- Volume 32 Issue 1
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 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 investigated the lower bound of $\eta_1$. In , , 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 .ed in .