ANALYTIC STABILITY OF THE CR CROSS-CAP

Adam Coffman

For m < n, any real analytic m-submanifold of complex n-space with a nondegenerate CR singularity is shown to be locally equivalent, under a holomorphic coordinate change, to a fixed real algebraic variety defined by linear and quadratic polynomials. The situation is analogous to Whitney’s stability theorem for cross-cap singularities of smooth maps. The complex analyticity of the normalizing transformation is proved using a rapid convergence argument.

1. Introduction

For m ≤ n, if a real m-manifold M is embedded in ℂ^{n}, then for each point x on M there are two possibilities: the tangent m-plane at x may contain a complex line, so M is said to be “CR singular” at x, or it may not, so M is said to be “totally real” at x. This article will consider the local extrinsic geometry of a real analytically embedded M near a CR singular point, in the case when the CR singularity satisfies some natural nondegeneracy properties and \frac{2}{3}(n + 1) ≤ m < n (so (m, n) = (4, 5) is the case of lowest dimension). The main result is an algebraizability property: there exists a holomorphic coordinate change in a neighborhood of x so that M is real algebraic in the new coordinate system. In fact, M will be biholomorphically equivalent to a fixed normal form variety, so that, unlike the well-known m = n case, nondegenerate CR singularities have no continuous invariants under biholomorphisms.

The analysis of normal forms near CR singular points is part of the program of studying the local equivalence problem for real m-submanifolds of ℂ^{n}, as described in the survey [BER]. Normal forms for CR singular real n-manifolds in ℂ^{n}, m = n ≥ 2, have been the subject of much study, e.g., by [Bishop], [M], [MW], [Webster]. Real surfaces in ℂ^{n} (m = 2, n ≥ 3) have been considered by [H_{1}], [H_{2}], [C_{4}], and real threefolds in ℂ^{4} by [C_{5}]. A formal normal form for a CR singular real 4-manifold in ℂ^{5} was found by [Beloshapka] and [C_{1}] — it was shown that there exists a transformation (not unique) defined by formal power series, taking M to the normal form.

Mathematics Subject Classification. 32V40, 32S05.

Key words and phrases. Normal form, CR singularity, real submanifold.

The new result here is the existence of a normalizing transformation defined by series that are convergent in a neighborhood of the singularity.

2. Topological considerations

We briefly recall some topological properties of CR singularities. We could consider real submanifolds of any complex manifold, but since the main result on the normalization is about the local geometry, we can begin by assuming M is a smoothly immersed real m-manifold in ℂ^{n}.

The most basic invariant of a CR singularity at a point x ∈ M is the number j(x) = dim_{ℂ} T_{x} ∩ J_{x}T_{x}, where T_{x} is the real tangent space of M at x and J_{x} is the complex structure operator corresponding to scalar multiplication by i on the tangent space of the ambient complex manifold. The number j(x) is the dimension of the largest complex subspace tangent to M at x, so 0 ≤ j(x) ≤ m/2.

One way to keep track of j(x) is the following construction. For m ≤ n, let G be the grassmannian variety of real m-subspaces in ℂ^{n} ≅ ℝ^{2n} (see [G], [C_{1}]), and then the real m-subspaces T such that dim_{ℂ} T ∩ iT ≥ j form a subvariety D_{j} of real codimension 2j(n − m + j) in G. The occurrence of complex tangents of an immersion corresponds to the intersection of D_{j} with the image of the Gauss map M → G : x ↦ T_{x}, and the immersion could be called “generic” if the Gauss map meets each stratum D_{j} \ D_{j+1} transversely. So, generic immersions of M in ℂ^{n} are totally real outside a codimension 2(n − m + 1) subset of M, and if m < \frac{2}{3}(n + 1), then a generic immersion of M is totally real everywhere. This resembles the bounds in Whitney’s embedding and immersion theorems ([W_{2}], [W_{3}]). In the range \frac{2}{3}(n+1) ≤ m ≤ n, CR singularities are topologically stable — small smooth perturbations of a generic immersion with a CR singular point will still have a CR singular point. For compact real submanifolds of complex manifolds, there are topological obstructions to the property of being totally real at every point, and the CR singularities can be enumerated by characteristic class formulas (see [D_{1}], [D_{2}], [C_{1}], and their references on this topic).

The case addressed by this paper is \frac{2}{3}(n+1) ≤ m < n, and j(x) = 1 — only points x where exactly one complex line is tangent at x will be considered, and only in dimension cases where the CR singularity is stable under smooth perturbations of the immersion. As mentioned in the Introduction, the m = n case has a qualitatively different local geometry than the m < n case and is not considered here. The cases (m, n) = (2, 3) or (3, 4) considered in [C_{4}], [C_{5}], fall outside the topological stability range. The case (m, n) = (4, 5), considered by [Beloshapka], [D_{1}], [C_{1}], [C_{2}], has the lowest dimensions in the range, and the generic singularity is isolated (codimension 4 in M).

3. The quadratic normal form

Let the ambient complex space be ℂ^{n}, with coordinates (z_{1}, …, z_{n}). The real and imaginary parts of the coordinate functions are labeled z_{j} = x_{j} +iy_{j} for j = 1, …, n. Let M be a real analytic m-dimensional submanifold embedded in ℂ^{n}, with m < n, and let x be a point on M at which M is tangent to a complex line but not to any complex 2-plane — in terms of the previous Section, j(x) = 1, which we regard as a nondegeneracy assumption, since for M in general position, the points where j(x) > 1 form a subset of higher codimension.

By a translation that moves x to the origin 0, and then a complex linear transformation of ℂ^{n}, the tangent space T = T0 of M can be assumed to be the (x_{1}, y_{1}, x_{2}, …, x_{m−1})-space, which contains the z_{1}-axis. Then there is some neighborhood Δ of the origin in ℂ^{n} so that the defining equations of M in Δ are in the form of a graph over a neighborhood of the origin in T:

(1)  y_{s} = H_{s}(z_{1}, z̄_{1}, x_{2}, …, x_{m−1})
z_{u} = h_{u}(z_{1}, z̄_{1}, x_{2}, …, x_{m−1}),

where for s = 2, …, m − 1, H_{s} is a real valued real analytic function, and for u = m, …, n, h_{u} is a complex valued real analytic function, with H_{s} and h_{u} defined in a neighborhood of the origin in T, and vanishing to second order at (x_{1}, y_{1}, x_{2}, …, x_{m−1}) = (0, …, 0). The expression “x_{2}, …, x_{m−1}” is abbreviated as just x. So, the defining functions are of the following form:

H_{s}(z_{1}, z̄_{1}, x) = α_{s}z_{1}^{2} + β_{s}z_{1}z̄_{1} + γ_{s}z̄_{1}^{2} + ∑ δ_{s}^{s_{1}}z_{1}x_{s_{1}}
+ ∑ ϵ_{s}^{s_{1}} z̄_{1}x_{s_{1}} + ∑ θ_{s}^{s_{1}s_{2}}x_{s_{1}}x_{s_{2}} + E_{s}(z_{1}, z̄_{1}, x)
h_{u}(z_{1}, z̄_{1}, x) = α_{u}z_{1}^{2} + β_{u}z_{1}z̄_{1} + γ_{u}z̄_{1}^{2} + ∑ δ_{u}^{s_{1}}z_{1}x_{s_{1}}
+ ∑ ϵ_{u}^{s_{1}} z̄_{1}x_{s_{1}} + ∑ θ_{u}^{s_{1}s_{2}}x_{s_{1}}x_{s_{2}} + e_{u}(z_{1}, z̄_{1}, x),

with E_{s}, e_{u} having terms of degree three or higher. Each of these functions can be expressed as the restriction to {(z_{1}, ζ, x) ∈ ℂ^{m} : ζ = z̄_{1}, x = x̄} of an m-variable series with complex coefficients:

H_{s}(z_{1}, ζ, x) = α_{s}z_{1}^{2} + β_{s}z_{1}ζ + γ_{s}ζ^{2} + ∑ δ_{s}^{s_{1}}z_{1}x_{s_{1}}

+ ∑ ϵ_{s}^{s_{1}}ζx_{s_{1}} + ∑ θ_{s}^{s_{1}s_{2}}x_{s_{1}}x_{s_{2}} + ∑_{a+b+I≥3} E_{s}^{abI}z_{1}^{a}ζ^{b}x^{I}

h_{u}(z_{1}, ζ, x) = α_{u}z_{1}^{2} + β_{u}z_{1}ζ + γ_{u}ζ^{2} + ∑ δ_{u}^{s_{1}}z_{1}x_{s_{1}}

+ ∑ ϵ_{u}^{s_{1}}ζx_{s_{1}} + ∑ θ_{u}^{s_{1}s_{2}}x_{s_{1}}x_{s_{2}} + ∑_{a+b+I≥3} e_{u}^{abI}z_{1}^{a}ζ^{b}x^{I},

where x^{I} abbreviates x_{2}^{i_{2}}x_{3}^{i_{3}} ⋯ x_{m−1}^{i_{m−1}} and a+b+I abbreviates a+b+i_{2}+…+ i_{m−1}. Each of the series in (z_{1}, ζ, x) converges on some set {(z_{1}, ζ, x) : |z_{1}| < r, |ζ| < r, |x_{s}| < r\}, r > 0, to a complex analytic function, with γ_{s} = \overline{α_{s}}, ϵ_{s}^{s_{1}} = \overline{δ_{2}^{s_{1}}}, etc., so that H_{s}(z_{1}, z̄_{1}, x) and E_{s}(z_{1}, z̄_{1}, x) are real valued.

Definition 3.1. A (formal, with complex coefficient C) monomial of the form Cz_{1}^{a}ζ^{b}x^{I} has “degree” a + b + I. A power series (convergent or formal) in m variables e(z_{1}, ζ, x) = ∑ e^{abI}z_{1}^{a}ζ^{b}x^{I} is said to have “degree” d if e^{abI} = 0 for all (a, b, I) such that a + b + I < d. Sometimes a series of degree d will be abbreviated O(d). An ordered k-tuple of series (e_{1}, …, e_{k}) has degree d if all its components have degree d.

Definition 3.2. Similarly for n variables, a monomial Cz_{1}^{a_{1}} ⋯ z_{n}^{a_{n}} has degree a_{1} + ⋯ + a_{n}, but we will also work with the “weight” a_{1} + ⋯ + a_{m−1} + 2a_{m} + ⋯ + 2a_{n}. A series p(z⃗) = ∑ p^{a_{1}...a_{n}}z_{1}^{a_{1}} … z_{n}^{a_{n}} has “weight” W if p^{a_{1}...a_{n}} = 0 when a_{1} + ⋯ + a_{m−1} + 2a_{m} + ⋯ + 2a_{n} < W.

We consider two coordinate systems for a neighborhood of the origin in ℂ^{n}: the previously mentioned z⃗ = (z_{1}, …, z_{n}), and a new system z̃ = (z̃_{1}, …, z̃_{n}), with z̃_{j} = x̃_{j} + ỹ_{j}. The two systems are related by the following “change of coordinates” transformation:

(2)  z̃ = z⃗ + p⃗(z⃗),

where p⃗(z⃗) = (p_{1}(z⃗), …, p_{n}(z⃗)) and each component p_{j} is a holomorphic function of z_{1}, …, z_{n}, whose series expansion has weight 2, and for j ≥ m, also degree 2. Such a transformation of ℂ^{n} has invertible linear part, so it is invertible on some neighborhood of 0. In the calculations of this Section, we will neglect considering the size of that neighborhood, and consider only points close enough to the origin, but the size of the domain of p⃗ will be important information in later Sections.

The goal of this Section is to establish some nondegeneracy conditions on the defining equations (1), by using complex linear transformations and nonlinear transformations of the form (2) to put the quadratic terms of (1) into a normal form. Similar calculations have already been done in the m = n case, and the (m, n) = (2, 3), (3, 4), and (4, 5) cases, by [Bishop], [C_{4}], [C_{5}], and [Beloshapka], respectively, so we will skip some of the computational details.

As the first special case of a transformation of the form (2) to be used, let p_{1} = 0 and let p_{2}, …, p_{n} be homogeneous quadratic polynomials in z_{1}, …, z_{m−1}. Using such a transformation, the quadratic terms in h_{u} that are products of z_{1} and x only, without a z̄_{1} factor, can be eliminated in the new coordinate system, or their coefficients (α_{u}, δ_{u}^{s_{1}}, θ_{u}^{s_{1}s_{2}}) can be altered to attain any complex values, by a suitable choice of p⃗. This transformation may also change some higher-degree terms but does not alter the coefficients β_{u}, γ_{u}, ϵ_{u}^{s_{1}}. Similarly, the quadratic terms without z̄_{1} in each H_{s} can also be eliminated by a transformation z̃ = z⃗ + p⃗, which simultaneously eliminates their conjugates (using γ_{s} = \overline{α_{s}}), leaving only the mixed term β_{s}z_{1}z̄_{1}.

The result of this preliminary normalization is that for any CR singular submanifold M of the form (1), there exists a quadratic coordinate transformation of the form (2) with p_{1} = 0, so that M has the following general normal form. In a local coordinate system z⃗ in some neighborhood of the CR singularity, the defining equations of M are of the form (1), with

y_{s} = H_{s}(z_{1}, z̄_{1}, x) = β_{s}z_{1}z̄_{1} + O(3)
z_{u} = h_{u}(z_{1}, z̄_{1}, x) = β_{u}z_{1}z̄_{1} + γ_{u}z̄_{1}^{2} + ∑ ϵ_{u}^{s_{1}} z̄_{1}x_{s_{1}} + O(3).

At this point we consider which invertible complex linear transformations of ℂ^{n} fix the tangent plane T with coordinates (z_{1}, x). The matrix representation of such a transformation must be of the form z̃ = Az, where:

A = ( \begin{matrix}a_{1} & a_{2} … a_{m−1} & a_{m} … a_{n} \\ 0 & R & ∗ \\ 0 & 0 & C \end{matrix} ) .

The entries a_{1}, …, a_{n} are complex, with a_{1} ≠ 0, the (m−2)×(m−2) block R has real entries and a nonzero determinant, and the (n−m+1)×(n−m+1) block C has complex entries and a nonzero determinant.

The \underline{first nondegeneracy condition} is that the (n − m + 1) × 2 block of coefficients β_{u}, γ_{u} in the functions h_{u} satisfies:

(3)  rank ( \begin{matrix}β_{m} & γ_{m} \\ ⋮ & ⋮ \\ β_{n} & γ_{n} \end{matrix} ) = 2.

In particular, this requires m < n. In this nondegenerate case, there is a linear transformation of ℂ^{n} which uses the block C in the above complex matrix to put these coefficients into a row echelon form:

z_{t} = h_{t}(z_{1}, z̄_{1}, x) = ∑ ϵ_{t}^{s_{1}} z̄_{1}x_{s_{1}} + O(3)
z_{n−1} = h_{n−1}(z_{1}, z̄_{1}, x) = z̄_{1}^{2} + ∑ ϵ_{n−1}^{s_{1}}z̄_{1}x_{s_{1}} + O(3)
z_{n} = h_{n}(z_{1}, z̄_{1}, x) = z_{1}z̄_{1} + ∑ ϵ_{n}^{s_{1}} z̄_{1}x_{s_{1}} + O(3),

for t = m …, n − 2, or there are no h_{t} expressions if m = n − 1.

As a consequence of the first nondegeneracy condition on the functions h_{u}, the functions H_{s} can also be simplified. There is a linear transformation with components z̃_{s} = z_{s} − iβ_{s}z_{n} (where for s = 2, …, m − 1, the complex coefficients −iβ_{s} are entries from the ∗ block of the matrix A) that eliminates the β_{s}z_{1}z̄_{1} terms from each function H_{s}. This may introduce more terms of the form z_{1}x_{s_{1}} or z̄_{1}x_{s_{1}} in the H_{s} functions, which can be eliminated by z̃_{s} = z_{s} + p_{s} quadratic transformations as done previously, without reintroducing any z_{1}z̄_{1} terms.

