Bloch's principle

Bloch's Principle is a philosophical principle in mathematics stated by André Bloch.^[1]

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,^[2] Hayman'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

Let $\{f_{n}\}$ be a sequence of meromorphic functions in a region D, which is not a normal family. Then there exist a sequence of points ${\displaystyle z_{n}}$ in D and positive numbers ${\displaystyle \rho _{n}}$ with ${\displaystyle \lim _{n\rightarrow \infty }\rho _{n}=0}$ such that

${\displaystyle f_{n}(z_{n}+\rho _{n}z)\to f,\,}$

where f is a non-constant meromorphic function in the complex plane.^[3]

Brody's lemma

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.^[4]

References

1. Bloch, A. (1926). "La conception actuelle de la theorie de fonctions entieres et meromorphes". Enseignement math. 25. pp. 83–103.
2. Lang, S. (1987). Introduction to complex hyperbolic spaces. Springer Verlag.
3. Zalcman, L. (1975). "Heuristic principle in complex function theory". Amer. Math. Monthly. 82: 813–817.
4. Lang (1987).