Poincaré’s Theorem: The Ball vs. The Polydisc in Several Complex Variables
Why can’t we generalize conformal mappings to higher dimensions? Explore Poincaré’s theorem, which launched the study of several complex variables as a field of its own.
The main result
Dear Reader, in today's brief, I would like to discuss a less widely known Theorem by Henri Poincaré. Informally speaking, it tells us that the generalisation of conformal mappings to higher (complex) dimensions is not straightforward at all. This theorem initiated the study of functions of multiple complex variables as an independent discipline in the beginning of the XXst century. Poincaré proved it in the case of two complex variables, but I will give it in a more general case. The idea of the proof consists of computing the group of automorphisms of each domain. Since they turn out to be quite different, the conclusion is obvious. Please enjoy this beautiful result.
Dr. Denys Dutykh
Associate Professor of Mathematics
Khalifa University of Science and Technology, Abu Dhabi, UAE
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