A notion of deformation is defined and studied for complete minimal surfaces in R} and R3/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 47r in R}/Z and R3/Z2 are studied. It is proved that the helicoid and Scherk's surface are isolated under periodic perturbations. Let M be a submanifold of a Riemannian manifold N and let Te(M) be a tubular

more »

... of M in N of radius e. A C*-e variation of M in N is a submanifold Mx c Te(M) which is a graph over M and is e-C^close to M. This means Mx is pointwise e-close to M in each fibre of Te(M) and the tangent planes as well. We are interested in complete minimal submanifolds M of N (abbreviated c.m.s.) and Cl-e variations which are also c.m.s.'s, and henceforth we always assume M and Mx are c.m.s.'s. We say M is isolated if for some e > 0, the only e-C1-variations of M differ from M by an ambient isometry of N. In this paper we shall study this question when N is R3 or a translation space: R3 modulo a group of translations. A flat plane in R3 is isolated. This follows immediately from Bernstein's theorem: a function of two variables on R2 whose graph is a c.m.s. is linear. We shall prove Enneper's surface, the catenoid, and a certain 3-punctured sphere, discovered by Meeks and Jorge, are isolated in R3. We prove the helicoid and Scherk's surface are isolated under periodic deformations, i.e., the helicoid of total curvature Atr in R3 modulo one translation is isolated in this translation space and Scherk's surface of total curvature Att is isolated in R3 modulo two translations. This Scherk surface is conformally a 4-punctured sphere and the helicoid a 2-punctured sphere. We shall study minimal submanifolds of translation spaces. Many of the techniques developed by R. Osserman can be adapted to this context and yield information about periodic minimal surfaces in R3. For example, we classify c.m.s.'s of total curvature Art in translation space. In R3, Osserman has proved the catenoid and Enneper's surface are the only c.m.s.'s of total curvature 4tt. We obtain an analogous classification in R?/G. In §VI, we describe these surfaces for G = Z. We study c.m.s.'s M of finite total curvature in translation spaces. We prove a deformation (i.e., an e-C^variation) Mx of a finite total curvature c.m.s. in a translation space is also of finite total curvature, and is conformally a compact