Examples of open manifolds with almost quadratic volume growth and infinite Betti numbers

2025/05/27 by Huihong Jiang, Jiang, Huihong
Mathematics · #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals

paper · pdf · doi:10.48550/arxiv.2505.20599

openalex publication_date 2025/05/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We construct a family of examples of complete (2+n)-dimensional (n≥ 2) open manifolds with positive Ricci curvature, sectional curvature bounded from below and infinite Betti numbers b₂,b_n, moreover its volume growth can be arbitrarily close to quadratic volume growth. Compared with some known result of finite topology for manifolds with nonnegative Ricci curvature and lower sectional curvature bound, it makes sense to ask whether complete manifolds with such curvature bounds must be of finite topological type or not provided with at most quadratic volume growth.

Citations

• The codimension one homology of a complete manifold with nonnegative Ricci curvature
• Manifolds with Quadratic Curvature Decay and Slow Volume Growth
• Differential equations on riemannian manifolds and their geometric applications

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