A linear transformation of the form z̃_{1} = z_{1} + ∑ a_{s}z_{s}, using the block a_{2}, …, a_{m−1} from the matrix A, can eliminate the terms of the form z̄_{1}x_{s_{1}} from either the h_{n−1} quantity or the h_{n} quantity, but generally not both at once — we make the choice to eliminate the terms ∑ ϵ_{n−1}^{s_{1}}z̄_{1}x_{s_{1}} from h_{n−1}. This may introduce more terms of the form z_{1}x_{s_{1}} or x_{s_{1}}x_{s_{2}} in the other h_{u} functions, which can be eliminated by z̃_{u} = z_{u}+p_{u} quadratic transformations as done previously.

The real and imaginary parts of the coefficients ϵ_{u}^{s_{1}}, for u = m, …, n − 2 and u = n, on the terms z̄_{1}x_{s_{1}}, s_{1} = 2, …, m − 1, form a real 2(n − m) × (m − 2) coefficient matrix, in this expression where the LHS is a column (n − m)-vector:

(∑ ϵ_{u}^{s_{1}} z̄_{1}x_{s_{1}})_{u=m,...,n−2,n} = (4)

( \begin{matrix}1 & i … 0 & 0 \\ ⋮ & & ⋮ \\ 0 & 0 … 1 & i \end{matrix} )(\begin{matrix}Re(ϵ_{m}^{2}) & Re(ϵ_{m}^{3}) … & Re(ϵ_{m}^{m−1}) \\ Im(ϵ_{m}^{2}) & Im(ϵ_{m}^{3}) … & Im(ϵ_{m}^{m−1}) \\ ⋮ & & ⋮ \\ Re(ϵ_{n−2}^{2}) & Re(ϵ_{n−2}^{3}) … & Re(ϵ_{n−2}^{m−1}) \\ Im(ϵ_{n−2}^{2}) & Im(ϵ_{n−2}^{3}) … & Im(ϵ_{n−2}^{m−1}) \\ Re(ϵ_{n}^{2}) & Re(ϵ_{n}^{3}) … & Re(ϵ_{n}^{m−1}) \\ Im(ϵ_{n}^{2}) & Im(ϵ_{n}^{3}) … & Im(ϵ_{n}^{m−1}) \end{matrix})( x_{2} ; ⋮ ; x_{m−1} )z̄_{1}

The \underline{second nondegeneracy condition} is that this real matrix has rank 2(n − m). It follows that the number of x_{s} directions, m − 2, must be greater than or equal to the number 2(n − m), and this is equivalent to m ≥ \frac{2}{3}(n + 1), exactly the lower bound of the dimensions of topological stability, as discussed in Section 2.

When the second nondegeneracy condition holds, the real R block of the matrix A can transform the x_{s} variables to put the above real matrix into echelon form, transforming the real and imaginary parts of the ϵ_{u}^{s_{1}} coefficients, without altering the z̄_{1}^{2} and z_{1}z̄_{1} terms. We get the following quadratic normal form for a nondegenerate CR singularity:

(5) y_{s} = H_{s}(z_{1}, z̄_{1}, x) = E_{s}(z_{1}, z̄_{1}, x) = O(3)
z_{t} = h_{t}(z_{1}, z̄_{1}, x) = z̄_{1}x_{2(t−m+2)} + iz̄_{1}x_{2(t−m+2)+1} + e_{t}(z_{1}, z̄_{1}, x)
z_{n−1} = h_{n−1}(z_{1}, z̄_{1}, x) = z̄_{1}^{2} + e_{n−1}(z_{1}, z̄_{1}, x)
z_{n} = h_{n}(z_{1}, z̄_{1}, x) = z_{1}z̄_{1} + z̄_{1}x_{2} + iz̄_{1}x_{3} + e_{n}(z_{1}, z̄_{1}, x),

with s = 2, …, m−1, t = m, …, n−2, or, again, there are no h_{t} expressions if m = n−1. If m > \frac{2}{3}(n+1), then the x_{2}, …, x_{2n−2m+1} variables appear in the quadratic part of the normal form but the variables x_{2n−2m+2}, …, x_{m−1} do not. In fact, near the origin, the locus of CR singularities (with j(x) = 1) is a codimension 2(n − m + 1) submanifold of M whose tangent space at the origin is the real subspace with coordinates x_{2n−2m+2}, …, x_{m−1}.

Having stated these two nondegeneracy conditions, we are now ready to state the main result:

Proposition 3.3. Given \frac{2}{3}(n + 1) ≤ m < n, let M be a real analytic msubmanifold of ℂ^{n} with a CR singularity at x, with j(x) = 1. If its local defining equations (of the form (1)) satisfy both nondegeneracy conditions (the full rank of coefficient matrices (3, 4)) so that they can be put into the form (5), then there exists a holomorphic coordinate change z̃ = z⃗ + p⃗ as in (2), in a neighborhood of 0 ∈ ℂ^{n}, transforming the equations (5) into the following real algebraic normal form:

(6)  ỹ_{s} = 0, s = 2 …, m − 1
z̃_{t} = ¯ z̃_{1}(x̃_{2(t−m+2)} + ix̃_{2(t−m+2)+1}), t = m …, n − 2
z̃_{n−1} = ¯ z̃_{1}^{2}
z̃_{n} = ¯ z̃_{1}(z̃_{1} + x̃_{2} + ix̃_{3}).

The real algebraic variety defined by (6) is denoted M̃^{m,n}, or more briefly M̃. The example M̃^{4,5} is exactly the normal form of [Beloshapka]. The Proposition states that any real analytic M satisfying only j(x) = 1 at a point and both quadratic nondegeneracy conditions is locally biholomorphically equivalent to the real algebraic model. This is the “analytic stability” mentioned in the title, and it is apparently analogous to stability theorems in the singularity theory of smooth maps, where any sufficiently nondegenerate singularity is equivalent under a change of coordinates to a unique polynomial model. The equations for M̃ resemble the normal forms for smooth maps with cross-cap (or “S_{1}”) singularities, as in [W_{4}], [Haefliger], [GG] §VII.4, and M̃ and the images of the singular maps also have similar structures as a cartesian product when the singularity is not isolated. The main difference between Whitney’s normal forms and (6) is that the quantities in (6) are not monomials, and cannot be simultaneously transformed into monomials by holomorphic coordinate transformations in the nondegenerate case. More will be said about the analogies with singularity theory in Section 8.

In the case m = \frac{2}{3}(n + 1) when the singularity is isolated, some of the topological invariants mentioned in Section 2 depend on an orientation of M, so it may be useful to consider normalizing transformations that fix a given orientation of the tangent plane T. This corresponds to the real block R of matrix A having a positive determinant, and the last equation of the normal form (5) falls into two cases: z_{n} = z̄_{1}(z_{1} +x_{2} ±ix_{3}). The two normal forms are equivalent under the biholomorphic transformation z̃_{3} = −z_{3}, but this reverses the orientation of T. In the remaining Sections we will not be concerned with the orientation.

4. A functional equation

To show the existence of a normalizing transformation, we will set up a system of nonlinear functional equations, so that any solution p⃗ of the system will define a normalizing transformation z̃ = z⃗ + p⃗ as in (2). In addition to finding a formal power series solution, we will also have to show that the solution is convergent in some neighborhood of the origin. The method of proof is the rapid convergence technique, as used in [M] and [C_{4}]. Rather than trying to solve the system of equations directly, we first find an approximate solution by solving a related system of linear equations. Iteration of this process gives a sequence of approximations that approach an exact solution. The issue of the domain of convergence of the exact solution was not addressed by [Beloshapka], and was left open by [C_{1}]. In [C_{1}], each approximate solution in the sequence was constructed only on a domain a fraction of the size of the previous one in the sequence — when the domains shrink to a point, the limit is an exact formal series solution, but no conclusion can be drawn about its analyticity. The new step here, which is crucial for the method of [M] to be applicable, is the construction of a sequence of approximate solutions whose domains shrink slowly enough so their diameters are bounded below by a positive constant.

Starting with the quadratic part of the defining equations in normal form (5), we consider the effect of a coordinate change (2). As previously mentioned, the z̃ = z⃗ + p⃗ transformation is (at least formally) invertible near 0, and it may be useful to think of z̃ = z⃗ + p⃗ as having identity linear part, although there could be linear terms with weight 2, for example, z̃_{1} = z_{1}+a_{n}z_{n}.

In terms of z̃ and z⃗, consider the system of equations

(7) 0 = Im(z̃_{s}) = Im(z_{s} + p_{s}(z⃗))
0 = z̃_{t} − (¯z̃_{1}x̃_{2(t−m+2)} + i¯z̃_{1}x̃_{2(t−m+2)+1})
= z_{t} + p_{t}(z⃗) − \overline{(z_{1} + p_{1}(z⃗))}Re(z_{2(t−m+2)} + p_{2(t−m+2)}(z⃗))
−i\overline{(z_{1} + p_{1}(z⃗))}Re(z_{2(t−m+2)+1} + p_{2(t−m+2)+1}(z⃗))
0 = z̃_{n−1} − ¯ z̃_{1}^{2}
= z_{n−1} + p_{n−1}(z⃗) − \overline{(z_{1} + p_{1}(z⃗))}^{2}
0 = z̃_{n} − ¯ z̃_{1}(z̃_{1} + x̃_{2} + ix̃_{3})
= z_{n} + p_{n}(z⃗)

−\overline{(z_{1} + p_{1}(z⃗))}(z_{1} + p_{1}(z⃗) + Re(z_{2} + p_{2}(z⃗)) + iRe(z_{3} + p_{3}(z⃗))).

In order to get (6) to be the defining equations for M in the z̃ coordinates, the above equalities must hold for points z⃗ on M and near 0. So, we can replace the z⃗ = (z_{1}, …, z_{n}) expressions in (7) by the defining functions (5):

(8)  z⃗ = (z_{1}, x_{2} + iH_{2}(z_{1}, z̄_{1}, x), …, h_{n}(z_{1}, z̄_{1}, x)),

to get a system of equations where the RHS functions depend only on z_{1}, z̄_{1}, x:

(9) 0 = Im(x_{s} + iH_{s} + p_{s}(z⃗)) = E_{s}(z_{1}, z̄_{1}, x) + Im(p_{s}(z⃗))
0 = e_{t}(z_{1}, z̄_{1}, x) + p_{t}(z⃗) − \overline{p_{1}(z⃗)}(x_{2(t−m+2)} + ix_{2(t−m+2)+1})
−z̄_{1}(Re(p_{2(t−m+2)}(z⃗)) + iRe(p_{2(t−m+2)+1}(z⃗)))
−\overline{p_{1}(z⃗)}(Re(p_{2(t−m+2)}(z⃗)) + iRe(p_{2(t−m+2)+1}(z⃗)))
0 = e_{n−1}(z_{1}, z̄_{1}, x) + p_{n−1}(z⃗) − 2z̄_{1}\overline{p_{1}(z⃗)} − \overline{p_{1}(z⃗)}^{2}
0 = e_{n}(z_{1}, z̄_{1}, x) + p_{n}(z⃗) − z̄_{1}(p_{1}(z⃗) + Re(p_{2}(z⃗)) + iRe(p_{3}(z⃗)))
−\overline{(p_{1}(z⃗))}(z_{1} + x_{2} + ix_{3})
−\overline{(p_{1}(z⃗))}(p_{1}(z⃗) + Re(p_{2}(z⃗)) + iRe(p_{3}(z⃗))).

The components of e⃗ = (E_{2}, …, E_{m−1}, e_{m}, …, e_{n}) appear in two ways — as terms in each equation of (9), and also in the z⃗ input (8) for each p_{j}(z⃗) in (9), j = 1, …, n. So, given e⃗, if we happen to have an exact solution p⃗ of the above system of functional equations, the conclusion of Proposition 3.3 holds and we are done. However, (9) is a nonlinear system in the unknown quantity p⃗, where in addition to the composition with the given defining functions (8), there are products of the components p_{j} and their complex conjugates.

As a first step in solving for p⃗ in terms of e⃗, consider the following system of simpler equations:

(10) 0 = E_{s}(z_{1}, z̄_{1}, x) + Im(p_{s}(z⃗))
0 = e_{t}(z_{1}, z̄_{1}, x) + p_{t}(z⃗) − \overline{p_{1}(z⃗)}(x_{2(t−m+2)} + ix_{2(t−m+2)+1})
−z̄_{1}(Re(p_{2(t−m+2)}(z⃗)) + iRe(p_{2(t−m+2)+1}(z⃗)))
0 = e_{n−1}(z_{1}, z̄_{1}, x) + p_{n−1}(z⃗) − 2z̄_{1}\overline{p_{1}(z⃗)}
0 = e_{n}(z_{1}, z̄_{1}, x) + p_{n}(z⃗)

−z̄_{1}(p_{1}(z⃗) + Re(p_{2}(z⃗)) + iRe(p_{3}(z⃗))) − \overline{(p_{1}(z⃗))}(z_{1} + x_{2} + ix_{3}),

where the z⃗ input for each p_{j} is:

(11)  z⃗ = (z_{1}, x_{2}, …, x_{m−1}, z̄_{1}(x_{4} + ix_{5}), …, z̄_{1}^{2}, z̄_{1}(z_{1} + x_{2} + ix_{3})).

This simplifies p_{j}(z⃗) by considering only the linear and quadratic parts of the input (8). Also, the products of p_{j} are dropped, so that these are (real) linear equations.

To see how the new equations are related to the original system, suppose e⃗ has degree d ≥ 3, and that p⃗ is a solution of (10–11) so that p_{1}, …, p_{2n−2m+1} have weight ≥ d − 1, and p_{2n−2m+2}, …, p_{n} have weight ≥ d. Evaluating the RHS of (9) with this solution for p⃗ evidently results in expressions of degree ≥ 2d − 2. Converting these expressions in z_{1}, z̄_{1}, x to z̃_{1}, ¯ z̃_{1}, x̃ and equating them to the z̃ expressions in (7) gives the higher-order terms of the new defining equations for M in the z̃ coordinate system. (It will be shown later (Theorem 6.5) that in fact for z⃗ ∈ M close enough to 0, z_{1}, z̄_{1}, x are real analytic functions of z̃_{1}, ¯ z̃_{1}, x̃.) So, while a solution p⃗ of the linearized equations is just an approximation to the solution of the original system, using such a p⃗ to define a coordinate transformation does have the effect of nearly doubling the order of vanishing of the e⃗ quantity.

5. A solution of the linear equation

The goal of this Section is to construct a solution p⃗ of the system of linear equations (10–11), given the higher-order terms of the defining equations, e⃗. Considering p⃗ and e⃗ as formal power series, such a solution exists but is not unique — this fact, together with the approximate doubling of the degree mentioned in the previous Section and iteration of the linearization procedure, is enough to show the (already known, as mentioned previously) formal equivalence of M and M̃. The solution p⃗ constructed here will be an n-tuple of series in z⃗ = (z_{1}, …, z_{n}) with the following properties: the size of the domain of convergence of p⃗ is comparable in a certain sense to the size of the domain of e⃗, and also a suitable norm of p⃗ is bounded in terms of a suitable norm of e⃗.

Notation 5.1. For r = (r_{1}, …, r_{N}) ∈ ℝ^{N}, with all r_{j} > 0, define a polydisc in ℂ^{N} by

𝔻_{r} = {(z_{1}, …, z_{N}) : |z_{j}| < r_{j}}.

As special cases, let

D_{r} = 𝔻_{(r,r,...,r)} ⊆ ℂ^{m}

and

Δ_{r} = 𝔻_{(r,...,r,2r^{2},...,2r^{2},r^{2},3r^{2})} ⊆ ℂ^{n},

where there are m − 1 radius lengths r and n − m − 1 radius lengths 2r^{2}, in the z_{m}, …, z_{n−2} coordinate directions.

The initial assumption on the defining equations is that e⃗(z_{1}, z̄_{1}, x) = (E_{2}, …, E_{m−1}, e_{m}, …, e_{n}) is real analytic, so there is some r > 0 so that each component of e⃗ is the restriction to {ζ = z̄_{1}, x = x̄} of a multivariable power series in (z_{1}, ζ, x) with center (0, 0, …, 0) and complex coefficients which converges on a complex polydisc D_{r} ⊆ ℂ^{m} (or, equivalently, a complex analytic function on D_{r}).

Notation 5.2. For a complex valued function e(z_{1}, ζ, x) of m complex variables, which is defined on some set containing the polydisc D_{r}, define the norm

|e|_{r} = sup_{(z_{1},ζ,x)∈D_{r}} |e(z_{1}, ζ, x)|.

