Variational Probiem Of Functional Of The Total Length Of Second Fundamental Form For Submanifolds In Space Forms And Its Applications

Suppose x : Mⁿ→ N^n+p(c) is an isometric immersion, N^n+P(c) is a space form with the constant sectional curvature c . B is the second foundamental form of x in N^n+p .In this paper we consider the variational problems of the functional:and obtain the Euler - Lagrange equation:We call the submanifold which satisfies the Euler — Lagrange equation is W- minimal submanifold. h_ij^a is the second foundamental form,|B|² is the square of the length of the vector of the second foundamental form, H0 is the component of the vector of the mean curvature, A is Laplace operator.As an application,we find one kind of W-minimal rotation surface,and get one of the important theorem in this paper. The theorem shows some examples of this kind of W-minimal rotation surface we mentioned above.