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

Abstract

We study Willmore immersed submanifoldsf: M^m→Sⁿ 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 intoSⁿ. We solve a Bernestein-type problem for compact Willmore hypersurfaces ofSⁿ, namely, if ∃a ∈ℝⁿ⁺² such that <γf, a > ≠ 0 onM, whereγ _f 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.