Analytic differential equations and spherical real hypersurfaces

Abstract

Abstract. We establish an injective correspondence M −→ E (M ) between real-analytic nonminimal hypersurfaces M ⊂ C 2 , 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 dim hol(M, p) ≤ 5 for the infinitesimal automorphism algebra of an arbitrary germ (M, p) ∼ (S 3 , p ′ ) of a real-analytic Levi nonflat hypersurface M ⊂ C 2 (the Dimension Conjecture). This bound gives the proof of the dimension gap dim hol(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.

Blow-ups and infinitesimal automorphisms of CR-manifolds

“…This result improves on the statement of Conjecture 1.1. Note that the result is global in M, even if one takes M " U to be a small fixed neighborhood of a point x P M. The proof of the theorem also leads to the following local version of the result, generalizing theorems from [KS2,IK1,IK2] for arbitrary n. Corollary 1.3 With the assumptions of Theorem 1.2 in the case n ě 3 the condition dim holpM, xq ą n 2`2 n`2 for x P M implies that M is spherical in a neighborhood of the point x, and this estimate is sharp.…”

“…Proof of Theorem 1.2. For n " 1 the theorem was obtained in [KS2,IK1], for n " 2 its stronger variant was proven in [IK2], so we assume that n ě 3.…”

“…38]. The authors of [KS2] argument that for n " 1 this is a version of Poincaré's problème local [Po].…”

Symmetries of second order ODEs

“…CR-structures in 3D without Levi-nondegeneracy constraint were considered in [6], and it was proved that if CR-manifold M 3 is not Levi-flat then its symmetry algebra has dimension 8, 5, 4, 3, 2, 1 or 0. This was done by exploiting the above Segre correspondence and studying the respective Fuchsian type second order equations.…”

Normal Form for Second Order Differential Equations

“…The approach is based on a certain similarity between real submanifolds in complex space and manifolds of solutions of completely integrable systems of PDEs. Application of Associated Differential Equations to studying CR-manifolds has recently led to important developments in CR-geometry (see, e.g., [14], [15], [16,17]). The present paper is probably the first work on applying the connection between CR-manifolds and Differential Equations in the other direction.…”