Schwarz lemma

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In mathematics, the Schwarz lemma, named after Hermann Amandus Schwarz, is a result in complex differential geometry that estimates the (squared) pointwise norm | ∂ f | 2 {\displaystyle |\partial f|^{2}} of a holomorphic map f : ( X , g X ) → ( Y , g Y ) {\displaystyle f:(X,g_{X})\to (Y,g_{Y})} between Hermitian manifolds under curvature assumptions on g X {\displaystyle g_{X}} and g Y {\displaystyle g_{Y}} . The classical Schwarz lemma is a result in complex analysis typically viewed to be about holomorphic functions from the open unit disk D := { z ∈ C : | z | < 1 } {\displaystyle \mathbb {D} :=\{z\in \mathbb {C} :|z|<1\}} to itself. The Schwarz lemma has opened several branches of complex geometry, and become an essential tool in the use of geometric PDE methods in complex geometry.