On the equation of an~improper convex affine sphere:

V. N. Kokarev

Sbornik. Mathematics, Tome 194 (2003) no. 11, pp. 1647-1663

Résumé

It is proved that if a function $\varphi(t)$ of variable $t>0$ belongs to the class $C^{3,\alpha}$ and satisfies for sufficiently small positive $\varepsilon$ ($\varepsilon10^{-4}$) the conditions \begin{gather*} 1-\varepsilon\leqslant\varphi(t)\leqslant1+\varepsilon,\qquad t>0, \\ \begin{alignedat}{2} |\varphi'(t)|\leqslant\varepsilon\frac{\varphi(t)}t\,,\qquad t\geqslant 2\sqrt{1-\varepsilon}, \\ |\varphi''(t)|\leqslant\varepsilon\frac{\varphi(t)}{t^2}\,,\qquad t\geqslant2\sqrt{1-\varepsilon}, \\ |\varphi'''(t)|\leqslant\varepsilon\frac{\varphi(t)}{t^3}\,,\qquad t\geqslant2\sqrt{1-\varepsilon}, \end{alignedat} \end{gather*} then every complete solution $z(x,y)$ of the equation $z_{xx}z_{yy}-z_{xy}^2=\varphi(z_{xx}+z_{yy})$ is a quadratic polynomial.