The existence of conformal metrics with constant scalar curvature and constant boundary mean curvature

Communications in Analysis and Geometry

Volumn 8, Issue 4, 2000, Pages 809-869

  Han, Zheng Chao ^a   Li, Yanyan ^a  

^a RUTGERS UNIVERSITY   (United States)

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[No Author keywords available]

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EID: 0000531720     PISSN: 10198385     EISSN: None     Source Type: Journal    
DOI: 10.4310/CAG.2000.v8.n4.a5     Document Type: Article

References (25)

1. A. Ambrosetti and P.H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal. 14 (1973), 349-381.
2. T. Aubin, Équations différentielles non linéaires et probléme de Yamabe concernant la courbure scalaire, J. Math. Pures Appl. 55 (1976), 269-296.
3. A. Bahri, Proof of the Yamabe conjecture without the positive mass conjecture for locally conformally flat manifolds, Nonlinear variational problems and partial differential equations (Isola d'Elba, 1990), 13-43, Pitman Res. Notes Math. Ser., 320, Longman Sci. Tech., Harlow, 1995.
4. A. Bahri and H. Brezis, Non-linear elliptic equations on Riemannian manifolds with the Sobolev critical exponent Topics in geometry, 1-100, Progr. Nonlinear Differential Equations Appl., 20, Birkhduser Boston, Boston, MA, 1996.
5. H. Brezis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure A ppl. Math. 36 (1983), 437-477.
6. L. Caffarelli, B. Gidas and J. Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth, Comm. Pure Appl. Math. 42 (1989), 271-297.
7. P. Cherrier, Problèmes de Neumann nonlinéaires sur les variétés Riemanniennes, J. Func. Anal. 57 (1984), 154-207.
8. J.F. Escobar, Uniqueness theorems on conformal deformation of metric, Sobolev inequalities, and an eigenvalue estimate, Comm. Pure Appl. Math. 43 (1990), 857-883.
9. J. Escobar, The Yamabe problem on manifolds with boundary, J. Diff. Geom. 35 (1992), 21-84.
10. J. Escobar, Conformal deformation of a Riemannian metric to a scalar flat metric with constant mean curvature, Ann. of Math. 136 (1992), 1-50.
11. J. Escobar, Conformal deformation of a Riemannian metric to a constant scalar curvature metric with constant mean curvature on the boundary, Indiana Univ. Math. J. 45 (1996), 917-943.
12. J. Escobar, Conformal metrics with prescribed mean curvature on the boundary, Calc. Var. and PDEs 4 (1996), 559-592.
13. D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2nd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] No. 224, Springer-Verlag, Berlin-New York, 1983.
14. Z.C. Han and Y.Y. Li, The Yamabe problem on manifolds with boundaries: Existence and compactness results, Duke Mathematical Journal, 99 (1999), 489-542.
15. Z.C. Han and Y.Y. Li, Further results on the Yamabe problem with boundary, in preparation.
16. J. Lee and T. Parker, The Yamabe problem, Bull. Amer. Math. Soc. 17 (1987), 37-91.
17. Y.Y. Li and M. Zhu, Uniqueness theorems through the method of moving spheres, Duke Math. J. 80 (1995), 383-417.
18. P. L. Lions, The concentration-compactness principle in the calculus of variations. The limit case, Part 2, Revista Matemática Iberoamericana 1 (1985), 45-121.
19. R. Schoen, Conformal deformation of a Riemannian metric to constant scalar curvature, J. Diff. Geom. 20 (1984), 479-495.
20. R. Schoen, Variational theory for the total scalar curvature functional for Riemannian metrics and related topics, in Topics in Calculus of Variations, Lecture notes in mathematics, No. 1365, edited by M. Giaquinta, Springer-Verlag, 1989, 120-154
21. R. Schoen, On the number of constant scalar curvature metrics in a conformal class, Differential Geometry: A symposium in honor of Manfredo Do Carmo (H.B. Lawson and K. Tenenblat, eds), Wiley, 1991, 311-320.
22. R. Schoen and S.T. Yau, On the proof of the positive mass conjecture in General Relativity, Comm. Math. Phys. 65 (1979), 45-76.
23. R. Schoen and S.T. Yau, Conformally flat manifolds, Kleinian groups, and scalar curvature, Invent. Math. 92 (1988), 47-71.
24. N. Trudinger, Remarks concerning the conformal deformation of Riemannian structures on compact manifolds, Ann. Scuola Norm. Sup. Cl. Sci. (3) 22 (1968), 265-274.
25. H. Yamabe, On a deformation of Riemannian structures on compact manifolds, Osaka Math. J. 12 (1960), 21-37.