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

TL;DR

This work analyzes manifolds with almost non-negative Ricci curvature together with integral lower bounds on the sum of the lowest $k$ Ricci eigenvalues, denoted $\mathsf{R}_k$. It establishes a continuity theorem (CT1) for integral functionals of $\mathsf{R}_k$ under Gromov–Hausdorff convergence and uses it to prove macroscopic 1D structure when $k=2$, with strong metric and topological consequences, and to bound large-scale behavior by $(k-1)$-dimensions for general $k\ge2$, improving to $(n-2)$ when $k=n$ under extra non-negativity assumptions on $(n-2)$-Ricci curvature. The manuscript then develops the theory of thin metric spaces (SNP framework) showing every thin space sits near a 1D spine with explicit width bounds and a uniform 1-Urysohn width; these ideas are extended to the large-scale geometry of manifolds, establishing how integral curvature bounds enforce dimension drops for tangent cones at infinity and yield Betti-number control. Overall, the paper links integral curvature bounds to macroscopic dimensional constraints, with consequences for volume growth, ends, fundamental groups, and universal covers, enriching the landscape around the Yau/Gromov-type conjectures on scalar curvature in non-collapsed and collapsed settings.

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 $(M^n,g)$ 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 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 $π_1(M)$. If $(M^n,g)$ is a Riemannian manifold satisfying such bounds for $k\geq 2$, 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 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.

Theorems & Definitions (105)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7: Jiang-Naber
• Definition 2.1: $\delta$-GH maps
• Definition 2.2: pGH-convergence
• Definition 2.3: pmGH-convergence