We derive a complete set of invariants for a formal Bishop surface near a point of complex tangent with a vanishing Bishop invariant under the action of formal transformations. We prove that the modular space of Bishop surfaces with a vanishing Bishop invariant and with a fixed Moser invariant s < ∞ is of infinite dimension. We also prove that the equivalence class of the germ of a generic real analytic Bishop surface near a complex tangent with a vanishing Bishop invariant can not be determined by a finite part of the Taylor expansion of its defining equation. This answers, in the negative, a problem raised by J. Moser in 1985 after his joint work with Webster in 1983 and his own work in 1985. Such a phenomenon is strikingly different from the celebrated theory of Moser-Webster for elliptic Bishop surfaces with non-vanishing Bishop invariants. We also show that a formal map between two real analytic Bishop surfaces with the Bishop invariant λ = 0 and with the Moser invariant s = ∞ is convergent. Hence, two real analytic Bishop surfaces with λ = 0 and s < ∞ are holomorphically equivalent if and only if they have the same formal normal form (up to a trivial rotation). Notice that there are many non-convergent formal transformations between Bishop surfaces with λ = 0 and s = ∞. Notice also that a generic formal map between two real analytic hyperbolic Bishop surfaces is divergent as shown by Moser-Webster and Gong. Hence, Bishop surfaces with a vanishing Bishop invariant and s = ∞ behave very differently, in this respect, from hyperbolic Bishop surfaces or elliptic Bishop surfaces with λ = 0 and s = ∞. We also show that a Bishop surface with λ = 0 and s < ∞ generically has a trivial automorphism group and has the largest possible automorphism group if and only if it is biholomorphic to the model surface M s = {(z, w) ∈ C 2 : w = |z| 2 +z s +z s }. Notice that, by the Moser-Webster theorem, an elliptic Bishop surface with λ = 0, always has automorphic group Z 2 . Hence, Bishop surfaces with λ = 0 and s = ∞ have the similar character as that of strongly pseudoconvex real hypersurfaces in the complex spaces of higher dimensions. * Supported in part by NSF-0500626
Let M ⊂ C n+1 (n ≥ 2) be a real analytic submanifold defined by an equation of the form: w = |z| 2 + O(|z| 3 ), where we use (z, w) ∈ C n × C for the coordinates of C n+1 . We first derive a pseudo-normal form for M near 0. We then use it to prove that (M, 0) is holomorphically equivalent to the quadric (M ∞ : w = |z| 2 , 0) if and only if it can be formally transformed to (M ∞ , 0). We also use it to give a necessary and sufficient condition when (M, 0) can be formally flattened. The result is due to Moser for the case of n = 1. * Supported in part by Recently, there appeared several papers, in which CR singular points in the non-critical dimensional case were considered (see [Sto], [DTZ], , to name a few). In [Sto], among other things, Stolovitch introduced a set of generalized Bishop invariants for a non-degnerate general CR singular point, and established some of the results of Moser-Webster [MW] to the case of dim R M > dim C C n+1 . In [DTZ], Dolbeault-Tomassini-Zaitsev introduced the concept of the elliptic flat CR singular points and studied global filling property by complex analytic varieties for a class of compact submanifold of real codimension two in C n+1 with exactly two elliptic flat CR singular points.In this paper, we study the local holomorphic structure of a manifold M near a CR singular point p, for which we can find a local holomorphic change of coordinates such that in the new coordinates system, p = 0 and M near p is defined by an equation of the form: w = |z| 2 +O(|z| 3 ).Here we use (z, w) ∈ C n × C for the coordinates of C n+1 . Such a non-degenerate CR singular point has an intriguing nature that its quadric model has the largest possible symmetry. We will first derive a pseudo-normal form for M near p (see Theorem 2.3). As expected, the holomorphic structure of M near p is influenced not only by the nature of the CR singularity, but also by the fact that (M, p) partially inherits the property of strongly pseudoconvex CR structures for n > 1. Unfortunately, as in the case of n = 1 first considered by Moser [Mos], our pseudo-normal form is still subject to the simplification of the complicated infinite dimensional formal automorphism group of the quadric aut 0 (M ∞ ), where M ∞ is defined by w = |z| 2 . Thus, our pseudo-normal form can not be used to solve the local equivalence problem. However, with the rapid iteration procedure, we will show in §4 that if all higher order terms in our pseudonormal form vanish, then M is biholomorphically equivalent to the model M ∞ . Namely, we have the following:Theorem 1: Let M ⊂ C n+1 (n ≥ 1) be a real analytic submanifold defined by an equation of the form: w = |z| 2 + O(|z| 3 ). Then (M, 0) is holomorphically equivalent to the quadric (M ∞ , 0) if and only if it can be formally transformed to (M ∞ , 0).One of the differences of our consideration here from the case of n = 1 is that a generic (M, 0) can not be formally mapped into the Levi-flat hypersurface Im(w) = 0. As another application of the pseudo-normal form to be obtained in §2, we will g...
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.
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