Estimates of an Eigen Value of Some Isoperimetric Inequalities

In this paper we give an overview of results about two types of an isoperimetric Inequalities on its eigenvalues and the eigenvalues of the laplacian. We estimate the isoperimetric type constant (M_m) of 2-dimentional Riemannian manifold M_m. if D be the diameter of compact Riemannian manifold M, (n) is the volume of the unit n-dimensional sphere, and i (M) be an injectivity radius of M, we prove that Vol (M) /Dgreater than ( (2) ^3) / (54 (3) ^2) i (M) ^2.