Abstract
This chapter introduces basic concepts and methods from analysis, in particular, the Laplace-Beltrami operator. The essential properties of its spectrum are shown and relationships with the underlying geometry are discussed. The discussion then turns to the operation of the Laplace operator on differential forms and de Rham cohomology groups, together with the essential tools from elliptic PDE for treating these groups. The existence of harmonic forms representing cohomology classes is proved both by a variational method and by the heat flow method. The spectrum of the Laplacian on differential forms is also discussed.
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Notes
- 1.
Please do not confuse the H of the Sobolev space H p 1,2(M) (which may stand for Hilbert) with the H of the cohomology group H p (which stands for “homology”).
- 2.
In the bibliography, a superscript will indicate the edition of a monograph. For instance,72017 means 7th edition, 2017.
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Jost, J. (2017). Chapter 3 The Laplace Operator and Harmonic Differential Forms. In: Riemannian Geometry and Geometric Analysis. Universitext. Springer, Cham. https://doi.org/10.1007/978-3-319-61860-9_3
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DOI: https://doi.org/10.1007/978-3-319-61860-9_3
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