Invariance properties of the Laplace operator

Proceedings of the Winter School "Geometry and Physics", GDML_Books, (1990), p. [35]-47 / Harvested from

Résumé

[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 ${\bar{𝒞}}_{P}^{k}$ 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 $\nabla^{ω}$ is proved to be a property of the whole component comp( $ω$ ) of a connection $ω \in 𝒞_{P}$ in the completion ${\bar{𝒞}}_{P}^{k}$ . The result follows from the fact that the essential spectrum of the Laplacian $Δ^{ω}$ is the same for all $ω$ lying in the mentioned component.

MR MR1061787 | Zbl 0717.53028

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/