On manifolds with nonnegative Ricci curvature and the infimum of volume growth order $<2$

TL;DR

This paper establishes rigidity theorems for noncompact manifolds with nonnegative Ricci curvature and volume growth order less than 2, characterizing their universal covers and harmonic functions.

Contribution

It proves new rigidity results linking volume growth, flatness, and harmonic functions on manifolds with nonnegative Ricci curvature.

Findings

Universal cover has Euclidean volume growth iff the manifold is flat with an (n-1)-dimensional soul.

Existence of a nonconstant linear growth harmonic function characterizes the manifold as a product with 7R.

Manifolds with volume growth order < 2 exhibit specific geometric and harmonic properties.

Abstract

We prove two rigidity theorems for open (complete and noncompact) $n$-manifolds $M$ with nonnegative Ricci curvature and the infimum of volume growth order $<2$. The first theorem asserts that the Riemannian universal cover of $M$ has Euclidean volume growth if and only if $M$ is flat with an $n-1$ dimensional soul. The second theorem asserts that there exists a nonconstant linear growth harmonic function on $M$ if and only if $M$ is isometric to the metric product $\mathbb{R}\times N$ for some compact manifold $N$.