitational waves, Ann. Physics 7 (1959), 466–483. MR 108340, DOI 10.1016/0003-4916(59)90055-7
Robert Brooks, A construction of metrics of negative Ricci curvature, J. Differential Geom. 29 (1989), no. 1, 85–94. MR 978077
J. David Brown and James W. York Jr., Quasilocal energy and conserved charges derived from the gravitational action, Phys. Rev. D (3) 47 (1993), no. 4, 1407–1419. MR 1211109, DOI 10.1103/PhysRevD.47.1407
Gary L. Bunting and A. K. M. Masood-ul-Alam, Nonexistence of multiple black holes in asymptotically Euclidean static vacuum space-time, Gen. Relativity Gravitation 19 (1987), no. 2, 147–154. MR 876598, DOI 10.1007/BF00770326
Mingliang Cai, Volume minimizing hypersurfaces in manifolds of nonnegative scalar curvature, Minimal surfaces, geometric analysis and symplectic geometry (Baltimore, MD, 1999) Adv. Stud. Pure Math., vol. 34, Math. Soc. Japan, Tokyo, 2002, pp. 1–7. MR 1925731, DOI 10.2969/aspm/03410001
Mingliang Cai and Gregory J. Galloway, Rigidity of area minimizing tori in 3-manifolds of nonnegative scalar curvature, Comm. Anal. Geom. 8 (2000), no. 3, 565–573. MR 1775139, DOI 10.4310/CAG.2000.v8.n3.a6
A. P. Calderon and A. Zygmund, On the existence of certain singular integrals, Acta Math. 88 (1952), 85–139. MR 52553, DOI 10.1007/BF02392130
M. Cantor, Spaces of functions with asymptotic conditions on $R^{n}$, Indiana Univ. Math. J. 24 (1974/75), 897–902. MR 365621, DOI 10.1512/iumj.1975.24.24072
Murray Cantor, Elliptic operators and the decomposition of tensor fields, Bull. Amer. Math. Soc. (N.S.) 5 (1981), no. 3, 235–262. MR 628659, DOI 10.1090/S0273-0979-1981-14934-X
Alessandro Carlotto, Otis Chodosh, and Michael Eichmair, Effective versions of the positive mass theorem, Invent. Math. 206 (2016), no. 3, 975–1016. MR 3573977, DOI 10.1007/s00222-016-0667-3
Alessandro Carlotto and Richard Schoen, Localizing solutions of the Einstein constraint equations, Invent. Math. 205 (2016), no. 3, 559–615. MR 3539922, DOI 10.1007/s00222-015-0642-4
Manfredo Perdigão do Carmo, Riemannian geometry, Mathematics: Theory & Applications, Birkhäuser Boston, Inc., Boston, MA, 1992. Translated from the second Portuguese edition by Francis Flaherty. MR 1138207, DOI 10.1007/978-1-4757-2201-7
Brandon Carter, Black hole equilibrium states, Black holes/Les astres occlus (École d’Été Phys. Théor., Les Houches, 1972) Gordon and Breach, New York, 1973, pp. 57–214. MR 0465047
Alice Chaljub-Simon and Yvonne Choquet-Bruhat, Problèmes elliptiques du second ordre sur une variété euclidienne à l’infini, Ann. Fac. Sci. Toulouse Math. (5) 1 (1979), no. 1, 9–25 (French, with English summary). MR 533596
Isaac Chavel, Riemannian geometry, 2nd ed., Cambridge Studies in Advanced Mathematics, vol. 98, Cambridge University Press, Cambridge, 2006. A modern introduction. MR 2229062, DOI 10.1017/CBO9780511616822
Bang-Yen Chen, Differential geometry of warped product manifolds and submanifolds, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2017. With a foreword by Leopold Verstraelen. MR 3699316, DOI 10.1142/10419
PoNing Chen, Mu-Tao Wang, and Shing-Tung Yau, Evaluating quasilocal energy and solving optimal embedding equation at null infinity, Comm. Math. Phys. 308 (2011), no. 3, 845–863. MR 2855542, DOI 10.1007/s00220-011-1362-2
Yun Gang Chen, Yoshikazu Giga, and Shun’ichi Goto, Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations, J. Differential Geom. 33 (1991), no. 3, 749–786. MR 1100211
Otis Chodosh, Michael Eichmair, and Vlad Moraru, A splitting theorem for scalar curvature, arXiv:1804.01751 (2018).
Otis Chodosh and Daniel Ketover, Asymptotically flat three-manifolds contain minimal planes, Adv. Math. 337 (2018), 171–192. MR 3853048, DOI 10.1016/j.aim.2018.08.010
Yvonne Choquet-Bruhat, General relativity and the Einstein equations, Oxford Mathematical Monographs, Oxford University Press, Oxford, 2009. MR 2473363
Yvonne Choquet-Bruhat, Introduction to general relativity, black holes, and cosmology, Oxford University Press, Oxford, 2015. With a foreword by Thibault Damour. MR 3379262
Y. Choquet-Bruhat and D. Christodoulou, Elliptic systems in $H_{s,\delta }$ spaces on manifolds which are Euclidean at infinity, Acta Math. 146 (1981), no. 1-2, 129–150. MR 594629, DOI 10.1007/BF02392460
Yvonne Choquet-Bruhat and Robert Geroch, Global aspects of the Cauchy problem in general relativity, Comm. Math. Phys. 14 (1969), 329–335. MR 250640
Bennett Chow, Peng Lu, and Lei Ni, Hamilton’s Ricci flow, Graduate Studies in Mathematics, vol. 77, American Mathematical Society, Providence, RI; Science Press Beijing, New York, 2006. MR 2274812, DOI 10.1090/gsm/077
Demetrios Christodoulou, The formation of black holes in general relativity, EMS Monographs in Mathematics, European Mathematical Society (EMS), Zürich, 2009. MR 2488976, DOI 10.4171/068
Demetrios Christodoulou and Sergiu Klainerman, The global nonlinear stability of the Minkowski space, Princeton Mathematical Series, vol. 41, Princeton University Press, Princeton, NJ, 1993. MR 1316662
Piotr T. Chruściel and Daniel Maerten, 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 (2006), no. 2, 022502, 10. MR 2208148, DOI 10.1063/1.2167809
D. Christodoulou and N. Ó Murchadha, The boost problem in general relativity, Comm. Math. Phys. 80 (1981), no. 2, 271–300. MR 623161
Piotr Chruściel, Boundary conditions at spatial infinity from a Hamiltonian point of view, Topological properties and global structure of space-time (Erice, 1985) NATO Adv. Sci. Inst. Ser. B: Phys., vol. 138, Plenum, New York, 1986, pp. 49–59. MR 1102938
Piotr T. Chruściel, Mass and angular-momentum inequalities for axi-symmetric initial data sets. I. Positivity of mass, Ann. Physics 323 (2008), no. 10, 2566–2590. MR 2454698, DOI 10.1016/j.aop.2007.12.010
Piotr T. Chruściel, The geometry of black holes, 2015. http://homepage.univie.ac.at/piotr.chrusciel/teaching/Black~Holes/BlackHolesViennaJanuary2015.pdf.
