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

Real-analytic Levi-flat codimension-two CR singular submanifolds are a natural generalization to m, m > 2, of Bishop surfaces in 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 zm = z̄1z2 + z̄12, 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 3 in the nondegenerate case that shows infinitely many formal invariants.

Normal form, Levi-flat, CR singular, codimension two, Bishop surface, mixed-holomorphic equations
Mathematical Subject Classification 2010
Primary: 32V40
Secondary: 53C12, 32S05
Received: 3 March 2014
Accepted: 15 October 2014
Published: 12 April 2015
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