Skip to main content
Log in

Compactness of solutions to the Yamabe problem. II

  • Published:
Calculus of Variations and Partial Differential Equations Aims and scope Submit manuscript

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

References

  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. 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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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. Arkinson, F.V., Peletier, L.A.: Elliptic equations with nearly critical growth. J. Differential Equations 70, 349-365 (1987)

    Article  Google Scholar 

  6. Aubin, T.: Some nonlinear problems in Riemannian geometry (Springer Monographs in Mathematics). Springer, Berlin 1998

  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)

    Article  Google Scholar 

  8. Aubin, T., Bahri, A.: Une hypothése topologique pour le probléme de la courbure scalaire prescrite. J. Math. Pures Appl. 76, 843-850 (1997)

    Article  Google Scholar 

  9. Bahri, A.: Another proof of the Yamabe conjecture for locally conformally flat manifolds. Nonlinear Anal. 20, 1261-1278 (1993)

    Article  Google Scholar 

  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 1996

  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. Bahri, A., Coron, J.-M.: The scalar-curvature problem on the standard three-dimensional sphere. J. Funct. Anal. 95, 106-172 (1991)

    Article  Google Scholar 

  13. Bartnik, R.: The mass of an asymptotically flat manifold. Comm. Pure Appl. Math. 39, 661-693 (1986)

    Google Scholar 

  14. Bartolucci, D.: A compactness result for periodic multivortices in the electroweak theory. Nonlinear Anal. 53, 277-297 (2003)

    Article  Google Scholar 

  15. Bartolucci, D., Chen, C.C., Lin, C.S., Tarantello, G.: Profile of blow-up solutions to mean field equations with singular data. Preprint

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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. Brendle, S.: Convergence of the Yamabe flow for arbitrary initial energy. Preprint

  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)

    Article  Google Scholar 

  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. 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. 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 1989

  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. Caffarelli, L., Hardt, R., Simon, L.: Minimal surfaces with isolated singularities. Manuscripta Math. 48, 1-18 (1984)

    Article  Google Scholar 

  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. Chang, K.C., Liu, J.: On Nirenberg’s problem. Int. J. Math. 4, 35-58 (1993)

    Article  Google Scholar 

  29. Chang, S.Y.A., Yang, P.: Prescribing Gaussian curvature on \(S\sp 2\). Acta Math. 159, 215-259 (1987)

    Google Scholar 

  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. Chang, S.Y.A., Yang, P.: A perturbation result in prescribing scalar curvature on Sn. Duke Math. J. 64, 27-69 (1991)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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. 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)

    Article  Google Scholar 

  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. 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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  38. Chen, W.: Scalar curvature on Sn. Math. Ann. 283, 353-365 (1989)

    Article  Google Scholar 

  39. Chen, W., Ding, W.: A problem concerning the scalar curvature on \(\Bbb S^2 \). Kexue Tongbao 33, 533-537 (1988)

    Google Scholar 

  40. Chen, W., Li, C.: A priori estimates for prescribing scalar curvature equations. Ann. of Math. 145, 547-564 (1997)

    Google Scholar 

  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. Cherrier, P.: Probléme de Neumann non linéaires sur les variétés Riemanniennes. J. Funct. Anal. 57, 154-206 (1984)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  44. Druet, O.: From one bubble to several bubbles. The low-dimensional case. Journal of Differential Geometry 63, 399-473 (2003)

    Google Scholar 

  45. Druet, O.: Compactness for Yamabe metrics in low dimensions. Int. Math. Res. Not. 23, 1143-1191 (2004)

    Article  Google Scholar 

  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. Escobar, J.: The Yamabe problem on manifolds with boundary. J. Diff. Geom. 35, 21-84 (1992)

    Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  50. Escobar, J., Schoen, R.: Conformal metrics with prescribed scalar curvature. Invent. Math. 86, 243-254 (1986)

    Article  Google Scholar 

  51. Felli, V., Ould Ahmedou, M.: Compactness results in conformal deformations of Riemannian metrics on manifolds with boundaries. Math. Z. 244, 175-210 (2003)

    Article  Google Scholar 

  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. Günther, M.: Conformal normal coordinates. Ann. Global Anal. Geom. 11, 173-184 (1993)

    Article  Google Scholar 

  54. Gidas, B., Ni, W.M., Nirenberg, L.: Symmetry and related properties via the maximum principle. Comm. Math. Phys. 68, 209-243 (1979)

