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Bloch's Principle is a philosophical principle in
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
stated by André Bloch. Bloch states the principle in Latin as: ''Nihil est in infinito quod non prius fuerit in finito,'' and explains this as follows: Every proposition in whose statement the actual infinity occurs can be always considered a consequence, almost immediate, of a proposition where it does not occur, a proposition in ''finite terms''. Bloch mainly applied this principle to the theory of functions of a complex variable. Thus, for example, according to this principle, Picard's theorem corresponds to Schottky's theorem, and Valiron's theorem corresponds to Bloch's theorem. Based on his Principle, Bloch was able to predict or conjecture several important results such as the Ahlfors's Five Islands theorem, Cartan's theorem on holomorphic curves omitting hyperplanes,
Hayman Hayman is both a surname and a given name. Notable people with the name include: Surname *Al Hayman (1847–1917), business partner of Charles Frohman in ''Theatrical Syndicate'' *Andy Hayman, CBE, QPM (born 1959), retired British police officer, ...
's result that an exceptional set of radii is unavoidable in Nevanlinna theory. In the more recent times several general theorems were proved which can be regarded as rigorous statements in the spirit of the Bloch Principle:


Zalcman's lemma

A family \mathcal F of functions meromorphic on the unit disc \Delta is not normal if and only if there exist: * a number 0 < r < 1 * points z_n, , z_n, * functions f_n \in \mathcal F * numbers \rho_n \to 0 + such that f_n(z_n+\rho_n \zeta)\to g(\zeta), spherically uniformly on compact subsets of C, where g is a nonconstant meromorphic function on C. Zalcman's lemma may be generalized to several complex variables. First, define the following: A family \mathcal F of holomorphic functions on a domain \Omega\subset C^n is normal in \Omega if every sequence of functions \\subseteq \mathcal F contains either a subsequence which converges to a limit function f \ne \infty uniformly on each compact subset of \Omega, or a subsequence which converges uniformly to \infty on each compact subset. For every function \varphi of class C^2(\Omega) define at each point z\in \Omega a Hermitian form L_z(\varphi, v):=\sum_^n \frac(z) v_k \overline_l \ \ (v\in C^n), and call it the Levi form of the function \varphi at z. If function f is holomorphic on \Omega, set f^\sharp (z):=\sup_\sqrt. This quantity is well defined since the Levi form L_z(\log(1+, f, ^2), v) is nonnegative for all z\in \Omega. In particular, for n = 1 the above formula takes the form f^\sharp (z):=\frac and z^\sharp coincides with the spherical metric on 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: Suppose that the family \mathcal F of functions holomorphic on \Omega\subset C^n is not normal at some point z_0\in \Omega. Then there exist sequences f_j\in \mathcal F, z_j\to z_0, \rho_j=1/f_j^\sharp(z_j)\to 0, such that the sequence g_j(z)=f_j(z_j+\rho_j z) converges locally uniformly in C^n to a non-constant entire function g satisfying g^\sharp(z)\leq g^\sharp(0)=1


Brody's lemma

Let ''X'' be a compact complex analytic manifold, such that every holomorphic map from the
complex plane In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by the ...
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.Lang (1987).


References

Mathematical principles Philosophy of mathematics