Skip to main content Accessibility help
×
Hostname: page-component-f554764f5-rvxtl Total loading time: 0 Render date: 2025-04-17T13:56:03.901Z Has data issue: false hasContentIssue false

References

Published online by Cambridge University Press:  03 April 2025

Justin Corvino
Affiliation:
Lafayette College, Pennsylvania
Pengzi Miao
Affiliation:
University of Miami
Get access
Type
Chapter
Information
Publisher: Cambridge University Press
Print publication year: 2025

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Book purchase

Temporarily unavailable

References

Abraham, R., Marsden, J. E., and Ratiu, T., Manifolds, tensor analysis, and applications, 2nd ed., Applied Math. Sciences 75, Springer, New York, 1988.CrossRefGoogle Scholar
Adams, R. A., Sobolev spaces, Pure and Applied Math. 65, Academic Press, San Diego, 1975.Google Scholar
Agmon, S., Douglis, A., and Nirenberg, L., “Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions, II”, Comm. Pure Appl. Math. 17 (1964), 35–92.CrossRefGoogle Scholar
Alexandrov, A. D., “On the theory of mixed volumes, II: New inequalities between mixed volumes and their applications”, Mat. Sbornik (N.S.) 2 (1937), 1205–1238. In Russian.Google Scholar
Alexandrov, A. D., “On the theory of mixed volumes, III: Extension of two theorems of Minkowski on convex polyhedra to arbitrary convex surfaces”, Mat. Sbornik (N.S.) 3 (1938), 27–46. In Russian.Google Scholar
Anderson, J., Corvino, J., and Pasqualotto, F., “Multi-localized time-symmetric initial data for the Einstein vacuum equations”, J. Reine Angew. Math. 808 (2024), 67–110.Google Scholar
Andersson, L. and Moncrief, V., “Elliptic-hyperbolic systems and the Einstein equations”, Ann. Henri Poincaré 4:1 (2003), 1–34.CrossRefGoogle Scholar
Arnold, V. I., Mathematical methods of classical mechanics, Graduate Texts in Math. 60, Springer, New York, 1978.CrossRefGoogle Scholar
Arnowitt, R., Deser, S., and Misner, C. W., “Coordinate invariance and energy expressions in general relativity”, Phys. Rev. (2) 122 (1961), 997–1006.CrossRefGoogle Scholar
Arnowitt, R., Deser, S., and Misner, C. W., “The dynamics of general relativity”, pp. 227–265 in Gravitation: an introduction to current research, edited by Witten, L., Wiley, New York, 1962.Google Scholar
Ashtekar, A., Berger, B. K., Isenberg, J., and MacCallum, M. (editors), General relativity and gravitation: a centennial perspective, Cambridge Univ. Press, Cambridge, 2015.CrossRefGoogle Scholar
Axler, S., Bourdon, P., and Ramey, W., Harmonic function theory, Graduate Texts in Mathe- matics 137, Springer, 1992.CrossRefGoogle Scholar
Barbosa, J. L. and Carmo, M. do, “Stability of hypersurfaces with constant mean curvature”, Math. Z. 185:3 (1984), 339–353.CrossRefGoogle Scholar
Barbosa, J. L., do Carmo, M., and Eschenburg, J., “Stability of hypersurfaces of constant mean curvature in Riemannian manifolds”, Math. Z. 197:1 (1988), 123–138.CrossRefGoogle Scholar
Bartnik, R., “The mass of an asymptotically flat manifold”, Comm. Pure Appl. Math. 39:5 (1986), 661–693.CrossRefGoogle Scholar
Bartnik, R., “Phase space for the Einstein equations”, Comm. Anal. Geom. 13:5 (2005), 845–885.CrossRefGoogle Scholar
Bartnik, R. and Isenberg, J., “The constraint equations”, pp. 1–38 in The Einstein equations and the large scale behavior of gravitational fields, edited by P. T. Chrus´ciel and H. Friedrich, Birkhäuser, Basel, 2004.Google Scholar
Beig, R. and Murchadha, N. Ó, “The Poincaré group as the symmetry group of canonical general relativity”, Ann. Physics 174:2 (1987), 463–498.CrossRefGoogle Scholar
Beig, R., Chrus´ciel, P. T., and Schoen, R., “KIDs are non-generic”, Ann. Henri Poincaré 6:1 (2005), 155–194.CrossRefGoogle Scholar
Berger, M. and Ebin, D., “Some decompositions of the space of symmetric tensors on a Riemannian manifold”, J. Differential Geometry 3 (1969), 379–392.CrossRefGoogle Scholar
Bernal, A. N. and Sánchez, M., “Globally hyperbolic spacetimes can be defined as ‘causal’ instead of ‘strongly causal’”, Classical Quantum Gravity 24:3 (2007), 745–749.CrossRefGoogle Scholar
Besse, A. L., Einstein manifolds, Ergebnisse der Math. (3) 10, Springer, Berlin, 1987.CrossRefGoogle Scholar
Bonning, E., Marronetti, P., Neilsen, D., and Matzner, R., “Physics and initial data for multiple black hole spacetimes”, Phys. Rev. D (3) 68:4 (2003), 044019, 17.CrossRefGoogle Scholar
Bourguignon, J.-P., Ebin, D. G., and Marsden, J. E., “Sur le noyau des opérateurs pseudo- différentiels à symbole surjectif et non injectif”, C. R. Acad. Sci. Paris Sér. A-B 282:16 (1976), Aii, A867–A870.Google Scholar
Bray, H. L., The Penrose inequality in general relativity and volume comparison theorems involving scalar curvature, Ph.D. thesis, Stanford University, 1997, https://www.proquest.com/ docview/304386501. arXiv 0902.3241Google Scholar
