[For the entire collection see Zbl 0699.00032.] The paper deals with a special problem of gauge theory. In his previous paper [The invariance of Sobolev spaces over noncompact manifolds, Partial differential equations, Proc. Symp., Holzhaus/GDR 1988, Teubner- Texte Math. 112, 73-107 (1989; Zbl 0681.58011)], the author introduced the Sobolev completions ${\overline{𝒞}}_P^k$ of the space ${𝒞}_P$ of all G-connections on a G-principal fibre bundle P. In the present paper, under the assumption of bounded curvatures and their covariant derivatives up to order k, the closedness of the subspace im ${\nabla }^{\omega }$ is proved to be a property of the whole component comp($\omega$) of a connection $\omega \in {𝒞}_P$ in the completion ${\overline{𝒞}}_P^k$. The result follows from the fact that the essential spectrum of the Laplacian ${\Delta }^{\omega }$ is the same for all $\omega$ lying in the mentioned component.

EUDML-ID : urn:eudml:doc:220093
Mots clés:

@article{701460,
     title = {Invariance properties of the Laplace operator},
     booktitle = {Proceedings of the Winter School "Geometry and Physics"},
     series = {GDML\_Books},
     publisher = {Circolo Matematico di Palermo},
     address = {Palermo},
     year = {1990},
     pages = {[35]-47},
     mrnumber = {MR1061787},
     zbl = {0717.53028},
     url = {http://dml.mathdoc.fr/item/701460}
}

Eichhorn, Jürgen. Invariance properties of the Laplace operator, dans Proceedings of the Winter School "Geometry and Physics", GDML_Books,  (1990), pp. [35]-47. http://gdmltest.u-ga.fr/item/701460/