A Liouville theorem on asymptotically Calabi spaces

Abstract

In this paper, we will study harmonic functions on the complete and incomplete spaces with nonnegative Ricci curvature which exhibit inhomogeneous collapsing behaviors at infinity. The main result states that any nonconstant harmonic function on such spaces yields a definite exponential growth rate which depends explicitly on the geometric data at infinity.

Minor updates. The post [arXiv:1906.03368v1] has been split into two papers. This one is essentially extracted from Section 5 and the Appendix in [arXiv:1906.03368v1]. To appear in Calculus of Variations and Partial Differential Equations