For an (n − 1)-tuple e⃗ = (E_{2}, …, e_{n}), define

|e⃗|_{r} = |E_{2}|_{r} + ⋯ + |e_{n}|_{r}.

For a complex valued function p(z_{1}, …, z_{n}) of n complex variables, which is defined on some set containing the polydisc Δ_{r}, define the norm

∥p∥_{r} = sup_{z⃗∈Δ_{r}} |p(z⃗)|.

With this notation, we can further assume r > 0 is small enough so that |e⃗(z_{1}, ζ, x)|_{r} is finite. Given e⃗ with degree ≥ 3, the eventual goal is to find some r̃, 0 < r̃ ≤ r, and a holomorphic map p⃗ : Δr̃ → ℂ^{n}, so that the transformation z̃ = z⃗ + p⃗(z⃗) is a biholomorphism with domain Δr̃ taking M to M̃. That is, if z⃗ ∈ M ∩ Δr̃, then z̃ satisfies (6). However, in this Section we are only looking for p⃗ that is a solution of (10–11).

Some steps of the Proof of Theorem 5.6 will decompose series into subseries and their complex conjugates, where these preliminary Lemmas on the |e|_{r} norm will be useful.

Lemma 5.3. Given 0 < R < r and complex coefficients a_{jkI}, b_{jkI}, if

|∑ a_{jkI}z_{1}^{j}ζ^{k}x^{I} |_{r} ≤ K

and for complex x with |x_{s}| < r, j, k = 0, 1, 2, 3, …,

|∑_{I} b_{jkI}x^{I} | ≤ |∑_{I} a_{jkI}x^{I} | ,

then

|∑ b_{jkI}z_{1}^{j}ζ^{k}x^{I} |_{R} ≤ \frac{Kr^{2}}{(r − R)^{2}} .

Proof. For (z_{1}, ζ, x) ∈ D_{r}, these series are absolutely convergent and equal:

∑ a_{jkI}z_{1}^{j}ζ^{k}x^{I} = ∑_{j=0}^{∞} (∑_{k=0}^{∞} (∑_{I} a_{jkI}x^{I}) ζ^{k}) z_{1}^{j}.

Using Cauchy’s Estimate ([A]) twice,

|∑_{k=0}^{∞} (∑_{I} a_{jkI}x^{I}) ζ^{k}| ≤ \frac{K}{r^{j}} , and |∑_{I} a_{jkI}x^{I} | ≤ \frac{K}{r^{k}r^{j}} .

For (z_{1}, ζ, x) ∈ D_{r}, the series ∑ b_{jkI}z_{1}^{j}ζ^{k}x^{I} is absolutely convergent, and for (z_{1}, ζ, x) ∈ D_{R}:

|∑ b_{jkI}z_{1}^{j}ζ^{k}x^{I} | = |∑_{j=0}^{∞} (∑_{k=0}^{∞} (∑_{I} b_{jkI}x^{I}) ζ^{k}) z_{1}^{j}|

≤ ∑_{j=0}^{∞} (∑_{k=0}^{∞} |∑_{I} b_{jkI}x^{I} | |ζ|^{k}) |z_{1}|^{j}

≤ ∑_{j=0}^{∞} (∑_{k=0}^{∞} |∑_{I} a_{jkI}x^{I} | |ζ|^{k}) |z_{1}|^{j}

≤ ∑_{j=0}^{∞} (∑_{k=0}^{∞} \frac{K}{r^{k}r^{j}} |ζ|^{k}) |z_{1}|^{j} = ∑_{j,k} K (\frac{|ζ|}{r})^{k} (\frac{|z_{1}|}{r})^{j}

= K \frac{1}{1 − \frac{|ζ|}{r}} \frac{1}{1 − \frac{|z_{1}|}{r}} = \frac{Kr^{2}}{(r − |ζ|)(r − |z_{1}|)} < \frac{Kr^{2}}{(r − R)^{2}} .

In the applications of the Lemma, for each pair (j, k), the coefficients b_{jkI} will either be zero for all I or equal to a_{jkI} for all I, so the estimate in the hypothesis is satisfied.

Notation 5.4. On the complex vector space of formal power series, define the following real structure operator:

e = ∑ e^{abI}z_{1}^{a}ζ^{b}x^{I} ↦ e′ = ∑ \overline{e^{abI}}ζ^{a}z_{1}^{b}x^{I}.

Lemma 5.5. Given r > 0, the restriction of the above map to the subspace {e : |e|_{r} < ∞} is an isometry.

Proof. The equality of norms uses a change of variables which does not change the radius length r.

|e′|_{r} = sup_{(z_{1},ζ,x)∈D_{r}} |∑ \overline{e^{abI}}ζ^{a}z_{1}^{b}x^{I} |
= sup_{(ζ′,z′_{1},x′)=(z̄_{1},ζ̄,x̄)∈D_{r}} |∑ \overline{e^{abI}} \overline{(z′_{1})}^{a}\overline{(ζ′)}^{b}\overline{(x′)}^{I} |

= sup_{(ζ′,z′_{1},x′)∈D_{r}} |\overline{∑ e^{abI}(z′_{1})^{a}(ζ′)^{b}(x′)^{I}} |
= sup_{(z′_{1},ζ′,x′)∈D_{r}} |∑ e^{abI}(z′_{1})^{a}(ζ′)^{b}(x′)^{I} | = |e|_{r}.

Of course, this map is a representation of complex conjugation: given a series e(z_{1}, z̄_{1}, x) for real x, which “complexifies” to e = e(z_{1}, ζ, x) for (z_{1}, ζ, x) ∈ D_{r} for the purposes of finding its norm as in Notation 5.2, expanding \overline{e(z_{1}, z̄_{1}, x)} as a series in (z_{1}, z̄_{1}, x) and then complexifying gives e′ = e′(z_{1}, ζ, x).

In an attempt to simplify the notation by avoiding an excess of indices in an already intricate calculation, the following Theorem will focus on one particular dimension pair (m, n). In order to represent the most general behavior, we want m < n − 1, so there is a z_{t} equation in (5), and also m > \frac{2}{3}(n + 1), so there is a variable x_{m−1} that does not appear in the quadratic part of the defining equations. The smallest pair where both conditions occur is m = 7, n = 9, so we will be considering a real 7-manifold in ℂ^{9}, where the coordinates of the tangent plane are z_{1}, x_{2}, …, x_{6}, and the CR singular locus in M near 0 is a real curve tangent to the x_{6} axis at the origin.

Theorem 5.6. Given r > 0 and e⃗(z_{1}, ζ, x) convergent on D_{r} with |e⃗|_{r} < ∞ and degree d ≥ 3, there exists p⃗ which is convergent on Δ_{r}, and which solves this case of the system of equations (10–11):

(12) 0 = E_{s}(z_{1}, z̄_{1}, x) + Im(p_{s}(z⃗)), s = 2, …, 6
(13) 0 = e_{7}(z_{1}, z̄_{1}, x) + p_{7}(z⃗)
−\overline{p_{1}(z⃗)}(x_{4} + ix_{5}) − z̄_{1}(Re(p_{4}(z⃗)) + iRe(p_{5}(z⃗)))
(14) 0 = e_{8}(z_{1}, z̄_{1}, x) + p_{8}(z⃗) − 2z̄_{1}\overline{p_{1}(z⃗)}
(15) 0 = e_{9}(z_{1}, z̄_{1}, x) + p_{9}(z⃗)
−z̄_{1}(p_{1}(z⃗) + Re(p_{2}(z⃗)) + iRe(p_{3}(z⃗))) − \overline{(p_{1}(z⃗))}(z_{1} + x_{2} + ix_{3}),

where

(16)  z⃗ = (z_{1}, x_{2}, x_{3}, x_{4}, x_{5}, x_{6}, z̄_{1}(x_{4} + ix_{5}), z̄_{1}^{2}, z̄_{1}(z_{1} + x_{2} + ix_{3})),

and which satisfies:

∥p_{1}∥_{r} ≤ \frac{3}{2r} |e_{8}|_{r}, ∥p_{8}∥_{r} ≤ 4|e_{8}|_{r},

and for any 0 < R < r,

∥p_{2}∥_{R} ≤ \frac{3|e_{9}|_{r} + 18|e_{8}|_{r}}{R} +(\frac{8r^{2}}{(r − R)^{2}} + 10)|E_{2}|_{r} + (\frac{4r^{2}}{(r − R)^{2}} + 4)|E_{3}|_{r}
∥p_{3}∥_{R} ≤ \frac{3|e_{9}|_{r} + 18|e_{8}|_{r}}{R} +(\frac{4r^{2}}{(r − R)^{2}} + 4)|E_{2}|_{r} + (\frac{8r^{2}}{(r − R)^{2}} + 10)|E_{3}|_{r}
∥p_{4}∥_{R} ≤ \frac{3|e_{7}|_{r} + 9|e_{8}|_{r}}{R} + (\frac{8r^{2}}{(r − R)^{2}} + 10)|E_{4}|_{r} + (\frac{4r^{2}}{(r − R)^{2}} + 4)|E_{5}|_{r}
∥p_{5}∥_{R} ≤ \frac{3|e_{7}|_{r} + 9|e_{8}|_{r}}{R} + (\frac{4r^{2}}{(r − R)^{2}} + 4)|E_{4}|_{r} + (\frac{8r^{2}}{(r − R)^{2}} + 10)|E_{5}|_{r}
∥p_{6}∥_{R} ≤ \frac{20r^{2}}{(r − R)^{2}} |E_{6}|_{r}
∥p_{7}∥_{R} ≤ 4|e_{7}|_{r} + 12|e_{8}|_{r} + 8R (\frac{r^{2}}{(r − R)^{2}} + 1) (|E_{4}|_{r} + |E_{5}|_{r})
∥p_{9}∥_{R} ≤ 4|e_{9}|_{r} + 24|e_{8}|_{r} + 8R (\frac{r^{2}}{(r − R)^{2}} + 1) (|E_{2}|_{r} + |E_{3}|_{r}).

Proof. First, notice that if p⃗(z_{1}, …, z_{9}) is a formal series solution of (12–16), it does not follow that p⃗ is convergent at any point (other than the origin). For example, with any component p_{j}, the series expressions p_{j}(z⃗) and

(17)  p_{j}(z⃗) + ((z_{1} + z_{2} + iz_{3})^{2}z_{8} − z_{9}^{2}) · Q(z⃗)

are formally the same when restricted to z⃗ as in (16), for any (possibly divergent) series Q. So, if one formal solution p⃗ exists, then there exist infinitely many divergent solutions. There may also exist formal series solutions that are convergent only on some neighborhood of the origin much smaller than that claimed in the Theorem.

Continuing with the abbreviation x = x_{2}, x_{3}, x_{4}, x_{5}, x_{6}, and also using z = z_{2}, z_{3}, z_{4}, z_{5}, z_{6}, the following choice of normalization will simplify the construction of the solution p⃗ satisfying the claimed convergence and bounds:

p_{1}(z⃗) = p_{1}(z_{1}, z, z_{8})
p_{j}(z⃗) = p_{j}^{E}(z_{1}, z, z_{8}) + z_{9}p_{j}^{O}(z_{1}, z, z_{8}),

for j = 2, …, 9. Note that p⃗ does not depend on z_{7}, and the first component p_{1} does not depend on z_{9}. We may make the further assumption that p_{1} is an even function of z_{1}: p_{1}(z_{1}, z, z_{8}) = p_{1}(−z_{1}, z, z_{8}). The remaining components, p_{j}, have some terms not depending on z_{9}, labeled p_{j}^{E}, and other terms which have exactly one linear factor of z_{9}. The p_{j}^{E} and p_{j}^{O} terminology corresponds to even and odd powers of z̄_{1} which appear after the substitution of (16) into p⃗. The choice that p⃗ has at most linear terms in z_{9} = z̄_{1}(z_{1} + x_{2} + ix_{3}) is made to avoid high powers of the non-monomial quantity z̄_{1}(z_{1} + x_{2} + ix_{3}), since as in [C_{1}], any multinomial coefficients in the series expansion of p⃗(z⃗) could be large enough to affect the size of the domain of convergence.

We begin with the e_{8} equation of the system, (14). If the series expansion of e_{8} had only even powers of z̄_{1}, then it would be a very simple matter to compare the coefficients of e_{8}(z_{1}, z̄_{1}, x) and p_{8}^{E}(z_{1}, x, z̄_{1}^{2}), and get a solution of the equation with p_{8}^{O} = p_{1} = 0. The odd powers of z̄_{1} in e_{8} make the p_{8}^{O} and p_{1} quantities necessary to solve the equation. The consideration of the terms of the components of the given quantity e⃗ which are even or odd in z̄_{1} was part of the analysis of [Beloshapka] and [C_{1}] of the formal normal form problem, and even/odd decompositions also appeared in analogous calculations in [W_{1}]. However, to deal with the non-monomial property of the quadratic normal form, there will be some re-arrangements of the terms in the series which were not required in [W_{1}]. First, decompose e_{8} into even and odd parts e_{8A}, e_{8B}, e_{8C}, and then apply an add-and-subtract trick to e_{8C}, as follows:

e_{8} = ∑ e_{8}^{abI}z_{1}^{a}z̄_{1}^{b}x^{I} = e_{8A} + e_{8B} + e_{8C}
e_{8A} = ∑_{b even} e_{8}^{abI}z_{1}^{a}z̄_{1}^{b}x^{I}
e_{8B} = ∑_{a even, b odd} e_{8}^{abI}z_{1}^{a}z̄_{1}^{b}x^{I}
e_{8C} = ∑_{a, b odd} e_{8}^{abI}z_{1}^{a}z̄_{1}^{b}x^{I} = e_{8D} + e_{8E}
e_{8D} = z̄_{1}(z_{1} + x_{2} + ix_{3}) ∑_{a, b odd} e_{8}^{abI}z_{1}^{a−1}z̄_{1}^{b−1}x^{I}
e_{8E} = −(x_{2} + ix_{3}) ∑_{a, b odd} e_{8}^{abI}z_{1}^{a−1}z̄_{1}^{b}x^{I}.

Let f_{8} = e_{8B} + e_{8E}, so f_{8}(z_{1}, z̄_{1}, x) is even in z_{1} and odd in z̄_{1}. Then, combining e_{8} = e_{8A} +e_{8D} +f_{8} with the normalization for p_{1} and p_{8} in the e_{8} equation (14), a straightforward (by construction) comparison of coefficients yields the following equalities:

0 = e_{8A}(z_{1}, z̄_{1}, x) + p_{8}^{E}(z_{1}, x, z̄_{1}^{2})
0 = e_{8D}(z_{1}, z̄_{1}, x) + z̄_{1}(z_{1} + x_{2} + ix_{3})p_{8}^{O}(z_{1}, x, z̄_{1}^{2})
0 = f_{8}(z_{1}, z̄_{1}, x) − 2z̄_{1}\overline{p_{1}(z_{1}, x, z̄_{1}^{2})}.

If p_{8}^{E}(z_{1}, z, z_{8}) = ∑ p_{8}^{acI}z_{1}^{a}z_{8}^{c}z^{I}, then the coefficient p_{8}^{acI} must be equal to −e_{8}^{a,2c,I}, and we get an estimate for the norm of p_{8}^{E} on the polydisc Δ_{r} ⊆ ℂ^{9}:

∥p_{8}^{E}∥_{r} = sup_{z⃗∈Δ_{r}} |p_{8}^{E}(z⃗)| = sup_{|z_{1}|<r,|x_{s}|<r,|ζ^{2}|<r^{2}} |p_{8}^{E}(z_{1}, x, ζ^{2})|
= sup_{(z_{1},ζ,x)∈D_{r}} | − e_{8A}(z_{1}, ζ, x)| = |e_{8A}|_{r}

= |\frac{1}{2}(e_{8}(z_{1}, ζ, x) + e_{8}(z_{1}, −ζ, x))|_{r} ≤ |e_{8}|_{r}.

