Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
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THE STRUCTURE OF THE REGULAR LEVEL SETS

  • Received : 2010.07.14
  • Published : 2011.11.30
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Abstract

Consider the $L^2$-adjoint $s_g^{'*}$ of the linearization of the scalar curvature $s_g$. If ker $s_g^{'*}{\neq}0$ on an n-dimensional compact manifold, it is well known that the scalar curvature $s_g$ is a non-negative constant. In this paper, we study the structure of the level set ${\varphi}^{-1}$(0) and find the behavior of Ricci tensor when ker $s_g^{'*}{\neq}0$ with $s_g$ > 0. Also for a nontrivial solution (g, f) of $z=s_g^{'*}(f)$ on an n-dimensional compact manifold, we analyze the structure of the regular level set $f^{-1}$(-1). These results give a good understanding of the given manifolds.

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Acknowledgement

Supported by : Chung-Ang University

References

  1. S. Agmon, The $L_p$ approach to the Dirichlet Problem, Ann. Scuola Norm. Sup. Pisa 13 (1959), 405-448.
  2. M. Berger and D. Ebin, Some decompositions of the space of symmetric tensors on a Riemannian manifold, J. Differential Geometry 3 (1969), 379-392. https://doi.org/10.4310/jdg/1214429060
  3. A. L. Besse, Einstein Manifolds, Springer-Verlag, New York, 1987.
  4. L. Bessieres, J. Lafontiane, and L. Rozoy, Scalar curvature and black holes, preprint.
  5. J. P. Bourguignon, Une stratification de l'espace des structures riemanniennes, Compositio Math. 30 (1975), 1-41.
  6. A. E. Fischer and J. E. Marsden, Manifolds of Riemannian metrics with prescribed scalar curvature, Bull. Amer. Math. Soc. 80 (1974), 479-484. https://doi.org/10.1090/S0002-9904-1974-13457-9
  7. J. Hempel, 3-manifolds, Princeton, 1976.
  8. S. Hwang, Critical points of the total scalar curvature functional on the space of metrics of constant scalar curvature, Manuscripta Math. 103 (2000), no. 2, 135-142. https://doi.org/10.1007/PL00005857
  9. S. Hwang, J. Chang, and G. Yun, Rigidity of the critical point equation, Math. Nachr. 283 (2010), no. 6, 846-853. https://doi.org/10.1002/mana.200710037
  10. O. Kobayashi, A differential equation arising from scalar curvature function, J. Math. Soc. Japan 34 (1982), no. 4, 665-675. https://doi.org/10.2969/jmsj/03440665
  11. J. Lafontaine, Sur la geometrie d'une generalisation de l'equation differentielle d'Obata, J. Math. Pures Appl. (9) 62 (1983), no. 1, 63-72.
  12. J. Lafontaine and L. Rozoy, Courure scalaire et trous noirs, Seminaire de Theorie Spectrale et Geometrie, Vol. 18, Annee 1999-2000, 69-76, Semin. Theor. Spectr. Geom., 18, Univ. Grenoble I, Saint-Martin-d'Heres, 2000.
  13. Y. Shen, A note on Fisher-Marsden's conjecture, Proc. Amer. Math. Soc. 125 (1997), no. 3, 901-905. https://doi.org/10.1090/S0002-9939-97-03635-6