Piotr T. Chruściel and João Lopes Costa, On uniqueness of stationary vacuum black holes, Astérisque 321 (2008), 195–265 (English, with English and French summaries). Géométrie différentielle, physique mathématique, mathématiques et société. I. MR 2521649
P. T. Chruściel, E. Delay, G. J. Galloway, and R. Howard, Regularity of horizons and the area theorem, Ann. Henri Poincaré 2 (2001), no. 1, 109–178. MR 1823836, DOI 10.1007/PL00001029
Piotr T. Chruściel, Gregory J. Galloway, and Daniel Pollack, Mathematical general relativity: a sampler, Bull. Amer. Math. Soc. (N.S.) 47 (2010), no. 4, 567–638. MR 2721040, DOI 10.1090/S0273-0979-2010-01304-5
Piotr T. Chruściel and Marc Herzlich, The mass of asymptotically hyperbolic Riemannian manifolds, Pacific J. Math. 212 (2003), no. 2, 231–264. MR 2038048, DOI 10.2140/pjm.2003.212.231
Tobias Holck Colding and William P. Minicozzi II, A course in minimal surfaces, Graduate Studies in Mathematics, vol. 121, American Mathematical Society, Providence, RI, 2011. MR 2780140, DOI 10.1090/gsm/121
Justin Corvino, Scalar curvature deformation and a gluing construction for the Einstein constraint equations, Comm. Math. Phys. 214 (2000), no. 1, 137–189. MR 1794269, DOI 10.1007/PL00005533
Justin Corvino, A note on asymptotically flat metrics on ${\Bbb R}^3$ which are scalar-flat and admit minimal spheres, Proc. Amer. Math. Soc. 133 (2005), no. 12, 3669–3678. MR 2163606, DOI 10.1090/S0002-9939-05-07926-8
Justin Corvino, A note on the Bartnik mass, Nonlinear analysis in geometry and applied mathematics, Harv. Univ. Cent. Math. Sci. Appl. Ser. Math., vol. 1, Int. Press, Somerville, MA, 2017, pp. 49–75. MR 3729084
Justin Corvino and Lan-Hsuan Huang, Localized deformation for initial data sets with the dominant energy condition, arXiv:1606.03078 (2016).
Justin Corvino and Richard M. Schoen, On the asymptotics for the vacuum Einstein constraint equations, J. Differential Geom. 73 (2006), no. 2, 185–217. MR 2225517
Richard Courant, Plateau’s problem and Dirichlet’s principle, Ann. of Math. (2) 38 (1937), no. 3, 679–724. MR 1503362, DOI 10.2307/1968610
Mihalis Dafermos and Jonathan Luk, The interior of dynamical vacuum black holes I: The $c^0$-stability of the Kerr Cauchy horizon, arXiv:1710.01722 (2017).
Mattias Dahl and Eric Larsson, Outermost apparent horizons diffeomorphic to unit normal bundles, arXiv:1606.0841 (2016).
Xianzhe Dai and Li Ma, Mass under the Ricci flow, Comm. Math. Phys. 274 (2007), no. 1, 65–80. MR 2318848, DOI 10.1007/s00220-007-0275-6
Ennio De Giorgi, Frontiere orientate di misura minima, Editrice Tecnico Scientifica, Pisa, 1961 (Italian). Seminario di Matematica della Scuola Normale Superiore di Pisa, 1960-61. MR 0179651
Camillo De Lellis, The size of the singular set of area-minimizing currents, Surveys in differential geometry 2016. Advances in geometry and mathematical physics, Surv. Differ. Geom., vol. 21, Int. Press, Somerville, MA, 2016, pp. 1–83. MR 3525093
Camillo De Lellis and Emanuele Spadaro, Regularity of area minimizing currents I: gradient $L^p$ estimates, Geom. Funct. Anal. 24 (2014), no. 6, 1831–1884. MR 3283929, DOI 10.1007/s00039-014-0306-3
V. I. Denisov and V. O. Solov′ev, Energy defined in general relativity on the basis of the traditional Hamiltonian approach has no physical meaning, Teoret. Mat. Fiz. 56 (1983), no. 2, 301–314 (Russian, with English summary). MR 718105
Jesse Douglas, Solution of the problem of Plateau, Trans. Amer. Math. Soc. 33 (1931), no. 1, 263–321. MR 1501590, DOI 10.1090/S0002-9947-1931-1501590-9
Michael Eichmair, The Plateau problem for marginally outer trapped surfaces, J. Differential Geom. 83 (2009), no. 3, 551–583. MR 2581357
Michael Eichmair, The Jang equation reduction of the spacetime positive energy theorem in dimensions less than eight, Comm. Math. Phys. 319 (2013), no. 3, 575–593. MR 3040369, DOI 10.1007/s00220-013-1700-7
Michael Eichmair, Gregory J. Galloway, and Daniel Pollack, Topological censorship from the initial data point of view, J. Differential Geom. 95 (2013), no. 3, 389–405. MR 3128989
Michael Eichmair, Lan-Hsuan Huang, Dan A. Lee, and Richard Schoen, The spacetime positive mass theorem in dimensions less than eight, J. Eur. Math. Soc. (JEMS) 18 (2016), no. 1, 83–121. MR 3438380, DOI 10.4171/JEMS/584
Michael Eichmair and Jan Metzger, Large isoperimetric surfaces in initial data sets, J. Differential Geom. 94 (2013), no. 1, 159–186. MR 3031863
Michael Eichmair and Jan Metzger, Jenkins-Serrin-type results for the Jang equation, J. Differential Geom. 102 (2016), no. 2, 207–242. MR 3454546
Michael Eichmair, Pengzi Miao, and Xiaodong Wang, Extension of a theorem of Shi and Tam, Calc. Var. Partial Differential Equations 43 (2012), no. 1-2, 45–56. MR 2860402, DOI 10.1007/s00526-011-0402-2
Jean Eisenstaedt, Lemaître and the Schwarzschild solution, The attraction of gravitation: new studies in the history of general relativity (Johnstown, PA, 1991) Einstein Stud., vol. 5, Birkhäuser Boston, Boston, MA, 1993, pp. 353–389. MR 1735388
Roberto Emparan and Harvey S. Reall, A rotating black ring solution in five dimensions, Phys. Rev. Lett. 88 (2002), no. 10, 101101, 4. MR 1901280, DOI 10.1103/PhysRevLett.88.101101
Roberto Emparan and Harvey S. Reall, Black rings, Classical Quantum Gravity 23 (2006), no. 20, R169–R197. MR 2270099, DOI 10.1088/0264-9381/23/20/R01
Lawrence C. Evans, Partial differential equations, 2nd ed., Graduate Studies in Mathematics, vol. 19, American Mathematical Society, Providence, RI, 2010. MR 2597943, DOI 10.1090/gsm/019
