Nonflat Metrics with Nonnegative Scalar Curvature That Are Flat at Infinity: A Counterexample to a Conjecture of Gromov

In his list of questions on scalar curvature, Gromov states that a complete Riemannian manifold X with Sc(X) ≥ 0 which is isometric at infinity to a complete flat manifold X_fl is flat, provided there is a homomorphism π₁(X) → π₁(X_fl) compatible with the isometry at infinity. He conjectures that this hypothesis is not needed when π₁(X_fl) acts on R^n by parallel translations, and remarks that X can then be simply connected. We show that this conjecture is false in every dimension n ≥ 3. On R² × S^{n−2} we write down a complete metric dr² + f(r)² dτ² + h(r)² g_{S^{n−2}}, determined by one smooth step function, whose scalar curvature is nonnegative, and positive on an open set, and which outside a compact set is isometric to the complement of a compact set in the flat manifold S¹ × R^{n−1} = R^n/Z. For n ≥ 4 these manifolds are simply connected and spin. The proof is elementary: the sign of the scalar curvature is read off from a one-line identity. Quotients of products with Euclidean spaces give such examples for all flat ends T^k × R^N with N ≥ 2, and a cut-and-paste gives examples that are isometric at infinity to T^{n−1} × R, for a class of flat tori T^{n−1} which includes the standard ones. The mechanism, that of Witten's Kaluza–Klein bubble, is not new: the circle at infinity bounds a disc. Hao, Hu, Liu and Shi refuted the hyperbolic analogue of the conjecture by a gluing construction of this kind, and for n = 4 the existence of a metric as above also follows by combining published results: the Lohkamp-type compactification step in the positive mass theorem of Chen, Liu, Shi and Zhu, which does not use their incompressibility hypothesis, applied to the Euclidean Reissner–Nordström metric of negative mass on R² × S² (Brill and Horowitz). We claim as new only the explicit elementary construction, which is valid for all n ≥ 3, including n = 3, the simply connected spin examples it gives for every n ≥ 4, and the observation that Gromov's conjecture is false. This is an unrefereed note.