Negative eigenvalues of the conformal Laplacian

Abstract

Let M M be a closed differentiable manifold of dimension at least 3 3 . Let Λ 0 ( M ) \Lambda _0 (M) be the minimum number of non-positive eigenvalues that the conformal Laplacian of a metric on M M can have. We prove that for any k k greater than or equal to Λ 0 ( M ) \Lambda _0 (M) , there exists a Riemannian metric on M M such that its conformal Laplacian has exactly k k negative eigenvalues. Also, we discuss upper bounds for Λ 0 ( M ) \Lambda _0 (M) .