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Analysis on Riemannian manifolds and geometric applications: an introduction. (English) Zbl 0617.53002

These notes were given by the author during the spring term, 1986 at IMPA for graduate students in the differential geometry group.
Chapter I is concerned with eigenvalue problems on compact Riemannian manifolds, applications and eigenvalue comparison theorems. Chapter II is concerned with isoperimetric inequalities and Sobolev spaces. The author starts with isoperimetric constants and considers three cases: 1) complete non-compact manifolds or compact manifolds with boundary and Dirichlet conditions on the boundary, 2) compact manifolds with boundary and Neumann condition on the boundary, 3) closed manifolds. The end of the chapter is dealing with Sobolev inequalities. The author uses symmetrization to estimate the Sobolev constants. Chapter III is devoted to applications of Sobolev inequalities. The first part of this chapter is dealing with P. Li’s method for bounding some geometric or topological invariants such as Betti numbers, while the second part deals with lower bounds for Sobolev constants S(m,g). Chapter IV is dealing with comparison theorems and some geometric applications which include isoperimetric inequalities. The author gives a proof of the volume comparison theorems using Bochner’s formula for the Laplacian 1-forms and he also gives the proof of a sharper isoperimetric inequality in dimension 2.
He mentions that a part of the lectures dealing with abstract spectral theory and its applications to the spectrum of the Laplacian on Riemannian manifolds has been omitted due to a lack of time. Also, he intends to write an appendix on counter-examples in spectral geometry in a further edition of these notes. These notes are presented in an agreeable style and contain a sequence of important propositions in this field of research.
Reviewer: M.Marzouk

MSC:

53-02 Research exposition (monographs, survey articles) pertaining to differential geometry
58-02 Research exposition (monographs, survey articles) pertaining to global analysis
53C20 Global Riemannian geometry, including pinching
58J50 Spectral problems; spectral geometry; scattering theory on manifolds
58J60 Relations of PDEs with special manifold structures (Riemannian, Finsler, etc.)