Bray, H. L., “Proof of the Riemannian Penrose inequality using the positive mass theorem”, J. Differential Geom. 59:2 (2001), 177–267.CrossRefGoogle Scholar
Bray, H. L., “On dark matter, spiral galaxies, and the axioms of general relativity”, pp. 1–64 in Geometric analysis, mathematical relativity, and nonlinear partial differential equations, Contemp. Math. 599, Amer. Math. Soc., Providence, RI, 2013.Google Scholar
Bray, H. L. and Lee, D. A., “On the Riemannian Penrose inequality in dimensions less than eight”, Duke Math. J. 148:1 (2009), 81–106.CrossRefGoogle Scholar
Bray, H. and Morgan, F., “An isoperimetric comparison theorem for Schwarzschild space and other manifolds”, Proc. Amer. Math. Soc. 130:5 (2002), 1467–1472.Google Scholar
Bray, H. L. and Parry, A. R., “Modeling wave dark matter in dwarf spheroidal galaxies”, J. Phy. Conf. Series 615 (2015), art. id. 012001.Google Scholar
Brendle, S., “Constant mean curvature surfaces in warped product manifolds”, Publ. Math. Inst. Hautes Études Sci. 117 (2013), 247–269.CrossRefGoogle Scholar
Brendle, S. and Eichmair, M., “Isoperimetric and Weingarten surfaces in the Schwarzschild manifold”, J. Differential Geom. 94:3 (2013), 387–407.CrossRefGoogle Scholar
Brendle, S. and Eichmair, M., “Large outlying stable constant mean curvature spheres in initial data sets”, Invent. Math. 197:3 (2014), 663–682.CrossRefGoogle Scholar
Brendle, S. and Marques, F. C., “Scalar curvature rigidity of geodesic balls in Sn”, J. Differential Geom. 88:3 (2011), 379–394.CrossRefGoogle Scholar
Brendle, S., Marques, F. C., and Neves, A., “Deformations of the hemisphere that increase scalar curvature”, Invent. Math. 185:1 (2011), 175–197.CrossRefGoogle Scholar
Bunting, G. L. and Masood-ul-Alam, A. K. M., “Nonexistence of multiple black holes in asymptotically Euclidean static vacuum space-time”, Gen. Relativity Gravitation 19:2 (1987), 147–154.CrossRefGoogle Scholar
Cai, M. and Galloway, G. J., “Rigidity of area minimizing tori in 3-manifolds of nonnegative scalar curvature”, Comm. Anal. Geom. 8:3 (2000), 565–573.CrossRefGoogle Scholar
Callahan, J. J., The geometry of spacetime: an introduction to special and general relativity, Springer, New York, 2000.CrossRefGoogle Scholar
Cantor, M., “Elliptic operators and the decomposition of tensor fields”, Bull. Amer. Math. Soc. (N.S.) 5:3 (1981), 235–262.CrossRefGoogle Scholar
Carfora, M. and Marzuoli, A., Einstein constraints and Ricci flow: a geometrical averaging of initial data sets, Springer, Singapore, 2023.CrossRefGoogle Scholar
do Carmo, M. P., Riemannian geometry, Birkhäuser, Boston, 1992.CrossRefGoogle Scholar
Carroll, S., Spacetime and geometry: An introduction to general relativity, Addison Wesley, San Francisco, 2004.Google Scholar
Cederbaum, C. and Nerz, C., “Explicit Riemannian manifolds with unexpectedly behaving center of mass”, Ann. Henri Poincaré 16:7 (2015), 1609–1631.CrossRefGoogle Scholar
Chaljub-Simon, A., “Systèmes elliptiques linéaires dans des espaces de fonctions höldériennes à poids”, Rend. Circ. Mat. Palermo (2) 30:2 (1981), 300–310.CrossRefGoogle Scholar
Chaljub-Simon, A. and Choquet-Bruhat, Y., “Problèmes elliptiques du second ordre sur une variété euclidienne à l’infini”, Ann. Fac. Sci. Toulouse Math. (5) 1:1 (1979), 9–25.CrossRefGoogle Scholar
Chan, P.-Y. and Tam, L.-F., “A note on center of mass”, Comm. Anal. Geom. 24:3 (2016), 471–486.CrossRefGoogle Scholar
Chen, P.-N., Huang, L.-H., Wang, M.-T., and Yau, S.-T., “On the validity of the definition of angular momentum in general relativity”, Ann. Henri Poincaré 17:2 (2016), 253–270.CrossRefGoogle Scholar
Chen, P.-N., Wang, M.-T., and Yau, S.-T., “Conserved quantities in general relativity: from the quasi-local level to spatial infinity”, Comm. Math. Phys. 338:1 (2015), 31–80.CrossRefGoogle Scholar
Choquet-Bruhat, Y. (as Fourès-Bruhat), “Théorème d’existence pour certains systèmes d’équations aux dérivées partielles non linéaires”, Acta Math. 88 (1952), 141–225.Google Scholar
Choquet-Bruhat, Y., “New elliptic system and global solutions for the constraints equations in general relativity”, Comm. Math. Phys. 21 (1971), 211–218.CrossRefGoogle Scholar
Choquet-Bruhat, Y., General relativity and the Einstein equations, Oxford Univ. Press, 2009.Google Scholar
Choquet-Bruhat, Y. and York, J. W., Jr., “The Cauchy problem”, pp. 99–172 in General relativity and gravitation, vol. 1, edited by Held, A., Press, Plenum, York, New, 1980.Google Scholar
Chow, B. and Gulliver, R., “Aleksandrov reflection and geometric evolution of hypersurfaces”, Comm. Anal. Geom. 9:2 (2001), 261–280.CrossRefGoogle Scholar
Christodoulou, D., “Global solutions of nonlinear hyperbolic equations for small initial data”, Comm. Pure Appl. Math. 39:2 (1986), 267–282.CrossRefGoogle Scholar
