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Asymptotic spectrum of the universal covering of tori. (Spectre asymptotique du revêtement universel des tores.) (French) Zbl 1008.58022
Séminaire de théorie spectrale et géométrie. Année 2000-2001. St. Martin d’Hères: Université de Grenoble I, Institut Fourier, Sémin. Théor. Spectr. Géom. 19, 67-75 (2001).
Let $$(T^n,g)$$ be an $$n$$-torus with a Riemannian metric $$g$$ and $$\widetilde g$$ be the lift of $$g$$ to the universal covering space $$R^n$$ of $$T^n$$. Denoting by $$d_g$$ the distance on $$R^n$$ induced by $$\widetilde g$$, we put $$d_\rho(x,y)= d(\rho x,\rho y)/ \rho$$ for $$\rho>0$$ and $$x,y\in R^n$$.
By a result of Pansu, we know that there is a norm $$\|\cdot \|_\infty$$ on $$^n$$ such that $$\lim_{\rho\to \infty}d_\rho (x,y)=\|x-y\|_\infty$$ for every $$x$$, $$y\in R^n$$.
Denoting by $$B_g(\rho)$$ the geodesic ball on $$(R^n, \widetilde g)$$ with radius $$\rho$$, by $$\Delta_g$$ the Laplacian on $$(R^n, \widetilde g)$$ and by $$\lambda_1(B_g(\rho))$$ the first eigenvalue of $$\Delta_g$$ for the Dirichlet problem on $$B_g(\rho)$$ the author asserts that $$\lim_{\rho\to \infty}\rho^2 \lambda_1(B_g(\rho)) =\lambda_1^\infty\leq \lambda_{e,n }$$, where $$\lambda_{e,n}$$ (resp. $$\lambda^\infty_1)$$ is the first eigenvalue of the Euclidean Laplacian (resp. of an elliptic operator) on the unit ball of the norm $$\|\cdot \|_\infty$$. For the $$i$$-th eigenvalues the author gives similar results.
An outline of the proof is given with some definitions of new notions of the convergence of sequences of operators.
For the entire collection see [Zbl 0981.00005].
##### MSC:
 58J50 Spectral problems; spectral geometry; scattering theory on manifolds 53C20 Global Riemannian geometry, including pinching 58C40 Spectral theory; eigenvalue problems on manifolds
##### Keywords:
eigenvalues of Laplacians; $$n$$-torus; geodesic ball
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