We give the necessary and sufficient conditions for Lagrangian submanifolds in Kähler manifolds to be biharmonic. We classify biharmonic PNMC Lagrangian H-umbilical submanifolds in the complex space forms. Furthermore, we classify biharmonic PNMC Lagrangian surfaces in the two-dimensional complex space forms.

In this paper, we consider the (essential) spectrum of the discrete Laplacian of an infinite graph. We introduce a new quantity for an infinite graph, in terms of which we give new lower bound estimates of the (essential) spectrum and give also upper bound estimates when the infinite graph is bipartite. We give sharp estimates of the (essential) spectrum for several examples of infinite graphs.

We give a discrete analogue of the harmonic morphism between two Riemannian manifolds. Roughly speaking, this is a mapping between two graphs preserving local harmonic functions. We characterize harmonic morphisms in terms of horizontal conformality. Many examples including coverings, non-complete extended p-sums and collapsings are given. Introducing the horizontal and vertical Laplacians, the Green kernel estimates are obtained for the harmonic morphism. As applications, a general and sharp estimate of the Green kernel for an infinite tree is obtained.

P. Lévy introduced a generalized notion of Brownian motion in his monograph “Processus stochastiques et mouvement brownien” by taking the time parameter space to be a general metric space. Let (M, d) be a metric space and let O be a fixed point of M called the origin. Following his definition, a Brownian motion parametrized with the metric space (M, d) is a Gaussian system ℬ = {B(m); m ∈ M} such that the difference B(m) − B(m′) is a random variable with mean zero and variance d(m, m′), and that B(O) = 0.

Let (, g) be the standard Euclidean space or a Riemannian symmetric space of non-compact type of rank one. Let G be the identity component of the Lie group of all isometries of (, g). Let Γ be a discrete subgroup of G acting fixed point freely on whose quotient manifold MΓ is compact.

Let M be a compact, oriented Riemannian manifold of dimension dr and let Γ be the fundamental group of M. For a finite dimensional representation ρ of Γ on a vector space F, Ray and Singer [10] have defined the analytic torsion T(M, ρ) as follows: We denote by E the vector bundle over M with typical fibre F defined by the representation ρ.