Note that by using the averaging formula to extract the even part of e_{8}, we can just apply the triangle inequality to get the estimate for the subseries instead of Lemma 5.3. There is a similar estimate for the other component p_{8}^{O}, but this time the Schwarz Lemma ([A]) is used in two steps:

∥z_{9}p_{8}^{O}∥_{r} ≤ ∥z_{9}∥_{r}∥p_{8}^{O}∥_{r} = 3r^{2} sup_{|z_{1}|<r,|x_{s}|<r,|ζ^{2}|<r^{2}} |p_{8}^{O}(z_{1}, x, ζ^{2})|

= 3r^{2} sup_{(z_{1},ζ,x)∈D_{r}} |\frac{−e_{8D}(z_{1}, ζ, x)}{ζ(z_{1} + x_{2} + ix_{3})}|

= 3r^{2} sup_{(z_{1},ζ,x)∈D_{r}^{∗}} |\frac{−e_{8C}(z_{1}, ζ, x)}{z_{1}ζ}| ≤ 3r^{2} sup_{(z_{1},ζ,x)∈D_{r}^{∗}} \frac{\frac{|z_{1}|}{r} sup_{|z_{1}|<r} |e_{8C}|}{|z_{1}||ζ|}

≤ 3r sup_{(z_{1},ζ,x)∈D_{r}^{∗}} \frac{sup_{|z_{1}|<r} |\frac{|ζ|}{r} sup_{|ζ|<r} |e_{8C}||}{|ζ|} = 3|e_{8C}|_{r}

= 3 |\frac{e_{8}(z_{1}, ζ, x) − e_{8}(z_{1}, −ζ, x) − e_{8}(−z_{1}, ζ, x) + e_{8}(−z_{1}, −ζ, x)}{4}|_{r}
≤ 3|e_{8}|_{r}.

In some of the above steps, we restricted to the open subset D_{r}^{∗} = D_{r}\({z_{1} = 0} ∪ {ζ = 0}), which avoids division by 0 but by the maximum principle, does not affect the sup.

From f_{8} = e_{8B} + e_{8E} and the Schwarz Lemma,

|f_{8}|_{r} ≤ |e_{8B}|_{r} + |e_{8E}|_{r}
= |\frac{e_{8}(z_{1}, ζ, x) − e_{8}(z_{1}, −ζ, x) + e_{8}(−z_{1}, ζ, x) − e_{8}(−z_{1}, −ζ, x)}{4}|_{r}

+ |−\frac{x_{2} + ix_{3}}{z_{1}} e_{8C} |_{r}

≤ |e_{8}|_{r} + |x_{2} + ix_{3}|_{r} · \frac{1}{r}|e_{8C}|_{r} ≤ 3|e_{8}|_{r}.

Solving for p_{1} involves complex conjugation, so we take care to work out a few steps. By a comparison of the coefficients of f_{8} and p_{1}, if

p_{1}(z_{1}, z, z_{8}) = ∑_{α even} p_{1}^{αβI}z_{1}^{α}z_{8}^{β}z^{I},

then p_{1}^{αβI} = \frac{1}{2} \overline{f_{8}^{2β,α+1,I}}. Using the Schwarz Lemma and Lemma 5.5:

∥p_{1}∥_{r} = sup_{(z_{1},ζ,x)∈D_{r}} | ∑_{α even} p_{1}^{αβI}z_{1}^{α}ζ^{2β}x^{I} | = sup_{(z_{1},ζ,x)∈D_{r}} | ∑_{α even} \frac{1}{2}\overline{f_{8}^{2β,α+1,I}}z_{1}^{α}ζ^{2β}x^{I} |

= sup_{(z_{1},ζ,x)∈D_{r}} | ∑_{a even, b odd} \frac{\overline{f_{8}^{abI}}ζ^{a}z_{1}^{b}x^{I}}{2z_{1}} |

= |\frac{f′_{8}(z_{1}, ζ, x)}{2z_{1}} |_{r} ≤ \frac{1}{2r} |f′_{8}|_{r} = \frac{1}{2r} |f_{8}|_{r} ≤ \frac{3}{2r} |e_{8}|_{r}.

By construction, p_{1} has weight d − 1 and p_{8} has weight d.

Moving next to the E_{6} equation of (12), split the real valued series E_{6} into subseries, some real and some in complex conjugate pairs:

E_{6} = e_{6A} + \overline{e_{6A}} + E_{6B} + e_{6C} + \overline{e_{6C}} + e_{6D} + \overline{e_{6D}} + E_{6E}
e_{6A} = ∑_{a > b, b even} E_{6}^{abI}z_{1}^{a}z̄_{1}^{b}x^{I}
E_{6B} = ∑_{a even} E_{6}^{aaI}z_{1}^{a}z̄_{1}^{a}x^{I}
e_{6C} = ∑_{a > b, a even, b odd} E_{6}^{abI}z_{1}^{a}z̄_{1}^{b}x^{I}
e_{6D} = ∑_{a > b, a, b odd} E_{6}^{abI}z_{1}^{a}z̄_{1}^{b}x^{I}
E_{6E} = ∑_{a odd} E_{6}^{aaI}z_{1}^{a}z̄_{1}^{a}x^{I}.

By Lemma 5.3, |e_{6A}|_{R} ≤ \frac{r^{2}}{(r−R)^{2}} |E_{6}|_{r}, and all the other subseries have the same bound. We rearrange two of these subseries in order to be able to

18 A. COFFMAN

compare coefficients with p_{6}:

e_{6D} = e_{6F} + e_{6G}
e_{6F} = (z_{1} + x_{2} + ix_{3})z̄_{1} ∑_{a > b, a, b odd} E_{6}^{abI}z_{1}^{a−1}z̄_{1}^{b−1}x^{I}
e_{6G} = −(x_{2} + ix_{3}) ∑_{a > b, a, b odd} E_{6}^{abI}z_{1}^{a−1}z̄_{1}^{b}x^{I}
E_{6E} = e_{6H} + \overline{e_{6H}} + e_{6I} + \overline{e_{6I}}
e_{6H} = \frac{1}{2}(z_{1} + x_{2} + ix_{3})z̄_{1} ∑_{a odd} E_{6}^{aaI}z_{1}^{a−1}z̄_{1}^{a−1}x^{I}
e_{6I} = −\frac{1}{2}(x_{2} + ix_{3}) ∑_{a odd} E_{6}^{aaI}z_{1}^{a−1}z̄_{1}^{a}x^{I},

and collect some of these subseries back together:

f_{6A}(z_{1}, z̄_{1}, x) = e_{6A} + \overline{e_{6I}} = ∑_{a > b, b even} f_{6A}^{abI}z_{1}^{a}z̄_{1}^{b}x^{I}
f_{6C}(z_{1}, z̄_{1}, x) = \overline{e_{6C}} + \overline{e_{6G}} = ∑_{a < b, a odd, b even} f_{6C}^{abI}z_{1}^{a}z̄_{1}^{b}x^{I},

so

E_{6} = f_{6A} + \overline{f_{6A}} + E_{6B} + f_{6C} + \overline{f_{6C}} + e_{6F} + \overline{e_{6F}} + e_{6H} + \overline{e_{6H}}.

The unknown p_{6} can also be expressed as a sum of subseries:

p_{6} = p_{6}^{E}(z_{1}, z, z_{8}) + z_{9}p_{6}^{O}(z_{1}, z, z_{8})
p_{6}^{E} = p_{6A} + p_{6B} + p_{6C}
p_{6A} = ∑_{α > 2γ} p_{6A}^{αγI}z_{1}^{α}z^{I}z_{8}^{γ}
p_{6B} = ∑ p_{6B}^{γI}z_{1}^{2γ}z^{I}z_{8}^{γ}
p_{6C} = ∑_{α < 2γ, α odd} p_{6C}^{αγI}z_{1}^{α}z^{I}z_{8}^{γ}
p_{6}^{O} = p_{6D} + p_{6E}
p_{6D} = ∑_{α > 2γ, α even} p_{6D}^{αγI}z_{1}^{α}z^{I}z_{8}^{γ}
p_{6E} = ∑ p_{6E}^{γI}z_{1}^{2γ}z^{I}z_{8}^{γ}.

Comparing coefficients, the equation 0 = E_{6} + \frac{1}{2i}(p_{6} − \overline{p_{6}}) from (12) turns into these five equations and their complex conjugates:

0 = f_{6A} + \frac{1}{2i}p_{6A}
0 = \frac{1}{2}E_{6B} + \frac{1}{2i}p_{6B}
0 = f_{6C} + \frac{1}{2i}p_{6C}
0 = e_{6F} + \frac{1}{2i}(z_{1} + x_{2} + ix_{3})z̄_{1}p_{6D}
0 = e_{6H} + \frac{1}{2i}(z_{1} + x_{2} + ix_{3})z̄_{1}p_{6E}.

Solving for each component of p_{6} gives a weight d quantity, and using Lemma 5.5, the Schwarz Lemma, and the previously mentioned estimates for the subseries of E_{6}, we get these estimates:

∥p_{6A}∥_{R} = | − 2if_{6A}|_{R} = 2|e_{6A} + e′_{6I}|_{R} ≤ 2(|e_{6A}|_{R} + |e_{6I}|_{R})
≤ 2 (|e_{6A}|_{R} + |\frac{−(x_{2} + ix_{3})E_{6E}}{2z_{1}} |_{R})
≤ 2 (\frac{r^{2}|E_{6}|_{r}}{(r − R)^{2}} + \frac{2R}{2} · \frac{1}{R} · \frac{r^{2}|E_{6}|_{r}}{(r − R)^{2}} ) = \frac{4r^{2}}{(r − R)^{2}} |E_{6}|_{r},
∥p_{6B}∥_{R} = | − 2i · \frac{1}{2}E_{6B}|_{R} ≤ \frac{r^{2}}{(r − R)^{2}} |E_{6}|_{r},
∥p_{6C}∥_{R} = | − 2if_{6C}|_{R} = 2|e′_{6C} + e′_{6G}|_{R}
≤ 2 (|e_{6C}|_{R} + |\frac{−(x_{2} + ix_{3})e_{6D}}{z_{1}} |_{R})
≤ 2 (\frac{r^{2}|E_{6}|_{r}}{(r − R)^{2}} + 2R · \frac{1}{R} · \frac{r^{2}|E_{6}|_{r}}{(r − R)^{2}} ) = \frac{6r^{2}}{(r − R)^{2}} |E_{6}|_{r},
∥z_{9}p_{6D}∥_{R} ≤ ∥z_{9}∥_{R}∥p_{6D}∥_{R} = 3R^{2} |−2i\frac{e_{6F}}{(z_{1} + x_{2} + ix_{3})ζ} |_{R} = 6R^{2} |\frac{e_{6D}}{z_{1}ζ} |_{R}
≤ 6|e_{6D}|_{R} ≤ \frac{6r^{2}}{(r − R)^{2}} |E_{6}|_{r},
∥z_{9}p_{6E}∥_{R} ≤ 3R^{2} |−2i\frac{e_{6H}}{(z_{1} + x_{2} + ix_{3})ζ} |_{R} = 6R^{2} |\frac{e_{6E}}{2z_{1}ζ} |_{R} ≤ \frac{3r^{2}}{(r − R)^{2}} |E_{6}|_{r}.

Finding p_{2}, p_{3}, p_{4}, p_{5}, is a little trickier since each appears in more than one equation of (12–15). We will simultaneously solve for p_{4}, p_{5}, p_{7}, using a more involved comparison of coefficients, and similarly but independently, also p_{2}, p_{3}, p_{9}.

To find p_{4}, p_{5}, and p_{7}, we consider the E_{4}, E_{5}, e_{7} equations of (12, 13), and use the previously found solution for p_{1} to get the system with the unknowns on the LHS and the known O(d) quantities on the RHS:

(18)  Im(p_{4}) = −E_{4}
(19)  Im(p_{5}) = −E_{5}
(20)  p_{7} − z̄_{1}Re(p_{4}) − iz̄_{1}Re(p_{5}) = −e_{7} + (x_{4} + ix_{5})\overline{p_{1}}

Starting with the RHS of (18), the following decomposition of E_{4} is different from that of E_{6}:

E_{4} = E_{4A} + e_{4B} + \overline{e_{4B}} + e_{4C} + \overline{e_{4C}} + E_{4D}
E_{4A} = ∑_{a, b even} E_{4}^{abI}z_{1}^{a}z̄_{1}^{b}x^{I}
e_{4B} = ∑_{a > b, a odd, b even} E_{4}^{abI}z_{1}^{a}z̄_{1}^{b}x^{I}
e_{4C} = ∑_{a > b, a even, b odd} E_{4}^{abI}z_{1}^{a}z̄_{1}^{b}x^{I}
E_{4D} = ∑_{a, b odd} E_{4}^{abI}z_{1}^{a}z̄_{1}^{b}x^{I}
= e_{4E} + e_{4F}
e_{4E} = (z_{1} + x_{2} + ix_{3})z̄_{1} ∑_{a, b odd} E_{4}^{abI}z_{1}^{a−1}z̄_{1}^{b−1}x^{I}
e_{4F} = −(x_{2} + ix_{3}) ∑_{a, b odd} E_{4}^{abI}z_{1}^{a−1}z̄_{1}^{b}x^{I}.

The E_{4A} piece is simply an even part, so |E_{4A}|_{r} ≤ |E_{4}|_{r}, and similarly for the odd part, |E_{4D}|_{r} ≤ |E_{4}|_{r}. The other two subseries satisfy the estimate from Lemma 5.3: |e_{4B}|_{R} ≤ \frac{r^{2}}{(r−R)^{2}} |E_{4}|_{r} and |e_{4C}|_{R} ≤ \frac{r^{2}}{(r−R)^{2}} |E_{4}|_{r}. We re-group some of these subseries:

m_{4}(z_{1}, z̄_{1}, x) = \overline{e_{4B}} + e_{4C} + e_{4F} = ∑_{a even, b odd} m_{4}^{abI}z_{1}^{a}z̄_{1}^{b}x^{I}
f_{4}(z_{1}, z̄_{1}, x) = E_{4A} + e_{4B} + \overline{e_{4C}} + \overline{m_{4}} = ∑_{b even} f_{4}^{abI}z_{1}^{a}z̄_{1}^{b}x^{I}
⟹ E_{4} = f_{4} + e_{4E} + m_{4} − \overline{m_{4}}.

Similarly, E_{5} = f_{5} + e_{5E} + m_{5} − \overline{m_{5}}, where f_{5} is even in z̄_{1} and m_{5} is even in z_{1} and odd in z̄_{1}.

ANALYTIC STABILITY OF THE CR CROSS-CAP 21

The estimates follow from Lemma 5.5 and the Schwarz Lemma:

|m_{4}|_{R} = |e′_{4B} + e_{4C} + e_{4F} |_{R} ≤ |e_{4B}|_{R} + |e_{4C}|_{R} + |e_{4F} |_{R}
≤ \frac{r^{2}}{(r − R)^{2}} |E_{4}|_{r} + \frac{r^{2}}{(r − R)^{2}} |E_{4}|_{r} + |x_{2} + ix_{3}|_{R} |\frac{E_{4D}}{z_{1}} |_{R}
≤ (\frac{2r^{2}}{(r − R)^{2}} + 2) |E_{4}|_{r}
|f_{4}|_{R} = |E_{4A} + 2e_{4B} + 2e′_{4C} + e′_{4F} |_{R} ≤ (\frac{4r^{2}}{(r − R)^{2}} + 3) |E_{4}|_{r},

and similarly for m_{5} and f_{5}.

From the RHS of (20), let f_{7}(z_{1}, z̄_{1}, x) = −e_{7} + (x_{4} + ix_{5})\overline{p_{1}(z_{1}, x, z̄_{1}^{2})}, so

(21)  |f_{7}|_{r} ≤ | − e_{7}|_{r} + |(x_{4} + ix_{5})\frac{f_{8}}{2ζ} |_{r} ≤ |e_{7}|_{r} + 3|e_{8}|_{r}.

