Journal of Differential Geometry

Analytic differential equations and spherical real hypersurfaces

Ilya Kossovskiy and Rasul Shafikov

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We establish an injective correspondence $M \to \mathcal{E}(M)$ 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 $\dim \mathfrak{hol}(M,p) \leq 5$ for the infinitesimal automorphism algebra of an arbitrary 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 proof of the dimension gap $\dim \mathfrak{hol}(M,p) = \{8, 5, 4, 3, 2, 1, 0 \}$ 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 Fuchsian type, and prove its optimality for the extension of CR-mappings to nonminimal points.

We also obtain an existence theorem for solutions of a class of singular complex ODEs.

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J. Differential Geom., Volume 102, Number 1 (2016), 67-126.

First available in Project Euclid: 5 January 2016

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Kossovskiy, Ilya; Shafikov, Rasul. Analytic differential equations and spherical real hypersurfaces. J. Differential Geom. 102 (2016), no. 1, 67--126. doi:10.4310/jdg/1452002878.

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