Analytic Differential Equations and Spherical Real Hypersurfaces

2014/01/19 by Kossovskiy, Ilya, Shafikov, Rasul · 2 citations
#Complex Variables (math.CV) #Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.1401.4724

Abstract

We establish an injective correspondence M\longrightarrow\mathcal E(M) between real-analytic nonminimal hypersurfaces M⊂ℂ², 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 \mathfrakhol(M,p)≤ 5 for the infinitesimal automorphism algebra of an \it arbitrary \rm germ (M,p)\not∼(S³,p') of a real-analytic Levi nonflat hypersurface M⊂ℂ² (the Dimension Conjecture). This bound gives the first proof of the dimension gap dim \mathfrakhol(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 \it Fuchsian type, \rm 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).

Cited by

• Strong Ramadanov Conjecture for Real Ellipsoids in \mathbb C²: two approaches
• Sphericity and Analyticity of a strictly pdeusoconvex hypersurface in low regularity I

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