L. C. Evans and J. Spruck, Motion of level sets by mean curvature. I, J. Differential Geom. 33 (1991), no. 3, 635–681. MR 1100206
Herbert Federer, The singular sets of area minimizing rectifiable currents with codimension one and of area minimizing flat chains modulo two with arbitrary codimension, Bull. Amer. Math. Soc. 76 (1970), 767–771. MR 260981, DOI 10.1090/S0002-9904-1970-12542-3
Herbert Federer and Wendell H. Fleming, Normal and integral currents, Ann. of Math. (2) 72 (1960), 458–520. MR 123260, DOI 10.2307/1970227
Felix Finster, A level set analysis of the Witten spinor with applications to curvature estimates, Math. Res. Lett. 16 (2009), no. 1, 41–55. MR 2480559, DOI 10.4310/MRL.2009.v16.n1.a5
Felix Finster and Ines Kath, Curvature estimates in asymptotically flat manifolds of positive scalar curvature, Comm. Anal. Geom. 10 (2002), no. 5, 1017–1031. MR 1957661, DOI 10.4310/CAG.2002.v10.n5.a6
Arthur E. Fischer and Jerrold E. Marsden, Deformations of the scalar curvature, Duke Math. J. 42 (1975), no. 3, 519–547. MR 380907
Doris Fischer-Colbrie and Richard Schoen, The structure of complete stable minimal surfaces in $3$-manifolds of nonnegative scalar curvature, Comm. Pure Appl. Math. 33 (1980), no. 2, 199–211. MR 562550, DOI 10.1002/cpa.3160330206
Wendell H. Fleming, On the oriented Plateau problem, Rend. Circ. Mat. Palermo (2) 11 (1962), 69–90. MR 157263, DOI 10.1007/BF02849427
Gerald B. Folland, Introduction to partial differential equations, 2nd ed., Princeton University Press, Princeton, NJ, 1995. MR 1357411
Y. Fourès-Bruhat, Théorème d’existence pour certains systèmes d’équations aux dérivées partielles non linéaires, Acta Math. 88 (1952), 141–225 (French). MR 53338, DOI 10.1007/BF02392131
Alexandre Freire and Fernando Schwartz, Mass-capacity inequalities for conformally flat manifolds with boundary, Comm. Partial Differential Equations 39 (2014), no. 1, 98–119. MR 3169780, DOI 10.1080/03605302.2013.851211
Gregory J. Galloway, Maximum principles for null hypersurfaces and null splitting theorems, Ann. Henri Poincaré 1 (2000), no. 3, 543–567. MR 1777311, DOI 10.1007/s000230050006
Gregory J. Galloway, Stability and rigidity of extremal surfaces in Riemannian geometry and general relativity, Surveys in geometric analysis and relativity, Adv. Lect. Math. (ALM), vol. 20, Int. Press, Somerville, MA, 2011, pp. 221–239. MR 2906927
Gregory J. Galloway, Notes on Lorentzian causality, 2014. http://www.math.miami.edu/~galloway/vienna-course-notes.pdf.
Gregory J. Galloway, Rigidity of outermost MOTS: the initial data version, Gen. Relativity Gravitation 50 (2018), no. 3, Paper No. 32, 7. MR 3768955, DOI 10.1007/s10714-018-2353-9
Gregory J. Galloway and Richard Schoen, A generalization of Hawking’s black hole topology theorem to higher dimensions, Comm. Math. Phys. 266 (2006), no. 2, 571–576. MR 2238889, DOI 10.1007/s00220-006-0019-z
Dennis Gannon, Singularities in nonsimply connected space-times, J. Mathematical Phys. 16 (1975), no. 12, 2364–2367. MR 389141, DOI 10.1063/1.522498
L. Zhiyong Gao and S.-T. Yau, The existence of negatively Ricci curved metrics on three-manifolds, Invent. Math. 85 (1986), no. 3, 637–652. MR 848687, DOI 10.1007/BF01390331
Robert Geroch, Domain of dependence, J. Mathematical Phys. 11 (1970), 437–449. MR 270697, DOI 10.1063/1.1665157
Robert Geroch, Energy extraction, Sixth Texas symposium on relativistic astrophysics, 1973, pp. 108.
Robert Geroch, General relativity, Differential geometry (Proc. Sympos. Pure Math., Vol. XXVII, Part 2, Stanford Univ., Stanford, Calif., 1973) Amer. Math. Soc., Providence, R.I., 1975, pp. 401–414. MR 0378703
Robert Geroch, General relativity: 1972 lecture notes, Minkowski Institute Press, Montreal, 2013.
G. W. Gibbons, S. W. Hawking, Gary T. Horowitz, and Malcolm J. Perry, Positive mass theorems for black holes, Comm. Math. Phys. 88 (1983), no. 3, 295–308. MR 701918
David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, Classics in Mathematics, Springer-Verlag, Berlin, 2001. Reprint of the 1998 edition. MR 1814364
James D. E. Grant and Nathalie Tassotti, A positive mass theorem for low-regularity metrics, arXiv:1205.1302 (2012).
Jeremy Gray and Mario Micallef, About the cover: the work of Jesse Douglas on minimal surfaces, Bull. Amer. Math. Soc. (N.S.) 45 (2008), no. 2, 293–302. MR 2383307, DOI 10.1090/S0273-0979-08-01192-0
L. W. Green, Auf Wiedersehensflächen, Ann. of Math. (2) 78 (1963), 289–299 (German). MR 155271, DOI 10.2307/1970344
Mikhael Gromov and H. Blaine Lawson Jr., Spin and scalar curvature in the presence of a fundamental group. I, Ann. of Math. (2) 111 (1980), no. 2, 209–230. MR 569070, DOI 10.2307/1971198
Mikhael Gromov and H. Blaine Lawson Jr., The classification of simply connected manifolds of positive scalar curvature, Ann. of Math. (2) 111 (1980), no. 3, 423–434. MR 577131, DOI 10.2307/1971103
Pengfei Guan and Yan Yan Li, The Weyl problem with nonnegative Gauss curvature, J. Differential Geom. 39 (1994), no. 2, 331–342. MR 1267893
Fengbo Hang and Xiaodong Wang, Rigidity theorems for compact manifolds with boundary and positive Ricci curvature, J. Geom. Anal. 19 (2009), no. 3, 628–642. MR 2496569, DOI 10.1007/s12220-009-9074-y
F. Reese Harvey, Spinors and calibrations, Perspectives in Mathematics, vol. 9, Academic Press, Inc., Boston, MA, 1990. MR 1045637
Allen Hatcher, Algebraic topology, Cambridge University Press, Cambridge, 2002. MR 1867354
