An almost rigidity Theorem and its applications to noncompact RCD(0,N) spaces with linear volume growth

TL;DR

This paper proves an almost rigidity theorem for RCD(0,N) spaces, showing near-cylindrical structure when volume conditions are nearly maximal, and applies it to analyze geometric and harmonic properties of noncompact spaces with linear volume growth.

Contribution

It introduces an almost rigidity theorem for RCD(0,N) spaces and applies it to understand geometric structure and harmonic functions in noncompact spaces with linear volume growth.

Findings

Domains with near-maximal volume are close to cylinders.

Diameter of geodesic spheres grows sublinearly.

Harmonic functions with polynomial growth do not exist.

Abstract

The main results of this paper consists of two parts. Firstly, we obtain an almost rigidity theorem which says that on a RCD(0, N) space, when a domain between two level sets of a distance function has almost maximal volume compared to that of a cylinder, then this portion is close to a cylinder as a metric space. Secondly, we apply this almost rigidity theorem to study noncompact RCD(0, N) spaces with linear volume growth. More precisely, we obtain the sublinear growth of diameter of geodesic spheres, and study the non-existence of harmonic functions with polynomial growth on such RCD(0,N) spaces.