A contractible 3-manifold with nonnegative scalar curvature and bounded geometry must be R^3, and an open handlebody with such a metric has genus at most 1.
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3-Manifolds with positive scalar curvature and bounded geometry
A contractible 3-manifold with nonnegative scalar curvature and bounded geometry must be R^3, and an open handlebody with such a metric has genus at most 1.