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Mat. Fiz. Anal. Geom., 1996, Volume 3, Number 1/2, Pages 125–130 (Mi jmag487)  

Closed convex surfaces in $E^3$ with given functions of curvatures

A. I. Medianik

B. Verkin Institute for Low Temperature Physics and Engineering, National Academy of Sciences of Ukraine, Khar'kov

Abstract: It is proved that there are a regular closed convex surface $S$ and a constant vector $c$ for which the equality
$$K^{-1}+H^{-\alpha}+c\mathbf n=\varphi(\mathbf n)$$
is realized at a point with external normal $\mathbf n$. Here $K$ and $H$ are the Gauss and mean curvatures of $S$ at the point with normal $\mathbf n$, $\varphi(\mathbf n)$ is a given regular function on sphere, which satisfies the closeness condition and the inequality
$$\operatorname{inf}\varphi>\frac9{32}[1+\sqrt{1+\frac{64}9(\operatorname{sup}\varphi)^{2-\alpha}}](\operatorname{sup}\varphi)^{\alpha-1},$$
$\alpha\in(0,1]$. The solution $(S,c)$ is unique with a translation.

Full text: PDF file (264 kB)
Full text: http:/.../abstract.php?uid=m03-0125r

Bibliographic databases:
Received: 09.06.1994

Citation: A. I. Medianik, “Closed convex surfaces in $E^3$ with given functions of curvatures”, Mat. Fiz. Anal. Geom., 3:1/2 (1996), 125–130

Citation in format AMSBIB
\Bibitem{Med96}
\by A.~I.~Medianik
\paper Closed convex surfaces in $E^3$ with given functions of curvatures
\jour Mat. Fiz. Anal. Geom.
\yr 1996
\vol 3
\issue 1/2
\pages 125--130
\mathnet{http://mi.mathnet.ru/jmag487}
\zmath{https://zbmath.org/?q=an:0869.53001}


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