Analytic Differential Equations and Spherical Real Hypersurfaces

الأصل · EN

We establish an injective correspondence M(M) between real-analytic nonminimal hypersurfaces M², spherical at a generic point, and a class of second order complex ODEs with a meromorphic singularity. We apply this result to the proof of the bound dimhol(M,p)≤ 5 for the infinitesimal automorphism algebra of an arbitrary germ (M,p) (S³,p') of a real-analytic Levi nonflat hypersurface M² (the Dimension Conjecture). This bound gives the first proof of the dimension gap dimhol(M,p)={8,5,4,3,2,1,0} for the dimension of the automorphism algebra of a real-analytic Levi nonflat hypersurface. As another application we obtain a new regularity condition for CR-mappings of nonminimal hypersurfaces, that we call Fuchsian type, and prove its optimality for extension of CR-mappings to nonminimal points. We also obtain an existence theorem for solutions of a class of singular complex ODEs (Theorem 3.5).