Let Mn= (M,g)
be a compact, connected, Riemannian manifold of dimension n. Let μ be
the measure μ = σdvolg, where σ ∈ C∞(M) is a nonnegative density. We
first show that, under some mild metric conditions that do not involve the
curvature, the presence of a large eigenvalue (or more precisely of a large
gap in the spectrum) for the Laplacian associated to the density σ on M
implies a strong concentration phenomenon for the measure μ. When the
density is positive, we show that our result is optimal. Then we investigate
the case of a Laplace-type operator D = ∇∗∇ + T on a vector bundle E
over M, and show that the presence of a large gap between the (k + 1)-st
eigenvalue λk+1 and the k-th eigenvalue λk implies a concentration phenomenon
for the eigensections associated to the eigenvalues λ1,…,λk of the operator
D.