/sci/ - Science & Math

>>9596221
This is a problem on my complex analysis class. Given a Riemann Surface W_0 and a compact subset K_0 such that the Riemann Surface W = W_0 \ K_0 there's a point in W say p_0 such that a local barrier exists. Show that Green's function exists.

I tried looking at the solution over the complex plane, Ahlfors creates these two functions that bound the supremum of harmonic functions (Lemma's 1 and 2 around page 240-250 3rd edition) but the problem is the corresponding class of harmonic functions over Riemann surfaces has this log(z(p)) part where z is a coordinate map so it's very difficult to bound. I think you'd have to use the fact K_0 is compact but I'm not sure how. Any anons have some tips?

Pic related #3