2025/04/09 by Cucinotta, Alessandro, Mondino, Andrea
#Differential Geometry (math.DG) #FOS: Mathematics #Metric Geometry (math.MG)
paper · doi:10.48550/arxiv.2504.06865
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 and additionally the Ricci curvature is asymptotically non-negative, then we show that M has at most (k-1)-dimensional behavior at large scales. If k=n=\rm dim(M), so that the integral lower bound is on the scalar curvature, assuming in addition that the (n-2)-Ricci curvature is asymptotically non-negative, then 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.