It splits into even and odd parts, f_{7} = f_{7A} + f_{7B} + f_{7C}, with:

f_{7A} = ∑_{b even} f_{7A}^{abI}z_{1}^{a}z̄_{1}^{b}x^{I}
f_{7B} = ∑_{a, b odd} f_{7B}^{abI}z_{1}^{a}z̄_{1}^{b}x^{I}
f_{7C} = ∑_{a even, b odd} f_{7C}^{abI}z_{1}^{a}z̄_{1}^{b}x^{I},

with |f_{7A}|_{r} ≤ |f_{7}|_{r} and the same bound for f_{7B}, f_{7C}. Let:

g_{7A} = f_{7A} + iz̄_{1}m_{4} − z̄_{1}m_{5} = ∑_{b even} g_{7A}^{abI}z_{1}^{a}z̄_{1}^{b}x^{I}
g_{7B} = f_{7B} − iz̄_{1}\overline{m_{4}} + z̄_{1}\overline{m_{5}} = ∑_{a, b odd} g_{7B}^{abI}z_{1}^{a}z̄_{1}^{b}x^{I} = g_{7C} + g_{7D}
g_{7C} = (z_{1} + x_{2} + ix_{3})z̄_{1} ∑_{a, b odd} g_{7B}^{abI}z_{1}^{a−1}z̄_{1}^{b−1}x^{I}
g_{7D} = −(x_{2} + ix_{3}) ∑_{a, b odd} g_{7B}^{abI}z_{1}^{a−1}z̄_{1}^{b}x^{I}.

Then

f_{7} = g_{7A} + g_{7C} + g_{7D} + f_{7C} − z̄_{1}(im_{4} − i\overline{m_{4}} − m_{5} + \overline{m_{5}})

is in a form that compares to the LHS of (20) to give three equations:

(22)  p_{7}^{E} = g_{7A}
(23) (z_{1} + x_{2} + ix_{3})z̄_{1}p_{7}^{O} = g_{7C}
(24) −z̄_{1}(Re(p_{4}) + iRe(p_{5})) = g_{7D} + f_{7C} − z̄_{1}(im_{4} − i\overline{m_{4}} − m_{5} + \overline{m_{5}}).

Equations (22), (23) determine p_{7}, with the estimates

∥p_{7}^{E}∥_{R} = |g_{7A}|_{R} = |f_{7A} + iζm_{4} − ζm_{5}|_{R}
≤ |f_{7}|_{r} + R · (\frac{2r^{2}}{(r − R)^{2}} + 2) |E_{4}|_{r} + R · (\frac{2r^{2}}{(r − R)^{2}} + 2) |E_{5}|_{r}
∥z_{9}p_{7}^{O}∥_{R} ≤ 3R^{2} |\frac{g_{7C}}{(z_{1} + x_{2} + ix_{3})ζ} |_{R} = 3R^{2} |\frac{g_{7B}}{z_{1}ζ} |_{R}
≤ 3|f_{7B} − iζm′_{4} + ζm′_{5}|_{R}
≤ 3|f_{7}|_{r} + 3R (\frac{2r^{2}}{(r − R)^{2}} + 2) |E_{4}|_{r} + 3R (\frac{2r^{2}}{(r − R)^{2}} + 2) |E_{5}|_{r}.

Dividing (24) by −z̄_{1}, then considering the real and imaginary parts and recalling (18), (19), we get the system:

Re(p_{4}) = Re (\frac{g_{7D} + f_{7C}}{−z̄_{1}} ) + im_{4} − i\overline{m_{4}}
Re(p_{5}) = Im (\frac{g_{7D} + f_{7C}}{−z̄_{1}} ) + im_{5} − i\overline{m_{5}}
Im(p_{4}) = −E_{4} = −f_{4} − e_{4E} − m_{4} + \overline{m_{4}}
Im(p_{5}) = −E_{5} = −f_{5} − e_{5E} − m_{5} + \overline{m_{5}}.

Note that it is at this point where the second nondegeneracy condition (the full rank of coefficient matrix (4)) is used — if the quadratic term ϵ_{7}^{5}z̄_{1}x_{5} in h_{7} had coefficient 0 instead of i, then Re(p_{5}) would not appear in (13), and (20) could not be solved this way.

Recombining the real and imaginary parts of p_{4}, p_{5}, there is (by construction) a convenient cancellation:

p_{4} = Re(p_{4}) + iIm(p_{4}) = Re (\frac{g_{7D} + f_{7C}}{−z̄_{1}} ) − if_{4} − ie_{4E}
p_{5} = Re(p_{5}) + iIm(p_{5}) = Im (\frac{g_{7D} + f_{7C}}{−z̄_{1}} ) − if_{5} − ie_{5E}.

These equations were set up so that e_{4E} and e_{5E} are the only RHS terms with odd powers of z̄_{1}, so p_{4} = p_{4}^{E} + z_{9}p_{4}^{O} and p_{5} = p_{5}^{E} + z_{9}p_{5}^{O} are each determined by a comparison of coefficients, and by construction, p_{4} and p_{5} have weight d − 1 and satisfy the estimates:

∥p_{4}^{E}∥_{R} = |\frac{g_{7D}}{−2ζ} + \frac{f_{7C}}{−2ζ} + \frac{g′_{7D}}{−2z_{1}} + \frac{f′_{7C}}{−2z_{1}} − if_{4}|_{R}
≤ \frac{1}{R}|g_{7D}|_{R} + \frac{1}{R}|f_{7C}|_{R} + |f_{4}|_{R}
≤ \frac{1}{R} |−(x_{2} + ix_{3})\frac{g_{7B}}{z_{1}} |_{R} + \frac{1}{R}|f_{7}|_{R} + |f_{4}|_{R}
≤ \frac{3|e_{7}|_{r} + 9|e_{8}|_{r}}{R} + (\frac{8r^{2}}{(r − R)^{2}} + 7) |E_{4}|_{r} + (\frac{4r^{2}}{(r − R)^{2}} + 4) |E_{5}|_{r}
∥p_{5}^{E}∥_{R} = |\frac{g_{7D}}{−2iζ} + \frac{f_{7C}}{−2iζ} − \frac{g′_{7D}}{−2iz_{1}} − \frac{f′_{7C}}{−2iz_{1}} − if_{5}|_{R}
≤ \frac{3|e_{7}|_{r} + 9|e_{8}|_{r}}{R} + (\frac{4r^{2}}{(r − R)^{2}} + 4) |E_{4}|_{r} + (\frac{8r^{2}}{(r − R)^{2}} + 7) |E_{5}|_{r}

∥z_{9}p_{4}^{O}∥_{R} ≤ ∥z_{9}p_{4}^{O}∥_{r} ≤ 3r^{2} |\frac{−ie_{4E}}{(z_{1} + x_{2} + ix_{3})ζ} |_{r} = 3r^{2} |\frac{E_{4D}}{z_{1}ζ} |_{r} ≤ 3|E_{4}|_{r}
∥z_{9}p_{5}^{O}∥_{R} ≤ ∥z_{9}p_{5}^{O}∥_{r} ≤ 3r^{2} |\frac{−ie_{5E}}{(z_{1} + x_{2} + ix_{3})ζ} |_{r} = 3r^{2} |\frac{E_{5D}}{z_{1}ζ} |_{r} ≤ 3|E_{5}|_{r}.

The method of finding p_{2}, p_{3}, p_{9} can be copied from the solution of p_{4}, p_{5}, p_{7}. In the place of (18), (19), (20), the system to be solved is

(25)  Im(p_{2}) = −E_{2}
Im(p_{3}) = −E_{3}
p_{9} − z̄_{1}Re(p_{2}) − iz̄_{1}Re(p_{3}) = −e_{9} + (z_{1} + x_{2} + ix_{3})\overline{p_{1}} + z̄_{1}p_{1},

and the RHS of the third equation can be abbreviated f_{9}, in analogy with f_{7}. The estimate (21) changes to:

|f_{9}|_{r} ≤ | − e_{9}|_{r} + |(z_{1} + x_{2} + ix_{3})\frac{f_{8}}{2ζ} |_{r} + |ζ \frac{f′_{8}}{2z_{1}} |_{r} ≤ |e_{9}|_{r} + 6|e_{8}|_{r}.

Both the construction of the solution and the estimates proceed by only changing the subscripts from 4, 5, 7 to 2, 3, 9, and adjusting the estimate for f_{9} to get the claimed results — the second nondegeneracy condition on the quadratic part of h_{9} is used here also in the same way.

Corollary 5.7. Given \frac{2}{3}(n + 1) ≤ m < n, r > 0, and e⃗(z_{1}, ζ, x) convergent on D_{r} with |e⃗|_{r} < ∞ and degree d ≥ 3, there exists p⃗ which is convergent on Δ_{r}, and which solves the system of equations (10–11), and which satisfies:

∥p_{1}∥_{r} ≤ \frac{3}{2r} |e_{n−1}|_{r}, ∥p_{n−1}∥_{r} ≤ 4|e_{n−1}|_{r},

and for any 0 < R < r,

∥p_{2}∥_{R} ≤ \frac{3|e_{n}|_{r} + 18|e_{n−1}|_{r}}{R}

+ (\frac{8r^{2}}{(r − R)^{2}} + 10) |E_{2}|_{r} + (\frac{4r^{2}}{(r − R)^{2}} + 4) |E_{3}|_{r}

∥p_{3}∥_{R} ≤ \frac{3|e_{n}|_{r} + 18|e_{n−1}|_{r}}{R}

+ (\frac{4r^{2}}{(r − R)^{2}} + 4) |E_{2}|_{r} + (\frac{8r^{2}}{(r − R)^{2}} + 10) |E_{3}|_{r}

∥p_{n}∥_{R} ≤ 4|e_{n}|_{r} + 24|e_{n−1}|_{r} + 8R (\frac{r^{2}}{(r − R)^{2}} + 1) (|E_{2}|_{r} + |E_{3}|_{r}).

Further, if m > \frac{2}{3}(n + 1), then for s = 2n − 2m + 2, …, m − 1,

∥p_{s}∥_{R} ≤ \frac{20r^{2}}{(r − R)^{2}} |E_{s}|_{r},

and if m < n − 1, then for t = m, …, n − 2,

∥p_{2(t−m+2)}∥_{R} ≤ \frac{3|e_{t}|_{r} + 9|e_{n−1}|_{r}}{R} + (\frac{8r^{2}}{(r − R)^{2}} + 10) |E_{2(t−m+2)}|_{r}

+ (\frac{4r^{2}}{(r − R)^{2}} + 4) |E_{2(t−m+2)+1}|_{r}

∥p_{2(t−m+2)+1}∥_{R} ≤ \frac{3|e_{t}|_{r} + 9|e_{n−1}|_{r}}{R} + (\frac{4r^{2}}{(r − R)^{2}} + 4) |E_{2(t−m+2)}|_{r}

+ (\frac{8r^{2}}{(r − R)^{2}} + 10) |E_{2(t−m+2)+1}|_{r}

∥p_{t}∥_{R} ≤ 4|e_{t}|_{r} + 12|e_{n−1}|_{r}

+8R (\frac{r^{2}}{(r − R)^{2}} + 1) (|E_{2(t−m+2)}|_{r} + |E_{2(t−m+2)+1}|_{r}).

Proof. The method of solution from the Proof of Theorem 5.6 groups the system of equations into smaller sub-systems that can be solved sequentially, so the generalization from (7, 9) to (m, n) can be accomplished by a straightforward re-labeling of subscripts (described below), resulting in similar estimates as claimed. The nondegeneracy conditions remain essential for any (m, n).

The solution claimed by the Corollary can be chosen to have the following form, where now z abbreviates z_{2}, …, z_{m−1} and again p_{1} is even in z_{1}:

p_{1}(z⃗) = p_{1}(z_{1}, z, z_{n−1})
p_{j}(z⃗) = p_{j}^{E}(z_{1}, z, z_{n−1}) + z_{n}p_{j}^{O}(z_{1}, z, z_{n−1}),

for j = 2, …, n. For m < n − 1, this p⃗ does not depend on z_{m}, …, z_{n−2}.

The e_{n−1} equation from (10) determines p_{n−1} and p_{1}, exactly as in the solution of (14), replacing the subscript 8 with n − 1 in the first part of the above Proof. The subscript 1 does not change.

If m > \frac{2}{3}(n + 1), then each of the 3m − 2n − 2 individual E_{s} equations, s = 2n − 2m + 2, …, m − 1, independently determines p_{s}, in analogy with the solution for p_{6} in terms of E_{6} in the second part of the above Proof. If m = \frac{2}{3}(n + 1) (the case of an isolated singularity), there are no equations analogous to the above Proof’s E_{6} equation.

The sub-system of three equations determining p_{2}, p_{3}, p_{n} in terms of E_{2}, E_{3}, e_{n}, and p_{1}, can be solved in analogy with the above E_{2}, E_{3}, e_{9} group of equations (25), only the subscript 9 needs to change to n. If m = n − 1, then those three equations are the only remaining ones in the system.

If m < n − 1, then there are n − m − 1 more sub-systems of three equations, to be solved for p_{2(t−m+2)}, p_{2(t−m+2)+1}, p_{t}, t = m, …, n − 2, in terms of E_{2(t−m+2)}, E_{2(t−m+2)+1}, e_{t}, and p_{1}, in analogy with equations (18–20). Solving each of these sub-systems depends only on having solved for p_{1}, and not any other equations in the system (10).

It is not yet claimed that using the solution p⃗ of Theorem 5.6 or Corollary 5.7 in (2) defines a local biholomorphism; this will be shown later (Theorem 6.4), under certain conditions on e⃗ and r. The most important property so far of the solution p⃗ is that the norms of its components can be estimated on Δ_{R} for R less than, but arbitrarily close to, r.

Corollary 5.8. Given \frac{2}{3}(n + 1) ≤ m < n, there is a constant c_{1} > 0 (depending only on m, n), such that for any p⃗ and e⃗ as in Corollary 5.7, and any radius lengths ρ, r with \frac{1}{2} < ρ < r ≤ 1, the following hold:

max_{j=1,...,n}\{∥p_{j}∥_{ρ}\} ≤ \frac{c_{1}|e⃗|_{r}}{(r − ρ)^{2}}

max_{j=1,...,n} \{∑_{k=1}^{n} ∥\frac{dp_{k}}{dz_{j}} ∥_{ρ}\} ≤ \frac{c_{1}|e⃗|_{r}}{(r − ρ)^{3}} .

Proof. Let R = \frac{1}{2}(ρ + r). The bound on each p_{j} follows from ∥p_{j}∥_{ρ} ≤ ∥p_{j}∥_{R} and the bounds from the previous Corollary, using \frac{1}{2} < R < r ≤ 1, and 16 < \frac{1}{(r−R)^{2}} = \frac{4}{(r−ρ)^{2}} . The bounds for the derivatives of p_{k} follow from this consequence ([A]) of Cauchy’s estimate:

If 0 < R_{2} < R_{1} and f(w) is holomorphic and bounded by K for |w| < R_{1}, then \frac{df}{dw} is bounded by \frac{K}{R_{1}−R_{2}} for |w| < R_{2}.

This fact can be applied with K = ∥p_{k}∥_{R} and R_{1} −R_{2} = R −ρ = \frac{1}{2}(r −ρ) for the z_{1}, …, z_{m−1} derivatives, R_{1} − R_{2} = R^{2} − ρ^{2} > R − ρ = \frac{1}{2}(r − ρ) for the z_{n−1} derivatives, and R_{1} − R_{2} = 3R^{2} − 3ρ^{2} > \frac{3}{2}(r − ρ) for the z_{n} derivatives. The z_{m}, …, z_{n−2} derivatives are zero by construction.

The lower bound r > \frac{1}{2} was important for the previous Corollary, but it is not a significant a priori restriction on the manifold M. By a real rescaling z⃗ ↦ (a_{1}z_{1}, …, a_{1}z_{m−1}, a_{1}^{2}z_{m}, …, a_{1}^{2}z_{n}), a_{1} > 0, the equations (5) can be assumed to define M for |z_{1}| < 1, |x_{s}| < 1, and for any η > 0, there is a rescaling making |e⃗|_{1} < η.

6. The new defining equations and some estimates

To get a solution of the nonlinear equation (9) by iterating the solution of the linear equation, the rapid convergence technique will apply, closely following the methods used by [M] on a different CR singularity problem. Each step along the way to a proof of Proposition 3.3 is stated as a Theorem.

