Travaux sur les symétries de Lie des équations aux ... - DMA - Ens

origin:{ ( ∣ ∣ )AJ 4 1∣∣∣ Θ(Θ) :=[Θ z Θ zw − Θ w Θ zz ] 3 Θ zzzz Θ w Θ z Θ w ∣∣∣w −Θ zz Θ zw( ∣ ∣ ) ( ∣ ∣)∣∣∣ Θ− 2Θ zzzw Θ z Θ z Θ w ∣∣∣ ∣∣∣ Θw + ΘΘ zz Θ zzww Θ z Θ z Θ w ∣∣∣z +zw Θ zz Θ zw( ∣ ∣ ∣ ∣ ∣ ∣ )∣∣∣ Θ+ Θ zzz Θ z Θ w Θ ww ∣∣∣ ∣∣∣ Θz − 2ΘΘ zw Θ z Θ w Θ zw ∣∣∣ ∣∣∣ Θw + Θzww Θ zw Θ w Θ w Θ zz ∣∣∣w +zzw Θ zw Θ zzz( ∣ ∣ ∣ ∣ ∣ ∣)}∣∣∣ Θ+ Θ zzw − Θ z Θ z Θ ww ∣∣∣ ∣∣∣ Θz + 2ΘΘ zz Θ z Θ z Θ zw ∣∣∣ ∣∣∣ Θw − Θzww Θ zz Θ w Θ z Θ zz ∣∣∣w .zzw Θ zz Θ zzzWe hope, then, that the following precise statement will fill a gap in ourunderstanding of the vanishing of CR curvature tensors.Main (and unique) theorem. An arbitrary, not necessarily rigid, real analytichypersurface M ⊂ C 2 which is Levi nondegenerate at one of its pointsp and has a complex defining equation of the form:w = Θ ( z, z, w )in some system of local holomorphic coordinates (z, w) ∈ C 2 centered at p,is spherical at p if and only if its graphing complex function Θ satisfies thefollowing explicit sixth-order algebraic partial differential equation:(0 ≡− Θ w ∂Θ z Θ zw − Θ w Θ zz ∂z +identically in C { z, z, w } .Θ zΘ z Θ zw − Θ w Θ zz∂∂w) 2[AJ 4 (Θ) ]Here, it is understood that the first-order derivation in parentheses is appliedtwice to the fourth-order rational differential expression AJ 4 (Θ). The1factor[Θ z Θ zw −Θ w Θ zzthen appears, and after clearing out this denominator,] 7one obtains a universal polynomial differential expression AJ 6 (Θ) dependingupon the sixth-order jet Jz,z,w 6 Θ and having integer coefficients. A partialexpansion is provided in Section 5, and the already formidable incompressiblelength of this expansion perhaps explains the reason why no referencein the literature provides the explicit expressions, in terms of some definingfunction for M, of Élie Cartan’s two fundamental differential invariants 7which can (in principle) be used to classify real analytic hypersurfaces ofC 2 up to biholomorphisms, and to at least characterize sphericality.Suppose in particular for instance that M is rigid, given by a complexequation of the form w = −w + Ξ(z, z), that is to say with Θ(z, z, w)of the form −w + Ξ(z, z), so that the reality condition simply reads here:Ξ(z, z) ≡ Ξ(z, z). Then as a corollary-exercise, sphericality is explicitly7 See [3] and also [23], where the tight analogy with second-order ordinary differentialequations is well explained.187