S. W. Hawking, Gravitational radiation in an expanding universe, J. Math. Phys. 9 (1968), 598–604.
S. W. Hawking, Black holes in general relativity, Comm. Math. Phys. 25 (1972), 152–166. MR 293962
S. W. Hawking and G. F. R. Ellis, The large scale structure of space-time, Cambridge Monographs on Mathematical Physics, No. 1, Cambridge University Press, London-New York, 1973. MR 0424186
Sean A. Hayward, Gravitational energy in spherical symmetry, Phys. Rev. D (3) 53 (1996), no. 4, 1938–1949. MR 1380012, DOI 10.1103/PhysRevD.53.1938
Emmanuel Hebey, Sobolev spaces on Riemannian manifolds, Lecture Notes in Mathematics, vol. 1635, Springer-Verlag, Berlin, 1996. MR 1481970, DOI 10.1007/BFb0092907
Hans-Joachim Hein and Claude LeBrun, Mass in Kähler geometry, Comm. Math. Phys. 347 (2016), no. 1, 183–221. MR 3543182, DOI 10.1007/s00220-016-2661-4
John Hempel, $3$-Manifolds, Annals of Mathematics Studies, No. 86, Princeton University Press, Princeton, N. J.; University of Tokyo Press, Tokyo, 1976. MR 0415619
Joseph Hersch, Quatre propriétés isopérimétriques de membranes sphériques homogènes, C. R. Acad. Sci. Paris Sér. A-B 270 (1970), A1645–A1648 (French). MR 292357
Marc Herzlich, A Penrose-like inequality for the mass of Riemannian asymptotically flat manifolds, Comm. Math. Phys. 188 (1997), no. 1, 121–133. MR 1471334, DOI 10.1007/s002200050159
Marc Herzlich, The positive mass theorem for black holes revisited, J. Geom. Phys. 26 (1998), no. 1-2, 97–111. MR 1626060, DOI 10.1016/S0393-0440(97)00040-5
Nigel Hitchin, Harmonic spinors, Advances in Math. 14 (1974), 1–55. MR 358873, DOI 10.1016/0001-8708(74)90021-8
Lan-Hsuan Huang, On the center of mass of isolated systems with general asymptotics, Classical Quantum Gravity 26 (2009), no. 1, 015012, 25. MR 2470255, DOI 10.1088/0264-9381/26/1/015012
Lan-Hsuan Huang, On the center of mass in general relativity, Fifth International Congress of Chinese Mathematicians. Part 1, 2, AMS/IP Stud. Adv. Math., 51, pt. 1, vol. 2, Amer. Math. Soc., Providence, RI, 2012, pp. 575–591. MR 2908093
Lan-Hsuan Huang and Dan A. Lee, Stability of the positive mass theorem for graphical hypersurfaces of Euclidean space, Comm. Math. Phys. 337 (2015), no. 1, 151–169. MR 3324159, DOI 10.1007/s00220-014-2265-9
Lan-Hsuan Huang and Dan A. Lee, Rigidity of the spacetime positive mass theorem, arXiv:1706.03732 (2017).
Lan-Hsuan Huang, Dan A. Lee, and Christina Sormani, Intrinsic flat stability of the positive mass theorem for graphical hypersurfaces of Euclidean space, J. Reine Angew. Math. 727 (2017), 269–299. MR 3652253, DOI 10.1515/crelle-2015-0051
Lan-Hsuan Huang, Daniel Martin, and Pengzi Miao, Static potentials and area minimizing hypersurfaces, Proc. Amer. Math. Soc. 146 (2018), no. 6, 2647–2661. MR 3778165, DOI 10.1090/proc/13936
Lan-Hsuan Huang and Damin Wu, Hypersurfaces with nonnegative scalar curvature, J. Differential Geom. 95 (2013), no. 2, 249–278. MR 3128984
Lan-Hsuan Huang and Damin Wu, The equality case of the Penrose inequality for asymptotically flat graphs, Trans. Amer. Math. Soc. 367 (2015), no. 1, 31–47. MR 3271252, DOI 10.1090/S0002-9947-2014-06090-X
Gerhard Huisken and Tom Ilmanen, The inverse mean curvature flow and the Riemannian Penrose inequality, J. Differential Geom. 59 (2001), no. 3, 353–437. MR 1916951
Gerhard Huisken and Shing-Tung Yau, Definition of center of mass for isolated physical systems and unique foliations by stable spheres with constant mean curvature, Invent. Math. 124 (1996), no. 1-3, 281–311. MR 1369419, DOI 10.1007/s002220050054
Werner Israel, Event horizons in static vacuum space-times, Phys. Rev. 164 (1967), 1776–1779.
Jeffrey L. Jauregui, Fill-ins of nonnegative scalar curvature, static metrics, and quasi-local mass, Pacific J. Math. 261 (2013), no. 2, 417–444. MR 3037574, DOI 10.2140/pjm.2013.261.417
Jeffrey L. Jauregui, On the lower semicontinuity of the ADM mass, Comm. Anal. Geom. 26 (2018), no. 1, 85–111. MR 3761654, DOI 10.4310/CAG.2018.v26.n1.a3
Jeffrey L. Jauregui, Smoothing the Bartnik boundary conditions and other results on Bartnik’s quasi-local mass, J. Geom. Phys. 136 (2019), 228–243. MR 3885243, DOI 10.1016/j.geomphys.2018.11.005
Jeffrey L. Jauregui and Dan A. Lee, Lower semicontinuity of mass under $c^0$ convergence and Huisken’s isoperimetric mass, arXiv:1602.00732 (2016).
Jeffrey L. Jauregui and Dan A. Lee, Lower semicontinuity of ADM mass under intrinsic flat convergence, arXiv:1903.00916 (2019).
Jürgen Jost, Riemannian geometry and geometric analysis, 6th ed., Universitext, Springer, Heidelberg, 2011. MR 2829653, DOI 10.1007/978-3-642-21298-7
Jürgen Jost, Partial differential equations, 3rd ed., Graduate Texts in Mathematics, vol. 214, Springer, New York, 2013. MR 3012036, DOI 10.1007/978-1-4614-4809-9
Jerry L. Kazdan and F. W. Warner, Existence and conformal deformation of metrics with prescribed Gaussian and scalar curvatures, Ann. of Math. (2) 101 (1975), 317–331. MR 375153, DOI 10.2307/1970993
Jerry L. Kazdan and F. W. Warner, Prescribing curvatures, Differential geometry (Proc. Sympos. Pure Math., Vol. XXVII, Part 2, Stanford Univ., Stanford, Calif., 1973) Amer. Math. Soc., Providence, R.I., 1975, pp. 309–319. MR 0394505
Marcus Khuri, Yukio Matsumoto, Gilbert Weinstein, and Sumio Yamada, Plumbing constructions and the domain of outer communication for 5-dimensional stationary black holes, arXiv:1807.03452 (2018).
