# THE EIGENVALUE ESTIMATE ON A COMPACT RIEMANNIAN MANIFOLD

• Kim, Bang-Ok (Department of Industrial Safety, Sunchon Junior Technical College, Sunchon 540-744) ;
• Kim, Kwon-Wook (Department of Mathematics Educational, Sunchon University, Sunchon 540-742)
• Published : 1995.02.01

#### Abstract

We will estimate the lower bound of the first nonzero Neumann and Dirichlet eigenvalue of Laplacian equation on compact Riemannian manifold M with boundary. In case that the boundary of M has positive second fundamental form elements, Ly-Yau[3] gave the lower bound of the first nonzero neumann eigenvalue $\eta_1$. In case that the second fundamental form elements of $\partial$M is bounded below by negative constant, Roger Chen[4] investigated the lower bound of $\eta_1$. In [1], [2], we obtained the lower bound of the first nonzero Neumann eigenvalue is estimated under the condtion that the second fundamental form elements of boundary is bounded below by zero. Moreover, I realize that "the interior rolling $\varepsilon$ - ball condition" is not necessary when the first Dirichlet eigenvalue was estimated in [1].ed in [1].