Apparaît dans la collection : Not Only Scalar Curvature Seminar

In the talk, I will present a joint work with Alessandro Cucinotta (University of Oxford), where 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 $(M^n,g)$ 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 submanifold. From this, we deduce several metric and topological consequences: $M$ has at most linear volume growth and at most two ends, the first Betti number of $M$ is bounded above by 1, and there is precise information on elements of infinite order in $π_1(M)$. If $(M^n,g)$ 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=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. Such results should be read in the broader framework of Gromov’s conjectures about 2-dimensional drop at large scale of manifolds with positive scalar curvature.