Normal forms for CR singular codimension two Levi-flat submanifolds

Abstract

Real-analytic Levi-flat codimension two CR singular submanifolds are a natural generalization to ${\mathbb{C}}^m$, $m>2$, of Bishop surfaces in ${\mathbb{C}}^2$. Such submanifolds for example arise as zero sets of mixed-holomorphic equations with one variable antiholomorphic. We classify the codimension two Levi-flat CR singular quadrics, and we notice that new types of submanifolds arise in dimension 3 or greater. In fact, the nondegenerate submanifolds, i.e. higher order purturbations of $z_m={\bar{z}}_1{z}_2+{\bar{z}}_1^2$, have no analogue in dimension 2. We prove that the Levi-foliation extends through the singularity in the real-analytic nondegenerate case. Furthermore, we prove that the quadric is a (convergent) normal form for a natural large class of such submanifolds, and we compute its automorphism group. In general, we find a formal normal form in ${\mathbb{C}}^3$ in the nondegenerate case that shows infinitely many formal invariants.

41 pages, accepted to Pacific Journal of Mathematics

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• Flattening a non-degenerate CR singular point of real codimension two
• A CR singular analogue of Severi's theorem
• Local normal forms of singular Levi-flat hypersurfaces
• On CR singular CR images
• CR Singularities and Generalizations of Moser's Theorem II
• Special Classes of CR Singularities I