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

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) ( M n , g ) is a Riemannian manifold satisfying such curvature bounds for $$k=2$$ 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 π ₁(M) π 1 ( M ) . If (Mⁿ,g) ( M n , g ) is a Riemannian manifold satisfying such bounds for k≥ 2 k ≥ 2 , then we show that M has at most $$(k-1)$$ ( k - 1 ) -dimensional behavior at large scales. If k=n=dim(M) k = n = dim ( M ) , so that the integral lower bound is on the scalar curvature, assuming in addition that the $$(n-2)$$ ( n - 2 ) -Ricci curvature is non-negative, we prove that the dimension drop at large scales improves to $$n-2$$ n - 2 . From the above results we deduce topological restrictions, such as upper bounds on the first Betti number.