Bloch's principle

Summary

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 lemmaEdit

A family   of functions meromorphic [analytic] on the unit disc   is not normal if and  only if there exist
(a) a number  
(b) points    
(c) functions  
(d) numbers  
such that
 
spherically uniformly [uniformly] on compact subsets of   where   is a nonconstant meromorphic  [entire] function on  [3]

Zalcman's lemma has the following generalization to several complex variables. The first thing to do is to make a precise definitions.

A family   of holomorphic functions on a domain   is normal in   if every sequence of functions   contains either a subsequence which converges to a limit function   uniformly on each compact subset of   or a subsequence which converges uniformly to   on each compact subset.

For every function   of class   we define at each point   a Hermitian form   and call it the Levi form of the function   at  

If function   is holomorphic on   set   This quantity is well defined since the Levi form   is nonnegative for all   In particular, for   the above formula takes the form   and   coincides with the spherical metric on  

Now, we have is the following important characterization of normality, based on Marty's theorem.[4]

Suppose that the  family   of functions holomorphic on   is  not normal at some point    Then there exist sequences        such that the sequence   converges locally uniformly in   to a non-constant entire function   satisfying  .[5]

See also [6] and.[7]

Brody's lemmaEdit

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.[8]

ReferencesEdit

  1. ^ Bloch, A. (1926). "La conception actuelle de la theorie de fonctions entieres et meromorphes". Enseignement Math. Vol. 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 (8): 813–817. doi:10.1080/00029890.1975.11993942.
  4. ^ P. V. Dovbush (2020). Zalcman's lemma in Cn, Complex Variables and Elliptic Equations, 65:5, 796-800, DOI: 10.1080/17476933.2019.1627529. doi:10.1080/17476933.2019.1627529. S2CID 198444355.
  5. ^ P. V. Dovbush (2020). Zalcman's lemma in Cn, Complex Variables and Elliptic Equations, 65:5, 796-800, DOI: 10.1080/17476933.2019.1627529. doi:10.1080/17476933.2019.1627529. S2CID 198444355.
  6. ^ P. V. Dovbush (2020). Zalcman–Pang's lemma in CN , Complex Variables and Elliptic Equations, DOI: 10.1080/17476933.2020.1797704. doi:10.1080/17476933.2020.1797704. S2CID 225403763.
  7. ^ P. V. Dovbush (2020). On normal families in Cn , Complex Variables and Elliptic Equations, DOI: 10.1080/17476933.2020.1797703. doi:10.1080/17476933.2020.1797703. S2CID 225426784.
  8. ^ Lang (1987).