Abstract
We consider local minimizers of the Ginzburg-Landau energy functional
∫12|∇u|2+14(1−u2)2dx
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
△u=u3−u
such that
|u|≤1,∂nu>0,limxn→±∞u(x′,xn)=±1
are hyperplanes in dimension n≤8.

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