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Compactness of solutions to the Yamabe problem. II

  • YanYan LiEmail author
  • Lei Zhang
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Keywords

System Theory Yamabe Problem 
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References

  1. 1.
    Alexandrov, A.D.: Uniqueness theorems for surfaces in the large. V, Vestnik Leningrad Univ. Mat. Mekh. Astronom 13, 5-8 (1958): Amer. Math. Soc. Transl. 21, 412-416 (1962)Google Scholar
  2. 2.
    Ambrosetti, A., Azorero, J.G., Peral, I.: Perturbation of \(\Delta u+u^{(N+2)/(N-2)}=0\), the scalar curvature problem in \({\mathbb R}^N\), and related topics. J. Funct. Anal. 165, 117-149 (1999)CrossRefGoogle Scholar
  3. 3.
    Ambrosetti, A., Li, Y.Y., Malchiodi, A.: On the Yamabe problem and the scalar curvature problems under boundary conditions. Math. Ann. 322, 667-699 (2002)CrossRefGoogle Scholar
  4. 4.
    Aubin, T.: Équations différentielles non linéaires et probléme de Yamabe concernant la courbure scalaire. J. Math. Pures Appl. 55, 269-296 (1976)Google Scholar
  5. 5.
    Arkinson, F.V., Peletier, L.A.: Elliptic equations with nearly critical growth. J. Differential Equations 70, 349-365 (1987)CrossRefGoogle Scholar
  6. 6.
    Aubin, T.: Some nonlinear problems in Riemannian geometry (Springer Monographs in Mathematics). Springer, Berlin 1998Google Scholar
  7. 7.
    Aubin, T., Bahri, A.: Méthodes de topologie algébrique pour le probléme de la courbure scalaire prescrite. J. Math. Pures Appl. 76, 525-549 (1997)CrossRefGoogle Scholar
  8. 8.
    Aubin, T., Bahri, A.: Une hypothése topologique pour le probléme de la courbure scalaire prescrite. J. Math. Pures Appl. 76, 843-850 (1997)CrossRefGoogle Scholar
  9. 9.
    Bahri, A.: Another proof of the Yamabe conjecture for locally conformally flat manifolds. Nonlinear Anal. 20, 1261-1278 (1993)CrossRefGoogle Scholar
  10. 10.
    Bahri, A., Brezis, H.: 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 1996Google Scholar
  11. 11.
    Bahri, A., Coron, J.-M.: On a nonlinear elliptic equation involving the critical Sobolev exponent: the effect of the topology of the domain. Comm. Pure Appl. Math. 41, 253-294 (1988)Google Scholar
  12. 12.
    Bahri, A., Coron, J.-M.: The scalar-curvature problem on the standard three-dimensional sphere. J. Funct. Anal. 95, 106-172 (1991)CrossRefGoogle Scholar
  13. 13.
    Bartnik, R.: The mass of an asymptotically flat manifold. Comm. Pure Appl. Math. 39, 661-693 (1986)Google Scholar
  14. 14.
    Bartolucci, D.: A compactness result for periodic multivortices in the electroweak theory. Nonlinear Anal. 53, 277-297 (2003)CrossRefGoogle Scholar
  15. 15.
    Bartolucci, D., Chen, C.C., Lin, C.S., Tarantello, G.: Profile of blow-up solutions to mean field equations with singular data. PreprintGoogle Scholar
  16. 16.
    Bartolucci, D., Tarantello, G.: Liouville type equations with singular data and their applications to periodic multivortices for the electroweak theory, Comm. Math. Phys. 229, 3-47 (2002)CrossRefGoogle Scholar
  17. 17.
    Bartolucci, D., Tarantello, G.: The Liouville equation with singular data: a concentration-compactness principle via a local representation formula. J. Differential Equations 185, 161-180 (2002)CrossRefGoogle Scholar
  18. 18.
