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

Abstract

For convex hypersurfaces in the affine space \({\mathbb {A}}^{n+1}\) (\(n\ge 2\)), A.-M. Li introduced the notion of \(\alpha \)-normal field as a generalization of the affine normal field. By studying a Monge–Ampère 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 \(n\ge 3\), 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.