Normal forms for CR singular codimension-two Levi-flat submanifolds

Xianghong Gong and Jiří Lebl

Vol. 275 (2015), No. 1, 115–165

DOI: 10.2140/pjm.2015.275.115

Abstract

Real-analytic Levi-flat codimension-two CR singular submanifolds are a natural generalization to ${ℂ}^m$, $m>2$, of Bishop surfaces in ${ℂ}^2$. 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 three or higher. In fact, the nondegenerate submanifolds, i.e., higher order perturbations of $z_m^{}={\bar{z}}_1^{}{z}_2^{}+{\bar{z}}_1^2$, have no analogue in dimension two. 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 ${ℂ}^3$ in the nondegenerate case that shows infinitely many formal invariants.

Keywords

Normal form, Levi-flat, CR singular, codimension two, Bishop surface, mixed-holomorphic equations

Mathematical Subject Classification 2010

Primary: 32V40

Secondary: 53C12, 32S05

Milestones

Received: 3 March 2014

Accepted: 15 October 2014

Published: 12 April 2015