The Hadamard Gap Theorem and Stein Manifolds
How do big gaps in power series relate to Stein manifolds? A look at the Hadamard Gap Theorem and why the unit disk is a Stein manifold.
The main result
Dear Reader, today's brief is a follow-up post on the topic of Stein manifolds that I mentioned some time ago. Basically, the Theorem we highlight today is probably the most straightforward way to see that the unit open disk is a Stein manifold. However, the real question is why this result holds. Informally speaking, we can say that big gaps imply that near the boundary, the series behaves almost like a single monomial term. The problem is that monomials of unbounded degree tend to oscillate too violently near the boundary, which prevents any possibility of analytic continuation.
Dr. Denys Dutykh
Associate Professor of Mathematics
Khalifa University of Science and Technology, Abu Dhabi, UAE
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