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Mat. Sb., 2003, Volume 194, Number 11, Pages 65–80 (Mi msb780)  

On the equation of an improper convex affine sphere: a generalization of a theorem of Jörgens

V. N. Kokarev

Samara State University

Abstract: 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.

DOI: https://doi.org/10.4213/sm780

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English version:
Sbornik: Mathematics, 2003, 194:11, 1647–1663

Bibliographic databases:

UDC: 513.0+517.946
MSC: Primary 53A05, 53C45; Secondary 35B99
Received: 12.11.2001 and 26.08.2002

Citation: V. N. Kokarev, “On the equation of an improper convex affine sphere: a generalization of a theorem of Jörgens”, Mat. Sb., 194:11 (2003), 65–80; Sb. Math., 194:11 (2003), 1647–1663

Citation in format AMSBIB
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\paper On the equation of an~improper convex affine sphere:
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\yr 2003
\vol 194
\issue 11
\pages 65--80
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\zmath{https://zbmath.org/?q=an:1080.53010}
\transl
\jour Sb. Math.
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\vol 194
\issue 11
\pages 1647--1663
\crossref{https://doi.org/10.1070/SM2003v194n11ABEH000780}
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\scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-1642408319}


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