Positive scalar curvature metrics and aspherical summands

We prove for $n\in{3,4,5}$ that the connected sum of a closed aspherical $n$-manifold with an arbitrary non-compact manifold does not admit a complete metric with nonnegative scalar curvature. In particular, a special case of our result answers a question of Gromov. More generally, we generalize the partial classification result of Chodosh, Li, and Liokumovich to the non-compact domination case with our newly-developed technique. Our result unifies all previous results of this type, and confirms the validity of Gromov’s non-compact domination conjecture for closed aspherical manifolds of dimensions 3, 4, and 5.

💡 Research Summary

The paper by Shuli Chen, Jianchun Chu, and Jintian Zhu addresses the existence of complete metrics with non‑negative (in particular, positive) scalar curvature on connected sums of closed aspherical manifolds with arbitrary non‑compact manifolds in dimensions three, four and five. Their main result, Theorem 1.2, states that if (N^n) is a closed aspherical manifold with (n\in{3,4,5}) and (X^n) is any (possibly non‑compact) (n)‑manifold, then the connected sum (Y=N# X) cannot carry a complete metric of positive scalar curvature. This directly answers a question posed by Gromov concerning punctured aspherical manifolds: a closed aspherical manifold with punctures is precisely such a connected sum with a punctured sphere, and the theorem shows no positive scalar curvature metric exists.

Beyond this specific obstruction, the authors extend the “non‑compact domination conjecture” formulated by Gromov. They prove (Theorem 1.8) that for the same dimensions, if a complete orientable manifold ((M,g)) with positive scalar curvature admits a quasi‑proper map of non‑zero degree onto a closed aspherical (N) whose universal cover is ((n-2))-connected, then a finite cover of (N) must be homotopy equivalent either to the sphere (S^n) or to a connected sum of copies of (S^{n-1}\times S^1). Consequently, Corollary 1.9 confirms Gromov’s non‑compact domination conjecture for closed aspherical manifolds in dimensions three through five. They also obtain a rigidity theorem (Theorem 1.11) for the case of non‑negative scalar curvature: either the same topological conclusion holds, or both (M) and (N) are closed aspherical and the metric on (M) is flat.

The proof strategy builds on the “chain‑closing” program previously used for the aspherical conjecture, which relies on constructing a non‑trivial homology class in the universal cover (\widetilde N) via a proper geodesic line, tubular neighborhoods, and µ‑bubble techniques. In the compact case, the universal cover has uniformly positive scalar curvature, allowing the use of Gromov’s µ‑bubble method and the slice‑and‑dice argument of Chodosh–Li to fill chains with uniformly bounded diameter. In the non‑compact setting, however, the scalar curvature of the connected sum may decay to zero at infinity, destroying the uniform positivity needed for the quantitative filling.

To overcome this, the authors introduce relative homology. They decompose the lifted space (\widetilde Y) into a core region (\widetilde N_\varepsilon) (a compact subset of the universal cover) and infinitely many copies of (\widetilde X) attached along their boundaries. By working with the pair ((\widetilde Y, \bigcup_i \widetilde X_{\varepsilon,i})) they obtain homological properties analogous to a contractible manifold, while effectively “ignoring” the uncontrolled pieces at infinity. They then repeat the chain‑closing construction: a proper line (\widetilde\sigma) in (\widetilde N_\varepsilon), a tubular neighborhood, cutting to produce a hypersurface (\widetilde M_{n-1}) with boundary, solving a Plateau problem, and applying the µ‑bubble method to obtain a codimension‑two submanifold (M_{n-2}). Although (M_{n-2}) may have regions of negative scalar curvature (since the ambient scalar curvature is not uniformly positive), the authors manage to keep its distance from (\widetilde\sigma) arbitrarily large and its stabilized scalar curvature arbitrarily close to the infimum of the ambient scalar curvature.

The crucial technical advance lies in a refined slice‑and‑dice argument adapted to the relative setting. The original Chodosh–Li method breaks a closed three‑manifold into small pieces using minimizing surfaces and then controls diameters via inradius estimates that rely on uniformly positive scalar curvature. In the present work, the authors must handle compact surfaces with boundary and possible negative curvature regions, for which inradius alone does not bound diameter. They develop new diameter estimates (Lemma 2.16) that combine inradius control with careful geometric decomposition, ensuring that each relative chain piece has uniformly bounded diameter and remains disjoint from (\widetilde\sigma). This yields a non‑trivial relative homology class contradicting the contractibility of the core region, thereby establishing the non‑existence of a positive scalar curvature metric on the connected sum.

The paper also discusses the implications for Gromov’s broader program, showing that the techniques developed here unify earlier results concerning SYS manifolds, torus‑type manifolds, and various partial classifications of manifolds admitting positive scalar curvature. By removing the compactness assumption on the dominating manifold, the authors extend the classification to the non‑compact domination setting, confirming Gromov’s conjecture in the low‑dimensional cases without requiring spin assumptions.

In summary, the authors provide a comprehensive treatment of scalar curvature obstructions for non‑compact connected sums involving aspherical summands, introduce innovative relative homology and refined slicing techniques to handle the lack of uniform curvature at infinity, and thereby settle several open questions posed by Gromov for dimensions three through five.