Noncompact RCD(0,N) spaces with linear volume growth

TL;DR

This paper investigates noncompact RCD(0,N) spaces with linear volume growth, establishing bounds on Busemann function level sets and a splitting theorem, thus extending classical geometric results to non-smooth metric measure spaces.

Contribution

It introduces new bounds on level set diameters and a splitting theorem for noncompact RCD(0,N) spaces with linear volume growth, generalizing manifold results to metric measure spaces.

Findings

Diameter of Busemann level sets grows at most linearly.

Splitting theorem at the noncompact end for spaces with minimal volume growth.

Extension of classical Ricci curvature results to non-smooth spaces.

Abstract

Since non-compact RCD(0, N) spaces have at least linear volume growth, we study noncompact RCD(0, N) spaces with linear volume growth in this paper. One of the main results is that the diameter of level sets of a Busemann function grow at most linearly on a noncompact RCD(0, N) space satisfying the linear volume growth condition. Another main result in this paper is a splitting theorem at the noncompact end for a RCD(0, N) space with strongly minimal volume growth. These results generalize some theorems on noncompact manifolds with nonnegative Ricci curvature to non-smooth settings.