Christodoulou, D. and Klainerman, S., The global nonlinear stability of the Minkowski space, Princeton Mathematical Series 41, Princeton University Press, 1993.Google Scholar
Christodoulou, D. and Murchadha, N. Ó, “The boost problem in general relativity”, Comm. Math. Phys. 80:2 (1981), 271–300.CrossRefGoogle Scholar
Chrus´ciel, P. T., “Boundary conditions at spatial infinity from a Hamiltonian point of view”, pp. 49–59 in Topological properties and global structure of space-time (Erice, 1985), NATO Adv. Sci. Inst. Ser. B Phys. 138, Plenum, New York, 1986.Google Scholar
Chrus´ciel, P. T., “Mass and angular-momentum inequalities for axi-symmetric initial data sets. I. Positivity of mass”, Ann. Physics 323:10 (2008), 2566–2590.Google Scholar
Chrus´ciel, P. T., “Elements of causality theory”, preprint, 2011. arXiv 1110.6706v1Google Scholar
Chrus´ciel, P. T. and Costa, J. L., “Mass, angular-momentum and charge inequalities for axisymmetric initial data”, Classical Quantum Gravity 26:23 (2009), 235013, 7.CrossRefGoogle Scholar
Chrus´ciel, P. T. and Delay, E., On mapping properties of the general relativistic constraints operator in weighted function spaces, with applications, Mém. Soc. Math. Fr. (N.S.) 94, 2003.Google Scholar
Chrus´ciel, P. T. and Grant, J. D. E., “On Lorentzian causality with continuous metrics”, Classical Quantum Gravity 29:14 (2012), art. id. 145001.CrossRefGoogle Scholar
Chrus´ciel, P. T., Li, Y., and Weinstein, G., “Mass and angular-momentum inequalities for axi-symmetric initial data sets. II. Angular momentum”, Ann. Physics 323:10 (2008), 2591–2613.Google Scholar
Chrus´ciel, P. T., Galloway, G. J., and Pollack, D., “Mathematical general relativity: a sampler”, Bull. Amer. Math. Soc. (N.S.) 47:4 (2010), 567–638.CrossRefGoogle Scholar
Cook, G. B., “Initial data for numerical relativity”, Living Rev. Relativ. 3 (2000), art. id. 2000–5.CrossRefGoogle ScholarPubMed
Corvino, J., “Scalar curvature deformation and a gluing construction for the Einstein constraint equations”, Comm. Math. Phys. 214:1 (2000), 137–189.CrossRefGoogle Scholar
Corvino, J. and Huang, L.-H., “Localized deformation for initial data sets with the dominant energy condition”, Calc. Var. Partial Differential Equations 59:1 (2020), Paper No. 42, 43.CrossRefGoogle Scholar
Corvino, J. and Pollack, D., “Scalar curvature and the Einstein constraint equations”, pp. 145–188 in Surveys in geometric analysis and relativity, Adv. Lect. Math. (ALM) 20, International Press, Somerville, MA, 2011.Google Scholar
Corvino, J. and Schoen, R. M., “On the asymptotics for the vacuum Einstein constraint equations”, J. Differential Geom. 73:2 (2006), 185–217.CrossRefGoogle Scholar
Corvino, J., Eichmair, M., and Miao, P., “Deformation of scalar curvature and volume”, Math. Ann. 357:2 (2013), 551–584.CrossRefGoogle Scholar
Dain, S., “Proof of the angular momentum-mass inequality for axisymmetric black holes”, J. Differential Geom. 79:1 (2008), 33–67.CrossRefGoogle Scholar
De Lellis, C. and Müller, S., “Optimal rigidity estimates for nearly umbilical surfaces”, J. Differential Geom. 69:1 (2005), 75–110.CrossRefGoogle Scholar
DeTurck, D. M., “Existence of metrics with prescribed Ricci curvature: local theory”, Invent. Math. 65:1 (1981/82), 179–207.CrossRefGoogle Scholar
DeTurck, D. M., “Deforming metrics in the direction of their Ricci tensors”, J. Differential Geom. 18:1 (1983), 157–162.CrossRefGoogle Scholar
DeWitt, B. S., “Quantum theory of gravity, I: The canonical theory”, Phys. Rev. 160:5 (1967), 1113–1148.CrossRefGoogle Scholar
Douglis, A. and Nirenberg, L., “Interior estimates for elliptic systems of partial differential equations”, Comm. Pure Appl. Math. 8 (1955), 503–538.CrossRefGoogle Scholar
Eichmair, M., “The Jang equation reduction of the spacetime positive energy theorem in dimensions less than eight”, Comm. Math. Phys. 319:3 (2013), 575–593.CrossRefGoogle Scholar
Eichmair, M., Huang, L.-H., Lee, D. A., and Schoen, R., “The spacetime positive mass theorem in dimensions less than eight”, J. Eur. Math. Soc. 18:1 (2016), 83–121.Google Scholar
Eichmair, M. and Metzger, J., “On large volume preserving stable CMC surfaces in initial data sets”, J. Differential Geom. 91:1 (2012), 81–102.CrossRefGoogle Scholar
Eichmair, M. and Metzger, J., “Large isoperimetric surfaces in initial data sets”, J. Differential Geom. 94:1 (2013), 159–186.CrossRefGoogle Scholar
Eichmair, M. and Metzger, J., “Unique isoperimetric foliations of asymptotically flat manifolds in all dimensions”, Invent. Math. 194:3 (2013), 591–630.CrossRefGoogle Scholar
