Flattening of CR singular points and analyticity of the local hull of holomorphy II

Abstract

This is the second article of the two papers, in which we investigate the holomorphic and formal flattening problem of a non-degenerate CR singular point of a codimension two real submanifold in C n with n ≥ 3. The problem is motivated from the study of the complex Plateau problem that looks for the Levi-flat hypersurface bounded by a given real submanifold and by the classical complex analysis problem of finding the local hull of holomorphy of a real submanifold in a complex space. The present article is focused on non-degenerate flat CR singular points with at least one non-parabolic Bishop invariant. We will solve the formal flattening problem in this setting. The results in this paper and those in [HY3] are taken from our earlier arxiv post [HY4]. We split [HY4] into two independent articles to avoid it being too long.

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

“…Our approach here is more along the lines of the geometric argument used in [HY3]. The great benefit of this argument, compared with the formal argument in [HY4], is that we do not need to know the precise structure of the quadratic normal form which is almost impossible to obtain when n + 1 > 3. The reason we want to have the set of CR singular points has real dimension less than (2n − 2) is because we need to find a good elliptic complex tangency for the sliced manifolds.…”

“…Indeed, one would suspect that (M, 0) might be not flattenable as suggested in Remark 6.5. Fortunately, the quadratic term in the defining equation of M in (55) now is in the simplest symmetric form, which made the formal normal form theory developed in [HY4] disposable here. And we are still able to produce a positive flattening result.…”

“…A solution in certain cases for this generalized Bishop problem was obtained in Huang-Yin [HY3]. In a subsequent paper of Huang-Yin [HY4], a formal version of problems similar to that considered in [HY3] has been studied.…”

“…A good thing about this exceptional case is that the associated quadratic term has the simplest possible symmetric form. This makes the formal argument developed in [HY4] to be very much adaptable to this setting. Indeed, we can first prove that in this case M can always be formally flattened by a special form of holomorphic transformations, which together with the Huang-Krantz construction of holomorphic disks give a convergent flattening as in the other cases considered in [HY4].…”

Convergence of the Chern–Moser–Beloshapka normal forms

“…Our paper takes up a very classical problem with a new tool, and gives a formal normal form for Levi-nondegenerat real analytic manifolds which under a rather simple condition (see (85)) can be shown to be convergent. Recent advances in normal forms for real submanifolds of complex spaces with respect to holomorphic transformations have been significant: We would like to cite in this context the recent works of Huang and Yin [HY09, HY16,HY17], the second author and Gong [GS16], and Gong and Lebl [GL15].…”