Deformations of Complete Minimal Surfaces

A notion of deformation is defined and studied for complete minimal surfaces in ${R^3}$ and ${R^3}/G,G$ a group of translations. The catenoid, Enneper’s surface, and the surface of Meeks-Jorge, modelled on a $3$-punctured sphere, are shown to be isolated. Minimal surfaces of total curvature $4\pi$ in ${R^3}/Z$ and ${R^3}/{Z^2}$ are studied. It is proved that the helicoid and Scherk’s surface are isolated under periodic perturbations.