Einstein, A., “Zur Elektrodynamik bewegter Körper”, Ann. der Physik 17 (1905), 891–921. Translated as “On the electrodynamics of moving bodies”, pp. 35–65 in The principle of relativity: a collection of original memoirs on the special and general theory of relativity, Methuen, London, 1923; reprinted Dover, New York, 1952. Essentially the same text is posted at http://www.fourmilab.ch/etexts/einstein/specrel/www/. A new translation appeared in pp. 123–160 of [212].Google Scholar
Einstein, A., “Die Grundlage der allgemeinen Relativitätstheorie”, Ann. der Physik 49:7 (1916), 769–822. Translated as “The foundation of the general theory of relativity”, pp. 109–164 in The principle of relativity: a collection of original memoirs on the special and general theory of relativity, Methuen, London, 1923; reprinted Dover, New York, 1952.Google Scholar
Einstein, A., Relativity: the special and the general theory, Methuen, London, 1920. Reprinted by Crown Publishers, New York, 1961.Google Scholar
Einstein, A., The meaning of relativity, Princeton Univ. Press, Princeton, 1922.Google Scholar
Evans, L. C., Partial differential equations, Graduate Studies in Math. 19, Amer. Math. Soc., Providence, RI, 1998.Google Scholar
Everitt, C. F. W. et al., “The Gravity Probe B test of general relativity”, Classical Quantum Gravity 32:22 (2015), art. id. 224001.CrossRefGoogle Scholar
Fischer, A. E. and Marsden, J. E., “Linearization stability of the Einstein equations”, Bull. Amer. Math. Soc. 79 (1973), 997–1003.CrossRefGoogle Scholar
Fischer, A. E. and Marsden, J. E., “Deformations of the scalar curvature”, Duke Math. J. 42:3 (1975), 519–547.CrossRefGoogle Scholar
Fischer, A. E. and Marsden, J. E., The initial value problem and the dynamical formulation of general relativity, edited by Hawking, S. W. and Israel, W., Univ, Cambridge. Press, Cambridge, 1979.Google Scholar
Fischer, A. E. and Wolf, J. A., “The structure of compact Ricci-flat Riemannian manifolds”, J. Differential Geometry 10 (1975), 277–288.CrossRefGoogle Scholar
Fischer-Colbrie, D. and Schoen, R., “The structure of complete stable minimal surfaces in 3-manifolds of nonnegative scalar curvature”, Comm. Pure Appl. Math. 33:2 (1980), 199–211.CrossRefGoogle Scholar
Folland, G. B., Introduction to partial differential equations, 2nd ed., Princeton University Press, 1995.Google Scholar
Frankel, T., Gravitational curvature: An introduction to Einstein’s theory, W. H. Freeman, San Francisco, 1979.Google Scholar
Freire, A. and Schwartz, F., “Mass-capacity inequalities for conformally flat manifolds with boundary”, Comm. PDE 39 (2014), 98–119.CrossRefGoogle Scholar
French, A. P., Special relativity, W. W. Norton, New York, 1968.Google Scholar
Friedrich, H., “On the existence of n-geodesically complete or future complete solutions of Einstein’s field equations with smooth asymptotic structure”, Comm. Math. Phys. 107:4 (1986), 587–609.CrossRefGoogle Scholar
Friedrichs, K. O., “The identity of weak and strong extensions of differential operators”, Trans. Amer. Math. Soc. 55 (1944), 132–151.CrossRefGoogle Scholar
Galloway, G. J., “Least area tori, black holes and topological censorship”, pp. 113–123 in Differential geometry and mathematical physics (Vancouver, 1993), edited by Beem, J. K. and Duggal, K. L., Contemp. Math. 170, Amer. Math. Soc., Providence, RI, 1994.Google Scholar
Galloway, G. J., “Stability and rigidity of extremal surfaces in Riemannian geometry and general relativity”, pp. 221–239 in Surveys in geometric analysis and relativity, edited by Bray, H. L. and Minicozzi, W. P., Adv. Lect. Math. (ALM) 20, International Press, Somerville, MA, 2011.Google Scholar
Galloway, G. J., Notes on Lorentzian causality: ESI-EMS-IAMP Summer School on Mathematical Relativity (Vienna, 2014), 2014. http://hdl.handle.net/10385/2167.Google Scholar
Gerhardt, C., “Flow of nonconvex hypersurfaces into spheres”, J. Diff. Geom. 32 (1990), 299–314.Google Scholar
Geroch, R., “What is a singularity in general relativity?”, Ann. Phys. 48:3 (1968), 526–540.CrossRefGoogle Scholar
Geroch, R., “Domain of dependence”, J. Mathematical Phys. 11 (1970), 437–449.CrossRefGoogle Scholar
Geroch, R., “Energy extraction”, Ann. New York Acad. Sci. 224 (1973), 108–117.CrossRefGoogle Scholar
Gibbons, G. W., “The time symmetric initial value problem for black holes”, Comm. Math. Phys. 27 (1972), 87–102.CrossRefGoogle Scholar
Gilbarg, D. and Trudinger, N. S., Elliptic partial differential equations of second order, 2nd ed., Grundlehren der Math. Wiss. 224, Springer, Berlin, 1983.Google Scholar
Giusti, E., Minimal surfaces and functions of bounded variation, Monographs in Math. 80, Birkhäuser, Basel, 1984.CrossRefGoogle Scholar
Gray, A., Tubes, Addison-Wesley, Redwood City, CA, 1990.Google Scholar
Grøn, O., “Space geometry in rotating frames: a historical appraisal”, pp. 285–334 in Relativity in rotating frames: relativistic physics in rotating reference frames, edited by Rizzi, G. and Ruggiero, M. L., Kluwer, Dordrecht, 2004.Google Scholar
