关于复射影空间中紧致极小子流形的特征值

Eigenvalues of compact Minimal Submanifolds in a Complex Projective Space

本文在推广Yang,P.C.和 Yau,S.T.关于球面中紧致极小子流形的特征值不等式的基础上,得到了复射影空间中紧致极小子流形的某些特征值不等式.

In 1980, P.C.Yang and S.T.Yau obtained the estimates of the con-sective difference of eigenvalues for compact minimal submanifolds of Sn(1). In this paper, listing the eigenvalue sequence of a compact manifold Mas 0 = λ0<λ1≤λ2≤… ↑ ∞, we generalize their result as follows: Theorem. Let Mm be an m-dimensional submanifold of SN(c)=EN+1 with constant mean curvature. Then we have, where H is the mean curvature vector of Mm in En+1, and Using the standard imbedding of a complex projective space into Euclidean space and the above theorem, we obtain Theorem. Let Mm be an m-dimensional minimal submanifold of CPn(1). Then the following inequality holds