The development and fusion on the geometry of submanifolds and theory for eigenvalues of the differential operators in Riemannian manifolds

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In this project, by making use of fusion on the research methods in the geometry of submanifolds and the research methods for eigenvalues of the differential operators on bounded domains in Riemannian manifolds, we can construct trial functions with good properties such that we obtain a sharp universal inequalityfor eigenvalues of the Dirichlet eigenvalue problem of the Laplacian on bounded domainsin Riemannian manifolds and a universal inequality for eigenvalues of the clamped plate problem. Furthermore, by combining the universal inequality for eigenvalues with therecursion formula of Cheng and Yang, we give the upper bounds for the k-th eigenvalueof the Dirichlet eigenvalue problem of the Laplacian on bounded domains in Riemannian manifolds. It is optimal in the sense of the order of k. By using a new and original method replacing the method of the Fourier transform, we obtain a Li-Yau type lower bound for eigenvalues of the Dirichlet eigenvalue problem of the Laplacian on bounded domains in Riemannian manifolds. About the study on lower bounds for eigenvalues of the clamped plate problem on bounded domains in a Euclidean space, we improve the inequality of Levine and Protter by using the Fourier transform and the symmetry rearrangement of a domain. On the other hand, we study structures of curvatures and topological structures of submanifolds according to several different view points. An optimal upper bound for the first eigenvalue of Jacobi operator of compact hypersurfaces with constant scalar curvature in the unit sphere is given. Many embedded compact hypersurfaces with constantk-th mean curvature in the unit sphere are constructed.