In this paper we consider eigenvalues of the Dirichlet biharmonic operator on compact Riemannian manifolds with boundary (possibly empty) and prove a general inequality for them. By using this inequality, we study eigenvalues of the Dirichlet biharmonic operator on compact domains in a Euclidean space or a minimal submanifold of it and a unit sphere. We obtain universal bounds on the (k + 1)th eigenvalue on such objects in terms of the first k eigenvalues independent of the domains. The estimate for the (k + 1)th eigenvalue of bounded domains in a Euclidean space improves an important inequality obtained recently by Cheng and Yang.

Let (M, , ) be an n( 2)-dimensional compact Riemannian manifold with boundary and non-negative Ricci curvature. Consider the following two Stekloff eigenvalue problemswhere is the Laplacian operator on M and ν denotes the outward unit normal on ∂M. The first nonzero eigenvalues of the above problems will be denoted by p 1 and q 1 , respectively. In the present paper, we prove that if the principle curvatures of the second fundamental form of ∂M are bounded below by a positive constant c, then p 1 √ λ 1 ( √ λ 1 + λ 1 − (n − 1)c 2 )/{(n − 1)c} with equality holding if and only if Ω is isometric to an n-dimensional Euclidean ball of radius 1 c , here λ 1 denotes the first non-zero eigenvalue of the Laplacian of ∂M. We also show that if the mean curvature of ∂M is bounded below by a positive constant c then q 1 nc with equality holding if and only if M is isometric to an n-dimensional Euclidean ball of radius 1 c . Finally, we show that q 1 A/V and that if the equality holds and if there is a point x 0 ∈ ∂M such that the mean curvature of ∂M at x 0 is no less than A/{nV }, then M is isometric to an n-dimensional Euclidean ball, being A and V the area of ∂M and the volume of M, respectively. ✩ Partially supported by CNPq.

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