全平均曲率的一阶变分问题

The problem of one-order variation for total mean curvature

设M是一个n维黎曼流形,考虑泛函τ(ft)=∫M︱H︱ndMt,其中H是F:M×I→的平均曲率,dMt为流形的体积元.用变分学知识,可推导出τ(ft)在t=0时的一阶变分公式,并得到Euler-Lagrange方程.

Let M be an n - dimensional Riemann manifold. Considerτ（ft）=∫M｜H｜ndMt , where H is the mean curvature of F：M ×1→-M , dM, is the manifold＇ s volume element. Then using the theory of variation, we can calculated the first variational formula of τ（ft） , when t = 0 and obtain the Euler - Lagrange equation.