theory of differential forms on manifolds

Author:
Chad Scott

Journal:
Trans. Amer. Math. Soc. **347** (1995), 2075-2096

MSC:
Primary 58A14; Secondary 58G03

DOI:
https://doi.org/10.1090/S0002-9947-1995-1297538-7

MathSciNet review:
1297538

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Abstract: In this paper, we establish a Hodge-type decomposition for the space of differential forms on closed (i.e., compact, oriented, smooth) Riemannian manifolds. Critical to the proof of this result is establishing an estimate which contains, as a special case, the result referred to by Morrey as Gaffney's inequality. This inequality helps us show the equivalence of the usual definition of Sobolev space with a more geometric formulation which we provide in the case of differential forms on manifolds. We also prove the boundedness of Green's operator which we use in developing the theory of the Hodge decomposition. For the calculus of variations, we rigorously verify that the spaces of exact and coexact forms are closed in the norm. For nonlinear analysis, we demonstrate the existence and uniqueness of a solution to the -harmonic equation.

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1995-1297538-7

Keywords:
Differential form,
Hodge decomposition,
harmonic integral,
Sobolev space

Article copyright:
© Copyright 1995
American Mathematical Society