Homogeneous C21 Models

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 Poincaré-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.