The Area-preserving Variation Of A Class Of Functionals Of The Submanifolds In Space Forms

Let x : Mⁿ→R(n + p)(c) be an n-dimensional compact without boundarysubmanifold in a (n + p)-dimensional space form R^n+p(c). Assume that r is evenand r∈{2, . . . , n - 1}. We call M to be a generalized r-minimal submanifold if(r + 1)S_r+1 +λS₁≡0 on M, whereλis a constant. In this paper, we define anArea-preserving functionalby calculation of the area-preserving first variational formula of A_r(x). We show thatx is generalized r-minimal, if and only if x is the critical point of (*). We calculatethe second variational formula of (*) and define the concepts of stability and strongstability. Finally, we prove that there exists no compact without boundary stronglystable generalized r-minimal submanifold with Sr > (-λnp)/((p+r)(n-r)) in the unit sphere S^n+p.