Strengthened injectivity radius bounds for manifolds with positive scalar curvature

2024/04/15 by Richard, Thomas
#Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2404.09573

Abstract

Green's inequality shows that a compact Riemannian manifold with scalar curvature at least n(n-1) has injectivity radius at most π, and that equality is achieved only for the radius 1 sphere. In this work we show how extra topological assumptions can lead to stronger upper bounds. The topologies we consider are \mathbbS²×\mathbbT^n-k-2×ℝ^k for n≤ 7 and 0≤ k≤ 2 and 3-manifolds with positive scalar curvature except lens spaces L(p,q) with p odd. We also prove a strengthened inequality for 3-manifolds with positive scalar curvature and large diameter. Our proof uses previous results of Gromov and Zhu.

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