    Berestycki, H., Nirenberg, L.: On the method of moving planes and the sliding method. Bol. Soc. Brasil. Mat. (N.S.) 22, 1-37 (1991)CrossRefGoogle Scholar
  19. 19.
    Bianchi, G., Egnell, H.: An ODE approach to the equation \(\Delta u+K u^{(n+2)/(n-2)}=0\) in Rn. Math. Z. 210, 137-166 (1992)Google Scholar
  20. 20.
    Brendle, S.: Convergence of the Yamabe flow for arbitrary initial energy. PreprintGoogle Scholar
  21. 21.
    Brezis, H., Li, Y.Y., Shafrir, I.: A \(\sup + \inf \) inequality for some nonlinear elliptic equations involving exponential nonlinearities. J. Functional Analysis 115, 344-358 (1993)CrossRefGoogle Scholar
  22. 22.
    Brezis, H., Merle, F.: Uniform estimates and blow-up behavior for solutions of \(-\Delta u=V(x)e^u\) in two dimension. Comm. Partial Differtial Equation 16, 1223-1253 (1991)Google Scholar
  23. 23.
    Brezis, H., Nirenberg, L.: Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents. Comm. Pure Appl. Math. 36, 437-477 (1983)Google Scholar
  24. 24.
    Brezis, H., Peletier, L.A.: Asymptotics for elliptic equations involving critical growth. In: Partial differential equations and the calculus of variations, Vol. I, pp. 149-192. Progr. Nonlinear Differential Equations Appl., 1, Birkhäuser Boston, Boston, MA 1989Google Scholar
  25. 25.
    Caffarelli, L., Gidas, B., Spruck, J.: Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth. Comm. Pure Appl. Math. 42, 271-297 (1989)Google Scholar
  26. 26.
    Caffarelli, L., Hardt, R., Simon, L.: Minimal surfaces with isolated singularities. Manuscripta Math. 48, 1-18 (1984)CrossRefGoogle Scholar
  27. 27.
    Cao, J.: The existence of generalized isothermal coordinates for higher-dimensional Riemannian manifolds. Trans. Amer. Math. Soc. 324, 901-920 (1991)Google Scholar
  28. 28.
    Chang, K.C., Liu, J.: On Nirenberg’s problem. Int. J. Math. 4, 35-58 (1993)CrossRefGoogle Scholar
  29. 29.
    Chang, S.Y.A., Yang, P.: Prescribing Gaussian curvature on \(S\sp 2\). Acta Math. 159, 215-259 (1987)Google Scholar
  30. 30.
    Chang, S.Y.A., Yang, P.: Conformal deformation of metrics on \(S\sp 2\). J. Differential Geom. 27, 259-296 (1988)Google Scholar
  31. 31.
    Chang, S.Y.A., Yang, P.: A perturbation result in prescribing scalar curvature on Sn. Duke Math. J. 64, 27-69 (1991)CrossRefGoogle Scholar
  32. 32.
    Chang, S.Y.A., Gursky, M., Yang, P.: The scalar curvature equation on 2- and 3-spheres. Calc. Var. Partial Differential Equations 1, 205-229 (1993)CrossRefGoogle Scholar
  33. 33.
    Chen, C.C., Lin, C.S.: A sharp \(\sup + \inf\) inequality for a nonlinear elliptic equation in \(\Bbb R^2\). Comm. Anal. Geom. 6, 1-19 (1998)Google Scholar
  34. 34.
    Chen, C.C., Lin, C.S.: Estimates of the conformal scalar curvature equation via the method of moving planes. Comm. Pure Appl. Math. 50, 971-1017 (1997)CrossRefGoogle Scholar
  35. 35.
    Chen, C.C., Lin, C.S.: Estimates of the conformal scalar curvature equation via the method of moving planes. II. J. Diff. Geom. 49, 115-178 (1998)Google Scholar
  36. 36.
