Atsushi Kasue

A convergence theorem for Riemannian manifolds and some applications

The purpose of the present paper is first to reformulate a Lipschitz convergence theorem for Riemannian manifolds originally introduced by Gromov [17] and secondly to give some applications of the theorem to a class of open Riemannian manifolds.

Random walks and Kuramochi boundaries of infinite networks

In this paper, we study a connected non-parabolic, or transient, network compactified with the Kuramochi boundary, and show that the rand om walk converges almost surely to a random variable valued in the harmonic boundary, and a function of finite Dirichlet energy converges along the random walk to a random variable almost surely and in L 2 . We also give integral representations of solutions of Pois son equations on the Kuramochi compactification.

Convergence of Riemannian manifolds and Albanese tori

We may view J4(M) as a map of the set of isometry classes of compact Riemannian manifolds M onto that of flat tori. Given a positive integer m, a nonnegative number x and a positive one D, we write ώ(m,x,ΰ) for the set of isometry classes of compact Riemmanian mmanifolds M such that the Ricci curvature of M is bounded from below by — (m — l)κ and the diameter of M is bounded from above by D. Then according to Gromov [7], for a Riemannian manifold M in sH>{rn, x, D), the dimension of the Albanese torus A(M), namely, the first Betti number bi(M), has an upper bound depending only on the dimension m of M and xD. Using this, we shall show the following