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