Substituting the linear equation’s normalized solution p⃗ from Corollary 5.7 into E_{2}, …, e_{n} in the RHS of the nonlinear equation (9) gives a quantity q⃗ depending on z_{1}, z̄_{1}, x. Let

(26) z⃗ = (z_{1}, x, …, z̄_{1}(x_{2(t−m+2)} + ix_{2(t−m+2)+1}), …, z̄_{1}^{2}, z̄_{1}(z_{1} + x_{2} + ix_{3})),

as in (11), let

(27)  z⃗ + e⃗ = (z_{1}, x_{2} + iE_{2}, …, x_{m−1} + iE_{m−1},
…, z̄_{1}(x_{2(t−m+2)} + ix_{2(t−m+2)+1}) + e_{t}, …,
z̄_{1}^{2} + e_{n−1}, z̄_{1}(z_{1} + x_{2} + ix_{3}) + e_{n}),

as in (8), and then define q⃗(z_{1}, z̄_{1}, x) = (Q_{2}, …, Q_{m−1}, q_{m}, …, q_{n}) by:

(28) Q_{s} = Im(p_{s}(z⃗ + e⃗) − p_{s}(z⃗))
(29) q_{t} = p_{t}(z⃗ + e⃗) − p_{t}(z⃗)
−(x_{2(t−m+2)} + ix_{2(t−m+2)+1})\overline{(p_{1}(z⃗ + e⃗) − p_{1}(z⃗))}
−z̄_{1} · Re(p_{2(t−m+2)}(z⃗ + e⃗) − p_{2(t−m+2)}(z⃗))
−iz̄_{1} · Re(p_{2(t−m+2)+1}(z⃗ + e⃗) − p_{2(t−m+2)+1}(z⃗))

−\overline{p_{1}(z⃗ + e⃗)}(Re(p_{2(t−m+2)}(z⃗ + e⃗)) + iRe(p_{2(t−m+2)+1}(z⃗ + e⃗)))

(30) q_{n−1} = p_{n−1}(z⃗ + e⃗) − p_{n−1}(z⃗)
−2z̄_{1}\overline{(p_{1}(z⃗ + e⃗) − p_{1}(z⃗))} − \overline{(p_{1}(z⃗ + e⃗))}^{2}
(31) q_{n} = p_{n}(z⃗ + e⃗) − p_{n}(z⃗) − z̄_{1}(p_{1}(z⃗ + e⃗) − p_{1}(z⃗))
−z̄_{1}(Re(p_{2}(z⃗ + e⃗) − p_{2}(z⃗)) + iRe(p_{3}(z⃗ + e⃗) − p_{3}(z⃗)))
−(z_{1} + x_{2} + ix_{3})\overline{(p_{1}(z⃗ + e⃗) − p_{1}(z⃗))}

−\overline{(p_{1}(z⃗ + e⃗))}(p_{1}(z⃗ + e⃗) + Re(p_{2}(z⃗ + e⃗)) + iRe(p_{3}(z⃗ + e⃗))).

To outline the role of q⃗ in the argument, the next step (Theorem 6.2) will suppose p⃗(z_{1}, …, z_{n}) is complex analytic on Δ_{ρ}, and |e⃗|_{σ} is small enough so that z⃗ ∈ Δ_{σ} ⟹ z⃗ + e⃗ ∈ Δ_{ρ}, and so q⃗ is a real analytic function for (z_{1}, z̄_{1}, x) ∈ D_{σ}. If q⃗(z_{1}, z̄_{1}, x) happens to be identically zero, then the manifold M has been brought to normal form by the functions p⃗. Otherwise, the degree of q⃗ is at least 2d − 2 by the construction of the solution p⃗, and defining q⃗(z_{1}, ζ, x) by (28–31), with ζ formally substituted for z̄_{1} and allowing complex x, the norm |q⃗|_{σ} can be bounded in terms of the norm of e⃗. Then later, in the Proof of Theorem 6.6, converting q⃗(z_{1}, z̄_{1}, x) into an expression in z̃_{1}, ¯ z̃_{1}, x̃ and equating it to the z̃ polynomial expression in (7) gives the defining equations of M in the z̃ coordinate system.

The Ñ = N case of the following Lemma is proved in [C_{4}] (Lemma 4.1.).

Lemma 6.1. Let f = (f_{1}, …, fÑ) : 𝔻_{r} → ℂÑ be a holomorphic map with

max_{j=1,...,N} \{∑_{k=1}^{Ñ} sup_{z⃗∈𝔻_{r}} |\frac{df_{k}}{dz_{j}} (z⃗)|\} ≤ K.

Then, for z⃗, z ′ ∈ 𝔻_{r},

∑_{k=1}^{Ñ} |f_{k}(z ′) − f_{k}(z⃗)| ≤ K ∑_{j=1}^{N} |z′_{j} − z_{j}|.

Theorem 6.2. There are some constants c_{2} > 0 and δ_{1} > 0 (depending on m, n) such that if \frac{1}{2} < σ < r ≤ 1, and e⃗ is as in Corollary 5.7, with |e⃗|_{r} ≤ δ_{1}(r − σ), then

|q⃗|_{σ} ≤ \frac{c_{2}|e⃗|_{r}^{2}}{(r − σ)^{3}} .

Proof. Let ρ = \frac{1}{2}(r + σ). Note that if δ_{1} ≤ \frac{1}{2}, the formal series for q⃗ is convergent on D_{σ}, since then, for (z_{1}, ζ, x) ∈ D_{r}, |x_{s} + iE_{s}| < σ + δ_{1}(r − σ) ≤ σ + (ρ − σ) = ρ, |ζ(x_{2(t−m+2)} + ix_{2(t−m+2)+1}) + e_{t}| < 2σ^{2} + (ρ − σ) < 2σ^{2} + (ρ − σ)(2(ρ + σ)) = 2ρ^{2}, and similarly |ζ^{2} + e_{n−1}| < ρ^{2} and |(z_{1} +x_{2} +ix_{3})ζ +e_{n}| < 3ρ^{2}, so z⃗ +e⃗ ∈ Δ_{ρ}, which is contained in the domain of p⃗ by Corollary 5.7. The N = n, Ñ = 1, 𝔻_{r} = Δ_{ρ} case of Lemma 6.1 applies to p_{k} : Δ_{ρ} → ℂ, with

max_{j=1,...,n} \{∥\frac{dp_{k}}{dz_{j}} ∥_{ρ}\} ≤ K = \frac{c_{1}|e⃗|_{r}}{(r − ρ)^{3}} ,

by Corollary 5.8, and z ′ = z⃗ + e⃗ ∈ Δ_{ρ}, so the conclusion is:

|p_{k}(z⃗ + e⃗) − p_{k}(z⃗)| ≤ K(|E_{2}|_{r} + ⋯ + |e_{n}|_{r}) = \frac{c_{1}|e⃗|_{r}}{(r − ρ)^{3}} |e⃗|_{r} = \frac{8c_{1}|e⃗|_{r}^{2}}{(r − σ)^{3}} .

This provides bounds for the differences that appear in (28–31), and the remaining terms are the products, where we can use \frac{1}{2} < σ < ρ < r ≤ 1, the bound of Theorem 5.6 on the p_{1} factor, and the bounds of Corollary 5.8 on the other factors. For example, in a case of (29) where t = m < n − 1, part of the expression is the product:

sup_{D_{σ}} |(p_{1}(z⃗ + e⃗))′ \frac{p_{4}(z⃗ + e⃗) + (p_{4}(z⃗ + e⃗))′ + ip_{5}(z⃗ + e⃗) + i(p_{5}(z⃗ + e⃗))′}{2} |
≤ ∥p_{1}∥_{ρ}(∥p_{4}∥_{ρ} + ∥p_{5}∥_{ρ}) ≤ \frac{3}{2r} |e_{n−1}|_{r} (\frac{2c_{1}|e⃗|_{r}}{(r − ρ)^{2}} ) < \frac{6c_{1}|e⃗|_{r}^{2}}{(r − ρ)^{2}} < \frac{12c_{1}|e⃗|_{r}^{2}}{(r − σ)^{3}} .

The following Lemma on inverse functions will be used twice, in the construction of the new coordinate system and the new defining equations; a proof by a standard iteration procedure is sketched in ([C_{4}]).

Lemma 6.3. Suppose 0 < R_{2,k} < R_{1,k} for k = 1, …, N, so that

𝔻^{2} = 𝔻_{(R_{2,1},...,R_{2,N})} ⊆ 𝔻^{1} = 𝔻_{(R_{1,1},...,R_{1,N})}.

Let f(z⃗) = (f_{1}(z_{1}, …, z_{N}), …, f_{N}(z_{1}, …, z_{N})) be holomorphic on 𝔻^{1}, with

max_{j=1,...,N} \{∑_{k=1}^{N} sup_{z⃗∈𝔻^{1}} |\frac{df_{k}}{dz_{j}} (z⃗)|\} ≤ K < 1,

and

∑_{k=1}^{N} sup_{z⃗∈𝔻^{2}} |f_{k}(z⃗)| ≤ (1 − K) min_{k=1,...,N}{R_{1,k} − R_{2,k}}.

Then, given w⃗ ∈ 𝔻^{2}, there exists a unique solution z⃗ ∈ 𝔻^{1} of the equation

w⃗ = z⃗ + f(z⃗),

and this solution satisfies

∑_{k=1}^{N} |z_{k} − w_{k}| ≤ \frac{1}{1 − K} ∑_{k=1}^{N} |f_{k}(w⃗)|.

Theorem 6.4. There is some constant δ_{2} > 0 (depending on m, n) so that for any radius lengths \frac{1}{2} < σ < r ≤ 1, and e⃗, p⃗ as in Corollary 5.7, with |e⃗|_{r} ≤ δ_{2}(r − σ)^{3} and ρ = \frac{1}{2}(r + σ), the transformation

Ψ : z⃗ = (z_{1}, …, z_{n}) ↦ z̃ = (z_{1} + p_{1}(z⃗), …, z_{n} + p_{n}(z⃗))

has a holomorphic inverse ψ(z̃) = z⃗ such that if z̃ ∈ Δ_{σ}, then ψ(z̃) ∈ Δ_{ρ}.

Proof. By Corollary 5.8,

max_{j=1,...,n} \{∑_{k=1}^{n} ∥\frac{dp_{k}}{dz_{j}} ∥_{ρ}\} ≤ \frac{c_{1}|e⃗|_{r}}{(r − ρ)^{3}} ≤ \frac{c_{1}δ_{2}(r − σ)^{3}}{(r − ρ)^{3}} = 8δ_{2}c_{1} ≤ \frac{1}{2} = K,

if δ_{2} ≤ \frac{1}{16c_{1}} . Also by Corollary 5.8,

∑_{k=1}^{n} ∥p_{k}∥_{σ} ≤ \frac{nc_{1}|e⃗|_{r}}{(r − σ)^{2}} ≤ nc_{1}δ_{2}(r − σ) ≤ (1 − K)(ρ − σ),

if δ_{2} ≤ \frac{1}{4nc_{1}} . The hypotheses of Lemma 6.3 are satisfied with Δ_{σ} ⊆ Δ_{ρ}, and R_{1,k} − R_{2,k} ≥ ρ − σ, so given z̃ ∈ Δ_{σ}, there exists a unique z⃗ ∈ Δ_{ρ} such that z̃ = (z_{1} + p_{1}(z⃗), …, z_{n} + p_{n}(z⃗)). This defines ψ so that Ψ ∘ ψ is the identity map on Δ_{σ}.

For (z_{1}, ζ, x) ∈ D_{R_{1}} ⊆ ℂ^{m}, define z^{c} ∈ ℂ^{n} by:

z^{c} = (z_{1}, x_{2} + iE_{2}(z_{1}, ζ, x), …, x_{m−1} + iE_{m−1}(z_{1}, ζ, x),
…, ζ(x_{2(t−m+2)} + ix_{2(t−m+2)+1}) + e_{t}(z_{1}, ζ, x), …,
ζ^{2} + e_{n−1}(z_{1}, ζ, x), ζ(z_{1} + x_{2} + ix_{3}) + e_{n}(z_{1}, ζ, x)),

and define a map τ : D_{R_{1}} → ℂ^{m} by

τ(z_{1}, ζ, x) = (τ_{1}(z_{1}, ζ, x), …, τ_{m}(z_{1}, ζ, x))
= (z_{1} + p_{1}(z^{c}), ζ + (p_{1}(z^{c}))′,

x_{2} + \frac{1}{2}(p_{2}(z^{c}) + (p_{2}(z^{c}))′), …,

x_{m−1} + \frac{1}{2}(p_{m−1}(z^{c}) + (p_{m−1}(z^{c}))′)).

Theorem 6.5. There is some constant δ_{3} > 0 (depending on m, n) so that for any radius lengths \frac{1}{2} < r′ < r ≤ 1, and e⃗, p⃗ as in Corollary 5.7, with |e⃗|_{r} ≤ δ_{3}(r − r′)^{3}, and σ = r′ + \frac{1}{3}(r − r′), the transformation τ : (z_{1}, ζ, x) ↦ (z̃_{1}, ζ̃, x̃) has a holomorphic inverse ϕ(z̃_{1}, ζ̃, x̃) = (z_{1}, ζ, x) such that if (z̃_{1}, ζ̃, x̃) ∈ D_{r′}, then ϕ(z̃_{1}, ζ̃, x̃) ∈ D_{σ}.

Proof. Let ρ = r′+ \frac{2}{3}(r−r′), so σ−r′ = ρ−σ = r−ρ = \frac{1}{3}(r−r′) < \frac{1}{6}, and let r̄ = \frac{1}{2}(r + r′), so \frac{1}{2} < r′ < σ < r̄ < ρ < r ≤ 1. If (z_{1}, ζ, x) ∈ Dr̄, and δ_{3} ≤ \frac{2}{3}, then |E_{2}(z_{1}, ζ, x)| ≤ δ_{3}(r − r′)^{3} = 216δ_{3}(ρ − r̄)^{3} < \frac{216}{12^{2}}δ_{3}(ρ − r̄) ≤ ρ − r̄, and similarly |e_{n−1}(z_{1}, ζ, x)| < ρ^{2} −r̄^{2}, etc., so z^{c} ∈ Δ_{ρ}, and p⃗(z^{c}) and τ are welldefined and holomorphic on Dr̄. Using Cauchy’s estimate as in Corollary 5.8, for (z_{1}, ζ, x) ∈ D_{σ},

|\frac{d}{dz_{1}} p_{2}(z^{c})| ≤ \frac{|p_{2}(z^{c})|r̄}{r̄ − σ} ≤ \frac{∥p_{2}∥_{ρ}}{\frac{1}{2}(ρ − σ)} ≤ \frac{2c_{1}|e⃗|_{r}}{(ρ − σ)(r − ρ)^{2}} = \frac{54c_{1}|e⃗|_{r}}{(r − r′)^{3}} .

Similarly, the derivative of each term, p_{1}(z^{c}), p_{s}(z^{c}), (p_{1}(z^{c}))′, (p_{s}(z^{c}))′, with respect to each variable z_{1}, ζ, x_{s}, is bounded by a comparable quantity, so there is some constant c_{3} > 0 (depending on m, n) so that

max_{j=2,...,m−1}\{|\frac{dp_{1}(z^{c})}{dz_{1}} |_{σ}+ |\frac{d((p_{1}(z^{c}))′)}{dz_{1}} |_{σ}+ ∑_{s=2}^{m−1} |\frac{d(\frac{1}{2}(p_{s}(z^{c}) + (p_{s}(z^{c}))′))}{dz_{1}} |_{σ},

|\frac{dp_{1}(z^{c})}{dζ}|_{σ} + |\frac{d((p_{1}(z^{c}))′)}{dζ}|_{σ} + ∑_{s=2}^{m−1} |\frac{d(\frac{1}{2}(p_{s}(z^{c}) + (p_{s}(z^{c}))′))}{dζ}|_{σ} ,

|\frac{dp_{1}(z^{c})}{dx_{j}} |_{σ} + |\frac{d((p_{1}(z^{c}))′)}{dx_{j}} |_{σ} + ∑_{s=2}^{m−1} |\frac{d(\frac{1}{2}(p_{s}(z^{c}) + (p_{s}(z^{c}))′))}{dx_{j}} |_{σ}\}

≤ \frac{c_{3}|e⃗|_{r}}{(r − r′)^{3}} ≤ c_{3}δ_{3} ≤ \frac{1}{2},

if δ_{3} ≤ \frac{1}{2c_{3}} . It also follows from Corollary 5.8 that

|p_{1}(z^{c})|_{r′} + |(p_{1}(z^{c}))′|_{r′} + ∑_{s=2}^{m−1} |\frac{1}{2}(p_{s}(z^{c}) + (p_{s}(z^{c}))′)|_{r′}

≤ 2∥p_{1}∥_{ρ} + ∑_{s=2}^{m−1} ∥p_{s}∥_{ρ}

≤ \frac{mc_{1}|e⃗|_{r}}{(r − ρ)^{2}} ≤ \frac{mc_{1}δ_{3}(r − r′)^{3}}{(r − ρ)^{2}} = 9mc_{1}δ_{3}(r − r′) ≤ \frac{1}{2}(σ − r′),

if δ_{3} ≤ \frac{1}{54mc_{1}} . So, by Lemma 6.3, given (z̃_{1}, ζ̃, x̃) ∈ D_{r′}, there exists a unique (z_{1}, ζ, x) ∈ D_{σ} such that (z̃_{1}, ζ̃, x̃) = τ(z_{1}, ζ, x).

