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

Then the Lie criterion states that a holomorphic vector field X = Q ∂ z +R ∂ w belongs to Sym(E M ) if and only if its second prolongation X (2) =Q ∂ z + R ∂ w + R 1 ∂ W 1 + R 2 ∂ W 2 is tangent to ∆ M , where the coefficientsR 1 and R 2 are given by the formulas (7.8) specified for n = 2, namely:(7.17) ⎧⎪⎨R 1 = ∂ z R + [∂ w R − ∂ z Q] W 1 − ∂ w Q (W 1 ) 2 .R 2 = ∂zz 2 ⎪⎩R + [2∂2 zw R − ∂2 zz Q] W 1 + [∂ww 2 R − 2∂2 zw Q] (W 1 ) 2 − ∂ww 2 Q (W 1 ) 3 ++ [∂ w R − 2∂ z Q] W 2 − 3∂ w Q W 1 W 2 .The tangency condition yields the following equation which is satisfied on∆ M , i.e. after replacing W 2 by its value given by (7.16):(7.18) R 2 + (15i/2 2 )R 1 (W 1 ) 3 + O((W 1 ) 6 ) = 0.By expanding equation (7.18) in powers of W 1 up to order five, we obtainthe following system of six linear partial differential equations which mustbe satisfied by the derivatives of Q and R up to order two:⎧(e 0 ) : ∂zz 2 R − i(∂ wR − 2∂ z Q) ≡ 0.(e 1 ) : 2∂ 2 zw R − ∂2 zz Q ≡ 0.137(7.19)⎪⎨(e 2 ) : ∂ 2 ww R − 2∂2 zw Q ≡ 0.(e 3 ) : −∂ 2 ww Q + 15i2 2 ∂ z R ≡ 0.⎪⎩(e 4 ) : − 15i2 4 (∂ w R − 2∂ z Q) + 15i2 2 (∂ w R − ∂ z Q) ≡ 0.(e 5 ) : − 15i2 4 (−3∂ w Q) − 15i2 2 (∂ w Q) ≡ 0.It follows from the equation (e 5 ) that ∂ w Q ≡ 0 which implies ∂ 2 ww Q ≡ 0.Then by equation (e 3 ) we obtain ∂ z R ≡ 0, implying ∂ 2 zzR ≡ 0. From equation(e 0 ) we get ∂ w R ≡ 2∂ z Q and, from equation (e 4 ), we get ∂ w R ≡ ∂ z Q.Consequently ∂ z R ≡ ∂ w R ≡ ∂ z Q ≡ ∂ w Q ≡ 0. Since the two vector fields∂ z and ∂ w evidently belong to Sym(E M ), it follows that dim C Sym(E M ) =2. Finally, this implies that dim R Aut CR (M) = 2 and that Aut CR (M) isgenerated by ∂ w + ∂ ¯w and ∂ z + ∂¯z .Next, let χ(y) and χ ′ (y ′ ) be two real analytic functions, and assume thatM χ and M ′ χ ′ are biholomorphically equivalent. Let t′ = h(t) be such anequivalence. Reasoning as in §4 and taking into account that both are strongtubes, we see that h ∗ (∂ z ) and h ∗ (∂ w ) must be linear combinations of ∂ z ′ and∂ w ′ with real coefficients. It follows that h must be linear, of the form z ′ =az +bw, w ′ = cz +dw, where a, b, c and d are real. Since T 0 M χ = {v = 0}