Abstract
Let 0 be a fixed point in (n+1)-space Rnlx r=r(p) the distance of p from 0, peRn+l, H, the vth mean curvature of an n-dimensional surface in Rn+l. In this paper it is proved that the only compact n-surfaces satisfying r'H, = 1 are the spheres with center 0. Generalizations of this proposition are given in b and c. ?a contains two lemmas: Lemma 1 asserts the ellipticity of certain differential equations; in Lemma 2 a strong form of the maximum principle (due to E. Hopf) is stated for solutions of elliptic differential equations. a. Let F be an n-surface in (n+1)-space, FnCRn+l, n>2, F of class C2.2 F has no singularities; self-intersections are allowed; F is connected. The vth mean curvature is the vth elementary symmetric function of the principal curvatures k1, k2, , k, (kl > k2 > . . .> kn) divided by Cn,,.
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