On Levi-flat Hypersurfaces with Prescribed Boundary

Abstract

Abstract:We address the problem of existence and uniqueness of a Leviflat hypersurface M in C n with prescribed compact boundary S for n ≥ 3. The situation for n ≥ 3 differs sharply from the well studied case n = 2. We first establish necessary conditions on S at both complex and CR points, needed for the existence of M . All CR points have to be nonminimal and all complex points have to be "flat". Then, adding a positivity condition at complex points, which is similar to the ellipticity for n = 2 and excluding the possibility of S to contain complex (n − 2)-dimensional submanifolds, we obtain a solution M to the above problem as a projection of a possibly singular Levi-flat hypersurface in R × C n . It turns out that S has to be a topological sphere with two complex points and with compact CR orbits, also topological spheres, serving as boundaries of the (possibly singular) complex leaves of M . There are no more global assumptions on S like being contained in the boundary of a strongly pseudoconvex domain, as it was in case n = 2. Furthermore, we show in our situation that any other Levi-flat hypersurface with boundary S must coincide with the constructed solution.

On a family of analytic discs attached to a real submanifold $M ⊂ \mathbb{C}^{N+1}$

“…Then M given by Theorem 1.1 is not necessary a Levi-flat hypersurface as in Kenig-Webster's case from [14] in C 2 . The existence problem of a Levi-flat hypersurface with prescribed boundary S in C N+1 with N ≥ 2, was studied by Dolbeault-Tomassini-Zaitsev in [2] under the following natural assumptions on S:…”

“…The CR singularity p = 0 is called elliptic if the quadratic part from (1.2) is positive definite. We say that p = 0 is a "flat" if Definition 2.1 from [2] is satisfied. Under the precedent natural assumptions, Dolbeault-Tomassini-Zaitsev proved the existence of a (possibly singular) Levi-flat hypersurface which bounds S in the sense of currents (see Theorem 1.3, [2]).…”

Boundaries of Levi-flat hypersurfaces: special hyperbolic points

“…Proposition 1. ( [DTZ05,DTZ10]). Assume that S ⊂ C n , (n ≥ 3) is nowhere minimal at all its CR points and has an elliptic flat complex point p. Then there exists a neighborhood V of p such that V \ {p} is foliated by compact real (2n − 3)-dimensional CR orbits diffeomorphic to the sphere S 2n−3 and there exists a smooth function ν, having the CR orbits as the level surfaces.…”

“…As in [DTZ10], in the neighborhood of 0, denote by E(q), q ∈ S\{0}, w < 0 the tangent space to the local CR orbit K on S through q, and by E 0 (q 0 ), q 0 ∈ S 0 \ {0}, w < 0 the analogous object for S 0 . We have :…”

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.…”