Abstract
A central problem in Mathematics is the classification problem. Given a set of objects and an equivalence relation, loosely speaking, the problem asks how to find an accessible way to tell whether two objects are in the same equivalence class. A general approach to this problem is to find a complete set of (geometric, analytic or algebraic) invariants. In the subject of Several Complex Variables and Complex Geometry, a fundamental problem is to classify complex manifolds or more generally, normal complex spaces under the action of biholomorphic transformations. When the normal complex spaces are open and have strongly pseudo-convex boundary, by the Fefferman-Bochner theorem, one needs only to classify the corresponding boundary strongly pseudoconvex CR manifolds under the application of CR diffeomorphisms. The celebrated Chern-Moser theory is a theory which gives two different constructions of a complete set of invariants for such a classification problem. Among various aspects of the Chern-Moser theory (especially the geometric aspect of the theory), the Chern-Moser-Weyl tensor plays a key role. However, this trace-free tensor is defined in a very complicated manner. This makes it hard to apply in the applications. The majority of first several sections in this article surveys some work done in papers of Chern-Moser [3], Huang-Zhang [14], Huang-Zaitsev [13]. Here, we give a simple and more accessible account on the Chern-Moser-Weyl tensor. We also make an immediate application of the monotonicity property for this tensor to the study of CR embedding problem for the positive signature case.
Dedicated to our friend Dick Wheeden
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Acknowledgements
Xiaojun Huang was Supported in part by NSF-1363418.
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Huang, X., Xiao, M. (2017). Chern-Moser-Weyl Tensor and Embeddings into Hyperquadrics. In: Chanillo, S., Franchi, B., Lu, G., Perez, C., Sawyer, E. (eds) Harmonic Analysis, Partial Differential Equations and Applications. Applied and Numerical Harmonic Analysis. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-52742-0_6
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