The space of properly embedded minimal surfaces with finite total curvature

Abstract

It is showed that the set of non degenerate properly embedded minimal surfaces with finite total curvature and fixed topology in ℝ3 has a structure of finite dimensional real-analytic manifold - the non degeneration is defined in terms of the space of Jacobi functions on the surface which have logarithmic growth at the ends. As application we show that if a non degenerate minimal surface has a symmetry which fixes its ends, then any nearby minimal surface has the same kind of symmetry. Finally, we construct a natural Lagrangian immersion of the space of non degenerate minimal surfaces, quotiented by certain group of rigid motions, into a Euclidean space with its standard symplectic structure.