Symmetric solutions of the equation for the scalar curvature under conformal deformation of a Riemannian metric

Shin KATO

doi:10.2969/jmsj/04340733

Shin KATO

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JOURNAL FREE ACCESS

1991 Volume 43 Issue 4 Pages 733-750

DOI https://doi.org/10.2969/jmsj/04340733

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Published: 1991 Received: October 25, 1990 Available on J-STAGE: October 20, 2006 Accepted: - Advance online publication: - Revised: -
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Date of correction: October 20, 2006 Reason for correction: - Correction: CITATION Details: Wrong : 1) T. Aubin, The scalar curvature: Differential Geometry and Relativity, ed. M. Cahen and M. Flato, Reidel, 1976, pp. 5-18.
2) J. P. Bourguignon and J. P. Ezin, Scalar curvature functions in a conformal class of metric and conformal transformations, Trans. Am. Math. Soc., 301 (1987), 723-736.
3) W. Chen, Scalar curvatures on Sⁿ, Math. Ann., 283 (1989), 353-365.
4) J. F. Escobar and R. M. Schoen, Conformal metrics with prescribed scalar curvature, Invent. Math., 86 (1986), 243-254.
5) J. L. Kazdan and F. W. Warner, Scalar curvature and conformal deformation of Riemannian structure, J. Differential Geom., 10 (1975), 113-134.
6) O. Kobayashi, On large scalar curvature, Research Report, Keio Univ., 1985 (unpublished).
7) P. L. Lions, The concentration-compactness principle in the calculus of variations, the limit case, part. 2, Rev. Mat. Ibero., 1 (1985), 45-121.
8) J.M. Lee and T.H. Parker, The Yamabe problem, Bull. Amer. Math. Soc, 17 (1987), 37-91.
9) R. S. Palais, The principle of symmetric criticality, Comm. Math. Phys., 69 (1979), 19-30.
10) R. Schoen, Conformal deformation of a Riemannian metric to constant scalar curvature, J. Differential Geom., 20 (1984), 479-495.
11) R. M. Schoen, Variational theory for the total scalar curvature functional for Riemannian metrics and related topics, Topics in Calculus of Variations, ed. M. Giaquinta, Lecture Notes in Math., 1365, Springer, 1989, pp. 120-154.
12) R. Schoen and S. T. Yau, On the proof of the positive mass conjecture in general relativity, Comm. Math. Phys., 65 (1979), 45-76.
13) R. Schoen and S. T. Yau, Conformally flat manifolds, Kleinian groups and scalar curvature, Invent. Math., 92 (1988), 47-71.
14) M. Spivak, A comprehensive introduction to differential geometry, 4, Publish or Perish, Inc., Berkeley, 1979.
15) N. S. Trudinger, Remarks concerning the conformal deformation of Riemannian structures on compact manifolds, Ann. Scuola Norm. Sup. Pisa, 22 (1968), 265-274.
16) M. Vaugon, Transformation conforme de la courbure scalaire, Analyse non lineaire, Ann. Inst. Henri Poincare, 3 (1986), 55-65.
17) H. Yamabe, On a deformation of Riemannian structures on compact manifolds, Osaka Math. J., 12 (1960), 21-37.

Right : [1] T. Aubin, The scalar curvature: Differential Geometry and Relativity, ed. M. Cahen and M. Flato, Reidel, 1976, pp. 5-18.
[2] J. P. Bourguignon and J. P. Ezin, Scalar curvature functions in a conformal class of metric and conformal transformations, Trans. Am. Math. Soc., 301 (1987), 723-736.
[3] W. Chen, Scalar curvatures on Sⁿ, Math. Ann., 283 (1989), 353-365.
[4] J. F. Escobar and R. M. Schoen, Conformal metrics with prescribed scalar curvature, Invent. Math., 86 (1986), 243-254.
[5] J. L. Kazdan and F. W. Warner, Scalar curvature and conformal deformation of Riemannian structure, J. Differential Geom., 10 (1975), 113-134.
[6] O. Kobayashi, On large scalar curvature, Research Report, Keio Univ., 1985 (unpublished).
[7] P. L. Lions, The concentration-compactness principle in the calculus of variations, the limit case, part. 2, Rev. Mat. Ibero., 1 (1985), 45-121.
[8] J. M. Lee and T. H. Parker, The Yamabe problem, Bull. Amer. Math. Soc, 17 (1987), 37-91.
[9] R. S. Palais, The principle of symmetric criticality, Comm. Math. Phys., 69 (1979), 19-30.
[10] R. Schoen, Conformal deformation of a Riemannian metric to constant scalar curvature, J. Differential Geom., 20 (1984), 479-495.
[11] R. M. Schoen, Variational theory for the total scalar curvature functional for Riemannian metrics and related topics, Topics in Calculus of Variations, ed. M. Giaquinta, Lecture Notes in Math., 1365, Springer, 1989, pp. 120-154.
[12] R. Schoen and S. T. Yau, On the proof of the positive mass conjecture in general relativity, Comm. Math. Phys., 65 (1979), 45-76.
[13] R. Schoen and S. T. Yau, Conformally flat manifolds, Kleinian groups and scalar curvature, Invent. Math., 92 (1988), 47-71.
[14] M. Spivak, A comprehensive introduction to differential geometry, 4, Publish or Perish, Inc., Berkeley, 1979.
[15] N. S. Trudinger, Remarks concerning the conformal deformation of Riemannian structures on compact manifolds, Ann. Scuola Norm. Sup. Pisa, 22 (1968), 265-274.
[16] M. Vaugon, Transformation conforme de la courbure scalaire, Analyse non lineaire, Ann. Inst. Henri Poincare, 3 (1986), 55-65.
[17] H. Yamabe, On a deformation of Riemannian structures on compact manifolds, Osaka Math. J., 12 (1960), 21-37.

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