The intention behind my question is to understand what are the kind of general problems of which the ideas of Riemann's dissertation (1851) lie at the heart of it's solution methods. In his dissertation, Riemann introduced his mapping theorem (and gave a flawed proof of it based on Dirichlet's principle) and "Riemann surfaces". According to the historical introduction to Carathéodory's "Conformal representation":

In 1822 Gauss (1777-1855) stated and completely solved the general problem of finding all conformal transformations which transform a sufficiently small neighbourhood of a point on an arbitrary analytic surface into a plane area. This work of Gauss appeared to give the whole inquiry its final solution; actually it left unanswered the much more difficult question whether and in what way a given finite portion of the surface can be represented on a portion of the plane. This was first pointed out by Riemann (1826-1866), whose Dissertation (1851) marks a turning-point in the history of the problem which has been decisive for its whole later development; Riemann not only introduced all the ideas which have been at the basis of all subsequent investigation of the problem of conformal representation, but also showed that the problem itself is of fundamental importance for the theory of functions.

What i'm trying to understand is the meaning of "conformally representing a finite (not infinitisimal) portion of a surface on a portion of the plane". As far as i understand, a conformal mapping can be visualized as acting (everywhere on the surface) like a composition of rotation (so angles are preserved) and rescaling (magnification of linear element). It should be mentioned that in general, the rotation angle might vary from place to place, as well as the scaling factor. So, conformal mappings already act on finite portions of a surfaces. So what is the meaning of all the references i find to much more difficult problems?

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    $\begingroup$ Gauss's result only applies when one can choose "sufficiently small" neighborhood of the point. If one is given some portion of the surface, existence of a conformal mapping is not settled by that. "Infinitesimal" = local, "finite" = global here. $\endgroup$ – Conifold Nov 15 '18 at 23:18
  • $\begingroup$ This is what I don't understand - take, for example, the mercator projection- it's a conformal mapping that applies to the whole sphere, and it can be derived by Gauss's result (by finding isothermal coordinates on the sphere). $\endgroup$ – user2554 Nov 16 '18 at 10:53
  • $\begingroup$ @user2554: Mercator (and stereographic) projections are only two examples of a global conformal mapping. Riemann proved the first general theorem, that an ARBITRARY simply-connected plane domain (different from the plane itself) can be mapped conformally onto the unit disk. $\endgroup$ – Alexandre Eremenko Nov 17 '18 at 14:27
  • $\begingroup$ There is a difference between considering global examples and developing a global theory. Gauss did not have the latter, isothermal coordinates are local coordinates, that they happen to extend globally on the sphere is just luck. And this change of viewpoint, and development of new methods it required, not only in conformal, but also in differential and algebraic geometry, and the theory of functions, is one of those things that make Riemann a transformational figure compared to his predecessors. $\endgroup$ – Conifold Nov 18 '18 at 1:36
  • $\begingroup$ @Conifold, (a) can you give a concrete example of a surface for which it's easy to introduce local conformal coordinates (isothermal coordinates) but very difficult to extend them globally (on the whole surface)? (b) can you comment on my question about Gauss's work on the problem of conformal image of the ellipse? It's one of those remarkable contributions of Gauss that surprisingly I didn't find comments about in the literature. The construction of the mapping from the ellipse to the circle was only made much later by Schwarz, a mathematician responsible for many detailed confirmations of .. $\endgroup$ – user2554 Nov 18 '18 at 11:33

A homeomorphism between two surfaces is called conformal if it preserves angles. Examples of conformal maps were known since antiquity: stereographic projection maps conformally the sphere without one point onto the plane (this map was known to the ancient Greeks, but conformality was stated and proved only in 17th century). Another example is Mercator's projection which maps the plane onto the sphere minus 2 points (It is not one-to-one, but rather a covering, using the modern terminology. To make it one-to-one, one has to restrict it to a strip in the plane and the image will be the sphere minus a meridian.)

Gauss as the first who stated a LOCAL existence theorem: given any surface and a point on it, there exists a region containing this point which can be mapped conformally onto a region in the plane. (From the modern point of view, Gauss neither proved this, nor even stated accurately).

Riemann discovered the first general GLOBAL result: for every simply connected region in the plane (other than the plane itself), there exists a conformal map of this region onto the unit disk in the plane. Riemann's proof of this contained gaps which were immediately found. As the theorem is really of fundamental importance, this triggered an intensive research, and he first complete proof for arbitrary regions was obtained by William Osgood in 1900.

Since that time, the theory of conformal mapping is a central part of the complex function theory. The main question of this theory is how the geometric properties of the region reflect in the properties of the mapping function. This is an important subject of research still at the present time.

An important generalization of Riemann's theorem is the Uniformization theorem proved independently by Poincare and Kobe in 1907. A simple example of uniformization is the Merfcator map from the plane to the sphere minus 2 points. The theorem of Poincare and Kobe says that a uniformization exists for every surface.


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