Non-compact $$\text {RCD}(0,N)$$ RCD ( 0 , N ) Spaces with Linear Volume Growth

Abstract

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