2
$\begingroup$

I apologise for the slightly technical question, and I am aware of the existence of the Mathoverflow question with the same title The existence of meromorphic functions on Riemann surfaces

Proofs of the existence of meromorphic functions on (compact) Riemann surfaces seem to all be based on some form of Dirichlet/energy minimisation principle.

  1. What I want to call the classical Dirichlet principle is the method which allows to solve the equation $\Delta f= 0$ on some open set $U \subset \Sigma$ having prescribed the value of $f$ on the boundary of $U$. In particular, this allows to build holomorphic functions with prescribed real parts on Riemann surfaces with boundary.

  2. It seems that most proofs of the existence of meromorphic functions on compact Riemann surfaces do not achieve a direct construction of meromorphic functions, but instead prove the existence of a lot holomorphic $1$-forms (via the same analytic methods than those involved in the proof of the "classical" Dirichlet principle); and then use for instance that the ratio of two holomorphic $1$-forms is a meromorphic function.

To me it seems that, once one knows of 1., 2. is a pretty convoluted way to go about proving the existence of meromorphic functions (having to use the more complicated concept of harmonic and holomorphic $1$-forms). My question is the following:

Can a proof of the existence of meromorphic functions be given directly from the "classical" Dirichlet principle for functions, and if not, is there a good reason why the use of harmonic/meromorphic forms should be central to a proof?

Improve this question
$\endgroup$

1 Answer 1

Reset to default
4
$\begingroup$

In particular, this allows to build holomorphic functions with prescribed real parts on Riemann surfaces with boundary.

The catch is that this is not quite correct. You can build a harmonic function with prescribed singularities, pass to a harmonic conjugate by integrating a closed form, but in a non-simply connected case, nothing guarantees that it yields a single-valued function. Making it single-valued amounts to subtracting an Abelian integral with the same periods, which boils down to an existence problem for holomorphic differentials that you are trying to avoid. Worse, this existence problem has no solution in general, as we are trying to fix $2g$ parameters in a $g$-dimensional space.

In other words, if you just arbitrarily pick pole locations and expansion at poles, a meromorphic function with these singularities and expansions may simply not exits. The question "when does it exist?" is much more intricate than the corresponding question for meromorphic differentials, where the only restriction is that the residues sum up to zero.

Improve this answer
$\endgroup$
0

You must log in to answer this question.

Start asking to get answers

Find the answer to your question by asking.

Ask question

Explore related questions

See similar questions with these tags.