Talk:Riemann mapping theorem

message: The article states that the theorem holds for domains with fractal (non-differentiable) boundaries. An appeal is made to the Dirichlet principle, but its then hinted that the Dirichlet principle is suspect. Thus, the proof for the general case of a fractal boundary seems uncertain ... can the status be clarified? linas 03:28, 26 August 2006 (UTC)
replies: text: It seems like Riemann's proof used the Dirichlet principle, which as far as I can tell is valid in only a subset of the cases which the theorem is actually valid (Hilbert claimed I think that it was valid for the case Riemann was considering). The later proofs seem to have used a different method which doesn't have this frailty.--24.24.142.225 (talk) 18:55, 29 October 2010 (UTC)
thread_title: Clarification for fractal boundaries

message: The title of this section doesn't seem very encyclopedic to me; in mathematics it is pretty common that people say things like this, but it is not the business of an encyclopedia to say what is "impressive" and what isn't.  (And speaking just for myself, I don't like it when a written reference tries to sell me on the beauty of something by making explicit aesthetic claims about it.  The math can speak for itself.)  Everything in the section seems very relevant to the entry and the appreciation of the Riemann mapping theorem, but perhaps it should be under a more neutral title.
75.167.198.67 (talk) 07:10, 25 March 2009 (UTC)
thread_title: Why is this theorem impressive?

message: Maybe this isn't the best place to ask this, but the phrase "doubly connected domain" is used here without comment, it's not a phrase I'm familiar with, and I can't find it defined properly anywhere.  I found one paper by Osserman and Schiffer that defines a "double connected minimal surface" in R^2 to be the image of an annulus under a conformal mapping.  Is this the most general definition for "doubly connected"?
99.231.110.182 (talk) 04:12, 11 June 2009 (UTC)
replies: text: In the complex plane, a doubly connected set is one that is topologically equivalent to an annulus. That is my understanding of it anyway.--24.24.142.225 (talk) 19:02, 29 October 2010 (UTC)
thread_title:  Doubly connected domain 

message: In top of the article, it is stated in a note that "the existence of f is equivalent to the existence of a Green’s function." Could someone explain what this means? A Green's function corresponds to an inhomogeneous linear differential operator. Given U (the subset of the complex  plane), what would that operator be? —Kri (talk) 22:46, 7 March 2021 (UTC)
replies: text: It's a Green's function for the Laplace operator. Although that answer alone may not help that much. The connection between Riemann mappings and Green's functions really deserves a paragraph or a section in the article, rather than just a sentence in the lead section. (I wish I could write such a section, but I don't think I'd be able to do it justice at the moment.) Some of the references in the article, e.g. Walsh, discuss the subject. Adumbrativus (talk) 05:01, 8 March 2021 (UTC)
thread_title:  Is the existence of f equivalent to the existence of a Green’s function?