Importance: Medium ✭✭
 Author(s): Elsner, B.
 Subject: Analysis
 Keywords: Essential singularity Holomorphic functions Picard's theorem Residue of 1-form Riemann surfaces
 Posted by: MathOMan on: February 26th, 2010
Conjecture   Let be the open unit disk in the complex plane and let be open sets such that . Suppose there are injective holomorphic functions such that for the differentials we have on any intersection . Then those differentials glue together to a meromorphic 1-form on .

It is an evidence that the 1-form is holomorphic on . In the case that its residue at the origin vanishes we can use Picard's big theorem.

## Bibliography

*B. Elsner: Hyperelliptic action integral, Annales de l'institut Fourier 49(1), p.303–331

* indicates original appearance(s) of problem.