Regularity of flat level sets in phase transitions

We consider local minimizers of the Ginzburg-Landau energy functional ∫ 1 2 |∇u| + 1 4 (1− u)dx and prove that, if the 0 level set is included in a flat cylinder then, in the interior, it is included in a flatter cylinder. As a consequence we prove a conjecture of De Giorgi which states that level sets of global solutions of 4u = u − u such that |u| ≤ 1, ∂nu > 0, lim xn→±∞ u(x′, xn) = ±1 are hyperplanes in dimension n ≤ 8.