Han, Q., A basic course in partial differential equations, Graduate Studies in Mathematics 120, American Mathematical Society, 2011.Google Scholar
Hawking, S. W. and Ellis, G. F. R., The large scale structure of space-time, Cambridge Monog. Math. Phys. 1, Cambridge Univ. Press, London, 1973.CrossRefGoogle Scholar
Hawking, S. W. and Horowitz, G. T., “The gravitational Hamiltonian, action, entropy and surface terms”, Classical Quantum Gravity 13:6 (1996), 1487–1498.CrossRefGoogle Scholar
Hörmander, L., “Pseudo-differential operators and non-elliptic boundary problems”, Ann. of Math. (2) 83 (1966), 129–209.CrossRefGoogle Scholar
Hörmander, L., The analysis of linear partial differential operators, I: Distribution theory and Fourier analysis, Grundlehren der Math. Wiss. 256, Springer, 1990.Google Scholar
Huang, L.-H., “On the center of mass of isolated systems with general asymptotics”, Classical Quantum Gravity 26:1 (2009), 015012, 25.CrossRefGoogle Scholar
Huang, L.-H., “Foliations by stable spheres with constant mean curvature for isolated systems with general asymptotics”, Comm. Math. Phys. 300:2 (2010), 331–373.CrossRefGoogle Scholar
Huang, L.-H., “Solutions of special asymptotics to the Einstein constraint equations”, Classical Quantum Gravity 27:24 (2010), 245002, 10.Google Scholar
Huang, L.-H., “On the center of mass in general relativity”, pp. 575–591 in Fifth International Congress of Chinese Mathematicians, AMS/IP Stud. Adv. Math. 51, pt. 1, Amer. Math. Soc., 2012.Google Scholar
Huang, L.-H. and Lee, D. A., “Equality in the spacetime positive mass theorem”, Comm. Math. Phys. 376:3 (2020), 2379–2407.CrossRefGoogle Scholar
Huang, L.-H. and Wu, D., “The equality case of the Penrose inequality for asymptotically flat graphs”, Trans. Amer. Math. Soc. 367:1 (2015), 31–47.Google Scholar
Huang, L.-H., Schoen, R., and Wang, M.-T., “Specifying angular momentum and center of mass for vacuum initial data sets”, Comm. Math. Phys. 306:3 (2011), 785–803.CrossRefGoogle Scholar
Huisken, G., “Flow by mean curvature of convex surfaces into spheres”, J. Differential Geom. 20:1 (1984), 237–266.Google Scholar
Huisken, G., “The volume preserving mean curvature flow”, J. Reine Angew. Math. 382 (1987), 35–48.Google Scholar
Huisken, G. and Ilmanen, T., “The inverse mean curvature flow and the Riemannian Penrose inequality”, J. Differential Geom. 59:3 (2001), 353–437.CrossRefGoogle Scholar
Huisken, G. and Yau, S.-T., “Definition of center of mass for isolated physical systems and unique foliations by stable spheres with constant mean curvature”, Invent. Math. 124:1-3 (1996), 281–311.CrossRefGoogle Scholar
Isenberg, J., “Constant mean curvature solutions of the Einstein constraint equations on closed manifolds”, Classical Quantum Gravity 12:9 (1995), 2249–2274.CrossRefGoogle Scholar
Isenberg, J. and Moncrief, V., “A set of nonconstant mean curvature solutions of the Einstein constraint equations on closed manifolds”, Classical Quantum Gravity 13:7 (1996), 1819–1847.CrossRefGoogle Scholar
Isenberg, J., Mazzeo, R., and Pollack, D., “On the topology of vacuum spacetimes”, Ann. Henri Poincaré 4:2 (2003), 369–383.CrossRefGoogle Scholar
Jang, P. S., “On the positive energy conjecture”, J. Mathematical Phys. 17:1 (1976), 141–145.CrossRefGoogle Scholar
Jang, P. S. and Wald, R. M., “The positive energy conjecture and the cosmic censor hypothesis”, J. Math. Phys. 18 (1977), 41–44.CrossRefGoogle Scholar
Kaluza, T., “Zur Relativitätstheorie”, Physikal. Z. 11 (1910), 977–978. Translated in https://en.wikisource.org/wiki/Translation:On_the_Theory_of_Relativity_(Kaluza).Google Scholar
Kapouleas, N., “Constant mean curvature surfaces constructed by fusing Wente tori”, Invent. Math. 119:3 (1995), 443–518.CrossRefGoogle Scholar
Kazdan, J. L. and Warner, F. W., “A direct approach to the determination of Gaussian and scalar curvature functions”, Invent. Math. 28 (1975), 227–230.CrossRefGoogle Scholar
Kazdan, J. L. and Warner, F. W., “Existence and conformal deformation of metrics with prescribed Gaussian and scalar curvatures”, Ann. of Math. (2) 101 (1975), 317–331.CrossRefGoogle Scholar
Kazdan, J. L. and Warner, F. W., “Prescribing curvatures”, pp. 309–319 in Differential geometry (Stanford, 1973), vol. 2, Proc. Sympos. Pure Math. 27, 1975.CrossRefGoogle Scholar
Kleppner, D. and Kolenkow, R. J., An introduction to mechanics, McGraw Hill, New York, 1973.Google Scholar
Lam, M.-K. G., “The graph cases of the Riemannian positive mass and Penrose inequalities in all dimensions”, preprint, 2010. arXiv 1010.4256v1Google Scholar
Lawson, H. B., Jr. and Michelsohn, M.-L., Spin geometry, Princeton Math. Series 38, Princeton Univ. Press, 1989.Google Scholar
