Laplacian On A Class Of Riemannian Manifolds Ms,

In this paper,we shall mainly study the essential spectrum of the laplacian on a Riemannian manifold with a pole.There are three sections in this paper.The first section is the introduction of this paper.The second section is the preliminary knowledge.In the last section,we perfect two theorems' proof in paper[21]firstly,they are the following results:for any n-dimensional complete Riemannian manifold with nonnegative Ricci curvature,if the Nash inequality is satisfied,then it is diffeomorphic to R~n.We also obtain that if the Nash inequality holds on the Riemannian manifold without any curvature assumption,then the geodesic ball has maximal volume growth.Secondly,on the base,we give an apphcation of Nash inequality,to discuss the question about the essential spectrum of the laplacian on a Riemannian manifold with a pole.We still obtain Ess Spect(△)=[0,+∞) under another condition instead of nonnegative Ricci curvature Finally,we depict the essential spectrum of the laplacian on a kind of Riemannian manifolds with a pole.