Some Properties of the Eigenfunctions of The Laplace-Operator on Riemannian Manifolds

Source abstract

Let V be a connected, compact, differentiable Riemannian manifold. If V is not closed we denote its boundary by S . In terms of local coordinates ( x i ), i = 1, 2, … Ν, the line-element dr is given by where gik (x 1 , x 2 , … x N ) are the components of the metric tensor on V We denote by Δ the Beltrami-Laplace-Operator and we consider on V the differential equation (1) Δu + λu = 0.