Homogeneous C21 Models

TL;DR

This paper advances the classification of homogeneous C21 hypersurfaces in complex space by applying the power series method, confirming previous results, and constructing an invariant branching tree through complex calculations.

Contribution

It introduces a detailed invariant branching tree for homogeneous C21 hypersurfaces, completing the classification with high-order power series computations.

Findings

Confirmed Fels-Kaup classification results.

Constructed a differential-invariant branching tree.

Performed high-order power series calculations by hand.

Abstract

Fels-Kaup (Acta Mathematica 2008) classified homogeneous $\mathfrak{C}_{2,1}$ hypersurfaces $M^5 \subset \mathbb{C}^3$ and discovered that they are all biholomorphic to tubes $S^2 \times i \mathbb{R}^3$ over some affinely homogeneous surface $S^2 \subset \mathbb{R}^3$. The second and third authors in 2003.08166, by performing highly non-straightforward calculations, conducted the Cartan method of equivalence to classify homogeneous models of PDE systems related to such $\mathfrak{C}^{2,1}$ hypersurfaces $M^5 \subset \mathbb{C}^3$. Kolar-Kossovskiy 1905.05629 and the authors 2003.01952 constructed a formal and a convergent Poincar\'e-Moser normal form for $\mathfrak{C}_{2,1}$ hypersurfaces $M^5 \subset \mathbb{C}^3$. But this was only a first, preliminary step. Indeed, the invariant branching tree underlying Fels-Kaup's classification was still missing in the literature, due to computational obstacles. The present work applies the power series method of equivalence, confirms Fels-Kaup 2008, and finds a differential-invariant tree. To terminate the middle (thickest) branch, it is necessary to compute up to order $10$ with $5$ variables. Again, calculations, done by hand, are non-straightforward.