    Chen, C.C., Lin, C.S.: Sharp estimates for solutions of multi-bubbles in compact Riemann surfaces. Comm. Pure Appl. Math. 55, 728-771 (2002)CrossRefGoogle Scholar
  37. 37.
    Chen, C.C., Lin, C.S.: Topological degree for a mean field equation on Riemann surfaces. Comm. Pure Appl. Math. 56, 1667-1727 (2003)CrossRefGoogle Scholar
  38. 38.
    Chen, W.: Scalar curvature on Sn. Math. Ann. 283, 353-365 (1989)CrossRefGoogle Scholar
  39. 39.
    Chen, W., Ding, W.: A problem concerning the scalar curvature on \(\Bbb S^2 \). Kexue Tongbao 33, 533-537 (1988)Google Scholar
  40. 40.
    Chen, W., Li, C.: A priori estimates for prescribing scalar curvature equations. Ann. of Math. 145, 547-564 (1997)Google Scholar
  41. 41.
    Chen, X.: Remarks on the existence of branch bubbles on the blowup analysis of equation \(-\Delta u=e\sp {2u}\) in dimension two. Comm. Anal. Geom. 7, 295-302 (1999)Google Scholar
  42. 42.
    Cherrier, P.: Probléme de Neumann non linéaires sur les variétés Riemanniennes. J. Funct. Anal. 57, 154-206 (1984)CrossRefGoogle Scholar
  43. 43.
    Ding, W., Ni, W.-M.: On the elliptic equation \(\Delta u+K u^{(n+2)/(n-2)}=0\) and related topics. Duke Math. J. 52, 485-506 (1985)CrossRefGoogle Scholar
  44. 44.
    Druet, O.: From one bubble to several bubbles. The low-dimensional case. Journal of Differential Geometry 63, 399-473 (2003)Google Scholar
  45. 45.
    Druet, O.: Compactness for Yamabe metrics in low dimensions. Int. Math. Res. Not. 23, 1143-1191 (2004)CrossRefGoogle Scholar
  46. 46.
    Escobar, J.: Conformal deformation of a Riemannian metric to a scalar flat metric with constant mean curvature. Ann. of Math. 136, 1-50 (1992)Google Scholar
  47. 47.
    Escobar, J.: The Yamabe problem on manifolds with boundary. J. Diff. Geom. 35, 21-84 (1992)Google Scholar
  48. 48.
    Escobar, J.: Conformal deformation of a Riemannian metric to a constant scalar curvature metric with constant mean curvature on the boundary. Indiana Univ. Math. J. 45, 917-943 (1996)CrossRefGoogle Scholar
  49. 49.
    Escobar, J., Garcia, G.: Conformal metrics on the ball with zero scalar curvature and prescribed mean curvature on the boundary. J. Funct. Anal. 211, 71-152 (2004)CrossRefGoogle Scholar
  50. 50.
    Escobar, J., Schoen, R.: Conformal metrics with prescribed scalar curvature. Invent. Math. 86, 243-254 (1986)CrossRefGoogle Scholar
  51. 51.
    Felli, V., Ould Ahmedou, M.: Compactness results in conformal deformations of Riemannian metrics on manifolds with boundaries. Math. Z. 244, 175-210 (2003)CrossRefGoogle Scholar
  52. 52.
    Gidas, B., Spruck, J.: Global and local behavior of positive solutions of nonlinear elliptic equations. Comm. Pure Appl. Math. 34, 525-598 (1981)Google Scholar
  53. 53.
    Günther, M.: Conformal normal coordinates. Ann. Global Anal. Geom. 11, 173-184 (1993)CrossRefGoogle Scholar
  54. 54.
    Gidas, B., Ni, W.M., Nirenberg, L.: Symmetry and related properties via the maximum principle. Comm. Math. Phys. 68, 209-243 (1979)CrossRefGoogle Scholar
  55. 55.
    Han, Z.C.: Prescribing Gaussian curvature on S2. Duke Math. J. 61, 679-703 (1990)CrossRefGoogle Scholar
  56. 56.
