Journal of Integrative Natural Science (통합자연과학논문집)

The Basic Science Institute Chosun University (조선대학교 기초과학연구원)
권호 표지
DOI QR코드

DOI QR Code

The Nonexistence of Conformal Deformations on Riemannian Warped Product Manifolds

  • Received : 2012.01.13
  • Accepted : 2012.03.27
  • Published : 2012.03.30
Download PDF
⟨ Previous Next ⟩

Abstract

In this paper, when N is a compact Riemannian manifold, we discuss the nonexistence of conformal deformations on Riemannian warped product manifold $M=({\alpha},\;{\infty}){\times}_fN$ with prescribed scalar curvature functions.

Keywords

References

  1. P. Aviles and R. McOwen, "Conformal deformation to constant negative scalar curvature on noncompact Riemannian manifolds", Diff. Geom., Vol. 27, pp. 225-239, 1998.
  2. J. K. Beem, P. E. Ehrlich and Th.G. Powell, "Warped product manifolds in relativity", Selected Studies (Th.M. Rassias, G.M. Rassias, eds.), pp, 41-56, 1982.
  3. Y. T. Jung, "The Nonexistence of Conformal Deformations on Space-Times(II)", J. Honam Mathematical, Vol. 33, No.1, pp. 121-127, 2011. https://doi.org/10.5831/HMJ.2011.33.1.121
  4. Y. T. Jung and S.C. Lee, "The nonexistence of conformal deformations on space-times", J. Honam Math., Vol. 3,2 pp. 85-89, 2010. https://doi.org/10.5831/HMJ.2010.32.1.085
  5. J. L. Kazdan and F.W. Warner, "Scalar curvature and conformal deformation of Riemannian structure", J. Diff. Geo., Vol. 10, pp. 113-134, 1975. https://doi.org/10.4310/jdg/1214432678
  6. J. L. Kazdan and F.W. Warner, "Existence and conformal deformation of metrics with prescribed Guassian and scalar curvature", Ann. of Math., Vol. 101, pp. 317-331, 1975. https://doi.org/10.2307/1970993
  7. M. C. Leung, "Conformal scalar curvature equations on complete manifolds", Comm. in P.D.E., Vol. 20, pp. 367-417, 1995. https://doi.org/10.1080/03605309508821100
  8. M. C. Leung, "Conformal deformation of warped products and scalar curvature functions on open manifolds", preprint.
  9. M. C. Leung, "Uniqueness of Positive Solutions of the Equation $\Delta_{g_{0}}+c_{n}u=c_{n}u^{n+2/n-2}$ and Applications to Conformal Transformations", preprint.
  10. D.S. Mitrinovic, "Analytic inequalities", Springer-Verlag, 1970.
  11. A. Ratto, M. Rigoli and G. Setti, "On the Omori-Yau maximum principle and its applications to differential equations and geometry", J. Functional Analysis, Vol. 134, pp. 486-510, 1995. https://doi.org/10.1006/jfan.1995.1154