Willmore submanifolds of the Möbius space and a Bernstein-type theorem

We study Willmore immersed submanifoldsf: M ^m →S ⁿ into then-Möbius space, withm≥2, as critical points of a conformally invariant functionalW. We compute the Euler-Lagrange equation and relate this functional with another one applied to the conformal Gauss map of immersions intoS ⁿ . We solve a Bernestein-type problem for compact Willmore hypersurfaces ofS ⁿ , namely, if ?a ∈? ⁿ⁺² such that <γf, a > ≠ 0 onM, whereγ _f is the hyperbolic conformal Gauss map and <, > is the Lorentz inner product of? ⁿ⁺², and iff satisfies an additional condition, thenf(M) is an (n?1)-sphere.