Liouville theorem for bounded harmonic functions on manifolds and graphs satisfying non-negative curvature dimension condition

Abstract

Brighton [Bri13] proved the Liouville theorem for bounded harmonic functions on weighted manifolds satisfying non-negative curvature dimension condition, i.e. $CD(0,\infty ).$ In this paper, we provide a new proof of this result by using the reverse Poincaré inequality. Moreover, we adopt this approach to prove the Liouville theorem for bounded harmonic functions on graphs satisfying the $CD(0,\infty )$ condition.

9 pages, we added the proof of Brighton's Liouville theorem. Any comment is welcome

References in corpus (1)

• Equivalent Properties of CD Inequality on Graph

Cited by in corpus (2)

• Liouville property and non-negative Ollivier curvature on graphs
• Li-Yau inequality for unbounded Laplacian on graphs