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

Definition 8.43. A real analytic hypersurface M ⊂ C n+1 is called rigid atone of its points p if there exists L ∈ hol(M) with(8.44) T p M = T c pM ⊕ R ( L (p) + L (p) ) .Similar elementary facts hold for general submanifolds of solutions.Lemma 8.45. With n 1 and m = 1, let M be a (connected) submanifoldof solutions that is solvable with respect to the parameters. If there exists anonzero L + L ∗ ∈ SYM(M ), then at Zariski-generic points p ∈ M , wehave L (p) ∉ F v (p) and there exist local coordinates centered at p in whichL = ∂ , L ∗ = ∂ , whence M has equation of the form∂y ∂b(8.46) y = b + Π(x, a),with Π independent of b.The associated system (E M ) has then equations F α that are all independentof y.8.47. Study of the Lie symmetries of (E 5 ). In Example 1.28, it is thusessentially no restriction to assume the hypersurface M ⊂ C 3 to be rigid.Theorem 8.48. ([GM2003b, FK2005a, FK2005b]) The model hypersurfaceM 0 of equation(8.49) w = ¯w + i 2 z1¯z 1 + z 1 z 1¯z 2 + ¯z 1¯z 1 z 21 − z 2¯z 2has transitive Lie symmetry algebra hol(M 0 ) isomorphic to so(3, 2) andis locally biholomorphic to a neighborhood of every geometrically smoothpoint of the tube(8.50) (Rew ′ ) 2 = (Rez ′ 1) 2 + (Rez ′ 1) 3over the standard cone of R 3 . Both are Levi-degenerate with Levi form ofrank 1 at every point and are 2-nondegenerate. The associated PDE system(E M0 )(8.51) y x 2 = 1 4 (y x 1)2 , y x 1 x 1 x 1 = 0(plus other equations obtained by cross differentiation) has infinitesimal Liesymmetry algebra isomorphic to so(5, C), the complexification so(3, 2)⊗C.Through tentative issues ([Eb2006, GM2006]), it has been suspected thatM 0 is the right model in the category of real analytic hypersurfaces M ⊂ C 3having Levi form of rank 1 that are 2-nondegenerate everywhere. Basedon the rigidity of the simple Lie algebra so(5, C) (Theorem 5.15), Theorem8.105 below will confirm this expectation.275