Bloch's principle

A family ${\displaystyle {\mathcal {F}}}$ of functions meromorphic on the unit disc ${\displaystyle \Delta }$ is not normal if and only if there exist:

• a number ${\displaystyle 0<r<1}$
• points ${\displaystyle z_{n},}$ ${\displaystyle |z_{n}|<r}$
• functions ${\displaystyle f_{n}\in {\mathcal {F}}}$
• numbers ${\displaystyle \rho _{n}\to 0+}$

such that ${\displaystyle f_{n}(z_{n}+\rho _{n}\zeta )\to g(\zeta ),}$ spherically uniformly on compact subsets of ${\displaystyle C,}$ where ${\displaystyle g}$ is a nonconstant meromorphic function on ${\displaystyle C.}$[3]

Zalcman's lemma may be generalized to several complex variables. First, define the following:

A family ${\displaystyle {\mathcal {F}}}$ of holomorphic functions on a domain ${\displaystyle \Omega \subset C^{n}}$ is normal in ${\displaystyle \Omega }$ if every sequence of functions ${\displaystyle \{f_{j}\}\subseteq {\mathcal {F}}}$ contains either a subsequence which converges to a limit function ${\displaystyle f\neq \infty }$ uniformly on each compact subset of ${\displaystyle \Omega ,}$ or a subsequence which converges uniformly to ${\displaystyle \infty }$ on each compact subset.

For every function ${\displaystyle \varphi }$ of class ${\displaystyle C^{2}(\Omega )}$ define at each point ${\displaystyle z\in \Omega }$ a Hermitian form ${\displaystyle L_{z}(\varphi ,v):=\sum _{k,l=1}^{n}{\frac {\partial ^{2}\varphi }{\partial z_{k}\partial {\overline {z}}_{l}}}(z)v_{k}{\overline {v}}_{l}\ \ (v\in C^{n}),}$ and call it the Levi form of the function ${\displaystyle \varphi }$ at ${\displaystyle z.}$

If function ${\displaystyle f}$ is holomorphic on ${\displaystyle \Omega ,}$ set ${\displaystyle f^{\sharp }(z):=\sup _{|v|=1}{\sqrt {L_{z}(\log(1+|f|^{2}),v)}}.}$ This quantity is well defined since the Levi form ${\displaystyle L_{z}(\log(1+|f|^{2}),v)}$ is nonnegative for all ${\displaystyle z\in \Omega .}$ In particular, for ${\displaystyle n=1}$ the above formula takes the form ${\displaystyle f^{\sharp }(z):={\frac {|f'(z)|}{1+|f(z)|^{2}}}}$ and ${\displaystyle z^{\sharp }}$ coincides with the spherical metric on ${\displaystyle C.}$

The following characterization of normality can be made based on Marty's theorem, which states that a family is normal if and only if the spherical derivatives are locally bounded:[4]

Suppose that the family ${\displaystyle {\mathcal {F}}}$ of functions holomorphic on ${\displaystyle \Omega \subset C^{n}}$ is not normal at some point ${\displaystyle z_{0}\in \Omega .}$ Then there exist sequences ${\displaystyle f_{j}\in {\mathcal {F}},}$ ${\displaystyle z_{j}\to z_{0},}$ ${\displaystyle \rho _{j}=1/f_{j}^{\sharp }(z_{j})\to 0,}$ such that the sequence ${\displaystyle g_{j}(z)=f_{j}(z_{j}+\rho _{j}z)}$ converges locally uniformly in ${\displaystyle C^{n}}$ to a non-constant entire function ${\displaystyle g}$ satisfying ${\displaystyle g^{\sharp }(z)\leq g^{\sharp }(0)=1}$

Let X be a compact complex analytic manifold, such that every holomorphic map from the complex plane to X is constant. Then there exists a metric on X such that every holomorphic map from the unit disc with the Poincaré metric to X does not increase distances.[5]