Variational Problems Of Total Mean Curvature Of Submanifolds

Let M be an x:Mâ†'N(c) is an isometric immersion from n-dimensional smooth manifold to n+p-dimensional Riemannian manifold N(c),N(c) is a space form with the constant sectional curvature c.We study the variational problems of total mean curvatue functional In section one,we make use of variational methods to get its Euler-Lagrange equationWe call the submanifolds which satisfied the Euler-Lagrange equation above Z-submanifolds. Where hijα is the second fundmantal form,H is the mean curvature vector of x and Δ the Laplassian operator of normal bundle.The volume element of x is dM.In section two,we consider the compact Z-surface in S3 and get an inequality the integral of a function with Euler characteristic number,and get the submanifold when the equality holds in Theorem3.1.