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Scalar curvature and conformal deformations of noncompact Riemannian manifolds. (English) Zbl 0899.53033
The main aim of this paper is to study the following question. Problem. Let \(K\in C^\infty (M)\), where \((M,g)\) is an \(m\)-dimensional, connected, complete, noncompact Riemannian manifold, \(m\geq 3\). Does there exist a conformal deformation \(g_u\) \((g_u= u^{4/(m-2)}g\), \(u\) a smooth, positive function on \(M)\) of \(g\) such that \(K\) is the scalar curvature of \(g_u\) and \(g_u\) is complete?
Essentially, the authors study in detail the case where \(K(x)\) is negative outside a compact set. They refine the results obtained by Avilès-McOwen, Jin, Gui-Wang and establish some uniqueness theorems as well.

MSC:
53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
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