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No abstract

For a compact spin manifold M isometrically embedded into Euclidean space, we derive the extrinsic estimates from above and below for eigenvalues of the square of the Dirac operator, which depend on the second fundamental form of the embedding. We also show the bounds of the ratio of the eigenvalues. Since the unit sphere and the projective spaces admit the standard embedding into Euclidean spaces, we also obtain the corresponding results for their compact spin submanifolds.

Let Ω be a bounded domain in an n-dimensional Euclidean space R n . We study eigenvalues of an eigenvalue problem of a system of elliptic equations:Estimates for eigenvalues of the above eigenvalue problem are obtained. Furthermore, we obtain an upper bound on the (k + 1) th eigenvalue σ k+1 . We also obtain sharp lower bound for the first eigenvalue of two kinds of eigenvalue problems of the biharmonic operator on compact manifolds with boundary and positive Ricci curvature.Key words and phrases: Universal bounds, eigenvalues, a system of elliptic equations, Cheng-Yang's inequality, biharmonic operator, positive Ricci curvature.