Eigenvalues of perturbed Laplace operators on compact manifolds

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We obtain upper bounds for the eigenvalues of the Schrödinger operator L=Δg+q depending on integral quantities of the potential q and a conformal invariant called the min-conformal volume. Moreover, when the Schrödinger operator L is positive, integral quantities of q which appear in upper bounds, can be replaced by the mean value of the potential q. The upper bounds we obtain are compatible with the asymptotic behavior of the eigenvalues. We also obtain upper bounds for the eigenvalues of the weighted Laplacian or the Bakry-Emery Laplacian Δϕ=Δg+∇gϕ·∇g using two approaches: First, we use the fact that Δϕ is unitarily equivalent to a Schrödinger operator and we get an upper bound in terms of the L²-norm of ∇gϕ and the min-conformal volume. Second, we use its variational characterization and we obtain upper bounds in terms of the L∞-norm of ∇gϕ and a new conformal invariant. The second approach leads to a Buser type upper bound and also gives upper bounds which do not depend on ϕ when the Bakry-Emery Ricci curvature is non-negative.