The Spectrum of the Laplacian in Riemannian Geometry

To any compact Riemannian manifold (M, g) (with or without boundary), we can associate a second-order partial differential operator, the Laplace operator ∆, defined by ∆(f) = −div(grad(f)) for f ∈ L(M, g). We will also sometimes write ∆g for ∆ if we want to emphasize which metric the Laplace operator is associated with. The set of eigenvalues of ∆ (the spectrum of ∆, or of M), which we will write as spec(∆) or spec(M, g), then forms a discrete sequence 0 = λ0 ≤ λ1 ≤ λ2 ≤ . . . → ∞. For simplicity, we will assume that M is connected; this will for example imply that the smallest eigenvalue, λ0, occurs with multiplicity 1. Note that the Laplacian also acts on p-forms in addition to functions via the definition ∆ = −(dδ + δd), where δ is the adjoint of d with respect to the Riemannian structure on the manifold. This aspect of the Laplacian will not be treated in this paper, the focus being the ordinary Laplacian acting on functions. With that in mind, there are two broad questions that are at the heart of spectral geometry: