Optimal asymptotic volume ratio for noncompact 3-manifolds with asymptotically nonnegative Ricci curvature and a uniformly positive scalar curvature lower bound

2024/05/15 by Xian-Tao Huang, Shuai Liu, Huang, Xian-Tao +1 · 1 citation
Mathematics · #Advanced Operator Algebra Research #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds

paper · pdf · doi:10.48550/arxiv.2405.09379

openalex publication_date 2024/05/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper, we study 3-dimensional complete non-compact Riemannian manifolds with asymptotically nonnegative Ricci curvature and a uniformly positive scalar curvature lower bound. Our main result is that, if this manifold has k ends and finite first Betti number, then it has at most linear volume growth, and furthermore, if the negative part of Ricci curvature decays sufficiently fast at infinity, then we have an optimal asymptotic volume ratio \limsup_r→∞(Vol(B(p, r)))/(r)≤4kπ. In particular, our results apply to 3-dimensional complete non-compact Riemannian manifolds with nonnegative Ricci curvature and a uniformly positive scalar curvature lower bound.

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