The Yamabe problem
HTML articles powered by AMS MathViewer
- by John M. Lee and Thomas H. Parker PDF
- Bull. Amer. Math. Soc. 17 (1987), 37-91
References
- R. Arnowitt, S. Deser, and C. W. Misner, Dynamical structure and definition of energy in general relativity, Phys. Rev. (2) 116 (1959), 1322–1330. MR 113667, DOI 10.1103/PhysRev.116.1322
- 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, DOI 10.1103/PhysRev.118.1100
- 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, DOI 10.1103/PhysRev.122.997
- Thierry Aubin, Problèmes isopérimétriques et espaces de Sobolev, J. Differential Geometry 11 (1976), no. 4, 573–598 (French). MR 448404
- 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
- Thierry Aubin, Nonlinear analysis on manifolds. Monge-Ampère equations, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 252, Springer-Verlag, New York, 1982. MR 681859, DOI 10.1007/978-1-4612-5734-9
- Patricio Aviles and Robert C. McOwen, Conformal deformation to constant negative scalar curvature on noncompact Riemannian manifolds, J. Differential Geom. 27 (1988), no. 2, 225–239. MR 925121
- 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
- Haïm Brézis and Elliott Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc. 88 (1983), no. 3, 486–490. MR 699419, DOI 10.1090/S0002-9939-1983-0699419-3
- Haïm Brézis and Louis Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math. 36 (1983), no. 4, 437–477. MR 709644, DOI 10.1002/cpa.3160360405
- 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
- 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, DOI 10.5802/afst.527
- 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
- Luther Pfahler Eisenhart, Riemannian Geometry, Princeton University Press, Princeton, N. J., 1949. 2d printing. MR 0035081
- G. W. Gibbons, S. W. Hawking, and M. J. Perry, Path integrals and the indefiniteness of the gravitational action, Nuclear Phys. B 138 (1978), no. 1, 141–150. MR 496312, DOI 10.1016/0550-3213(78)90161-X
- David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, 2nd ed., Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 224, Springer-Verlag, Berlin, 1983. MR 737190, DOI 10.1007/978-3-642-61798-0 [G] R. Graham, A conformal normal form, in preparation.
- 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, DOI 10.1017/CBO9780511524646
- David Jerison and John M. Lee, A subelliptic, nonlinear eigenvalue problem and scalar curvature on CR manifolds, Microlocal analysis (Boulder, Colo., 1983) Contemp. Math., vol. 27, Amer. Math. Soc., Providence, RI, 1984, pp. 57–63. MR 741039, DOI 10.1090/conm/027/741039
- David Jerison and John M. Lee, The Yamabe problem on CR manifolds, J. Differential Geom. 25 (1987), no. 2, 167–197. MR 880182
- David Jerison and John M. Lee, Extremals for the Sobolev inequality on the Heisenberg group and the CR Yamabe problem, J. Amer. Math. Soc. 1 (1988), no. 1, 1–13. MR 924699, DOI 10.1090/S0894-0347-1988-0924699-9 [JL4] D. Jerison and J. Lee, Intrinsic CR normal coordinates and the CR Yamabe problem, in preparation. [K] J. Kazdan, Some applications of partial differential equations to problems in geometry, Surveys in Geometry, Japan, 1983 (preprint). [KN] S. Kobayashi and N. Nomizu, Foundations of differential geometry. I, Interscience, New York, 1963.
- Pierre-Louis Lions, Applications de la méthode de concentration-compacité à l’existence de fonctions extrémales, C. R. Acad. Sci. Paris Sér. I Math. 296 (1983), no. 15, 645–648 (French, with English summary). MR 705681
- 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 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
- 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
- Morio Obata, The conjectures on conformal transformations of Riemannian manifolds, J. Differential Geometry 6 (1971/72), 247–258. MR 303464
- Thomas Parker and Steven Rosenberg, Invariants of conformal Laplacians, J. Differential Geom. 25 (1987), no. 2, 199–222. MR 880183
- 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, DOI 10.1007/BF01208569
- 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
- Richard Schoen, Conformal deformation of a Riemannian metric to constant scalar curvature, J. Differential Geom. 20 (1984), no. 2, 479–495. MR 788292
- 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, DOI 10.1007/BF01940959 [SY2] R. Schoen and S.-T. Yau, Proof of the positive action conjecture in quantum relativity, Phys. Rev. Lett. 42 (1979), 547-548.
- 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, DOI 10.1007/BF01208285
- Richard Schoen and Shing Tung Yau, Proof of the positive mass theorem. II, Comm. Math. Phys. 79 (1981), no. 2, 231–260. MR 612249, DOI 10.1007/BF01942062 [SY5] R. Schoen and S.-T. Yau, The geometry and topology of manifolds of positive scalar curvature, in preparation.
- 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
- Giorgio Talenti, Best constant in Sobolev inequality, Ann. Mat. Pura Appl. (4) 110 (1976), 353–372. MR 463908, DOI 10.1007/BF02418013
- 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
- S. M. Webster, Pseudo-Hermitian structures on a real hypersurface, J. Differential Geometry 13 (1978), no. 1, 25–41. MR 520599, DOI 10.4310/jdg/1214434345
- Wei Yue Ding, On a conformally invariant elliptic equation on $\textbf {R}^n$, Comm. Math. Phys. 107 (1986), no. 2, 331–335. MR 863646, DOI 10.1007/BF01209398
- Edward Witten, A new proof of the positive energy theorem, Comm. Math. Phys. 80 (1981), no. 3, 381–402. MR 626707, DOI 10.1007/BF01208277
- Hidehiko Yamabe, On a deformation of Riemannian structures on compact manifolds, Osaka Math. J. 12 (1960), 21–37. MR 125546
- Shing Tung Yau (ed.), Seminar on Differential Geometry, Annals of Mathematics Studies, No. 102, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1982. Papers presented at seminars held during the academic year 1979–1980. MR 645728
Additional Information
- Journal: Bull. Amer. Math. Soc. 17 (1987), 37-91
- MSC (1980): Primary 53A30; Secondary 35J20, 35J60
- DOI: https://doi.org/10.1090/S0273-0979-1987-15514-5
- MathSciNet review: 888880