Lee, J. M., Riemannian manifolds: an introduction to curvature, Graduate Texts in Math. 176, Springer, New York, 1997.CrossRefGoogle Scholar
Lee, J. M., Introduction to smooth manifolds, Graduate Texts in Math. 218, Springer, New York, 2003.CrossRefGoogle Scholar
Lee, D. A., Geometric relativity, Graduate Studies in Mathematics 201, Amer. Math. Soc., Providence, RI, 2019.CrossRefGoogle Scholar
Lee, J. M. and Parker, T. H., “The Yamabe problem”, Bull. Amer. Math. Soc. (N.S.) 17:1 (1987), 37–91.Google Scholar
Leoni, G., A first course in Sobolev spaces, Graduate Studies in Math. 105, Amer. Math. Soc., Providence, RI, 2009.Google Scholar
Lichnerowicz, A., “L’intégration des équations de la gravitation relativiste et le problème des n corps”, J. Math. Pures Appl. (9) 23 (1944), 37–63.Google Scholar
Lohkamp, J., “Scalar curvature and hammocks”, Math. Ann. 313:3 (1999), 385–407.CrossRefGoogle Scholar
Lohkamp, J., “The higher dimensional positive mass theorem, II”, preprint, 2017. arXiv 1612.07505v2Google Scholar
Lovelock, D. and Rund, H., Tensor, differential forms, and variational principles, Wiley, New York, 1975.Google Scholar
Marquardt, T., The inverse mean curvature flow for hypersurfaces with boundary, Ph.D thesis, Freie Universität Berlin, 2012.Google Scholar
Mars, M., “Present status of the Penrose inequality”, Proc. Amer. Math. Soc. 132:1 (2004), 217–222.Google Scholar
Maxwell, D., “Rough solutions of the Einstein constraint equations on compact manifolds”, J. Hyperbolic Differ. Equ. 2:2 (2005), 521–546.Google Scholar
Maxwell, D., “Solutions of the Einstein constraint equations with apparent horizon boundaries”, Comm. Math. Phys. 253:3 (2005), 561–583.CrossRefGoogle Scholar
Maxwell, D., “The conformal method and the conformal thin-sandwich method are the same”, Classical Quantum Gravity 31:14 (2014), 145006, 34.CrossRefGoogle Scholar
Maxwell, D., “Initial data in general relativity described by expansion, conformal deformation and drift”, Comm. Anal. Geom. 29:1 (2021), 207–281.CrossRefGoogle Scholar
McOwen, R. C., “The behavior of the Laplacian on weighted Sobolev spaces”, Comm. Pure Appl. Math. 32:6 (1979), 783–795.CrossRefGoogle Scholar
Metzger, J., “Foliations of asymptotically flat 3-manifolds by 2-surfaces of prescribed mean curvature”, J. Differential Geom. 77:2 (2007), 201–236.Google Scholar
Meyers, N., “An expansion about infinity for solutions of linear elliptic equations”, J. Math. Mech. 12 (1963), 247–264.Google Scholar
Miao, P., “Quasi-local mass via isometric embeddings: a review from a geometric perspective”, Classical Quantum Gravity 32:23 (2015), art. id. 233001.CrossRefGoogle Scholar
Miao, P. and Tam, L.-F., “Evaluation of the ADM mass and center of mass via the Ricci tensor”, Proc. Amer. Math. Soc. 144:2 (2016), 753–761.Google Scholar
Mirandola, H. and Vitório, F., “The positive mass theorem and Penrose inequality for graphical manifolds”, Comm. Anal. Geom. 23:2 (2015), 273–292.CrossRefGoogle Scholar
Misner, C. W., Thorne, K. S., and Wheeler, J. A., Gravitation, Freeman, San Francisco, 1973.Google Scholar
Møller, C., The theory of relativity, Clarendon Press, Oxford, 1952.Google Scholar
Moncrief, V., “Spacetime symmetries and linearization stability of the Einstein equations, I”, J. Mathematical Phys. 16 (1975), 493–498.CrossRefGoogle Scholar
Montiel, S. and Ros, A., Curves and surfaces, vol. 69, Second ed., Graduate Studies in Mathematics, American Mathematical Society, Providence, RI; Real Sociedad Matemática Española, Madrid, 2009. Translated from the 1998 Spanish original by Montiel and edited by Babbitt, Donald.Google Scholar
Munkres, J. R., Elements of algebraic topology (Menlo Park, CA), Addison-Wesley, 1984.Google Scholar
Nerz, C., “Foliations by stable spheres with constant mean curvature for isolated systems without asymptotic symmetry”, Calc. Var. Partial Differential Equations 54:2 (2015), 1911–1946.CrossRefGoogle Scholar
Neves, A., “Insufficient convergence of inverse mean curvature flow on asymptotically hyberbolic manifolds”, J. Differential Geom. 84:1 (2010), 191–229.CrossRefGoogle Scholar
Nirenberg, L. and Walker, H. F., “The null spaces of elliptic partial differential operators in Rn”, J. Math. Anal. Appl. 42 (1973), 271–301. Collection of articles dedicated to Salomon Bochner.Google Scholar
Norton, J., “What was Einstein’s principle of equivalence?”, Stud. Hist. Philos. Sci. 16:3 (1985), 203–246.CrossRefGoogle Scholar
Norton, J. D., “General covariance and the foundations of general relativity: eight decades of dispute”, Rep. Progr. Phys. 56:7 (1993), 791–858.CrossRefGoogle Scholar
Norton, J. D., “Mach’s principle before Einstein”, pp. 9–57 in Mach’s principle: from Newton’s bucket to quantum gravity, edited by Barbour, J. and Pfister, H., Einstein Studies 6, Birkhäuser, 1995.Google Scholar
