| Résume | We will present some aspects of the geometry of surfaces which are critical points of the area among Lagrangian immersions in a Kähler manifold, the so called Lagrangian and Hamiltonian stationary surfaces. We will go through the questions which have motivated this variational problem and describe some geometric conjectures attached to it. Then we will move to the analytical difficulties of studying the variation of area under pointwize Lagrangian or Legendrian constraints. We will then come to a recent work in collaboration with Alessandro Pigati in which we are proving that surfaces which are critical points of the area under Legendrian constraint, that is surfaces which are everywhere tangent to a non integrable plane distributions and critical point of the area under this sub-riemannian constraint, are smooth away from isolated conical singularities. We will then come to the still mysterious problem relative to the location of these conical singularities and explain why it is relevant to fundamental questions in differential geometry which have to do with the realization or not of Lagrangian homology classes by calibrated surfaces. Finally, if time permits, we will explain another motivation to study this problem in relation to the Willmore conjecture in arbitrary co-dimension. |
