Geodesic
On the equation of an improper convex affine sphere:
Sbornik. Mathematics, Tome 194 (2003) no. 11, pp. 1647-1663 Cet article a éte moissonné depuis la source Math-Net.Ru

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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 $\varepsilon$ ($\varepsilon<10^{-4}$) 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. 
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     title = {On the equation of an~improper convex affine sphere:},
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V. N. Kokarev. On the equation of an improper convex affine sphere:. Sbornik. Mathematics, Tome 194 (2003) no. 11, pp. 1647-1663. http://geodesic.mathdoc.fr/item/SM_2003_194_11_a2/

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