Murchadha, N. Ó and York, J. W., Jr., “Initial-value problem of general relativity, I: General formulation and physical interpretation”, Phys. Rev. D (3) 10 (1974), 428–436.CrossRefGoogle Scholar
Obata, M., “Certain conditions for a Riemannian manifold to be isometric with a sphere”, J. Math. Soc. Japan 14 (1962), 333–340.CrossRefGoogle Scholar
O’Neill, B., Semi-Riemannian geometry, with applications to relativity, Pure and Applied Math. 103, Academic Press, New York, 1983.Google Scholar
Parker, T. and Taubes, C. H., “On Witten’s proof of the positive energy theorem”, Comm. Math. Phys. 84:2 (1982), 223–238.CrossRefGoogle Scholar
Penrose, R., “Asymptotic properties of fields and space-times”, Phys. Rev. Lett. 10 (1963), 66–68.CrossRefGoogle Scholar
Penrose, R., “Gravitational collapse and space-time singularities”, Phys. Rev. Lett. 14 (1965), 57–59.CrossRefGoogle Scholar
Penrose, R., “Gravitational collapse: the role of general relativity”, Nuovo Cimento 1:(numero speciale) (1969), 252–276.Google Scholar
Penrose, R., Techniques of differential topology in relativity, CBMS Regional Conf. Series Appl. Math. 7, Society for Industrial and Applied Mathematics, Philadelphia, 1972.CrossRefGoogle Scholar
Penrose, R., “Naked singularities”, Ann. New York Acad. Sci. 224 (1973), 125–134.CrossRefGoogle Scholar
Penrose, R., “Some unresolved problems in classical general relativity”, pp. 631–668 in Seminar on Differential Geometry, edited by Yau, S.-T., Annals of Math. Stud. 102, Princeton University Press, Princeton, NJ, 1982.Google Scholar
Petersen, P., Riemannian geometry, 2nd ed., Graduate Texts in Math. 171, Springer, 2006.Google Scholar
Pólya, G. and Szegö, G., Isoperimetric inequalities in mathematical physics, Annals of Mathematics Studies 27, Princeton University Press, Princeton, NJ, 1951.Google Scholar
Qing, J. and Tian, G., “On the uniqueness of the foliation of spheres of constant mean curvature in asymptotically flat 3-manifolds”, J. Amer. Math. Soc. 20:4 (2007), 1091–1110 (electronic).CrossRefGoogle Scholar
Qing, J. and Yuan, W., “On scalar curvature rigidity of vacuum static spaces”, Math. Ann. 365:3-4 (2016), 1257–1277.CrossRefGoogle Scholar
Qiu, W., “Interior regularity of solutions to the isotropically constrained Plateau problem”, Comm. Anal. Geom. 11:5 (2003), 945–986.CrossRefGoogle Scholar
Regge, T. and Teitelboim, C., “Role of surface integrals in the Hamiltonian formulation of general relativity”, Ann. Physics 88 (1974), 286–318.CrossRefGoogle Scholar
Resnick, R., Introduction to special relativity, Wiley, 1968.Google Scholar
Rindler, W., Introduction to special relativity, 2nd ed., Oxford Univ. Press, New York, 1991.Google Scholar
Ringström, H., The Cauchy problem in general relativity, European Mathematical Society, Zürich, 2009. Errata at https://people.kth.se/∼hansr/errata.html.CrossRefGoogle Scholar
Ringström, H., On the topology and future stability of the universe, Oxford Univ. Press, 2013.CrossRefGoogle Scholar
Robertson, H. P., “Postulate versus observation in the special theory of relativity”, Rev. Modern Phys. 21 (1949), 378–382.CrossRefGoogle Scholar
Royden, H. L., Real analysis, 3rd ed., Macmillan, New York, 1988.Google Scholar
Rudin, W., Real and complex analysis, 3rd ed., McGraw-Hill, New York, 1987.Google Scholar
Rudin, W., Functional analysis, 2nd ed., McGraw-Hill, New York, 1991.Google Scholar
Schoen, R., “Conformal deformation of a Riemannian metric to constant scalar curvature”, J. Differential Geom. 20:2 (1984), 479–495.CrossRefGoogle Scholar
Schoen, R. M., “Variational theory for the total scalar curvature functional for Riemannian metrics and related topics”, pp. 120–154 in Topics in calculus of variations (Montecatini Terme, 1987), Lecture Notes in Math. 1365, Springer, 1989.Google Scholar
Schoen, R. and Yau, S. T., “Existence of incompressible minimal surfaces and the topology of three-dimensional manifolds with nonnegative scalar curvature”, Ann. of Math. (2) 110:1 (1979), 127–142.CrossRefGoogle Scholar
Schoen, R. and Yau, S. T., “On the proof of the positive mass conjecture in general relativity”, Comm. Math. Phys. 65:1 (1979), 45–76.CrossRefGoogle Scholar
Schoen, R. and Yau, S. T., “On the structure of manifolds with positive scalar curvature”, Manuscripta Math. 28:1-3 (1979), 159–183.CrossRefGoogle Scholar
Schoen, R. M. and Yau, S. T., “Complete manifolds with nonnegative scalar curvature and the positive action conjecture in general relativity”, Proc. Nat. Acad. Sci. U.S.A. 76:3 (1979), 1024–1025.CrossRefGoogle ScholarPubMed
Schoen, R. and Yau, S. T., “The energy and the linear momentum of space-times in general relativity”, Comm. Math. Phys. 79:1 (1981), 47–51.CrossRefGoogle Scholar
