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

y k ′ = λk k y k and w ′ = µ w. Finally, λ k k = 1 and µ = 1, which completes theproof.7.5. Families of strongly rigid hypersurfaces. Alongside the same recipe,we can study some classes of hypersurfaces of the form v = ϕ(z¯z).Lemma 7.3. The Lie algebra Hol(M χ ) of the rigid real analytic hypersurfacesM χ in C 2 of equation v = ϕ(z¯z) = z¯z + z 5¯z 5 + z 7¯z 7 + z 8¯z 8 χ(z¯z)is two-dimensional and generated by ∂ w and iz∂ z . Furthermore, M χ is biholomorphicallyequivalent to M ′ χ ′ if and only if χ = χ′ .Proof. The associated differential equation is of the form(7.25) ∂ 2 zz w = [5z3 /4](∂ z w) 5 − [21z 5 /32](∂ z w) 7 + O((∂ z w) 9 ).Extracting from the associated Lie equations (7.10) the coefficients of themonomials (W 1 ) 4 , (W 1 ) 5 , (W 1 ) 6 and (W 1 ) 7 , we obtain four equationswhich are solved by z∂ z Q−Q ≡ 0, ∂ w Q ≡ 0, ∂ z R ≡ 0 and ∂ w R ≡ 0. Next,if M χ and M χ ′ ′ are biholomorphically equivalent, reasoning as in §4, takinginto account that h ∗ (iz∂ z ) and h ∗ (∂ w ) are linear combinations of iz ′ ∂ z ′ and∂ w ′ with real coefficients, we see first that z ′ = λ z e γw/2i and w ′ = µ w forsome three real constants γ, λ ≠ 0 and µ ≠ 0. Replacing z ′ and w ′ in theequation of M χ ′ ′, we get γ = 0, µ = 1 and λ ± 1. In other words, z′ = ±zand w ′ = w, which entails χ ′ (z ′¯z ′ ) ≡ χ(z¯z), as claimed.Perturbing this family we may exhibit other strongly rigid hypersurfaces :Lemma 7.4. The Lie algebra Hol(M χ ) of the real analytic hypersurfacesM χ in C 2 of equation v = ϕ(z, ¯z) = z¯z + z 5¯z 5 + z 7¯z 7 + z 8¯z 8 (z + ¯z) +z 10¯z 10 χ(z, ¯z) is one-dimensional and generated by ∂ w . Furthermore, M χ isbiholomorphically equivalent to M ′ χ ′ if and only if χ = χ′ .Proof. We already know that z∂ z Q − Q ≡ ∂ w Q ≡ ∂ z R ≡ ∂ w R ≡ 0.Extracting from the associated Lie equations (7.10) the coefficient of themonomials (W 1 ) 8 , we also get Q ≡ 0. Next, let M χ and M χ ′ ′ be biholomorphicallyequivalent. Let t ′ = h(t) be such an equivalence. Usingh ∗ (∂ w ) = µ ∂ w ′, where µ ∈ R is nonzero, we get z ′ = f(z) andw ′ = µw + g(z). Next from the equation(7.26) {µ(z¯z + z5¯z 5 + z 7¯z 7 + z 8¯z 8 (z + ¯z) + O(z 9¯z 9 )) + [g(z) − ḡ(¯z)]/2i ≡≡ f(z) ¯f(¯z) + f(z) 5 ¯f(¯z) 5 + f(z) 7 ¯f(¯z) 7 + f(z) 8 ¯f(¯z) 8 (f(z) + ¯f(¯z)) + O(z 9¯z 9 ),we get firstly f(z) = √ |µ|e iθ z by differentiating with respect to ¯z at ¯z = 0and secondly µ = e iθ = 1, which completes the proof.We provide a second family of strongly rigid hypersurfaces in C 2 with aone-dimensional Lie algebra :139