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Compactness theorems for differential forms. (English) Zbl 0649.58002
This paper proves that the set of forms \(\omega\) on a Riemannian manifold such that \(\| \omega \| _{L\quad p}+\| d\omega \| _{L\quad p}\leq 1,\) \((1<p<\infty)\), although not compact in L p in general, is compact for an L p analogue of the Whitney flat norm. The corresponding result with L p replaced by the space of currents with finite mass is also proved. Both follow from the Hodge decomposition and elliptic estimates. Special cases yield very simple proofs of the Federer-Fleming compactness theorem for normal currents and of an L p- version of the div-curl lemma (“compensated compactness”). The relation of the new flat norm to the L p-norm is investigated, as well as its behavior under smoothing by the heat kernel. In this connection, an example is given of a mass minimizing current which is not “smooth” (i.e., not supported by a submanifold).
Reviewer: S.Kichenassamy

MSC:
58A10 Differential forms in global analysis
58J10 Differential complexes
49Q20 Variational problems in a geometric measure-theoretic setting
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