  We obtain inequalities for all Laplace eigenvalues of Riemannian manifolds
with an upper sectional curvature bound, whose rudiment version for the first
Laplace eigenvalue was discovered by Berger in 1979. We show that our
inequalities continue to hold for conformal metrics, and moreover, extend
naturally to minimal submanifolds. In addition, we obtain explicit estimates
for Laplace eigenvalues of minimal submanifolds in terms of geometric
quantities of the ambient space.
