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

Abstract

We study Willmore immersed submanifoldsf: Mm→Sn into then-Mobius 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 intoSn. We solve a Bernestein-type problem for compact Willmore hypersurfaces ofSn, namely, if ∃a ∈ℝn+2 such that ≠ 0 onM, whereγf is the hyperbolic conformal Gauss map and is the Lorentz inner product ofℝn+2, and iff satisfies an additional condition, thenf(M) is an (n−1)-sphere.