Affine minimal hypersurfaces of rotation

We give a complete list of affine minimal surfaces inA³ with Euclidean rotational symmetry, completing the treatise given in [1] and prove that these surfaces have maximal affine surface area within the class of all affine surfaces of rotation satisfying suitable boundary conditions. Besides we show that for rotationally symmetric locally strongly convex affine minimal hypersurfaces inAⁿ,n4, the second variation of the affine surface area is negative definite under certain conditions on the meridian.