Graduate Lecture

Graduate Lecture: Existence of non-constant integrable harmonic functions on Riemannian manifolds

A Lioville property (of harmonic functions) refers to a property of a space not admitting a non-constant harmonic function belonging to a certain class of functions. Since Liouville’s discovery for a Euclidean space, various types of Liouville properties have been investigated for a Riemannian manifold. In particular, a celebrated work of Yau shows that a complete Riemannian manifold enjoys the L^p Liouville property with finite p>1; however, it fails for p=1. The L^1 Liouville property has remained as an important open issue in Geometric Analysis until now. In this lecture serie, we first review various Liouville type properties of a Riemannian manifold and then learn that the L^1 Liouville property is closely related to long-term behaviors of Brownian motion of the manifolds. Moreover, we will see how it fails and holds through a new theory and by examples. Those new results were obtained in an on-going collaborative effort by A. Grigoryan, M. Murata, and the speaker.

Course material

• poster

Schedule

• Monday, May 30th, 11:10-12:40, WIL B 321
• Monday, June 6th, 11:10-12:40, WIL C 129
• Tuesday, June 7th, 13:00-14:30, HSZ/403/H

Course information

• Speaker:
  Prof. Dr. Jun Masamune
  (Tohoku University Japan)