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We consider local minimizers of the Ginzburg-Landau energy functionaland 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 ofare hyperplanes in dimension n ≤ 8.
Ovidiu Savin (Thu,) studied this question.