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

The space M of nondegenerate, properly embedded minimal surfaces in R-3 wit h finite total curvature and fixed topology is an analytic lagrangian subma nifold of 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 seco nd fundamental form of this submanifold. We also consider the compact bound ary case, and we show that the space of stable non at minimal annuli that b ound a fixed convex curve in a horizontal plane, having a horizontal end of finite total curvature, is a locally convex curve in the plane C.