Convex affine maximal surfaces (Q1107836)

This paper deals with affine maximal surfaces. The author derives the fundamental formulas of affine differential geometry for locally strictly convex surfaces, expressed in terms of complex isothermal parameters with respect to the Blaschke metric, and proves that an affine maximal surface can be generated by a holomorphic immersion \(Z: M\to C^ 3\) satisfying \(\det(Z-\bar Z,Z',\bar Z')>0\) under the relation \[ x=-i(Zx\bar Z+\int ZxdZ-\int \bar Zxd\bar Z). \] This representation was also obtained by \textit{C.-L. Terng} [Ann. Math. Stud. 103, 207-216 (1983; Zbl 0533.53004)]. Using the complex representation the author proves the following theorem. Let \(x: M\to A^ 3\) be locally strictly convex, maximal, and complete with respect to the Blaschke metric. If the image of the holomorphic curve Z lies in a half space of \(C^ 3\) then x(M) lies in an elliptic paraboloid. This theorem is an improvement of his previous result in Am. J. Math. 104, 91-126 (1982; Zbl 0501.53037).