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

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Abstract: In this paper, we study noncompact complete Riemannian -manifolds with which are not pointwise conformal to subdomains of any compact Riemannian -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.

**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**

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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