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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Scalar curvature and warped products of Riemann manifolds

Authors: F. Dobarro and E. Lami Dozo
Journal: Trans. Amer. Math. Soc. 303 (1987), 161-168
MSC: Primary 53C20; Secondary 58G30
MathSciNet review: 896013
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Abstract: We establish the relationship between the scalar curvature of a warped product $ M \times {}_fN$ of Riemann manifolds and those ones of $ M$ and $ N$. Then we search for weights $ f$ to obtain constant scalar curvature on $ M \times {}_fN$ when $ M$ is compact.

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  • [Am] H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM Rev. 18 (1976), 620-709. MR 0415432 (54:3519)
  • [Au1] T. Aubin, Nonlinear analysis on manifolds. Monge-Ampère equations, A Series of Comprehensive Studies in Mathematics 252, Springer, Berlin. MR 681859 (85j:58002)
  • [Au2] -, Métriques Riemanniennes et courbure, J. Differential Geom. 4 (1970), 383-424. MR 0279731 (43:5452)
  • [Bere] H. Berestycki, Le nombre de solutions de certains problèmes semi-linéaires elliptiques, J. Funct. Anal. 40 (1981), 1-29. MR 607588 (82k:35033)
  • [B-G-M] M. Berger, P. Gauduchon, and E. Mazet, Le spectre d'une variété Riemannienne, Lecture Notes in Math., vol. 194, Springer, 1971. MR 0282313 (43:8025)
  • [B-O] R. L. Bishop and B. O'Neill, Manifolds of negative curvature, Trans. Amer. Math. Soc. 145 (1969), 1-49. MR 0251664 (40:4891)
  • [C-P] Ph. Clément and L. A. Peletier, An anti-maximum principle for second-order elliptic operators, J. Differential Equations 34 (1979), 218-229. MR 550042 (83c:35034)
  • [C-R-T] M. G. Crandall, P. H. Rabinowitz, and L. Tartar, On a Dirichlet problem with a singular nonlinearity, MRC Report 1680, 1976. MR 0427826 (55:856)
  • [E-1] N. Ejiri, A negative answer to a conjecture of conformal transformations of Riemann manifolds, J. Math. Soc. Japan 33 (1981), 261-266. MR 607943 (82e:53064)
  • [E-2] -, Some compact hypersurfaces of constant scalar curvature in a sphere, J. Geom. 19 (1982), 197-199. MR 695711 (84f:53047)
  • [L] P. L. Lions, On the existence of positive solutions of semilinear elliptic equations, SIAM Rev. 24 (1982), 441-467. MR 678562 (84a:35093)
  • [M] J. Moser, On a nonlinear problem in differential geometry, Dynamical Systems, M. M. Peixoto (ed), Academic Press, New York, 1973, pp. 273-280. MR 0339258 (49:4018)
  • [R1] P. H. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Funct. Anal. 7 (1971), 487-513. MR 0301587 (46:745)
  • [R2] -, On bifurcation from infinity, J. Differential Equations 14 (1973), 462-475. MR 0328705 (48:7047)
  • [T] E. E. L. Turner, Positive solutions of nonlinear eigenvalue problems, C.I.M.E. Varenna, 1974, pp. 212-239.
  • [Y] H. Yamabe, On a deformation of Riemannian structures on compact manifolds, Osaka Math. J. 12 (1960), 21-37. MR 0125546 (23:A2847)
  • [Z] S. Zucker, $ {L_2}$ cohomology of warped products and arithmetic groups, Invent. Math. 70 (1982), 169-218. MR 684171 (86j:32063)

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Keywords: Nonlinear eigenvalue problem, principal eigenvalue and eigenfunction, bifurcation from infinity
Article copyright: © Copyright 1987 American Mathematical Society

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