A variational problem for submanifolds of Euclidean space

Abstract:

Let ${M^n}$ be a compact differentiable manifold and ${R^{n + k}}$ Euclidean space. A necessary and sufficient condition is given for an immersion $\psi :{M^n} \to {R^{n + k}}$ to be a stationary immersion for $J = \smallint M_\psi ^n\langle x - {x_c},x - {x_c}\rangle$ dv subject to the side condition $V = \smallint M_\psi ^n$ dv= a fixed constant, where ${x_c}$ is the center of mass. In particular, minimal submanifolds of spheres satisfy this condition.