Berger’s Inequality in the Presence of Upper Sectional Curvature Bound

We obtain inequalities for all Laplace eigenvalues of Riemannian manifolds with an upper sectional curvature bound, whose rudiment version for the 1st 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 upper bounds for Laplace eigenvalues of minimal submanifolds in terms of geometric quantities of the ambient space.