Does anyone have any recommendations for, as the title suggests, a book from which to learn about the Laplacian on Riemanian manifolds, or even just on smooth manifolds? I found this presentation, which deals with it in cursorily and without proofs - in the style of a presentation - and at a deeper-than-introductory level. I do not know much about the Laplacian at all (I'm familiar with smooth manifold theory and Riemannian geometry through Lee's and do Carmo's books, respectively, but none of Lee's books cover the Laplacian and neither does do Carmo's). My main goal is to learn about the isospectral problem, that is, "Can one hear the shape of a drum?"
This (great) book of Peter Buser is about the spectrum of the Laplace Beltrami operator over manifolds. He also gives methods to build isospectral but not isometric manifold based on the Sunada Theorem (which answers the question: "Can one hear the shape of a drum?").
Here is the book entitled "The Laplacian on a Riemannian Manifold". But I did not study it in details. Instead, I would recommend Chapter III of Schoen and Yau's book "Lectures on Differential Geometry, which I have studied in details. It introduces many results about eigenvalues of Riemannian manifolds.
Here's a set of lecture notes by Michael E. Taylor: http://mtaylor.web.unc.edu/notes/functional-analysis-course/
Chapter 8 "Spectral theory" contains some results about spectral theory of the Laplacian on complete Riemannian manifolds: Its extension to a self-adjoint operator, and compactness of the resolvent.
He gives many references, but I haven't found which of them talks about Riemannian manifolds.
He doesn't mention the isospectral problem, if that is what you are looking for.