The Variation For The Functional Of The Total Length Of The Second Fundamental Form Of A Submanifold

Let x:Mn→Nn+p(c)be an isometic immersion,Nn+p(c)is a space form with the constant sectional curvature c.We define functional: By calculating the first variation of Z(x),we get the equations(Euler-Lagrange equation) of the critical submanifolds as follows: ‖h‖n-2(ncHα-‖h‖2Hα+hijβhkjαhjiβ)+Tijα(‖h‖n-2),ij+n△⊥(‖h‖n-2Hα)=0, where Tijα=hijα-nHαδij,h is the second foundamental form of x in Nn+p(c),hijα is the second foundamental form,‖h‖2 is the square of the length of the second foundamental form,Hα is the component of the vector of the mean curvature,and△is Laplace operator.In this paper,we get a kind of Simons-type integral inequality on a critical submanifold of Z(x) as follows: where f=‖h‖n-2[(2-1/p)‖h‖4-n‖h‖2|H|2-nc‖h‖2+2n2c|H|2].By using the integral inequaliti above:For a compact critical submanifold of Z(x),we have sup M f≥0,when a the equal sign holds if and only if (1)p=1, the submanifold is isometrialy equivalent to torus where λ2 is root of the following equation: k(n-k)y3+k(2n-k)y2-(n2-k2)y-k(n-k)=0; and λμ=-1,respectively.(2)p=2,the submanifold is Veronese surface in S4.