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

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.

arXiv comment40 pages

References in corpus (7)

• Metric measure spaces with Riemannian Ricci curvature bounded from below
• The splitting theorem in non-smooth context
• Nonlinear diffusion equations and curvature conditions in metric measure spaces
• Sobolev spaces on warped products
• Local spectral convergence in $RC{D}^{\ast }(K,N)$ spaces
• Weyl's law on $RC{D}^{\ast }(K,N)$ metric measure spaces
• Noncompact RCD(0,N) spaces with linear volume growth