Approximation and convergence of formal CR-mappings
Approximation and convergence of formal CR-mappings
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DOI：
10.1155/s1073792803205146
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发表时间：
2002-05
影响因子：
1
通讯作者：
F. Meylan;N. Mir;D. Zaitsev
中科院分区：
文献类型：
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作者：
F. Meylan;N. Mir;D. Zaitsev
理解具有某些性质的解析对象的存在性的一个重要步骤是在形式幂级数的层次上理解同一个问题。后一个问题可以简化为一个未知幂级数系数的代数方程序列，并且通常比原问题更简单，因为原问题要求幂级数收敛。因此，我们有兴趣知道这样的幂级数是否自动收敛，或者是否可以被满足相同性质的其他收敛幂级数所取代。这类著名的结果是Artin的近似定理[1]，该定理指出，解析方程系统的形式解可以被同一系统的收敛解所取代，该解近似于任意规定阶的原始解。在本文中，我们研究了形式（全纯）映射的收敛性和逼近性（以[1]的精神），它们将实解析子流形M∧C和M′∧C′彼此相接，N,N′≥2。在这种情况下，不能直接应用上述Artin定理。此外，如果没有对子流形的附加假设，类比近似陈述甚至不成立。的确，鉴于Moser-Webster[23]的一个例子，存在实代数曲面M，M′C，它们在形式上是等价的，但不是生物全纯等价的。然而，我们的第一个主要结果表明，如果M是C中的最小cr -子流形（不一定是代数的），则不会发生这种现象（见2.1节的符号和定义）。定理1.1。设M´C是一个实解析极小cr子流形，M´C´是一个实代数子集，其中p∈M， p ‘∈M ’。那么对于任何形式（全纯）
An important step in understanding the existence of analytic objectswith certain properties consists of understanding the same problem at the level of formal power series. The latter problem can be reduced to a sequence of algebraic equations for the coefficients of the unknown power series and is often simpler than the original problem, where the power series are required to be convergent. It is therefore of interest to know whether such power series are automatically convergent or can possibly be replaced by other convergent power series satisfying the same properties. A celebrated result of this kind is Artin’s approximation theorem [1] which states that a formal solution of a system of analytic equations can be replaced by a convergent solution of the same system that approximates the original solution at any prescribed order. In this paper, we study convergence and approximation properties (in the spirit of [1]) of formal (holomorphic) mappings sending real-analytic submanifolds M ⊂ C and M ′ ⊂ C ′ into each other, N,N ′ ≥ 2. In this situation, the above theorem of Artin cannot be applied directly. Moreover, without additional assumptions on the submanifolds, the analogous approximation statement is not even true. Indeed, in view of an example of Moser-Webster [23], there exist real-algebraic surfaces M,M ′ ⊂ C that are formally but not biholomorphically equivalent. However, our firstmain result shows that this phenomenon cannot happen if M is a minimal CR-submanifold (not necessarily algebraic) in C (see Section 2.1 for notation and definitions). Theorem 1.1. Let M ⊂ C be a real-analytic minimal CR-submanifold and M ′ ⊂ C ′ a real-algebraic subset with p ∈ M and p ′ ∈ M ′. Then for any formal (holomorphic)