    Han, Z.C.: Asymptotic approach to singular solutions for nonlinear elliptic equations involving critical Sobolev exponent. Ann. Inst. H. Poincaré Anal. Non Linéaire 8, 159-174 (1991)Google Scholar
  57. 57.
    Han, Z.C., Li, Y.Y.: The Yamabe problem on manifolds with boundaries: Existence and compactness results. Duke Math. J. 99, 489-542 (1999)CrossRefGoogle Scholar
  58. 58.
    Han, Z.C., Li, Y.Y.: The existence of conformal metrics with constant scalar curvature and constant boundary mean curvature. Comm. Anal. Geom. 8, 809-869 (2000)Google Scholar
  59. 59.
    Hebey, E.: Changements de métriques conformes sur la sphére, Le probléme de Nirenberg. Bull. Sci. Math. 114, 215-242 (1990)Google Scholar
  60. 60.
    Hebey, E., Vaugon, M.: Le probliéme de Yamabe équivariant. Bull. Sci. Math. 117, 241-286 (1993)Google Scholar
  61. 61.
    Hong, C.W.: A best constant and the Gaussian curvature. Proc. Amer. Math. Soc. 97, 737-747 (1986)Google Scholar
  62. 62.
    Kazdan, J., Warner, F.: Scalar curvature and conformal deformations of Riemannian structure. J. Diff. Geom. 10, 113-134 (1975)Google Scholar
  63. 63.
    Lee, J., Parker, T.: The Yamabe problem. Bull. Amer. Math. Soc. (N.S.) 17, 37-91 (1987)Google Scholar
  64. 64.
    Li, A., Li, Y.Y.: On some conformally invariant fully nonlinear equations. Comm. Pure Appl. Math. 56, 1414-1464 (2003)Google Scholar
  65. 65.
    Li, A., Li, Y.Y.: On some conformally invariant fully nonlinear equations, Part II: Liouville, Harnack and Yamabe. arXiv:math.AP/0403442 v1 25 Mar 2004Google Scholar
  66. 66.
    Li, Y.Y.: Prescribing scalar curvature on Sn and related problems, Part I. J. Diff. Equations 120, 319-410 (1995)CrossRefGoogle Scholar
  67. 67.
    Li, Y.Y.: Prescribing scalar curvature on Sn and related problems, Part II: Existence and compactness. Comm. Pure Appl. Math. 49, 541-597 (1996)CrossRefGoogle Scholar
  68. 68.
    Li, Y.Y.: A Harnack type inequality: the method of moving planes. Comm. Math. Phys. 200, 421-444 (1999)CrossRefGoogle Scholar
  69. 69.
    Li, Y.Y., Zhang, L.: Liouville type theorems and Harnack type inequalities for semilinear elliptic equations. Journal d’Analyse Mathematique 90, 27-87 (2003)Google Scholar
  70. 70.
    Li, Y.Y., Zhang, L.: A Harnack type inequality for the Yamabe equation in low dimensions. Calc. Var. and PDEs 20, 133-151 (2004)CrossRefGoogle Scholar
  71. 71.
    Li, Y.Y., Zhang, L.: Compactness of solutions to the Yamabe problem. C. R. Math. Acad. Sci. Paris 338, 693-695 (2004)Google Scholar
  72. 72.
    Li, Y.Y., Zhang, L.: Compactness of solutions to the Yamabe problem. III. In preparationGoogle Scholar
  73. 73.
    Li, Y.Y., Zhu, M.: Yamabe type equations on three dimensional Riemannian manifolds. Communications in Contemporary Math. 1, 1-50 (1999)CrossRefGoogle Scholar
  74. 74.
    Li, Y.Y., Shafrir, I.: Blow up analysis for solutions of \(-\Delta u = Ve^u\) in dimension two. Indiana Univ. Math. J. 43, 1255-1270 (1994)CrossRefGoogle Scholar
  75. 75.
