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

Bobo Hua

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

Metric Geometry 2018-06-15 v2 Combinatorics Functional Analysis Probability

Authors: Bobo Hua

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 Poincar\'e inequality. Moreover, we adopt this approach to prove the Liouville theorem for bounded harmonic functions on graphs satisfying the $CD(0,\infty)$ condition.

Keywords

Liouville theorem bounded harmonic functions weighted manifolds graphs curvature dimension condition reverse Poincaré inequality

Cite

@article{arxiv.1702.00961,
  title  = {Liouville theorem for bounded harmonic functions on manifolds and graphs satisfying non-negative curvature dimension condition},
  author = {Bobo Hua},
  journal= {arXiv preprint arXiv:1702.00961},
  year   = {2018}
}

Comments

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

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

Abstract

Keywords

Cite

Comments

Related papers