#### Vol. 275, No. 1, 2015

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

### Xianghong Gong and Jiří Lebl

Vol. 275 (2015), No. 1, 115–165
##### 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 ${z}_{m}^{}={\stackrel{̄}{z}}_{1}^{}{z}_{2}^{}+{\stackrel{̄}{z}}_{1}^{2}$, 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
##### Authors
 Xianghong Gong Department of Mathematics University of Wisconsin - Madison Madison, WI 53706-1388 United States Jiří Lebl Department of Mathematics Oklahoma State University Stillwater, OK 74078 United States