By inspection of the form of τ, if (z_{1}, ζ, x) ∈ D_{σ}, and τ(z_{1}, ζ, x) = (z̃_{1}, ζ̃, x̃), then τ(ζ̄, z̄_{1}, x̄) = (\overline{ζ̃}, ¯ z̃_{1}, ¯ x̃). If, further, (z̃_{1}, ζ̃, x̃) = (\overline{ζ̃}, ¯ z̃_{1}, ¯ x̃) ∈ D_{r′}, then (z_{1}, ζ, x) = (ζ̄, z̄_{1}, x̄) by uniqueness of the inverse. In particular, if |z̃_{1}| < r′ and for s = 2, …, m − 1, x̃_{s} is real and |x̃_{s}| < r′, then ϕ(z̃_{1}, ¯ z̃_{1}, x̃) is of the form (z_{1}, z̄_{1}, x) for some z_{1} with |z_{1}| < σ and x real with |x_{s}| < σ. Such (z_{1}, x) is unique, given (z̃_{1}, x̃): suppose there were (z_{1}^{0}, x^{0}) with |z_{1}^{0}| < σ, |x_{s}^{0}| < σ, x^{0} real, such that

z̃_{1} = τ_{1}(z_{1}^{0}, \overline{z_{1}^{0}}, x^{0})
x̃_{s} = τ_{s+1}(z_{1}^{0}, \overline{z_{1}^{0}}, x^{0}), s = 2, …, m − 1.

Then the second component τ_{2}(z_{1}^{0}, \overline{z_{1}^{0}}, x^{0}) can be calculated to have some value ζ̃, so τ(z_{1}^{0}, \overline{z_{1}^{0}}, x^{0}) = (z̃_{1}, ζ̃, x̃). By the formula for τ, ζ̃ = ¯ z̃_{1}, so (z̃_{1}, ζ̃, x̃) ∈ D_{r′} and (z_{1}^{0}, \overline{z_{1}^{0}}, x^{0}) = ϕ(z̃_{1}, ζ̃, x̃) = ϕ(z̃_{1}, ¯ z̃_{1}, x̃) = (z_{1}, z̄_{1}, x), so we can conclude from the uniqueness of Lemma 6.3 that z_{1}^{0} = z_{1} and x^{0} = x.

Theorem 6.6. There exist constants c_{4} > 0 and δ_{4} > 0 (depending on m, n) such that for any \frac{1}{2} < r′ < r ≤ 1 (with σ, ρ as in the previous Theorem), and any e⃗ as in Corollary 5.7 with |e⃗|_{r} ≤ δ_{4}(r − r′)^{3}, there exist a holomorphic map Ψ : Δ_{ρ} → ℂ^{n} : (z_{1}, …, z_{n}) ↦ (z̃_{1}, …, z̃_{n}), with a holomorphic inverse ψ : Δ_{σ} → Δ_{ρ}, and a holomorphic map ẽ = (Ẽ_{2}, …, ẽ_{n}) : D_{r′} → ℂ^{n−1}, such that the defining equations for M are

ỹ_{s} = Ẽ_{s}(z̃_{1}, ¯ z̃_{1}, x̃)
z̃_{t} = ¯ z̃_{1}(x̃_{2(t−m+2)} + ix̃_{2(t−m+2)+1}) + ẽ_{t}(z̃_{1}, ¯ z̃_{1}, x̃)
z̃_{n−1} = ¯ z̃_{1}^{2} + ẽ_{n−1}(z̃_{1}, ¯ z̃_{1}, x̃)
z̃_{n} = ¯ z̃_{1}(z̃_{1} + x̃_{2} + ix̃_{3}) + ẽ_{n}(z̃_{1}, ¯ z̃_{1}, x̃),

for |z̃_{1}| < r′, |x̃_{s}| < r′. Further, the degree of ẽ is at least 2d − 2, and

|ẽ|_{r′} ≤ \frac{c_{4}|e⃗|_{r}^{2}}{(r − r′)^{3}} .

Proof. Initially, choose δ_{4} ≤ min\{\frac{8}{3}δ_{1}, \frac{8}{27}δ_{2}, δ_{3}\}, so that Theorems 6.2, 6.4, 6.5 apply, and define Ψ, ψ, q⃗, and ϕ in terms of the given e⃗ and the functions p⃗ constructed in Corollary 5.7. Define ẽ to be the composite of holomorphic maps q⃗ ∘ ϕ : D_{r′} → ℂ^{n−1}, so that by Theorem 6.2,

|ẽ|_{r′} ≤ |q⃗|_{σ} ≤ \frac{c_{2}|e⃗|_{r}^{2}}{(r − σ)^{3}} = \frac{c_{2}|e⃗|_{r}^{2}}{(\frac{2}{3}(r − r′))^{3}} .

Since ϕ(z̃_{1}, ¯ z̃_{1}, x̃) has no constant terms, and q⃗ has degree ≥ 2d − 2 by construction, ẽ(z̃_{1}, ¯ z̃_{1}, x̃) also has degree at least 2d − 2.

Given z̃_{1}, x̃ such that |z̃_{1}| < r′, and x̃ is real with |x̃_{s}| < r′, define quantities z̃_{2}, …, z̃_{n} by:

(32)  z̃_{s} = x̃_{s} + iẼ_{s}(z̃_{1}, ¯ z̃_{1}, x̃)
z̃_{t} = ¯ z̃_{1}(x̃_{2(t−m+2)} + ix̃_{2(t−m+2)+1}) + ẽ_{t}(z̃_{1}, ¯ z̃_{1}, x̃)
z̃_{n−1} = ¯ z̃_{1}^{2} + ẽ_{n−1}(z̃_{1}, ¯ z̃_{1}, x̃)
z̃_{n} = ¯ z̃_{1}(z̃_{1} + x̃_{2} + ix̃_{3}) + ẽ_{n}(z̃_{1}, ¯ z̃_{1}, x̃),

and denote z̃ = (z̃_{1}, z̃_{2}, …, z̃_{n}). The claim of the Theorem is that ψ(z̃) ∈ M. If (δ_{4})^{2} ≤ \frac{32}{81c_{2}} , then

|ẽ|_{r′} ≤ \frac{c_{2}(δ_{4}(r − r′)^{3})^{2}}{(r − σ)^{3}} = c_{2}(δ_{4})^{2} \frac{3^{6}}{2^{3}} (σ − r′)^{3} ≤ c_{2}(δ_{4})^{2} \frac{3^{6}}{2^{3}6^{2}} (σ − r′) ≤ σ − r′,

so z̃ ∈ Δ_{σ}, the domain of ψ.

By Theorem 6.5, there exists a unique (z_{1}, x) (the first and last components of (z_{1}, z̄_{1}, x) = ϕ(z̃_{1}, ¯ z̃_{1}, x̃)) such that |z_{1}| < σ, x is real with |x_{s}| < σ, and

z̃_{1} = z_{1} + p_{1}(z_{1}, x_{2} + iE_{2}(z_{1}, z̄_{1}, x), …, z̄_{1}(z_{1} + x_{2} + ix_{3}) + e_{n}(z_{1}, z̄_{1}, x))
x̃_{s} = x_{s}+Re(p_{s}(z_{1}, x_{2} + iE_{2}(z_{1}, z̄_{1}, x), …, z̄_{1}(z_{1} + x_{2} + ix_{3}) + e_{n}(z_{1}, z̄_{1}, x))).

Then, define quantities z_{2}, …, z_{n} by:

z_{s} = x_{s} + iE_{s}(z_{1}, z̄_{1}, x)
z_{t} = z̄_{1}(x_{2(t−m+2)} + ix_{2(t−m+2)+1}) + e_{t}(z_{1}, z̄_{1}, x)
z_{n−1} = z̄_{1}^{2} + e_{n−1}(z_{1}, z̄_{1}, x)
z_{n} = z̄_{1}(z_{1} + x_{2} + ix_{3}) + e_{n}(z_{1}, z̄_{1}, x),

and denote, as in (11) and (26), z⃗ = (z_{1}, x, …, z̄_{1}(z_{1} +x_{2} +ix_{3})) and z⃗ +e⃗ = (z_{1}, z_{2}, …, z_{n}) as in (8) and (27). Since |z_{1}| < σ < r and |x_{s}| < σ < r, z⃗ + e⃗ ∈ M, and if δ_{4} ≤ \frac{4}{3}, then

|e⃗|_{σ} ≤ |e⃗|_{r} ≤ δ_{4}(r − r′)^{3} = δ_{4} · 27(ρ − σ)^{3} < δ_{4} \frac{27}{6^{2}}(ρ − σ) ≤ (ρ − σ),

so z⃗ + e⃗ ∈ Δ_{ρ}, which is contained in the domain of p⃗.

Ψ(z⃗ + e⃗)

= (z_{1} + p_{1}(z⃗ + e⃗), …, z_{n} + p_{n}(z⃗ + e⃗)),
= (z̃_{1}, …, x̃_{s} + iE_{s}(z_{1}, z̄_{1}, x) + iIm(p_{s}(z⃗ + e⃗)), …,

\overline{(z̃_{1} − p_{1}(z⃗ + e⃗))}(x̃_{2(t−m+2)} − Re(p_{2(t−m+2)}(z⃗ + e⃗)))

+i\overline{(z̃_{1} − p_{1}(z⃗ + e⃗))}(x̃_{2(t−m+2)+1} − Re(p_{2(t−m+2)+1}(z⃗ + e⃗))) +e_{t}(z_{1}, z̄_{1}, x) + p_{t}(z⃗ + e⃗), …

\overline{(z̃_{1} − p_{1}(z⃗ + e⃗))}^{2} + e_{n−1}(z_{1}, z̄_{1}, x) + p_{n−1}(z⃗ + e⃗),

\overline{(z̃_{1}−p_{1}(z⃗ + e⃗))}(z̃_{1}−p_{1}(z⃗ + e⃗) + x̃_{2}−Re(p_{2}(z⃗ + e⃗)) + ix̃_{3}−iRe(p_{3}(z⃗ + e⃗))) +e_{n}(z_{1}, z̄_{1}, x) + p_{n}(z⃗ + e⃗)),

= (z̃_{1}, …, x̃_{s} + iQ_{s}(z_{1}, z̄_{1}, x), …,

¯z̃_{1}(x̃_{2(t−m+2)} + ix̃_{2(t−m+2)+1}) + q_{t}(z_{1}, z̄_{1}, x), …,

¯z̃_{1}^{2} + q_{n−1}(z_{1}, z̄_{1}, x), ¯ z̃_{1}(z̃_{1} + x̃_{2} + ix̃_{3}) + q_{n}(z_{1}, z̄_{1}, x))

= (z̃_{1}, …, x̃_{s} + iQ_{s}(ϕ(z̃_{1}, ¯ z̃_{1}, x̃)), …,

¯z̃_{1}(x̃_{2(t−m+2)} + ix̃_{2(t−m+2)+1}) + q_{t}(ϕ(z̃_{1}, ¯ z̃_{1}, x̃)), …,

¯z̃_{1}^{2} + q_{n−1}(ϕ(z̃_{1}, ¯ z̃_{1}, x̃)), ¯ z̃_{1}(z̃_{1} + x̃_{2} + ix̃_{3}) + q_{n}(ϕ(z̃_{1}, ¯ z̃_{1}, x̃)))

= (z̃_{1}, …, x̃_{s} + iẼ_{s}(z̃_{1}, ¯ z̃_{1}, x̃), …,

¯z̃_{1}(x̃_{2(t−m+2)} + ix̃_{2(t−m+2)+1}) + ẽ_{t}(z̃_{1}, ¯ z̃_{1}, x̃), …,

¯z̃_{1}^{2} + ẽ_{n−1}(z̃_{1}, ¯ z̃_{1}, x̃), ¯ z̃_{1}(z̃_{1} + x̃_{2} + ix̃_{3}) + ẽ_{n}(z̃_{1}, ¯ z̃_{1}, x̃)) = z̃,

by construction of q⃗, ẽ, and z̃, and using the fact that p⃗ is a solution of (10–11). By the uniqueness of Theorem 6.4, ψ(z̃) = z⃗ + e⃗ ∈ M.

7. Composition of approximate solutions

The previous Theorem’s quadratic estimate on the size of ẽ in terms of e⃗ allows for the rapid convergence of a sequence of approximations. A couple technical Lemmas will be needed to measure the behavior of composite mappings. Theorem 7.7, which is the last step in proving Proposition 3.3, uses these Lemmas and the estimates of the previous Section to prove convergence of a sequence of transformations, following the ideas of [M].

Notation 7.1. For R_{1} > 0 and a n × n matrix of complex valued functions F = (F_{kj}(z⃗)) on Δ_{R_{1}}, define

|||F |||_{R_{1}} = max_{j=1,...,n} {∑_{k=1}^{n} sup_{z⃗∈Δ_{R_{1}}} |F_{kj}(z⃗)|} .

This “maximum column sum” norm appeared already, in Corollary 5.8 and Lemmas 6.1, 6.3, in the case where F = Df = Dz⃗f, the Jacobian matrix of some map f : Δ_{R_{1}} → ℂ^{n} at z⃗ ∈ Δ_{R_{1}}. The 3 × 3 case of the following Lemma was proved in [C_{4}].

Lemma 7.2. If |||A|||_{R_{1}} < 1, then 𝕀 + A is invertible (where 𝕀 is the n × n identity matrix), and

|||(𝕀 + A)^{−1}|||_{R_{1}} ≤ \frac{1}{1 − |||A|||_{R_{1}}} .

Also, the following elementary fact from the calculus of one real variable will be used.

Lemma 7.3. If μ_{k} is a sequence such that 0 ≤ μ_{k} < 1 and ∑_{k=0}^{∞} μ_{k} is a

convergent series, then the sequence of partial products

∏_{k=0}^{N} \frac{1}{1 − μ_{k}}

is bounded above by some positive limit.

Notation 7.4. For ν = 0, 1, 2, …, define a sequence \{1, \frac{3}{4}, \frac{4}{6}, \frac{5}{8}, …\} by the formula

r_{ν} = \frac{1}{2} (1 + \frac{1}{ν + 1}) .

Note that \frac{1}{2} < r_{ν} ≤ 1, and the sequence is decreasing, with

r_{ν} − r_{ν+1} = \frac{1}{2(ν + 1)(ν + 2)},
\frac{r_{ν+1} − r_{ν+2}}{r_{ν} − r_{ν+1}} = \frac{ν + 1}{ν + 3} ≥ \frac{1}{3}.

Notation 7.5. Define σ_{ν} = r_{ν+1} + \frac{1}{3}(r_{ν} − r_{ν+1}), ρ_{ν} = r_{ν+1} + \frac{2}{3}(r_{ν} − r_{ν+1}), as in Theorem 6.5.

Recall that given η > 0, there is some scaling transformation so that M ∩ Δ_{1} is defined by (5), with e⃗ holomorphic on D_{1}, degree d ≥ 3, and |e⃗|_{1} ≤ η.

Notation 7.6. Denote e⃗_{0} = e⃗ (so |e⃗_{0}|_{r_{0}} = |e⃗|_{1} ≤ η), and inductively define the formal series e⃗_{ν+1}(z_{1}, ζ, x) in terms of e⃗_{ν}(z_{1}, ζ, x), by the e⃗ ↦ ẽ procedure of Theorem 6.6, with r = r_{ν}, r′ = r_{ν+1}. Each e⃗_{ν} defines, as in the previous Theorems, functions p⃗_{ν}, q⃗_{ν}, Ψ_{ν}, ψ_{ν}, ϕ_{ν}, and the degree of e⃗_{ν} is denoted d_{ν}.

