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

Abstract 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 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.