2021/04/19 by Foo, Wei-Guo, Merker, Joel, Nurowski, Pawel +1
#Complex Variables (math.CV) #Differential Geometry (math.DG) #FOS: Mathematics
paper · doi:10.48550/arxiv.2104.09608
Fels-Kaup (Acta Mathematica 2008) classified homogeneous \mathfrakC2,1 hypersurfaces M5 ⊂ ℂ3 and discovered that they are all biholomorphic to tubes S2 × i ℝ3 over some affinely homogeneous surface S2 ⊂ ℝ3. 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 \mathfrakC2,1 hypersurfaces M5 ⊂ ℂ3. Kolar-Kossovskiy 1905.05629 and the authors 2003.01952 constructed a formal and a convergent Poincaré-Moser normal form for \mathfrakC2,1 hypersurfaces M5 ⊂ ℂ3. 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.