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Invariance properties of the Laplace operator. (English) Zbl 0717.53028
Geometry and physics, Proc. 9th Winter Sch., Srní/Czech. 1989, Suppl. Rend. Circ. Mat. Palermo, II. Ser. 22, 35-47 (1990).
[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 \(\bar {\mathcal C}^ k_ P\) of the space \({\mathcal C}_ P\) 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 \(\omega\in {\mathcal C}_ P\) in the completion \(\bar {\mathcal C}^ k_ P\). 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.
Reviewer: J.Chrastina

MSC:
53C20 Global Riemannian geometry, including pinching
58J50 Spectral problems; spectral geometry; scattering theory on manifolds