Classification of Hessian Rank 1 Affinely Homogeneous Hypersurfaces H^n in R^{n+1} in Dimensions n = 2, 3, 4

TL;DR

This paper classifies affinely homogeneous hypersurfaces with constant Hessian rank 1 in dimensions 2, 3, and 4, revealing unique models and illustrating a novel power series method of equivalence.

Contribution

It determines all non-product constant Hessian rank 1 affinely homogeneous hypersurfaces in dimensions 2, 3, and 4, introducing the Power Series Method of Equivalence for invariant analysis.

Findings

Dimension 3 has a unique homogeneous model with a specific formula.

Dimension 4 has two simply homogeneous models differing by a sign.

Complete classification is achieved for dimensions 2, 3, and 4.

Abstract

In a previous memoir 2202.03030, we showed that in every dimension $n \geq 5$, there exists (unexpectedly) no affinely homogeneous hypersurface $H^n \subset \mathbb{R}^{n+1}$ having Hessian of constant rank 1 (and not being affinely equivalent to a product with $\mathbb{R}^{m \geqslant 1}$). The present work is devoted to determine all non-product constant Hessian rank 1 affinely homogeneous hypersurfaces $H^n \subset \mathbb{R}^{n+1}$ in dimensions $n = 2, 3, 4$, the cases $n = 1, 2$ being known. With complete details in the case $n = 2$, we illustrate the main features of what can be termed the "Power Series Method of Equivalence". The gist is to capture invariants at the origin only, to create branches, and to infinitesimalize calculations. In dimension $n = 3$, we find a single homogeneous model: \[ u \,=\, \frac{1}{3\,z^2} \Big\{ \big( 1-2\,y+y^2-2\,xz \big)^{3/2} - (1-y)\, \big( 1-2\,y+y^2-3\,xz \big) \Big\}, \] the singularity $\frac{1}{3z^2}$ being illusory. In dimension $n = 4$, without reaching closed forms, we find two simply homogeneous models, differing by some $\pm$ sign.