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

TL;DR

This paper studies the geometric and topological properties of manifolds with specific curvature conditions, revealing constraints on their volume, structure, and topology.

Contribution

The paper introduces new curvature bounds and derives novel topological and metric consequences for manifolds under these conditions.

Findings

Manifolds with curvature bounds for k=2 are near a 1-dimensional sub-manifold.

Such manifolds have linear volume growth and at most two ends.

Topological restrictions include an upper bound on the first Betti number.

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 \documentclass[12pt]{minimal}
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\begin{document}$$(M^n,g)$$\end{document}(Mn,g) is a Riemannian manifold satisfying such curvature bounds for \documentclass[12pt]{minimal}
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\begin{document}$$k=2$$\end{document}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 \documentclass[12pt]{minimal}
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\begin{document}$$\pi _1(M)$$\end{document}π1(M). If \documentclass[12pt]{minimal}
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\begin{document}$$(M^n,g)$$\end{document}(Mn,g) is a Riemannian manifold satisfying such bounds for \documentclass[12pt]{minimal}
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\begin{document}$$k\ge 2$$\end{document}k≥2, then we show that M has at most \documentclass[12pt]{minimal}
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\begin{document}$$(k-1)$$\end{document}(k-1)-dimensional behavior at large scales. If \documentclass[12pt]{minimal}
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\begin{document}$$k=n=\textrm{dim}(M)$$\end{document}k=n=dim(M), so that the integral lower bound is on the scalar curvature, assuming in addition that the \documentclass[12pt]{minimal}
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\begin{document}$$(n-2)$$\end{document}(n-2)-Ricci curvature is non-negative, we prove that the dimension drop at large scales improves to \documentclass[12pt]{minimal}
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\begin{document}$$n-2$$\end{document}n-2. From the above results we deduce topological restrictions, such as upper bounds on the first Betti number.