 

WeiGuo Foo,
Joël Merker, and
TheAnh Ta
On convergent PoincaréMoser reduction for Levi degenerate embedded 5dimensional CR manifolds
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Published: 
January 28, 2022. 
Keywords: 
Normal forms, Power series method, Explicit differential invariants, Levi degenerate CR manifolds, Holomorphic mappings, Classification of hypersurfaces, Pocchiola's invariants. 
Subject: 
Primary: 32V40, 58K50, 32V35, 53A55, 5308.
Secondary: 58A15, 32A05, 53A07, 53B25, 22E05, 22E60, 58A30. 


Abstract
Applying Lie's elementary theory for appropriate prolongations to jet spaces of orders 1 and 2, we show that
any real analytic hypersurface M^{5} in C^{3} which is 2nondegenerate of constant Levi rank 1
carries two sorts of CartanMoser chains, that are of orders 1 and 2.
Integrating and straightening any given order 2 chain passing through any point p in M to be the vaxis in coordinates
(z, s, w = u + i v) centered at p, without setting up the formal theory in advance, we show that there exists a convergent
change of complex coordinates (z, s, w) > (z', s', w') fixing the origin in which the chain is the vaxis and in which
M has a certain explicit PoincaréMoser reduced equation.
The values at the origin of Pocchiola's two primary invariants are proportional to two specific Taylor coefficients of this normal form.


Acknowledgements
Zhangchi Chen provided the Maple figures of Sections 8 and 9.


Author information
WeiGuo Foo:
Nanyang Technological University
50 Nanyang Ave, Singapore 639798
weiguo.foo@ntu.edu.sg
Joël Merker:
Département de Mathématiques d'Orsay
CNRS, Université ParisSaclay
91405 Orsay Cedex, France
joel.merker@universiteparissaclay.fr
TheAnh Ta:
Département de Mathématiques d'Orsay
CNRS, Université ParisSaclay
91405 Orsay Cedex, France
tatheanhdtvt@gmail.com

