Local and Global Analysis of Eigenfunctions on Riemannian Manifolds
نویسنده
چکیده
This is a survey on eigenfunctions of the Laplacian on Riemannian manifolds (mainly compact and without boundary). We discuss both local results obtained by analyzing eigenfunctions on small balls, and global results obtained by wave equation methods. Among the main topics are nodal sets, quantum limits, and L norms of global eigenfunctions. The emphasis is on the connection between the behavior of eigenfunctions and the dynamics of the geodesic flow, reflecting the relation between quantum mechanics and the underlying classical mechanics. We also discuss the analytic continuation of eigenfunctions of real analytic Riemannian manifolds (M, g) to the complexification of M and its applications to nodal geometry. Besides eigenfunctions, we also consider quasi-modes and random linear combinations of eigenfunctions with close eigenvalues. Many examples are discussed.
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