Flattening a non-degenerate CR singular point of real codimension two

Abstract

This paper continues the previous studies in two papers of Huang-Yin [HY3-4] on the flattening problem of a CR singular point of real codimension two sitting in a submanifold in C n+1 with n + 1 ≥ 3, whose CR points are non-minimal. Partially based on the geometric approach initiated in [HY3] and a formal theory approach used in [HY4], we are able to provide a very general flattening theorem for a non-degenerate CR singular point. As an application, we provide a solution to the local complex Plateau problem and obtain the analyticity of the local hull of holomorphy near a real analytic definite CR singular point in a general setting.

On the Levi-flat Plateau problem

“…Proof. Fang-Huang [6] prove that such an M can be holomorphically flattened near p, that is, realized as a subset of C n × R. Then the authors' result from [12], which is stated in C n × R, obtains the holomorphic extension Φ.…”

“…Next we use a theorem of Fang-Huang [6]: When n ≥ 2, a real-analytic CR singular submanifold of C n+1 with an A-nondegenerate CR singularity, except for one exceptional case, can be "flattened" locally if it is flat at the CR points. Using this flattening and the previous extension result of the authors [12] on the inverse, if we further assume that f (∂Ω) has only A-nondegenerate CR singularities as above, then F becomes an immersion on all of Ω, including the CR singularities.…”

On normal forms of complex points of small C2-perturbations of real 4-manifolds embedded in a complex 3-manifold

“…Note that using the list of normal forms in dimension 2, recently a result on holomorphical flattenability of CR-nonminimal codimension 2 real analytic submanifold near a complex point in C n , n ≥ 2, was obtained through the works of Huang and Yin [26,27], Fang and Huang [28].…”

On normal forms of complex points of codimension 2 submanifolds

“…Global theory (filling spheres with holomorphic discs) was studied by Bedford and Gaveau [1] and Bedford and Klingenberg [2], and has later resulted in many important theorems in symplectic and contact geometry. In higher dimensions, assuming real analyticity, a similar problem of finding an appropriate Levi flat hypersurface that is bounded by the submanifold near the complex point, is treated first in the papers of Dolbeault, Tomasini and Zaitsev [7,8] and to a greater generality by Huang and Yin [19,20] and Fang and Huang [9]. The problem is equivalent to understanding when the manifold can be holomorphically flattened near the complex point.…”