Eigenvalue Estimates For Manifolds With Bounded Sectional Curvature And Its Submanifolds

In this paper, we firstly obtain some integral inequalities for sub-domains and sub-manifolds of the manifolds with bounded sectional curvature, stable minimal hypersurfacesof manifolds with negative curvature and so on, combining Rayleigh or Min-Max principle,we get the estimates of the first eigenvalue.The paper divided into four sections:In section one, the basic problems and facts about the first eigenvalue are presented.In section two, we consider the lower bounds of the first eigenvalue for sub-domainsand submanifolds of the manifolds with bounded curvature. As a special case, we alsoobtain the corresponding results for Euclidean unit sphere.In section three, for the stable minimal hypersurfaces of manifolds with negativecurvature, we get the upper and lower bounds of the first eigenvalue provided that theYamabe invariant of the hypersurfaces is negative.In section four, for the submanifolds of codimension p≥1 in the Riemannian man-ifolds with bounded curvature, we obtain a proposition concerning the upper bounds ofthe first eigenvalue by selecting certain functions defined on R1 as the test functions. Asapplications, we get the upper bounds of the first eigenvalue for closed hypersurfaces ofmanifolds with bounded curvature.