Colloquium MASAMUNE, Jun “Constructions of non-trivial integrable harmonic functions on narrow or ample manifolds (joint with A. Grigoryan and M. Murata)”

Date
2016-11-2 16:30 - 18:00
Place
Room 501, Faculty of Science Bld. #4, Hokkaido University
Speaker/Organizer
MASAMUNE, Jun
 
16:30–17:00 Tea time
17:00–18:00 MASAMUNE, Jun

Title:Constructions of non-trivial integrable harmonic functions on narrow or ample manifolds (joint with A. Grigoryan and M. Murata)

Abstract:We say a Riemannian manifold $M$ enjoys $\mathcal F$-Liouville property when the set of functions $\mathcal F$ on $M$ is trivial. It is known that certain $\mathcal F$-Liouville properties manifest analytic properties of $M$; for instance, $L^2$-Liouville property and the essential self-adjointness of the Laplacian on $M$, and $L^\infty$-Liouville property and the recurrence of Brownian motion of $M$, etc. On the other hand, the meaning of the $L^1$-Liouville property, the most fundamental one, has been unclear. In this talk, we will learn that $L^1$-Liouville property is closely related with the behavior of Brownian motion at infinity, and arrive at two new classes of manifolds "narrow" and "ample". I will keep the talk simple and accessible also for non experts.