Felix Klein, Vorlesungen über höhere Geometrie, Die Grundlehren der mathematischen Wissenschaften, Band 22, Springer-Verlag, Berlin, 1968 (German). Dritte Auflage; Bearbeitet und herausgegeben von W. Blaschke. MR 0226476
M. G. Kreĭn and M. A. Rutman, Linear operators leaving invariant a cone in a Banach space, Amer. Math. Soc. Translation 1950 (1950), no. 26, 128. MR 0038008
Marcus Kriele and Sean A. Hayward, Outer trapped surfaces and their apparent horizon, J. Math. Phys. 38 (1997), no. 3, 1593–1604. MR 1435684, DOI 10.1063/1.532010
Hari K. Kunduri and James Lucietti, Supersymmetric black holes with lens-space topology, Phys. Rev. Lett. 113 (2014), 211101, 5.
Mau-Kwong George Lam, The Graph Cases of the Riemannian Positive Mass and Penrose Inequalities in All Dimensions, ProQuest LLC, Ann Arbor, MI, 2011. Thesis (Ph.D.)–Duke University. MR 2873434
H. Blaine Lawson Jr. and Marie-Louise Michelsohn, Spin geometry, Princeton Mathematical Series, vol. 38, Princeton University Press, Princeton, NJ, 1989. MR 1031992
C. W. Lee, A restriction on the topology of Cauchy surfaces in general relativity, Comm. Math. Phys. 51 (1976), no. 2, 157–162. MR 426805
Dan A. Lee, On the near-equality case of the positive mass theorem, Duke Math. J. 148 (2009), no. 1, 63–80. MR 2515100, DOI 10.1215/00127094-2009-021
Dan A. Lee, A positive mass theorem for Lipschitz metrics with small singular sets, Proc. Amer. Math. Soc. 141 (2013), no. 11, 3997–4004. MR 3091790, DOI 10.1090/S0002-9939-2013-11871-X
Dan A. Lee and Philippe G. LeFloch, The positive mass theorem for manifolds with distributional curvature, Comm. Math. Phys. 339 (2015), no. 1, 99–120. MR 3366052, DOI 10.1007/s00220-015-2414-9
Dan A. Lee and André Neves, The Penrose inequality for asymptotically locally hyperbolic spaces with nonpositive mass, Comm. Math. Phys. 339 (2015), no. 2, 327–352. MR 3370607, DOI 10.1007/s00220-015-2421-x
Dan A. Lee and Christina Sormani, Near-equality of the Penrose inequality for rotationally symmetric Riemannian manifolds, Ann. Henri Poincaré 13 (2012), no. 7, 1537–1556. MR 2982632, DOI 10.1007/s00023-012-0172-1
Dan A. Lee and Christina Sormani, Stability of the positive mass theorem for rotationally symmetric Riemannian manifolds, J. Reine Angew. Math. 686 (2014), 187–220. MR 3176604, DOI 10.1515/crelle-2012-0094
John M. Lee, Riemannian manifolds, Graduate Texts in Mathematics, vol. 176, Springer-Verlag, New York, 1997. An introduction to curvature. MR 1468735, DOI 10.1007/b98852
John M. Lee and Thomas H. Parker, The Yamabe problem, Bull. Amer. Math. Soc. (N.S.) 17 (1987), no. 1, 37–91. MR 888880, DOI 10.1090/S0273-0979-1987-15514-5
Philippe G. LeFloch and Christina Sormani, The nonlinear stability of rotationally symmetric spaces with low regularity, J. Funct. Anal. 268 (2015), no. 7, 2005–2065. MR 3315585, DOI 10.1016/j.jfa.2014.12.012
Yanyan Li and Gilbert Weinstein, A priori bounds for co-dimension one isometric embeddings, Amer. J. Math. 121 (1999), no. 5, 945–965. MR 1713298
Yu Li, Ricci flow on asymptotically Euclidean manifolds, Geom. Topol. 22 (2018), no. 3, 1837–1891. MR 3780446, DOI 10.2140/gt.2018.22.1837
André Lichnerowicz, Spineurs harmoniques, C. R. Acad. Sci. Paris 257 (1963), 7–9 (French). MR 156292
Chen-Yun Lin, Parabolic constructions of asymptotically flat 3-metrics of prescribed scalar curvature, Calc. Var. Partial Differential Equations 49 (2014), no. 3-4, 1309–1335. MR 3168634, DOI 10.1007/s00526-013-0623-7
Fanghua Lin and Xiaoping Yang, Geometric measure theory—an introduction, Advanced Mathematics (Beijing/Boston), vol. 1, Science Press Beijing, Beijing; International Press, Boston, MA, 2002. MR 2030862
Hans Lindblad and Igor Rodnianski, The global stability of Minkowski space-time in harmonic gauge, Ann. of Math. (2) 171 (2010), no. 3, 1401–1477. MR 2680391, DOI 10.4007/annals.2010.171.1401
Chiu-Chu Melissa Liu and Shing-Tung Yau, Positivity of quasi-local mass. II, J. Amer. Math. Soc. 19 (2006), no. 1, 181–204. MR 2169046, DOI 10.1090/S0894-0347-05-00497-2
Robert B. Lockhart, Fredholm properties of a class of elliptic operators on noncompact manifolds, Duke Math. J. 48 (1981), no. 1, 289–312. MR 610188
Robert B. Lockhart and Robert C. McOwen, On elliptic systems in $\textbf {R}^{n}$, Acta Math. 150 (1983), no. 1-2, 125–135. MR 697610, DOI 10.1007/BF02392969
Joachim Lohkamp, Metrics of negative Ricci curvature, Ann. of Math. (2) 140 (1994), no. 3, 655–683. MR 1307899, DOI 10.2307/2118620
Joachim Lohkamp, Scalar curvature and hammocks, Math. Ann. 313 (1999), no. 3, 385–407. MR 1678604, DOI 10.1007/s002080050266
Joachim Lohkamp, The higher dimensional positive mass theorem I, arXiv:0608795 (2006).
Joachim Lohkamp, Hyperbolic geometry and potential theory on minimal hypersurfaces, arXiv:1512.08251 (2015).
Joachim Lohkamp, Skin structures in scalar curvature geometry, arXiv:1512.08251 (2015).
Joachim Lohkamp, Skin structures on minimal hypersurfaces, arXiv:1512.08249 (2015).
Joachim Lohkamp, The higher dimensional positive mass theorem II, arXiv:1612.07505 (2016).
Siyuan Lu and Pengzi Miao, Minimal hypersurfaces and boundary behavior of compact manifolds with nonnegative scalar curvature, arXiv:1703.08164 (2017).
