Differential and Integral Equations

Prescribing Gauss-Kronecker curvature on group invariant convex hypersurfaces

Richard Mikula

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We consider the problem of prescribing Gauss-Kronecker curvature in Euclidean space. In particular, by a degree theory argument, we prove the existence of a closed convex hypersurface in $\mathbb{R}^{3}$ which has its Gauss-Kronecker curvature equal to $F$, a prescribed positive function, which is invariant under a fixed-point free subgroup $G$ of the orthogonal group $O(3)$, requiring that $F$ satisfy natural growth assumptions near the origin and at infinity.

Article information

Differential Integral Equations, Volume 19, Number 10 (2006), 1103-1128.

First available in Project Euclid: 21 December 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 53C42: Immersions (minimal, prescribed curvature, tight, etc.) [See also 49Q05, 49Q10, 53A10, 57R40, 57R42]
Secondary: 35J60: Nonlinear elliptic equations 58J05: Elliptic equations on manifolds, general theory [See also 35-XX]


Mikula, Richard. Prescribing Gauss-Kronecker curvature on group invariant convex hypersurfaces. Differential Integral Equations 19 (2006), no. 10, 1103--1128. https://projecteuclid.org/euclid.die/1356050311

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