Analytic differential equations and spherical real hypersurfaces

Abstract

We establish an injective correspondence $M\longrightarrow\mathcal E(M)$ between real-analytic nonminimal hypersurfaces $M\subset\mathbb{C}^{2}$, 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 $\mbox{dim}\,\mathfrak{hol}(M,p)\leq 5$ for the infinitesimal automorphism algebra of an \it arbitrary \rm germ $(M,p)\not\sim(S^3,p')$ of a real-analytic Levi nonflat hypersurface $M\subset\mathbb{C}^2$ (the Dimension Conjecture). This bound gives the first proof of the dimension gap $\mbox{dim}\,\mathfrak{hol}(M,p)=\{8,5,4,3,2,1,0\}$ 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 \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).