A class of new complete affine maximal type hypersurfaces
Yalin Sun, Ruiwei Xu
TL;DR
This work advances affine differential geometry by classifying a family of Calabi hypersurfaces with negative constant sectional curvature and constructing new Euclidean complete and Calabi complete solutions to the affine maximal type and Abreu equations with negative curvature parameters. It introduces a special Ejiri frame, derives conditions on the theta function and curvature, and reveals a warped-product structure that enables explicit local classifications under theta=2. It then presents three explicit models corresponding to the sign of a curvature parameter and completes the local classification for Calabi hypersurfaces with the specified frame. Finally, it provides two non-quadratic complete examples solving the affine maximal type equation for $a=-\frac{n}{n+1}$ and $a=-\frac{1}{n+1}$, together with Legendre-dual descriptions and completeness analyses, highlighting implications for Bernstein-type problems in affine geometry.
Abstract
In this paper we classify a kind of special Calabi hypersurfaces with negative constant sectional curvature in Calabi affine geometry. Meanwhile, we find a class of new Euclidean complete and Calabi complete affine hypersurfaces, which satisfy the affine maximal type equation and the Abreu equation with negative constant scalar curvatures.