Also recall that the degree d_{ν+1} of e⃗_{ν+1} is at least 2d_{ν} − 2; it can be checked that this, together with d_{0} = d ≥ 3, implies d_{ν} ≥ 2^{ν} + 2.

The plan is to show that the bound for e⃗_{ν} in the hypothesis of Theorem 6.6 holds for all ν, to get a sequence of transformations ψ_{ν} : Δ_{σ_{ν}} → Δ_{ρ_{ν}}, so that the composition ψ_{0} ∘ … ∘ ψ_{ν−1} ∘ ψ_{ν} : Δ_{σ_{ν}} → Δ_{ρ_{0}} is well-defined, e⃗_{ν} is holomorphic on D_{r_{ν}}, and lim_{ν→∞} |e⃗_{ν}|_{r_{ν}} = 0.

Theorem 7.7. There exists η > 0 (depending on m, n) so that if e⃗_{0} and M are as described above, then there exists a holomorphic transformation ψ : Δ_{\frac{1}{2}} → ℂ^{n}, with a holomorphic inverse Ψ, and such that if z̃ ∈ M̃ ∩ Δ_{\frac{1}{2}} , then ψ(z̃) ∈ M.

Proof. Let δ_{5} = min\{δ_{4}, \frac{1}{27c_{4}} \}, and choose 0 < η < min\{\frac{δ_{5}}{64}, \frac{1}{1728c_{1}} \}. It will be shown that

|e⃗_{ν}|_{r_{ν}} ≤ δ_{5}(r_{ν} − r_{ν+1})^{3} ⟹ |e⃗_{ν+1}|_{r_{ν+1}} ≤ δ_{5}(r_{ν+1} − r_{ν+2})^{3}.

By Theorem 6.6, |e⃗_{ν}|_{r_{ν}} ≤ δ_{4}(r_{ν} − r_{ν+1})^{3} and |e⃗_{ν}|_{r_{ν}} ≤ \frac{1}{27c_{4}} (r_{ν} − r_{ν+1})^{3} imply

|e⃗_{ν+1}|_{r_{ν+1}} ≤ \frac{c_{4}|e⃗_{ν}|_{r_{ν}}^{2}}{(r_{ν} − r_{ν+1})^{3}} ≤ \frac{1}{27}|e⃗_{ν}|_{r_{ν}};

this already suggests a geometric decrease in the sequence of norms. Then, using the properties of the sequence r_{ν},

\frac{1}{27}|e⃗_{ν}|_{r_{ν}} ≤ \frac{1}{27}δ_{5}(r_{ν} − r_{ν+1})^{3} ≤ δ_{5}(r_{ν+1} − r_{ν+2})^{3},

which proves the claimed implication. Using this as an inductive step, and starting the induction with |e⃗_{0}|_{r_{0}} ≤ η < \frac{1}{64}δ_{5} = δ_{5}(r_{0} − r_{1})^{3}, the hypothesis of Theorem 6.6 is satisfied for all ν. The first of three conclusions from Theorem 6.6 is that e⃗_{ν} is holomorphic on D_{r_{ν}}, with degree d_{ν} ≥ 2^{ν} + 2, and |e⃗_{ν}|_{r_{ν}} ≤ 27^{−ν}η. Secondly, ψ_{0} ∘ … ∘ ψ_{ν} is a well-defined holomorphic map Δ_{σ_{ν}} → Δ_{ρ_{0}}, and Ψ_{ν} ∘ … ∘ Ψ_{0} is well-defined and holomorphic on the image (ψ_{0} ∘ … ∘ ψ_{ν})(Δ_{σ_{ν}}), so that Ψ_{ν} ∘ … ∘ Ψ_{0} ∘ ψ_{0} ∘ … ∘ ψ_{ν} is the identity on Δ_{σ_{ν}}. The third conclusion is that if |z̃_{1}| < r_{ν+1} and |x̃_{s}| < r_{ν+1}, and z̃ is defined as in (32) with ẽ = e⃗_{ν+1}, then (ψ_{0} ∘ … ∘ ψ_{ν})(z̃) ∈ M. For any z⃗ = (z_{1}, …, z_{n}) ∈ Δ_{\frac{1}{2}} , the sequence (depending on ν) (ψ_{0}∘…∘ψ_{ν−1}∘ψ_{ν})(z⃗) is contained in Δ_{ρ_{0}} = Δ_{11/12}. The following argument, beginning with several applications of Lemma 6.1, shows this sequence is a Cauchy sequence, and converges to some value ψ(z⃗).

∑_{k=1}^{n} |(ψ_{0} ∘ … ∘ ψ_{ν+1})_{k}(z⃗) − (ψ_{0} ∘ … ∘ ψ_{ν})_{k}(z⃗)|

= ∑_{k=1}^{n} |(ψ_{0})_{k}((ψ_{1} ∘ … ∘ ψ_{ν+1})(z⃗)) − (ψ_{0})_{k}((ψ_{1} ∘ … ∘ ψ_{ν})(z⃗))|

≤ |||Dψ_{0}|||_{ρ_{1}} · ∑_{j=1}^{n} |(ψ_{1} ∘ … ∘ ψ_{ν+1})_{j}(z⃗) − (ψ_{1} ∘ … ∘ ψ_{ν})_{j}(z⃗)|

(33) ≤ (∏_{ℓ=0}^{ν} |||Dψ_{ℓ}|||_{ρ_{ℓ+1}}) · ∑_{j=1}^{n} |(ψ_{ν+1})_{j}(z⃗) − z_{j}|.

By the estimate from Lemma 6.3, with f = p⃗_{ν+1} and K = \frac{1}{2} from the Proof of Theorem 6.4, and then using the bound for p⃗ from Corollary 5.8,

∑_{j=1}^{n} |(ψ_{ν+1})_{j}(z⃗) − z_{j}| ≤ \frac{1}{1 − \frac{1}{2}} ∑_{j=1}^{n} |(p⃗_{ν+1})_{j}(z⃗)| ≤ 2 ∑_{j=1}^{n} ∥(p⃗_{ν+1})_{j}∥_{\frac{1}{2}}

≤ 2 ∑_{j=1}^{n} ∥(p⃗_{ν+1})_{j}∥_{ρ_{ν+1}} ≤ 2n\frac{c_{1}|e⃗_{ν+1}|_{r_{ν+1}}}{(r_{ν+1} − ρ_{ν+1})^{2}} = 18n\frac{c_{1}|e⃗_{ν+1}|_{r_{ν+1}}}{(r_{ν+1} − r_{ν+2})^{2}}

= 72nc_{1}(ν + 2)^{2}(ν + 3)^{2}|e⃗_{ν+1}|_{r_{ν+1}} ≤ \frac{72nc_{1}(ν + 2)^{2}(ν + 3)^{2}η}{27^{ν+1}} .

It follows from Dz⃗ψ_{ℓ} = (𝕀 + D_{ψ_{ℓ}(z⃗)}p⃗_{ℓ})^{−1} and Lemma 7.2 that:

|||Dψ_{ℓ}|||_{ρ_{ℓ+1}} = |||(𝕀 + D_{ψ_{ℓ}(z⃗)}p⃗_{ℓ})^{−1}|||_{ρ_{ℓ+1}} ≤ |||(𝕀 + Dp⃗_{ℓ})^{−1}|||_{ρ_{ℓ}} ≤ \frac{1}{1 − |||Dp⃗_{ℓ}|||_{ρ_{ℓ}}} .

Then, by Lemma 7.3, the product from (33) is bounded above by some constant c_{5} > 0, since by Corollary 5.8,

∑_{ℓ=0}^{∞} |||Dp⃗_{ℓ}|||_{ρ_{ℓ}} ≤ ∑_{ℓ=0}^{∞} \frac{c_{1}|e⃗_{ℓ}|_{r_{ℓ}}}{(r_{ℓ} − ρ_{ℓ})^{3}} = ∑_{ℓ=0}^{∞} \frac{27c_{1}|e⃗_{ℓ}|_{r_{ℓ}}}{(r_{ℓ} − r_{ℓ+1})^{3}}

= ∑_{ℓ=0}^{∞} 216(ℓ + 1)^{3}(ℓ + 2)^{3}c_{1}|e⃗_{ℓ}|_{r_{ℓ}} ≤ ∑_{ℓ=0}^{∞} \frac{216(ℓ + 1)^{3}(ℓ + 2)^{3}c_{1}η}{27^{ℓ}} ,

a convergent infinite series with terms < 1.

The inequality

∑_{k=1}^{n} |(ψ_{0} ∘ … ∘ ψ_{ν+1})_{k}(z⃗) − (ψ_{0} ∘ … ∘ ψ_{ν})_{k}(z⃗)| ≤ \frac{72nc_{1}c_{5}(ν + 2)^{2}(ν + 3)^{2}η}{27^{ν+1}}

is enough to show that the sequence of composite functions converges pointwise and uniformly to a function ψ on Δ_{\frac{1}{2}} .

Remark. Although some details remain to be checked, it seems plausible that a similar rapid convergence argument could be used to prove an analogous analytic stability property for a nondegenerate CR singularity of a real 3-manifold in ℂ^{4}, as conjectured in [C_{5}].

8. Analogy with singularity theory

To continue with the theme of analogies between the normal form result and the properties of Whitney’s cross-cap singularity, we briefly consider the notion of complexification. If the defining equations of a real m-submanifold M in ℂ^{n} with a CR singularity at 0 are given as a graph over the tangent space as in (1), then M can also be considered as the image of a real analytic parametrization π : ℝ^{m} → ℝ^{2n} : (z, x) ↦ (z, x, H_{s}(z, x), h_{u}(z, x)). Then the spaces ℝ^{m}, ℝ^{2n} can be embedded as totally real subspaces of ℂ^{m}, ℂ^{2n}, and there is a a complex analytic map π_{c} : ℂ^{m} → ℂ^{2n} which restricts to π on the totally real subspaces. In the following examples, composing with a projection P : ℂ^{2n} → ℂ^{n} gives a holomorphic map P ∘π_{c} which restricts to π on the totally real ℝ^{m} subspace, and its image is a complex subvariety of ℂ^{n} containing M. Even though π_{c} is an embedding, the composite P ∘π_{c} can be singular, and the image of its critical point set contains the CR singular locus of M. For details and more examples of this construction, see [Webster], [C_{2}], [C_{3}], and to be more precise, these maps should be considered only in some neighborhood of the origin in the domain and target.

Example 8.1. In the m = n = 2 case ([Bishop]), the local defining equation of a real surface with a nondegenerate CR singularity in ℂ^{2} can be normalized to z_{2} = β(z_{1}^{2} + z̄_{1}^{2}) + z_{1}z̄_{1} + O(3), where the coefficient β ≥ 0 is a biholomorphic invariant. Considering the real embedding’s quadratic part,

π : (z_{1}, z̄_{1}) ↦ (z_{1}, z̄_{1}, z_{2} = β(z_{1}^{2} + z̄_{1}^{2}) + z_{1}z̄_{1}, z̄_{2} = \overline{β(z_{1}^{2} + z̄_{1}^{2}) + z_{1}z̄_{1}})

is a real analytic map from the totally real subspace {(z_{1}, w_{1}) : w_{1} = z̄_{1}} of ℂ^{2} to the totally real subspace {(z_{1}, w_{1}, z_{2}, w_{2}) : w_{1} = z̄_{1}, w_{2} = z̄_{2}} of ℂ^{4}, which extends to a complex analytic embedding π_{c} : (z_{1}, w_{1}) ↦ (z_{1}, w_{1}, β(z_{1}^{2} + w_{1}^{2}) + z_{1}w_{1}, β(w_{1}^{2} + z_{1}^{2}) + w_{1}z_{1}). Then composing with the projection P : ℂ^{4} → ℂ^{2} that forgets the w_{1}, w_{2} variables in the target gives a map P ∘ π_{c} : (z, w) ↦ (z, β(z^{2} + w^{2}) + zw). For β > 0, this is a ramified two-to-one map onto ℂ^{2} ([MW], [Webster]), and is analogous to Whitney’s “fold” singularity (x, y) ↦ (x, y^{2}).

Example 8.2. An example of a cubic normal form for a CR singular surface in ℂ^{2} in the β = 0 case is z_{2} = z_{1}z̄_{1} + z̄_{1}^{3} ([M]). The map P ∘ π_{c} : (z_{1}, w_{1}) ↦ (z_{1}, z_{1}w_{1} + w_{1}^{3}) is analogous to Whitney’s “cusp”, (x, y) ↦ (x, xy + y^{3}).

Example 8.3. An example of a surface M in ℂ^{3} with a topologically unstable CR singularity, considered by [C_{4}], has real equations z_{2} = z̄_{1}^{2}, z_{3} = z_{1}z̄_{1}, which complexify to P ∘π_{c} : (z_{1}, w_{1}) ↦ (z_{1}, w_{1}^{2}, z_{1}w_{1}), exactly Whitney’s normal form for the parametrization of the cross-cap singularity. The image of P ∘ π_{c} in ℂ^{3} is {z_{1}^{2}z_{2} − z_{3}^{2} = 0}, a singular complex hypersurface (Whitney’s “umbrella” surface), and the smallest complex variety containing M.

Example 8.4. For the normal form variety M̃^{4,5}, a parametrization ℂ^{4} → ℂ^{10} → ℂ^{5} of the complexification is

(z_{1}, w_{1}, z_{2}, z_{3}) ↦ (z_{1}, z_{2}, z_{3}, w_{1}^{2}, w_{1}(z_{1} + z_{2} + iz_{3})).

The real manifold M̃^{4,5} is the image of the restriction of this map to the totally real subspace {w_{1} = z̄_{1}, z_{2} = z̄_{2}, z_{3} = z̄_{3}} in the domain. The holomorphic map ℂ^{4} → ℂ^{5} parametrizes a singular complex hypersurface H, which is the product of Whitney’s cross-cap surface and a complex 2-plane, and the image {(z_{1} + z_{2} + iz_{3})^{2}z_{4} − z_{5}^{2} = 0} is the smallest complex variety in ℂ^{5} containing M̃^{4,5}; a similar expression appeared in (17). The geometry of M̃^{4,5} ⊆ H is considered in [C_{3}] §8, but with a different expression for the quadratic normal form.

Example 8.5. In general, the real variety M̃^{m,n} is contained in a singular subvariety of complex dimension m in ℂ^{n}, the defining ideal of which contains, for example, (z_{1} + z_{2} + iz_{3})^{2}z_{n−1} − z_{n}^{2}. As a consequence of Proposition 3.3, any real analytic M is not a local uniqueness set for holomorphic functions in a neighborhood of a nondegenerate CR singularity (cf [H_{2}]).

For surfaces in ℂ^{2}, the two-to-one nature of the complexification ℂ^{2} → ℂ^{2} as in Example 8.1 was used by [MW] to solve a normal form problem in the 0 < β < \frac{1}{2} case. Their methods are different from that of this paper; for example in the (m, n) = (4, 5) case, the map ℂ^{4} → ℂ^{5} from Example 8.4 is generally one-to-one, the two-to-one locus being contained in a complex subvariety in the domain as shown in [C_{3}].

Normal forms for the complexifications that look more like Whitney’s monomial normal forms would be possible using a larger group, where the z and w variables could be transformed independently. Under the subgroup used to normalize the CR singularity, one expects equivalence classes of maps to be smaller, and continuous parameters (“moduli”) to appear sooner (for more and for lower-order terms). However, invariants which distinguish maps under the larger group will still distinguish them under the smaller group. One may speculate that invariants of the complexification, such as the intrinsic derivative, the Boardman sequence, Jacobian extensions, etc., could provide a coarse but general beginning to the development of a CR singularity theory analogous to the singularity theory of maps ([GG], [P]).

Acknowledgments. The author thanks the referee(s) for comments on this, and also on an earlier version, which circulated as a preprint entitled Formal stability of the CR cross-cap, and which was based on [C_{1}]. The question of convergence answered by Proposition 3.3 was posed to the author by S. Webster.

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Received Received date / Revised version date

Department of Mathematical Sciences Indiana University - Purdue University Fort Wayne Fort Wayne, IN 46805-1499

E-mail address: CoffmanA@ipfw.edu
http://www.ipfw.edu/math/Coffman/