# Article

 Title: Invariance properties of the Laplace operator  (English) Author: Eichhorn, Jürgen Language: English Journal: Proceedings of the Winter School "Geometry and Physics" Volume: Issue: 1989 Year: Pages: [35]-47 . Category: math . Summary: [For the entire collection see Zbl 0699.00032.] \par 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 $\bar {\cal C}\sp k\sb P$ of the space ${\cal C}\sb P$ 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 $\nabla\sp{\omega}$ is proved to be a property of the whole component comp($\omega$) of a connection $\omega\in {\cal C}\sb P$ in the completion $\bar {\cal C}\sp k\sb P$. The result follows from the fact that the essential spectrum of the Laplacian $\Delta\sp{\omega}$ is the same for all $\omega$ lying in the mentioned component. MSC: 53C20 MSC: 53C21 MSC: 58G25 MSC: 58J50 idZBL: Zbl 0717.53028 idMR: MR1061787 . Date available: 2009-07-13T21:24:11Z Last updated: 2012-09-18 Stable URL: http://hdl.handle.net/10338.dmlcz/701460 .

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