Finite jet determination of local analytic CR automorphisms and their parametrization by 2-jets in the finite type case
Finite jet determination of local analytic CR automorphisms and their parametrization by 2-jets in the finite type case
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有限类型情况下局部解析 CR 自同构的有限射流确定及其 2 射流参数化
DOI：
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发表时间：
2001
期刊：
影响因子：
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通讯作者：
D. Zaitsev
中科院分区：
文献类型：
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作者：
P. Ebenfelt;B. Lamel;D. Zaitsev
本文证明了一类局部实解析CR自同构的芽
实解析超曲面M
$mathbb{C}每次2美元
p ∈ M
是由它们的喷流决定的，
p当且仅当M是
这似乎是第一个必要的和充分的
有限喷流确定的结果和这类的第一个结果，
无限型情形。若M在p处为有限型，
我们证明了一个更强的断言：局部实解析
M不动p的CR自同构
解析参数化（因此唯一确定）
在P的两个喷气机。
这个结果是最优的，因为单位球面的自同构是
而不是由它们在球的一点处的1-喷射决定的。有限
类型条件是必要的，因为否则所需的喷射顺序可以
是任意高的[Kow 1，2]，[Z2]。此外，我们通过一个例子表明，
对于有限型超曲面，2-jet的判定已经失败，
在$mathbb（C）$3中，我们也给出了局部系统芽动力学的应用
$mathbb（C）$2的双全纯。
AbstractWe show that germs of local real-analytic CR automorphisms of a
real-analytic hypersurface M in
$mathbb{C}$2 at a point
p ∈ M
are uniquelydetermined by their jets of some finite order at
p if and only if M is
not Levi-flat near p. This seems to be the first necessary and sufficient
result on finite jet determination and the first result of this kind in
the infinite type case.If M is of finite type at p,
we prove a stronger assertion: the local real-analytic
CR automorphisms of M fixing p
are analytically parametrized (and hence uniquely determined)
by their 2-jets at p.
This result is optimal since the automorphisms of the unit sphere are
not determined by their 1-jets at a point of the sphere. The finite
type condition is necessary since otherwise the needed jet order can
be arbitrarily high [Kow1,2], [Z2]. Moreover, we show, by an example,
that determination by 2-jets fails for finite type hypersurfaces already
in $mathbb{C)$3.We also give an application to the dynamics of germs of local
biholomorphisms of $mathbb{C)$2.