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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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