Edward Malec and Niall Ó Murchadha, Trapped surfaces and the Penrose inequality in spherically symmetric geometries, Phys. Rev. D (3) 49 (1994), no. 12, 6931–6934. MR 1278625, DOI 10.1103/PhysRevD.49.6931
Christos Mantoulidis and Pengzi Miao, Total mean curvature, scalar curvature, and a variational analog of Brown-York mass, Comm. Math. Phys. 352 (2017), no. 2, 703–718. MR 3627410, DOI 10.1007/s00220-016-2767-8
Christos Mantoulidis and Richard Schoen, On the Bartnik mass of apparent horizons, Classical Quantum Gravity 32 (2015), no. 20, 205002, 16. MR 3406373, DOI 10.1088/0264-9381/32/20/205002
Fernando C. Marques and André Neves, Rigidity of min-max minimal spheres in three-manifolds, Duke Math. J. 161 (2012), no. 14, 2725–2752. MR 2993139, DOI 10.1215/00127094-1813410
Fernando C. Marques and André Neves, Min-max theory and the Willmore conjecture, Ann. of Math. (2) 179 (2014), no. 2, 683–782. MR 3152944, DOI 10.4007/annals.2014.179.2.6
Marc Mars, Present status of the Penrose inequality, Classical Quantum Gravity 26 (2009), no. 19, 193001, 59. MR 2545137, DOI 10.1088/0264-9381/26/19/193001
Umberto Massari and Mario Miranda, Minimal surfaces of codimension one, North-Holland Mathematics Studies, vol. 91, North-Holland Publishing Co., Amsterdam, 1984. Notas de Matemática [Mathematical Notes], 95. MR 795963
Donovan McFeron and Gábor Székelyhidi, On the positive mass theorem for manifolds with corners, Comm. Math. Phys. 313 (2012), no. 2, 425–443. MR 2942956, DOI 10.1007/s00220-012-1498-8
Robert C. McOwen, The behavior of the Laplacian on weighted Sobolev spaces, Comm. Pure Appl. Math. 32 (1979), no. 6, 783–795. MR 539158, DOI 10.1002/cpa.3160320604
Robert C. McOwen, On elliptic operators in $\textbf {R}^{n}$, Comm. Partial Differential Equations 5 (1980), no. 9, 913–933. MR 584101, DOI 10.1080/03605308008820158
William Meeks III, Leon Simon, and Shing Tung Yau, Embedded minimal surfaces, exotic spheres, and manifolds with positive Ricci curvature, Ann. of Math. (2) 116 (1982), no. 3, 621–659. MR 678484, DOI 10.2307/2007026
William H. Meeks III and Shing Tung Yau, Topology of three-dimensional manifolds and the embedding problems in minimal surface theory, Ann. of Math. (2) 112 (1980), no. 3, 441–484. MR 595203, DOI 10.2307/1971088
Norman Meyers, An expansion about infinity for solutions of linear elliptic equations. , J. Math. Mech. 12 (1963), 247–264. MR 0149072
Pengzi Miao, Positive mass theorem on manifolds admitting corners along a hypersurface, Adv. Theor. Math. Phys. 6 (2002), no. 6, 1163–1182 (2003). MR 1982695, DOI 10.4310/ATMP.2002.v6.n6.a4
Pengzi Miao, Variational effect of boundary mean curvature on ADM mass in general relativity, Mathematical physics research on the leading edge, Nova Sci. Publ., Hauppauge, NY, 2004, pp. 145–171. MR 2068577
Pengzi Miao, On a localized Riemannian Penrose inequality, Comm. Math. Phys. 292 (2009), no. 1, 271–284. MR 2540078, DOI 10.1007/s00220-009-0834-0
Pengzi Miao and Luen-Fai Tam, Static potentials on asymptotically flat manifolds, Ann. Henri Poincaré 16 (2015), no. 10, 2239–2264. MR 3385979, DOI 10.1007/s00023-014-0373-x
Pengzi Miao and Luen-Fai Tam, Evaluation of the ADM mass and center of mass via the Ricci tensor, Proc. Amer. Math. Soc. 144 (2016), no. 2, 753–761. MR 3430851, DOI 10.1090/proc12726
Mario Micallef and Vlad Moraru, Splitting of 3-manifolds and rigidity of area-minimising surfaces, Proc. Amer. Math. Soc. 143 (2015), no. 7, 2865–2872. MR 3336611, DOI 10.1090/S0002-9939-2015-12137-5
Maung Min-Oo, Scalar curvature rigidity of asymptotically hyperbolic spin manifolds, Math. Ann. 285 (1989), no. 4, 527–539. MR 1027758, DOI 10.1007/BF01452046
Charles W. Misner, Kip S. Thorne, and John Archibald Wheeler, Gravitation, W. H. Freeman and Co., San Francisco, Calif., 1973. MR 0418833
Frank Morgan, Geometric measure theory, 5th ed., Elsevier/Academic Press, Amsterdam, 2016. A beginner’s guide; Illustrated by James F. Bredt. MR 3497381
John Morgan and Gang Tian, Ricci flow and the Poincaré conjecture, Clay Mathematics Monographs, vol. 3, American Mathematical Society, Providence, RI; Clay Mathematics Institute, Cambridge, MA, 2007. MR 2334563, DOI 10.1305/ndjfl/1193667709
John Morgan and Gang Tian, The geometrization conjecture, Clay Mathematics Monographs, vol. 5, American Mathematical Society, Providence, RI; Clay Mathematics Institute, Cambridge, MA, 2014. MR 3186136
Charles B. Morrey Jr., The problem of Plateau on a Riemannian manifold, Ann. of Math. (2) 49 (1948), 807–851. MR 27137, DOI 10.2307/1969401
H. Müller zum Hagen, David C. Robinson, and H. J. Seifert, Black holes in static vacuum space-times, Gen. Relativity Gravitation 4 (1973), 53–78. MR 398432, DOI 10.1007/bf00769760
R. C. Myers and M. J. Perry, Black holes in higher-dimensional space-times, Ann. Physics 172 (1986), no. 2, 304–347. MR 868295, DOI 10.1016/0003-4916(86)90186-7
Louis Nirenberg, The Weyl and Minkowski problems in differential geometry in the large, Comm. Pure Appl. Math. 6 (1953), 337–394. MR 58265, DOI 10.1002/cpa.3160060303
Louis Nirenberg and Homer F. Walker, The null spaces of elliptic partial differential operators in $\textbf {R}^{n}$, J. Math. Anal. Appl. 42 (1973), 271–301. Collection of articles dedicated to Salomon Bochner. MR 320821, DOI 10.1016/0022-247X(73)90138-8
Ivaldo Nunes, Rigidity of area-minimizing hyperbolic surfaces in three-manifolds, J. Geom. Anal. 23 (2013), no. 3, 1290–1302. MR 3078354, DOI 10.1007/s12220-011-9287-8
Todd A. Oliynyk and Eric Woolgar, Rotationally symmetric Ricci flow on asymptotically flat manifolds, Comm. Anal. Geom. 15 (2007), no. 3, 535–568. MR 2379804
Barrett O’Neill, Semi-Riemannian geometry, Pure and Applied Mathematics, vol. 103, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1983. With applications to relativity. MR 719023
Thomas Parker and Clifford Henry Taubes, On Witten’s proof of the positive energy theorem, Comm. Math. Phys. 84 (1982), no. 2, 223–238. MR 661134
Roger Penrose, Gravitational collapse and space-time singularities, Phys. Rev. Lett. 14 (1965), 57–59. MR 172678, DOI 10.1103/PhysRevLett.14.57
Roger Penrose, Naked singularities, Annals N. Y. Acad. Sci. 224 (1973), 125–134.
