On the essential spectrum of the Laplacian and vague convergence of the curvature at infinity

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We shall prove that under some volume growth condition, the essential spectrum of the Laplacian contains the interval [(n-1)²K/4, ∞) if an n-dimensional Riemannian manifold has an end and the average of the part of the Ricci curvature on the end which lies below a nonpositive constant (n-1)K converges to zero at infinity.