Skip to main content
MathOverflow

Return to Answer

added 16 characters in body
Source Link
Alexandre Eremenko
Alexandre Eremenko
  • 91.8k
  • 9
  • 259
  • 429

For open surfaces, there are counterexamples. The first one was constructed by P. Myrberg:

Ueber die analytische Fortsetzung von beschrankten Funktionen, Ann. Acad. Sci. Fenn., Ser. A. I N:o 58 (1949)

Since this paper is difficult to obtain (and written in German), I refer to another paper

Heins, Maurice, Riemann surfaces of infinite genus. Ann. of Math. (2) 55 (1952), 296–317,

Which proves an even stronger result: there is an open Riemann surface, such that if you remove a disk from it, then on the remaining surface every non-constant meromorphic function takes all complex values, except at most two of them.

For open surfaces, there are counterexamples. The first one was constructed by P. Myrberg:

Ueber die analytische Fortsetzung von beschrankten Funktionen, Ann. Acad. Sci. Fenn., Ser. A. I N:o 58 (1949)

Since this paper is difficult to obtain (and written in German), I refer to another paper

Heins, Maurice, Riemann surfaces of infinite genus. Ann. of Math. (2) 55 (1952), 296–317,

Which proves an even stronger result: there is an open Riemann surface, such that if you remove a disk from it, then on the remaining surface every non-constant meromorphic function takes all complex values, except two.

For open surfaces, there are counterexamples. The first one was constructed by P. Myrberg:

Ueber die analytische Fortsetzung von beschrankten Funktionen, Ann. Acad. Sci. Fenn., Ser. A. I N:o 58 (1949)

Since this paper is difficult to obtain (and written in German), I refer to another paper

Heins, Maurice, Riemann surfaces of infinite genus. Ann. of Math. (2) 55 (1952), 296–317,

Which proves an even stronger result: there is an open Riemann surface, such that if you remove a disk from it, then on the remaining surface every non-constant meromorphic function takes all complex values, except at most two of them.

added 2 characters in body
Source Link
Alexandre Eremenko
Alexandre Eremenko
  • 91.8k
  • 9
  • 259
  • 429

For open surfaces, there are counterexamples. The first one was constructed by P. Myrberg:

Ueber die analytische Fortsetzung von beschrdnktenbeschrankten Funktionen, Ann. Acad. Sci. Fenn., Ser. A. I N:o 58 (1949)

Since this paper is difficult to obtain (and written in German), I refer to another paper

Heins, Maurice, Riemann surfaces of infinite genus. Ann. of Math. (2) 55 (1952), 296–317,

Which proves an even stronger result: there areis an open Riemann surface, such that if you remove a disk from it, then on the remaining surface every non-constant meromorphic function takes all complex values, except two.

For open surfaces, there are counterexamples. The first one was constructed by P. Myrberg:

Ueber die analytische Fortsetzung von beschrdnkten Funktionen, Ann. Acad. Sci. Fenn., Ser. A. I N:o 58 (1949)

Since this paper is difficult to obtain (and written in German), I refer to another paper

Heins, Maurice Riemann surfaces of infinite genus. Ann. of Math. (2) 55 (1952), 296–317,

Which proves an even stronger result: there are open Riemann surface, such that if you remove a disk from it, then on the remaining surface every non-constant meromorphic function takes all complex values, except two.

For open surfaces, there are counterexamples. The first one was constructed by P. Myrberg:

Ueber die analytische Fortsetzung von beschrankten Funktionen, Ann. Acad. Sci. Fenn., Ser. A. I N:o 58 (1949)

Since this paper is difficult to obtain (and written in German), I refer to another paper

Heins, Maurice, Riemann surfaces of infinite genus. Ann. of Math. (2) 55 (1952), 296–317,

Which proves an even stronger result: there is an open Riemann surface, such that if you remove a disk from it, then on the remaining surface every non-constant meromorphic function takes all complex values, except two.

Source Link
Alexandre Eremenko
Alexandre Eremenko
  • 91.8k
  • 9
  • 259
  • 429

For open surfaces, there are counterexamples. The first one was constructed by P. Myrberg:

Ueber die analytische Fortsetzung von beschrdnkten Funktionen, Ann. Acad. Sci. Fenn., Ser. A. I N:o 58 (1949)

Since this paper is difficult to obtain (and written in German), I refer to another paper

Heins, Maurice Riemann surfaces of infinite genus. Ann. of Math. (2) 55 (1952), 296–317,

Which proves an even stronger result: there are open Riemann surface, such that if you remove a disk from it, then on the remaining surface every non-constant meromorphic function takes all complex values, except two.