On manifolds with almost non-negative Ricci curvature and integrally-positive k th -scalar curvature

Journal article

We consider manifolds with almost non-negative Ricci curvature and strictly positive integral lower bounds on the sum of the lowest k eigenvalues of the Ricci tensor. If (Mn, g) is a Riemannian manifold satisfying such curvature bounds for k=2, then we show that M is contained in a neighbourhood of controlled width of an isometrically embedded 1-dimensional sub-manifold. From this, we deduce several metric and topological consequences: M has at most linear volume growth and at most two ends, it has bounded 1-Urysohn width, the first Betti number of M is bounded above by 1, and there is precise information on elements of infinite order in π1(M). If (Mn, g) is a Riemannian manifold satisfying such bounds for k≥2, then we show that M has at most (k-1)-dimensional behavior at large scales. If k=n=dim(M), so that the integral lower bound is on the scalar curvature, assuming in addition that the (n-2)-Ricci curvature is non-negative, we prove that the dimension drop at large scales improves to n-2. From the above results we deduce topological restrictions, such as upper bounds on the first Betti number.

Authors: Cucinotta, A; Mondino, A

• Alternative title: On manifolds with almost non-negative Ricci curvature..
• Publication status: Published
• Peer review status: Peer reviewed
• Publisher: Springer
• Journal: Mathematische Annalen
• Volume: 394
• Issue: 2
• Article number: 49
• Publication date: 2026-02-15
• Acceptance date: 2026-01-29
• DOI: 10.1007/s00208-026-03406-8
• EISSN: 1432-1807
• ISSN: 0025-5831
• Language: English