1993
DOI: 10.1017/s0027763000004694
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Complete conformal metrics with prescribed scalar curvature on subdomains of a compact manifold

Abstract: 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 equationwhere ▽u denotes the Laplacian of u, defined with respect to the metric g.

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Cited by 7 publications
(4 citation statements)
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“…Plugging (6. 19), (6.20) and (6.21) into (6.16) yields In what follows, we frequently use the following type of the change of variables such as…”
Section: 2mentioning
confidence: 99%
See 1 more Smart Citation
“…Plugging (6. 19), (6.20) and (6.21) into (6.16) yields In what follows, we frequently use the following type of the change of variables such as…”
Section: 2mentioning
confidence: 99%
“…where M ⊂ R n is a compact m-dimensional submanifold and Γ R n is the fundamental solution of Laplace's equation on R n . Functions of type U M were employed by Delanoë [10] and Kato and Nayatani [19] for studying the singular Yamabe problem.…”
Section: Introductionmentioning
confidence: 99%
“…for some positive constants $C_{3},$ $C_{4}$ and $\alpha:=l-2+4d/(n-2)$ . In [10], we assumed $d\leq(n-2)/2$ for this fact. However, if we do not assert the completeness of the metric $u_{1}^{q-1}g$ , then the same consequence as above holds also when $(n-2)/2<d<n-2$.…”
Section: The Uniqueness Of Solutions Of the Maximal Ordermentioning
confidence: 99%
“…Under stronger structural assumptions on K := S n \ Ω, one can show the converse statement. In particular, if K = S n \ K is a finite union of Lipschitz submanifolds of dimension k ≤ (n − 2)/2 then there is a g, solving the Yamabe problem, such that R(g) = 0, see [Del92], [KN93], [MM92]. See also [BPS16] for the periodic setting with equator as singular set in n ≥ 5 sphere.…”
Section: Introductionmentioning
confidence: 99%