WILLMORE SUBMANIFOLDS OF THE MOBIUS SPACE AND A BERNSTEIN-TYPE THEOREM

We study Willmore immersed submanifolds f : M(m) --> S-n into the n-Mo bius space, with m greater than or equal to 2, as critical points of a conformally invariant functional W. We compute the Euler-Lagrange equ ation and relate this functional with another one applied to the confo rmal Gauss map of immersions into S-n. We solve a Bernestein-type prob lem for compact Willmore hypersurfaces of Sn, namely, if There Exists a is an element of IR(n+2) such that <gamma f, a>not equal 0 on M, whe re gamma f is the hyperbolic conformal Gauss map and <,> is the Lorent z inner product of IR(n+2), and if f satisfies an additional condition , then f(M) is an (n - 1)-sphere.