Abstract
In this paper, the authors consider the 3-dimensional local Calabi flow on noncompact 3-manifolds with a complete metric of nonpositive scalar curvature and negative somewhere. Then long-time existence and asymptotic convergence of a subsequence of solutions of the flow are claimed. For its applications, it is proved that every noncompact 3-manifold admits a complete extremal metric. In particular, there exists a complete metric on a noncompact 3-manifold, a counterexample to the Yamabe problem, which is conformal to a complete extremal metric of non-constant scalar curvature.
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