R. Penrose, Gravitational collapse: the role of general relativity, Gen. Relativity Gravitation 34 (2002), no. 7, 1141–1165. Reprinted from Rivista del Nuovo Cimento 1969, Numero Speziale I, 252–276. MR 1915236, DOI 10.1023/A:1016578408204
Grisha Perelman, The entropy formula for the Ricci flow and its geometric applications, arXiv:math (2002).
Grisha Perelman, Finite extinction time for the solutions to the Ricci flow on certain three-manifolds, arXiv:math (2003).
Grisha Perelman, Ricci flow with surgery on three-manifolds, arXiv:math (2003).
Peter Petersen, Riemannian geometry, 3rd ed., Graduate Texts in Mathematics, vol. 171, Springer, Cham, 2016. MR 3469435, DOI 10.1007/978-3-319-26654-1
A. V. Pogorelov, Regularity of a convex surface with given Gaussian curvature, Mat. Sbornik N.S. 31(73) (1952), 88–103 (Russian). MR 0052807
Jie Qing and Gang Tian, On the uniqueness of the foliation of spheres of constant mean curvature in asymptotically flat 3-manifolds, J. Amer. Math. Soc. 20 (2007), no. 4, 1091–1110. MR 2328717, DOI 10.1090/S0894-0347-07-00560-7
Tibor Radó, On Plateau’s problem, Ann. of Math. (2) 31 (1930), no. 3, 457–469. MR 1502955, DOI 10.2307/1968237
Tullio Regge and Claudio Teitelboim, Role of surface integrals in the Hamiltonian formulation of general relativity, Ann. Physics 88 (1974), 286–318. MR 359663, DOI 10.1016/0003-4916(74)90404-7
E. R. Reifenberg, Solution of the Plateau Problem for $m$-dimensional surfaces of varying topological type, Acta Math. 104 (1960), 1–92. MR 114145, DOI 10.1007/BF02547186
Robert C. Reilly, Variational properties of functions of the mean curvatures for hypersurfaces in space forms, J. Differential Geometry 8 (1973), 465–477. MR 341351
D. C. Robinson, Uniqueness of the Kerr black hole, Phys. Rev. Lett. 34 (1975), 905–906.
D. C. Robinson, A simple proof of the generalization of Israel’s theorem, General Relativity and Gravitation 8 (1977-08), 695–698.
David C. Robinson, Four decades of black hole uniqueness theorems, The Kerr spacetime, Cambridge Univ. Press, Cambridge, 2009, pp. 115–143. MR 2789145
Jonathan Rosenberg, Manifolds of positive scalar curvature: a progress report, Surveys in differential geometry. Vol. XI, Surv. Differ. Geom., vol. 11, Int. Press, Somerville, MA, 2007, pp. 259–294. MR 2408269, DOI 10.4310/SDG.2006.v11.n1.a9
Jonathan Rosenberg and Stephan Stolz, Metrics of positive scalar curvature and connections with surgery, Surveys on surgery theory, Vol. 2, Ann. of Math. Stud., vol. 149, Princeton Univ. Press, Princeton, NJ, 2001, pp. 353–386. MR 1818778
R. K. Sachs, Gravitational waves in general relativity. VIII. Waves in asymptotically flat space-time, Proc. Roy. Soc. London Ser. A 270 (1962), 103–126. MR 149908, DOI 10.1098/rspa.1962.0206
J. Sacks and K. Uhlenbeck, The existence of minimal immersions of $2$-spheres, Ann. of Math. (2) 113 (1981), no. 1, 1–24. MR 604040, DOI 10.2307/1971131
J. Sacks and K. Uhlenbeck, Minimal immersions of closed Riemann surfaces, Trans. Amer. Math. Soc. 271 (1982), no. 2, 639–652. MR 654854, DOI 10.1090/S0002-9947-1982-0654854-8
J. Schauder, Über lineare elliptische Differentialgleichungen zweiter Ordnung, Math. Z. 38 (1934), no. 1, 257–282 (German). MR 1545448, DOI 10.1007/BF01170635
Richard M. Schoen, Uniqueness, symmetry, and embeddedness of minimal surfaces, J. Differential Geom. 18 (1983), no. 4, 791–809 (1984). MR 730928
Richard Schoen, Conformal deformation of a Riemannian metric to constant scalar curvature, J. Differential Geom. 20 (1984), no. 2, 479–495. MR 788292
Richard M. Schoen, The existence of weak solutions with prescribed singular behavior for a conformally invariant scalar equation, Comm. Pure Appl. Math. 41 (1988), no. 3, 317–392. MR 929283, DOI 10.1002/cpa.3160410305
Richard M. Schoen, Variational theory for the total scalar curvature functional for Riemannian metrics and related topics, Topics in calculus of variations (Montecatini Terme, 1987) Lecture Notes in Math., vol. 1365, Springer, Berlin, 1989, pp. 120–154. MR 994021, DOI 10.1007/BFb0089180
Richard Schoen and Leon Simon, Regularity of stable minimal hypersurfaces, Comm. Pure Appl. Math. 34 (1981), no. 6, 741–797. MR 634285, DOI 10.1002/cpa.3160340603
Richard M. Schoen and Shing Tung Yau, Complete manifolds with nonnegative scalar curvature and the positive action conjecture in general relativity, Proc. Nat. Acad. Sci. U.S.A. 76 (1979), no. 3, 1024–1025. MR 524327, DOI 10.1073/pnas.76.3.1024
R. Schoen and Shing Tung Yau, Existence of incompressible minimal surfaces and the topology of three-dimensional manifolds with nonnegative scalar curvature, Ann. of Math. (2) 110 (1979), no. 1, 127–142. MR 541332, DOI 10.2307/1971247
Richard Schoen and Shing Tung Yau, On the proof of the positive mass conjecture in general relativity, Comm. Math. Phys. 65 (1979), no. 1, 45–76. MR 526976
R. Schoen and S. T. Yau, On the structure of manifolds with positive scalar curvature, Manuscripta Math. 28 (1979), no. 1-3, 159–183. MR 535700, DOI 10.1007/BF01647970
Richard Schoen and Shing Tung Yau, The energy and the linear momentum of space-times in general relativity, Comm. Math. Phys. 79 (1981), no. 1, 47–51. MR 609227
Richard Schoen and Shing Tung Yau, Proof of the positive mass theorem. II, Comm. Math. Phys. 79 (1981), no. 2, 231–260. MR 612249
Richard Schoen and S. T. Yau, The existence of a black hole due to condensation of matter, Comm. Math. Phys. 90 (1983), no. 4, 575–579. MR 719436
R. Schoen and S.-T. Yau, Conformally flat manifolds, Kleinian groups and scalar curvature, Invent. Math. 92 (1988), no. 1, 47–71. MR 931204, DOI 10.1007/BF01393992
R. Schoen and S.-T. Yau, Lectures on differential geometry, Conference Proceedings and Lecture Notes in Geometry and Topology, I, International Press, Cambridge, MA, 1994. Lecture notes prepared by Wei Yue Ding, Kung Ching Chang [Gong Qing Zhang], Jia Qing Zhong and Yi Chao Xu; Translated from the Chinese by Ding and S. Y. Cheng; With a preface translated from the Chinese by Kaising Tso. MR 1333601
Richard Schoen and Shing-Tung Yau, Positive scalar curvature and minimal hypersurface singularities, arXiv:1704.05490 (2017).
