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

206This observation has essentially no practical interest, because the computationof F ∗ in terms of F relies upon the composition of three maps . . .except notably in the CR case, since the duality in this case is complex conjugation:Φ ∗ = Φ. In summary, we have established the following.Proposition. An arbitrary, not necessarily rigid, real analytic hypersurfaceM ⊂ C 2 which is Levi nondegenerate at one of its points p and has a complexdefinining 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 the right-hand side Φ of its uniquely associatedsecond-order ordinary complex differential equation:w zz (z) = Φ ( z, w(z), w z (w) )satisfies the single fourth-order partial differential equation:0 ≡ Φ wzw zw zw z(z, w, wz).It now only remains to re-express this fourth-order partial differentialequation in terms of the complex graphing function Θ(z, z, w) for M. Wewill achieve this more generally for F yxy xy xy x.§4. EFFECTIVE DIFFERENTIAL CHARACTERIZATIONOF SPHERICALITY IN C 2Reminding the reasonings and notations introduced in a neighborhood ofequation (7.28), the transformation:(x, y, yx)↦−→(x, a, b)and its inverse are given by the two triples of functions:⎡⎡x = xx = x⎢⎣ a = A(x, y, y x )b = B(x, y, y x )and⎢⎣ y = Q(x, a, b)y x = Q x (x, a, b).Equivalently, one has the two pairs of identically satisfied equations:a ≡ A ( x, Q(x,a,b), Q x (x,a,b) )b ≡ B ( x, Q(x,a,b), Q x (x,a,b) ) and y ≡ Q ( x, A(x,y,y x ), B(x,y,y x ) )y x ≡ Q x(x, A(x,y,yx ), B(x,y,y x ) ) .Differentiating the second column of equations with respect to x, to y and toy x yields:0 = Q x + Q a A x + Q b B x 0 = Q xx + Q xa A x + Q xb B x1 = Q a A y + Q b B y 0 = Q xa A y + Q xb B y0 = Q a A yx + Q b B yx 1 = Q xa A yx + Q xb B yx .