Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

An obstruction to the conformal compactification of Riemannian manifolds


Author: Seongtag Kim
Journal: Proc. Amer. Math. Soc. 128 (2000), 1833-1838
MSC (1991): Primary 53C21; Secondary 58G30
DOI: https://doi.org/10.1090/S0002-9939-99-05207-7
Published electronically: September 30, 1999
MathSciNet review: 1646195
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper, we study noncompact complete Riemannian $n$-manifolds with $n\ge 3$ which are not pointwise conformal to subdomains of any compact Riemannian $n$-manifold. For this, we compare the Sobolev Quotient at infinity of a noncompact complete Riemannian manifold with that of the singular set in a compact Riemannian manifold using the method for the Yamabe problem.


References [Enhancements On Off] (What's this?)

  • 1. T. Aubin, Equations différentielles non linéaires et problème de Yamabe concernant la courbure scalaire , J. Math. Pures Appl. n 55 (1976) 269-296. MR 55:4288
  • 2. P. Aviles and R. McOwen, Complete conformal metrics with negative scalar curvature in compact Riemannian manifolds, Duke Math. J. 56 (1988) 225-239. MR 89b:58224
  • 3. P. Delanoe, Generalized stereographic projections with prescribed scalar curvature, Geometry and nonlinear partial differential equations, (Contemp. Math., vol 127 pp. 17-25) Providence, A.M.S. 1992 MR 93e:53045
  • 4. S. Kim, Scalar curvature on noncompact complete Riemannian manifolds, Nonlinear Analysis 26 (1996) 1985-1993. MR 97a:53056
  • 5. R. McOwen, Singularities and the conformal scalar curvature equation, Geometry and nonlinear partial differential equations (Contemp. Math., vol 127 pp. 221-233) Providence, A.M.S. 1992 MR 94b:53076
  • 6. K. Nomizu and H. Ozeki, The existence of complete Riemannian metrics, Proc. Amer. Math. Soc. 12 (1961) 889-891. MR 24:A3610
  • 7. R. Schoen, Conformal deformation of a Riemannian metric to constant scalar curvature, J. Diff. Geom. 20 (1984) 479-495. MR 86i:58137
  • 8. R. Schoen and S. Yau, Conformally flat manifolds, Kleinian groups and scalar curvature, Invent. Math. 92 (1988) 47-71. MR 89c:58139
  • 9. H. Yamabe, On a deformation of Riemannian structures on compact manifolds, Osaka Math. J. 12 (1960) 21-37. MR 23:A2847

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 53C21, 58G30

Retrieve articles in all journals with MSC (1991): 53C21, 58G30


Additional Information

Seongtag Kim
Affiliation: Department of Mathematics, Sungkyunkwan University, Suwon 440-746, South Korea
Email: stkim@yurim.skku.ac.kr

DOI: https://doi.org/10.1090/S0002-9939-99-05207-7
Keywords: Scalar curvature, Yamabe problem, conformal metric
Received by editor(s): July 22, 1998
Published electronically: September 30, 1999
Additional Notes: The author was supported in part by KOSEF96070102013 and BSRI 97-1419 Ministry of Education.
Communicated by: Christopher Croke
Article copyright: © Copyright 2000 American Mathematical Society

American Mathematical Society