A sextic holomorphic form of affine surfaces with constant affine mean curvature (Q1888692)

The author extends an original idea of Calabi for affine maximal surfaces and defines a sextic holomorphic differential form for affine surfaces with constant affine mean curvature. He gets some rigidity results for affine complete surfaces by using this sextic holomorphic form. In fact, he proves Theorem 3.1: Let \(x:M\to A^3\) be an affine surface with constant mean curvature \(L_1\). Then the following sextic form \(\Phi= \varphi (dz)^6\), \(\varphi=\frac 12 L_1\alpha^2+\beta \nabla^1\alpha-\alpha\nabla^2\beta\) is holomorphic on \(M\). Theorem 4.1: Let \(x:M\to A^3\) be an affine surface which is complete with respect to the Blaschke metric. Assume that \(2\|\varphi\|\leq CJ^2+2L_1J\), then \(x(M)\) is a paraboloid or ellipsoid, where, \(C(<3)\) is a positive constant. Theorem 4.2: Let \(x:M\to A^3\) be an affine surface with constant mean curvature \(L_1<0\) which is complete with respect to the Blaschke metric. If \(2\| \varphi\|\leq (K-L_1)(-L_1)\), then \(K\leq 0\), where \(K\) is the Gauss curvature with respect to the Blaschke metric. The author also gives a short and transparent proof of a result of Martinez-Milan.