The associated family of an elliptic surface and applications to minimal submanifolds

Theodoros Vlachos

Abstract:  We discuss the reformability of elliptic surfaces in space forms for which all curvature
ellipses of a certain order are circles. We show that elliptic surfaces in space forms for which all
curvature ellipses of a certain order are circles allow a family of isometric deformations preserving
the second fundamental form and the normal curvature tensor. For minimal surfaces this generates
a one-parameter family of minimal isometric deformations that adds to the standard associated
family. We also show how the associated family of a minimal Euclidean submanifold of rank two is
determined by the associated family of an elliptic surface clarifying the geometry around the
associated family of these higher dimensional submanifolds.