Schwarz lemma

In mathematics, the Schwarz lemma, named after Hermann Amandus Schwarz, is a result in complex differential geometry that estimates the (squared) pointwise norm ${\displaystyle |\partial f|^{2}}$ of a holomorphic map ${\displaystyle f:(X,g_{X})\to (Y,g_{Y})}$ between Hermitian manifolds under curvature assumptions on ${\displaystyle g_{X}}$ and ${\displaystyle g_{Y}}$.

The classical Schwarz lemma is a result in complex analysis typically viewed to be about holomorphic functions from the open unit disk ${\displaystyle \mathbb {D}$ :=\{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.