The space of complete minimal surfaces with finite total curvature as lagrangian submanifold

The space ${\cal M}$ of nondegenerate, properly embedded minimal surfaces in ${\mathbb R}^3$ with finite total curvature and fixed topology is an analytic lagrangian submanifold of ${\mathbb C}^n$, where $n$ is the number of ends of the surface. In this paper we give two expressions, one integral and the other pointwise, for the second fundamental form of this submanifold. We also consider the compact boundary case, and we show that the space of stable nonflat minimal annuli that bound a fixed convex curve in a horizontal plane, having a horizontal end of finite total curvature, is a locally convex curve in the plane ${\mathbb C}$.