Fernando Schwartz, Existence of outermost apparent horizons with product of spheres topology, Comm. Anal. Geom. 16 (2008), no. 4, 799–817. MR 2471370, DOI 10.4310/CAG.2008.v16.n4.a3
Ying Shen and Shunhui Zhu, Rigidity of stable minimal hypersurfaces, Math. Ann. 309 (1997), no. 1, 107–116. MR 1467649, DOI 10.1007/s002080050105
Wan-Xiong Shi, Ricci deformation of the metric on complete noncompact Riemannian manifolds, J. Differential Geom. 30 (1989), no. 2, 303–394. MR 1010165
Yuguang Shi and Luen-Fai Tam, Positive mass theorem and the boundary behaviors of compact manifolds with nonnegative scalar curvature, J. Differential Geom. 62 (2002), no. 1, 79–125. MR 1987378
James Simons, Minimal varieties in riemannian manifolds, Ann. of Math. (2) 88 (1968), 62–105. MR 233295, DOI 10.2307/1970556
Leon Simon, Lectures on geometric measure theory, Proceedings of the Centre for Mathematical Analysis, Australian National University, vol. 3, Australian National University, Centre for Mathematical Analysis, Canberra, 1983. MR 756417
Leon Simon, Rectifiability of the singular sets of multiplicity $1$ minimal surfaces and energy minimizing maps, Surveys in differential geometry, Vol. II (Cambridge, MA, 1993) Int. Press, Cambridge, MA, 1995, pp. 246–305. MR 1375258
Leon Simon, Schauder estimates by scaling, Calc. Var. Partial Differential Equations 5 (1997), no. 5, 391–407. MR 1459795, DOI 10.1007/s005260050072
Miles Simon, Deformation of $C^0$ Riemannian metrics in the direction of their Ricci curvature, Comm. Anal. Geom. 10 (2002), no. 5, 1033–1074. MR 1957662, DOI 10.4310/CAG.2002.v10.n5.a7
Nathan Smale, Generic regularity of homologically area minimizing hypersurfaces in eight-dimensional manifolds, Comm. Anal. Geom. 1 (1993), no. 2, 217–228. MR 1243523, DOI 10.4310/CAG.1993.v1.n2.a2
Francis R. Smith, On the existence of embedded minimal 2-spheres in the 3-sphere, endowed with an arbitrary Riemannian metric, 1982. Thesis (Ph.D.)–University of Melbourne.
Christina Sormani and Iva Stavrov Allen, Geometrostatic manifolds of small ADM mass, arXiv:1707.03008 (2017).
Michael Spivak, A comprehensive introduction to differential geometry. Vol. I, 2nd ed., Publish or Perish, Inc., Wilmington, Del., 1979. MR 532830
Stephan Stolz, Simply connected manifolds of positive scalar curvature, Ann. of Math. (2) 136 (1992), no. 3, 511–540. MR 1189863, DOI 10.2307/2946598
Peter Takáč, A short elementary proof of the Kreĭn-Rutman theorem, Houston J. Math. 20 (1994), no. 1, 93–98. MR 1272563
I. Tamanini, On the sphericity of liquid droplets, Astérisque 118 (1984), 235–241 (English, with French summary). Variational methods for equilibrium problems of fluids (Trento, 1983). MR 761754
V. A. Toponogov, Evaluation of the length of a closed geodesic on a convex surface, Dokl. Akad. Nauk SSSR 124 (1959), 282–284 (Russian). MR 0102055
A. Trautman, Radiation and boundary conditions in the theory of gravitation, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astr. Phys. 6 (1958), 407–412. MR 0097266
Neil S. Trudinger, Remarks concerning the conformal deformation of Riemannian structures on compact manifolds, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 22 (1968), 265–274. MR 240748
Wilderich Tuschmann and Michael Wiemeler, Differentiable stability and sphere theorems for manifolds and einstein manifolds with positive scalar curvature, arXiv:1408.3006 (2014).
Willie Wai-Yeung Wong, A positive mass theorem for two spatial dimensions, arXiv:1202.6279 (2012).
Robert M. Wald, General relativity, University of Chicago Press, Chicago, IL, 1984. MR 757180, DOI 10.7208/chicago/9780226870373.001.0001
Li He Wang, A geometric approach to the Calderón-Zygmund estimates, Acta Math. Sin. (Engl. Ser.) 19 (2003), no. 2, 381–396. MR 1987802, DOI 10.1007/s10114-003-0264-4
Mu-Tao Wang, Four lectures on quasi-local mass, arXiv:1510.02931 (2015).
Mu-Tao Wang and Shing-Tung Yau, Isometric embeddings into the Minkowski space and new quasi-local mass, Comm. Math. Phys. 288 (2009), no. 3, 919–942. MR 2504860, DOI 10.1007/s00220-009-0745-0
Xiaodong Wang, The mass of asymptotically hyperbolic manifolds, J. Differential Geom. 57 (2001), no. 2, 273–299. MR 1879228
Brian White, The space of $m$-dimensional surfaces that are stationary for a parametric elliptic functional, Indiana Univ. Math. J. 36 (1987), no. 3, 567–602. MR 905611, DOI 10.1512/iumj.1987.36.36031
Wikipedia contributors, Wikipedia, the free encyclopedia. [Online].
T. J. Willmore, Note on embedded surfaces, An. Şti. Univ. “Al. I. Cuza" Iaşi Secţ. I a Mat. (N.S.) 11B (1965), 493–496 (English, with Romanian and Russian summaries). MR 202066
Edward Witten, A new proof of the positive energy theorem, Comm. Math. Phys. 80 (1981), no. 3, 381–402. MR 626707
J. Wloka, Partial differential equations, Cambridge University Press, Cambridge, 1987. Translated from the German by C. B. Thomas and M. J. Thomas. MR 895589, DOI 10.1017/CBO9781139171755
Hidehiko Yamabe, On a deformation of Riemannian structures on compact manifolds, Osaka Math. J. 12 (1960), 21–37. MR 125546

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Geometric Relativity

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