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On the first eigenvalue of non-orientable closed surfaces. (English) Zbl 0932.58035

Let \( (M, g) \) be a 2-dimensional non-orientable closed Riemannian manifold and \( (\widetilde{M}, \widetilde{g}) \) its orientable Riemannian double cover. Let \(\lambda_1(M,g)\) (resp. \(\lambda_1(\widetilde{M},\widetilde{g})\)) denote the first nonzero eigenvalues of the Laplacian for functions on \((M,g)\) (resp. on \((\widetilde{M},\widetilde{g})\)).
The author proves the following two results:
(i) If \( M \) is homeomorphic to \( \mathbb{R} P ^2 , \) then \( \lambda _1 (M, g) > \lambda _1 ( \widetilde{M}, \widetilde{g}) \) for every metric \( g \) on \( M .\)
(ii) If \( M \) is homeomorphic to \( \# ^n \mathbb{R} P ^2\) (the connected sum of \(n\) copies of the real projective plane) \((n \geq 2)\), there exists a metric \( g \) on \( M \) such that \( \lambda _1 (M, g) = \lambda _1 ( \widetilde{M}, \widetilde{g})\).

MSC:

58J50 Spectral problems; spectral geometry; scattering theory on manifolds
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