Unstability problem of real analytic maps

IF 0.8 3区数学 Q2 MATHEMATICS Bulletin of the London Mathematical Society Pub Date : 2024-07-22 DOI:10.1112/blms.13124

Karim Bekka, Satoshi Koike, Toru Ohmoto, Masahiro Shiota, Masato Tanabe

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Abstract

As well known, the $C^{\infty}$ stability of proper $C^{\infty}$ maps is characterized by the infinitesimal $C^{\infty}$ stability. In the present paper, we study the counterpart in real analytic context. In particular, we show that the infinitesimal $C^{ω}$ stability does not imply $C^{ω}$ stability; for instance, a Whitney umbrella $R^{2} \to R^{3}$ is not $C^{ω}$ stable. A main tool for the proof is a relative version of Whitney's analytic approximation theorem that is shown by using H. Cartan's Theorems A and B.

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实解析映射的不稳定性问题

众所周知，适当 C ∞ $C^infty$ 映射的 C ∞ $C^infty$ 稳定性是以无穷小 C ∞ $C^infty$ 稳定性为特征的。在本文中，我们研究了实解析背景下的对应关系。特别是，我们证明了无穷小 C ω $C^\omega$ 稳定性并不意味着 C ω $C^\omega$ 稳定性；例如，惠特尼伞 R 2 → R 3 $\mathbb {R}^2 \rightarrow \mathbb {R}^3$ 不是 C ω $C^\omega$ 稳定性。证明的一个主要工具是惠特尼解析近似定理的一个相对版本，它是通过 H. Cartan 的定理 A 和 B 来证明的。

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