Variational Problem Of Mean Curvature's Square

Let∑be a compact,possibly with boundary,curved surface in 3-dimensional space form E³. Conside Willmore functional as to mean curvature'ssquare , where H is the mean curvature of curvedsurface cluster X:∑×(-1,1)→R³, and dA_t=θ₁(t)(?)θ₂(t) is the volume element of it. Then using the basic theory as to variation of curved surface, we can calculate the firstvariational formula of F(t) when t = 0,F'(0) = ,where ais normal components of variational vector field (?)X/(?)t|_t=0. When F'(0) = 0,we canfigure out the second variational formula of F(t), for t = 0Where h_ij denote the components of the second fundamental form of curved suuface X( ,0) .Specially,if we choose that∑be torus in R³ , that is0≤(?)≤2Ï€, we obtain that the torus in R³ satisify the Euler-Lagrange equationâ–³H+2H³-2KH=0 if and only if a/r=2^1/2...