Schoen, R. and Yau, S. T., “Proof of the positive mass theorem, II”, Comm. Math. Phys. 79:2 (1981), 231–260.CrossRefGoogle Scholar
Schoen, R. and Yau, S.-T., Lectures on differential geometry, Conference Proceedings and Lecture Notes in Geometry and Topology, I, International Press, Cambridge, MA, 1994.Google Scholar
Schoen, R. and Yau, S.-T., “Positive scalar curvature and minimal hypersurface singularities”, pp. 441–480 in Surveys in differential geometry, 2019: Differential geometry, Calabi–Yau theory, and general relativity, vol. 2, Surv. Differ. Geom. 24, International Press, Boston, 2022. arXiv 1704.05490Google Scholar
Schoen, R. and Zhou, X., “Convexity of reduced energy and mass angular momentum inequalities”, Ann. Henri Poincaré 14:7 (2013), 1747–1773.CrossRefGoogle Scholar
Schutz, B. F., A first course in general relativity, Cambridge Univ. Press, Cambridge, 1990.Google Scholar
Senovilla, J. M. M. and Garfinkle, D., “The 1965 Penrose singularity theorem”, Classical Quantum Gravity 32:12 (2015), 124008, 45.CrossRefGoogle Scholar
Simon, L., Theorems on regularity and singularity of energy minimizing maps, Birkhäuser, Basel, 1996.CrossRefGoogle Scholar
Simon, L., “Schauder estimates by scaling”, Calc. Var. Partial Differential Equations 5:5 (1997), 391–407.CrossRefGoogle Scholar
Smith, B. and Weinstein, G., “Quasiconvex foliations and asymptotically flat metrics of non-negative scalar curvature”, Comm. Anal. Geom. 12:3 (2004), 511–551.Google Scholar
Stachel, J., Einstein’s miraculous year: five papers that changed the face of physics, Princeton Univ. Press, 1998.Google Scholar
Streets, J. D., “Quasi-local mass functionals and generalized inverse mean curvature flow”, Comm. Anal. Geom. 16:3 (2008), 495–537.CrossRefGoogle Scholar
Taylor, M. E., Partial differential equations, III: Nonlinear equations, Applied Math. Sciences 117, Springer, New York, 1997.Google Scholar
Topping, P., “Relating diameter and mean curvature for submanifolds of Euclidean space”, Comment. Math. Helv. 83:3 (2008), 539–546.CrossRefGoogle Scholar
Urbas, J., “On the expansion of starshaped hypersurfaces by symmetric functions of their principal curvatures”, Math. Z. 205 (1990), 355–372.CrossRefGoogle Scholar
Vick, J. W., Homology theory: an introduction to algebraic topology, Pure and Applied Math. 53, Academic Press, New York, 1973.Google Scholar
Wald, R. M., General relativity, University of Chicago Press, 1984.CrossRefGoogle Scholar
Warner, F. W., Foundations of differentiable manifolds and Lie groups, Graduate Texts in Mathematics 94, Springer-Verlag, New York-Berlin, 1983. Corrected reprint of the 1971 edition.CrossRefGoogle Scholar
Wasserman, R. H., Tensors and manifolds, with applications to mechanics and relativity, Oxford Univ. Press, New York, 1992.Google Scholar
Weinberg, S., Gravitation and cosmology, Wiley, New York, 1972.Google Scholar
Wente, H. C., “Counterexample to a conjecture of H. Hopf”, Pacific J. Math. 121:1 (1986), 193–243.CrossRefGoogle Scholar
Willmore, T., Total curvature in Riemannian geometry, Wiley, 1982.Google Scholar
Witten, E., “A new proof of the positive energy theorem”, Comm. Math. Phys. 80:3 (1981), 381–402.CrossRefGoogle Scholar
Ye, R., “Foliation by constant mean curvature spheres on asymptotically flat manifolds”, pp. 369–383 in Geometric analysis and the calculus of variations, International Press, Cambridge, MA, 1996.Google Scholar
York, J. W., Jr., “Gravitational degrees of freedom and the initial-value problem”, Phys. Rev. Lett. 26 (1971), 1656–1658.CrossRefGoogle Scholar
York, J. W., Jr., “Role of conformal three-geometry in the dynamics of gravitation”, Phys. Rev. Lett. 28:16 (1972), 1082–1085.CrossRefGoogle Scholar
York, J. W., Jr., “Conformally invariant orthogonal decomposition of symmetric tensors on Riemannian manifolds and the initial-value problem of general relativity”, J. Mathematical Phys. 14 (1973), 456–464.CrossRefGoogle Scholar
York, J. W., Jr., “Boundary terms in the action principles of general relativity”, Found. Phys. 16:3 (1986), 249–257.CrossRefGoogle Scholar
Yuan, W., “Brown-York mass and compactly supported conformal deformations of scalar curvature”, J. Geom. Anal. 27:1 (2017), 797–816.CrossRefGoogle Scholar
Zhang, X., “Angular momentum and positive mass theorem”, Comm. Math. Phys. 206:1 (1999), 137–155.CrossRefGoogle Scholar
Zhou, X., “Mass angular momentum inequality for axisymmetric vacuum data with small trace”, Comm. Anal. Geom. 22:3 (2014), 519–571.CrossRefGoogle Scholar

Save book to Kindle

To save this book to your Kindle, first ensure [email protected] is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Available formats
×

Save book to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

Available formats
×