The Schwarz Lemma -- Complex Analysis
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Part 1 -- The Maximum Principle: • MAXIMUM PRINCIPLE -- Part 1 -- Core T...
Part 3 -- Liouville's theorem: • Liouville's theorem -- Complex Analysis
In today's video, we want to take a look at the Schwarz lemma — this is a monumental result in the subject of one complex variable, and has lead to many developments in various areas of mathematics. The so-called Ahlfors—Schwarz lemma, and the Yau—Schwarz lemma are extremely powerful tools in the area of complex differential geometry, and are complex analogues of the Laplacian of the energy density in the study of harmonic maps. The Schwarz lemma is also a motivating result and the initial kindling for the study of Kobayashi hyperbolic manifolds, which are objects of great interest today.
The Schwarz lemma describes the behaviour of holomorphic maps from the unit disk to itself. It says that if f is a holomorphic map from the unit disk to itself, then it is distance decreasing. On the face of it, this does not sound too interesting, but it turns out to have powerful applications. In particular, the Schwarz lemma can be used to give a proof of the famous Liouville theorem, which states that a bounded entire function is constant.
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