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Complete conformal metrics with prescribed scalar curvature on subdomains of a compact manifold

Published online by Cambridge University Press:  22 January 2016

Shin Kato  and
Shin Nayatani
Shin Kato
Affiliation:
Department of Mathematics, Nara Women’s University, Nara 630, Japan
Shin Nayatani
Affiliation:
Mathematical Institute, Tohoku University, Sendai 980, Japan
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Let (M, g) be a Riemannian manifold of dimension n≥ 3 and ĝanother metric on M which is pointwise conformai to g. It can be written where u is a positive smooth function on M. Then the curvature of g is computable in terms of that of g and the derivatives of u up to second order. In particular, if S and S denote the scalar curvature of g and g respectively, they are related by the equation

where ▽u denotes the Laplacian of u, defined with respect to the metric g.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 132 , December 1993 , pp. 155 - 173
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1993

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