Compactness theorems for differential forms.

*(English)*Zbl 0649.58002This 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 |

##### Keywords:

Whitney flat norm; Hodge decomposition and elliptic estimates; Federer- Fleming compactness; heat kernel
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\textit{S. Kichenassamy}, Commun. Pure Appl. Math. 42, No. 1, 47--53 (1989; Zbl 0649.58002)

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