A minimizing deformation of Legendrian submanifolds in the standard sphere

In this note we study the moduli space of minimal Legendrian submanifolds in the standard sphere S^2n-1. We show that new examples of minimal Legendrian submanifolds can be constructed, if we can solve a certain equation for a function on a nearby glued Legendrian submanifold. As a step toward solving this equation, we prove shorttime existence for a particular gradient flow on the space of immersed Legendrian submanifolds. A new necessary condition for a Lagrangian embedding into CP^n is given.