Hypersurfaces of constant Gauss-Kronecker curvature with Li-normalization in affine space

TL;DR

This paper studies convex hypersurfaces in affine space with constant Gauss-Kronecker curvature under Li-normalization, showing they form foliations of regular domains via solutions to a Monge-Ampère equation, extending previous results.

Contribution

It introduces a new approach to characterize hypersurfaces with constant Gauss-Kronecker curvature using Li-normalization and Monge-Ampère equations, generalizing prior work for different dimensions.

Findings

Hypersurfaces form foliations of convex domains with constant curvature.

The approach applies to all dimensions with regularity assumptions for n≥3.

No regularity assumption is needed when n=2.

Abstract

For convex hypersurfaces in the affine space $\mathbb{A}^{n+1}$ ($n\geq2$), A.-M.\ Li introduced the notion of $\alpha$-normal field as a generalization of the affine normal field. By studying a Monge-Amp\`ere equation with gradient blowup boundary condition, we show that regular domains in $\mathbb{A}^{n+1}$, defined with respect to a proper convex cone and satisfying some regularity assumption if $n\geq3$, are foliated by complete convex hypersurfaces with constant Gauss-Kronecker curvature relative to the Li-normalization. When $n=2$, a key feature is that no regularity assumption is required, and the result extends our recent work about the $\alpha=1$ case.