On the geometry at infinity of manifolds with linear volume growth and nonnegative Ricci curvature

2023/10/01 by Xingyu Zhu, Zhu, Xingyu · 1 citation
Mathematics · Physics and Astronomy · #53C21 #53C23 #Advanced Differential Geometry Research #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds

paper · pdf · doi:10.48550/arxiv.2310.00640

openalex publication_date 2023/10/01 · openalex created_date 2023/10/04 · openalex updated_date 2026/07/28

Abstract

We prove that an open manifold with nonnegative Ricci curvature, linear volume growth and noncollapsed ends always splits off a line at infinity. This completes the final step to prove the existence of isoperimetric sets given large volumes in the above setting. We also find that under our assumptions, the diameter of the level sets of any Busemann function are uniformly bounded as opposed to a classical result stating that they can have sublinear growth when ends are collapsing. Moreover, some equivalent characterizations of linear volume growth are given. Finally, we construct an example to show that for manifolds in our setting, although their limit spaces at infinity are always cylinders, the cross sections can be nonhomeomorphic.

Cited by

• Uniqueness of the asymptotic limits for Ricci-flat manifolds with linear volume growth

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