Quadratic Scalar Curvature Decay and Uniform Positivity in Dimensions Four through Seven

Zhehui Wang; Jintian Zhu

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arXiv · 2609.03676

Quadratic Scalar Curvature Decay and Uniform Positivity in Dimensions Four through Seven

Zhehui Wang·Jintian Zhu

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Abstract

Let $4\le n\le7$ and let $(M^n,g)$ be a complete, connected, orientable, noncompact Riemannian manifold of positive scalar curvature. We prove that if the asymptotic quadratic scalar curvature coefficient of $g$ is greater than $(n-1)/n$, then $M$ carries a complete smooth metric whose scalar curvature is at least one. The threshold $(n-1)/n$ and the strict inequality are optimal. This confirms the second part of Gromov's critical rate of decay conjecture in dimensions four through seven.

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Zhehui Wang, Jintian Zhu. 2026-09-03. Quadratic Scalar Curvature Decay and Uniform Positivity in Dimensions Four through Seven. https://arxiv.org/abs/2609.03676

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