Compact Riemann surfaces with boundary

Question

I am familiar (to an extent) with the theory of compact Riemann surfaces without boundary, but I am interested in the theory of compact Riemann surfaces with boundary.

To what extent do the following ideas/results generalize?

• Riemann-Roch theorem
• Abel's theorem
• The Riemann-Hurwitz formula
• The classification of holomorphic line bundles

The literature seems to be a little sparse on this particular subject, so any suggestions would be appreciated.

Answer 1

A Riemann surface with boundary is not an algebraic-geometric object, but a real two dimensional manifold, so most of the things you mention do not make sense in that setting. Riemann surfaces with cusps are probably the closest you can come (and there the theory works fine - read Forster's book).

Answer 2

I think, your question is too broad. Part 3 has an easy answer since the R-H formula for $n$-fold ramified coverings of compact surfaces with boundary, $f: S'\to S$, will also read $$ \chi(S')= n\chi(S)- \sum_{p\in S'}(e_p-1), $$ where $e_p$ is the ramification index of the ramification point $p\in S'$. The proof is the same as the one sketched here.

As for other questions, if you do not impose any boundary conditions and your surface is connected with nonempty boundary, you will get very boring answers, e.g. the dimension in the R-R theorem is infinite, all (discrete) divisors are principal, etc. It is possible you get some interesting results by imposing some boundary conditions. But, in the case of, say, R-R theorem you have to ensure that the result is a finite-dimensional vector space. It is very much unclear (to me) how this can be accomplished. But the complex analysis literature is vast and it is not impossible that some clever people came up with some creative answers. In my mind, one should start with a specific analytical problem involving holomorphic functions on surfaces with boundary, before trying to figure out what are the known tools.

Here is something to ponder: Even in the case when your surface is the closed unit disk $D$, there is no "easy" topological condition for an immersion $f: \partial D\to {\mathbb C}$ to extend to an immersion $F: D\to {\mathbb C}$. Even if you relax the restriction on $F$ and assume that it is a composition of a self-diffeomorphism $D\to D$ with a holomorphic map $D\to {\mathbb C}$, the condition for the existence of an extension is quite complicated. If you fix $f$ and require a holomorphic extension $F$, then there is always either no extension or exactly one extension.