A contractible complete 3-manifold with nonnegative scalar curvature is diffeomorphic to R³, and an open handlebody interior admits such a metric only if its genus is at most 1.
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Topological Rigidity of Contractible 3-Manifolds and Handlebody Interiors under Nonnegative Scalar Curvature
A contractible complete 3-manifold with nonnegative scalar curvature is diffeomorphic to R³, and an open handlebody interior admits such a metric only if its genus is at most 1.