Contractible 3-manifold and positive scalar curvature (I) (Q6593658)

The renowned Whitehead 3-manifold \(\mathrm{Wh}^3\) is the main example of an open 3-manifold which is contractible but not homeomorphic to the Euclidean 3-space \(\mathbb R^3\).\N\NThis interesting manifold was discovered by \textit{J. H. C. Whitehead} around 1934 [Q. J. Math., Oxf. Ser. 6, 268--279 (1935; JFM 61.0607.01)], actually as a counterexample to his attempted proof of the Poincaré Conjecture. Since then, it represents the prototype of an open 3-manifold with exotic topological behavior at infinity.\N\NThe main features of the Whitehead manifold are that: it is not simply connected at infinity (intuitively, this means that there are loops at infinity that cannot be killed staying close to infinity); it is not tame (which means that it is not homeomorphic to a compact manifold with a closed subset of the boundary removed); and the product \(\mathrm{Wh}^3 \times \mathbb R\) is actually diffeomorphic to the standard \(\mathbb R^4\).\N\NHence, if a condition seems to hold for every contractible manifold, the first test to do is to check its validity for Whitehead's manifold.\N\NThe question the author of the paper studied in the last years is the following one: ``Is any complete contractible 3-manifold with positive scalar curvature homeomorphic to \(\mathbb R^3\)?\N\NIn the paper under consideration, he shows that the Whitehead manifold does not admit any complete metric with positive scalar curvature.