Affine variation formulas and affine minimal surfaces (Q1123411)

Let x: \(M\to A^ 3\) be a locally strongly convex affine minimal surface. \textit{E. Calabi} proved that the second variation of the affine area is negative. He also proved that if M is a graph and is metrically complete, then M is an elliptic paraboloid [Am. J. Math. 104, 91-126 (1982; Zbl 0501.53037)]. In the present paper the authors give examples which show that the corresponding results in the nonconvex case are not possible. They also completely classify all affine minimal translation surfaces.