    Article  Google Scholar 

  55. Han, Z.C.: Prescribing Gaussian curvature on S2. Duke Math. J. 61, 679-703 (1990)

    Article  Google Scholar 

  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. Han, Z.C., Li, Y.Y.: The Yamabe problem on manifolds with boundaries: Existence and compactness results. Duke Math. J. 99, 489-542 (1999)

    Article  Google Scholar 

  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. Hebey, E.: Changements de métriques conformes sur la sphére, Le probléme de Nirenberg. Bull. Sci. Math. 114, 215-242 (1990)

    Google Scholar 

  60. Hebey, E., Vaugon, M.: Le probliéme de Yamabe équivariant. Bull. Sci. Math. 117, 241-286 (1993)

    Google Scholar 

  61. Hong, C.W.: A best constant and the Gaussian curvature. Proc. Amer. Math. Soc. 97, 737-747 (1986)

    Google Scholar 

  62. Kazdan, J., Warner, F.: Scalar curvature and conformal deformations of Riemannian structure. J. Diff. Geom. 10, 113-134 (1975)

    Google Scholar 

  63. Lee, J., Parker, T.: The Yamabe problem. Bull. Amer. Math. Soc. (N.S.) 17, 37-91 (1987)

    Google Scholar 

  64. Li, A., Li, Y.Y.: On some conformally invariant fully nonlinear equations. Comm. Pure Appl. Math. 56, 1414-1464 (2003)

    Google Scholar 

  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 2004

  66. Li, Y.Y.: Prescribing scalar curvature on Sn and related problems, Part I. J. Diff. Equations 120, 319-410 (1995)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  68. Li, Y.Y.: A Harnack type inequality: the method of moving planes. Comm. Math. Phys. 200, 421-444 (1999)

    Article  Google Scholar 

  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. Li, Y.Y., Zhang, L.: A Harnack type inequality for the Yamabe equation in low dimensions. Calc. Var. and PDEs 20, 133-151 (2004)

    Article  Google Scholar 

  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. Li, Y.Y., Zhang, L.: Compactness of solutions to the Yamabe problem. III. In preparation

  73. Li, Y.Y., Zhu, M.: Yamabe type equations on three dimensional Riemannian manifolds. Communications in Contemporary Math. 1, 1-50 (1999)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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 2004

  76. Moser, J.: On a nonlinear problem in differential geometry. In: Peixoto, M. (ed.) Dynamical systems, pp, 273-280. Academic Press, New York 1973

  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, 2001

  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. Schoen, R.: Conformal deformation of a Riemannian metric to constant scalar curvature. J. Differential Geom. 20, 479-495 (1984)

    Google Scholar 

  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 1989

  81. Schoen, R.: Courses at Stanford University, 1988, and New York University, 1989

  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 1991

  83. Schoen, R., Yau, S.T.: On the proof of the positive mass conjecture in General Relativity. Comm. Math. Phys. 65, 45-76 (1979)

    Article  Google Scholar 

  84. Schoen, R., Zhang, D.: Prescribed scalar curvature on the n-sphere. Calc. Var. Partial Differential Equations 4, 1-25 (1996)

    Google Scholar 

  85. Serrin, J.: A symmetry problem in potential theory. Arch. Rational Mech. Anal. 43, 304-318 (1971)

    Article  Google Scholar 

  86. Schwetlick, H., Struwe, M.: Convergence of the Yamabe flow for “large” energies. J. Reine Angew. Math. 562, 59-100 (2003)

    Google Scholar 

  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. Tarantello, G.: An Harnack inequality for Liouville-type equations with singular sources. Preprint

  89. Tarantello, G.: A quantization property for blow up solutions of singular Liouville-type equation. Preprint

  90. Tian, G.: A Harnack type inequality for certain complex Monge-Ampére equations. J. Differ. Geom. 29, 481-488 (1989)

    Google Scholar 

  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)

  92. Yamabe, H.: On a deformation of Riemannian structures on compact manifolds. Osaka Math. J. 12, 21-37 (1960)

    Google Scholar 

  93. Ye, R.: Global existence and convergence of Yamabe flow. J. Diff. Geom. 39, 35-50 (1994)

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to YanYan Li.

Additional information

Received: 2 November 2004, Accepted: 15 November 2004, Published online: 22 December 2004

Rights and permissions

Reprints and permissions

About this article

Cite this article

Li, Y., Zhang, L. Compactness of solutions to the Yamabe problem. II. Calc. Var. 24, 185–237 (2005). https://doi.org/10.1007/s00526-004-0320-7

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00526-004-0320-7

Keywords

Navigation