Picard Theorem - Generalization and Current Research

Generalization and Current Research

Big Picard is true in a slightly more general form that also applies to meromorphic functions:

Big Picard Theorem (meromorphic version): If M is a Riemann surface, w a point on M, P1(C) = C ∪ {∞} denotes the Riemann sphere and f: M \ {w} → P1(C) is a holomorphic function with essential singularity at w, then on any open subset of M that contains w the function f(z) attains all but at most two points of P1(C) infinitely often.

Example: The meromorphic function f(z) = 1/(1 − e1/z) has an essential singularity at z = 0 and attains the value ∞ infinitely often in any neighborhood of 0; however it does not attain the values 0 or 1.

With this generalization, Little Picard Theorem follows from Big Picard Theorem because an entire function is either a polynomial or it has an essential singularity at infinity.

The following conjecture is related to "Big Picard Theorem":

Conjecture: Let U₁, U₂, ..., U_n be a finite cover of the punctured unit disk D\{0} in the complex plane by open connected sets U_j. Suppose that on each U_j there is an injective holomorphic function f_j, such that df_j = df_k on each intersection U_j ∩ U_k. Then the differentials glue together to a meromorphic 1-form on the unit disk D.

It is clear that the differentials glue together to a holomorphic 1-form g dz on D \ {0}. In the special case where the residue of g at 0 is zero, then the conjecture follows from the "Big Picard Theorem".