空间型中紧致无边子流形上的第一特征值

First Eigenvalue of Compact Submanifolds Without Boundary in Space Forms

得到了两个关于空间形式中紧致无边子流形的广义位置向量场和其上Laplace算子第一特征值λ_1的积分不等式。并由此首先给出了λ_1与其上界间的间隔估计,其次得到了此紧致无边子流形等距浸入在空间形式的测地超球面或等距于测地超球面的充分条件,推广了Deshmukh[6]在欧氏空间中的相应结论。

This paper derives two sharp integral inequalities for the generalized position vector field and the first nonzero eigenvalue λ1 of the laplacian operator △ of a compact submanifold without boundary isometrically immersed into a space form. By using these inequalities, first an estimate of the gap between λ1 and its upper bound is obtained. Further more, the author obtains a theorem of the submanifold be immersed into a geodesic hypersphere, or isometric to a geodesic sphere in the space form, which generalized the corresponding result in Eucliclean space given by Deshmukh .