Resumen de One dimensional symmetry in the Heisenberg group

Isabeau Birindell, Jyotshana Prajapat

• Let HI denote the Heisenberg space and let u be a solution of + M(l - M~) = 0 in H" satisfying 1. Let x be any variable orthogonal to the anisotropic direction t. Assume that for xl going to plus or minus infinity u converges uniformly to 1 and -1 respectively. Under these assumptions we prove that u is a function depending only on xl and that it is monotone increasing. , This result, which is the analogue for the Heisenberg space of the weak formulation of a conjecture by De Giorgi, is obtained for a wider class of equations;

  it is a consequence of the invariance of the Heisenberg Laplacian with respect to Heisenberg group. The proof requires a Maximum Principle for unbounded domains which is interesting by itself. We also consider the case when u satisfies the limit condition in the t direction and we conclude that the solution is monotone in t.