Prescribing Gauss-Kronecker curvature on group invariant convex hypersurfaces

Abstract

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.