    Marques, F.C.: A priori estimates for the Yamabe problem in the non-locally conformally flat case. arXiv:math.DG/0408063 v1 4 Aug 2004Google Scholar
  76. 76.
    Moser, J.: On a nonlinear problem in differential geometry. In: Peixoto, M. (ed.) Dynamical systems, pp, 273-280. Academic Press, New York 1973Google Scholar
  77. 77.
    Nirenberg, L.: Topics in nonlinear functional analysis. Courant Lecture Notes in Mathematics 6, New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2001Google Scholar
  78. 78.
    Prajapat, J., Ramaswamy, M.: A priori estimates for solutions of “sub-critical” equations on CR sphere. Adv. Nonlinear Stud. 3, 355-395 (2003)Google Scholar
  79. 79.
    Schoen, R.: Conformal deformation of a Riemannian metric to constant scalar curvature. J. Differential Geom. 20, 479-495 (1984)Google Scholar
  80. 80.
    Schoen, R.: Variational theory for the total scalar curvature functional for Riemannian metrics and related topics. In: Giaquinta, M. (ed.) Topics in Calculus of Variations. Lecture Notes in Mathematics, Vol. 1365 120154. Springer, Berlin Heidelberg New York 1989Google Scholar
  81. 81.
    Schoen, R.: Courses at Stanford University, 1988, and New York University, 1989Google Scholar
  82. 82.
    Schoen, R.: On the number of constant scalar curvature metrics in a conformal class. In: Lawson, H.B., Tenenblat, K. (eds.) Differential geometry: a symposium in honor of Manfredo Do Carmo, pp. 311-320. Wiley, New York 1991Google Scholar
  83. 83.
    Schoen, R., Yau, S.T.: On the proof of the positive mass conjecture in General Relativity. Comm. Math. Phys. 65, 45-76 (1979)CrossRefGoogle Scholar
  84. 84.
    Schoen, R., Zhang, D.: Prescribed scalar curvature on the n-sphere. Calc. Var. Partial Differential Equations 4, 1-25 (1996)Google Scholar
  85. 85.
    Serrin, J.: A symmetry problem in potential theory. Arch. Rational Mech. Anal. 43, 304-318 (1971)CrossRefGoogle Scholar
  86. 86.
    Schwetlick, H., Struwe, M.: Convergence of the Yamabe flow for “large” energies. J. Reine Angew. Math. 562, 59-100 (2003)Google Scholar
  87. 87.
    Siu, Y-T.: The existence of K\”shler-Einstein metrics on manifolds with positive anticanonical line bundle and a suitable finite symmetry group. Ann. Math. 127, 585-627 (1988)Google Scholar
  88. 88.
    Tarantello, G.: An Harnack inequality for Liouville-type equations with singular sources. PreprintGoogle Scholar
  89. 89.
    Tarantello, G.: A quantization property for blow up solutions of singular Liouville-type equation. PreprintGoogle Scholar
  90. 90.
    Tian, G.: A Harnack type inequality for certain complex Monge-Ampére equations. J. Differ. Geom. 29, 481-488 (1989)Google Scholar
  91. 91.
    Trudinger, N.: Remarks concerning the conformal deformation of Riemannian structures on compact manifolds. Ann. Scuola Norm. Sup. Cl. Sci. (3) 22, 265-274 (1968)Google Scholar
  92. 92.
    Yamabe, H.: On a deformation of Riemannian structures on compact manifolds. Osaka Math. J. 12, 21-37 (1960)Google Scholar
  93. 93.
    Ye, R.: Global existence and convergence of Yamabe flow. J. Diff. Geom. 39, 35-50 (1994)Google Scholar

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© Springer-Verlag Berlin/Heidelberg 2005

Authors and Affiliations

  1. 1.Department of MathematicsRutgers UniversityPiscatawayUSA
  2. 2.Department of MathematicsUniversity of FloridaGainesvilleUSA

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