$C^0$-limits of Legendrian Submanifolds

Abstract

Laudenbach and Sikorav proved that closed, half-dimensional non-Lagrangian submanifolds of symplectic manifolds are immediately displaceable as long as there is no topological obstruction. From this they deduced that the $C^0$-limit of a sequence of Lagrangian submanifolds of certain manifolds is again Lagrangian, provided that the limit is smooth. In this note we extend Laudenbach and Sikorav's ideas to contact manifolds. We prove correspondingly that certain non-Legendrian submanifolds of contact manifolds can be displaced immediately without creating short Reeb chords as long as there is no topological obstruction. From this it will follow that under certain assumptions the $C^0$-limit of a sequence of Legendrian submanifolds with uniformly bounded Reeb chords is again Legendrian, provided that the limit is smooth.