Explanation of a remark of Gauss on exceptional cases of conformal mappings

Question

In his letter to Hansen (december 11, 1825), Gauss responds to several questions asked by Hansen about his recent 1822 publication ("A general solution to the problem of mapping the parts of a given surface onto another surface such that the image and the mapped part are similar in the smallest parts"), and writes:

You are quite right, that the essential condition in every map-projection is the infinitesimal similarity, a condition which should be discarded only in very special cases of need.

Reading more into this letter, i saw that Gauss gives some details about this remark, and says that the exceptions mentioned by him arise when there are singular points on the surfaces (i should have included a translation of Gauss's letter, but google translate gives too bad translation). Unfortunately, and i know this might look like a basic question to some readers here, i don't understand why exceptions arise near singular points.

In my reading of articles about the ideas of Gauss and his followers, I had seen that this letter to Hansen is mentioned and cited many times, not in the context of my question, but as a kind of historical document that traces his own perspective of many of his discoveries (issues mentioned in this letter include the meaning of $\sqrt{-1}$ and the metaphysics of space). Since this letter belongs to a period of intensive occupation with differential geometry, and since it was around this time that Gauss tried to classify "tract figures" (singular curves that arise as projections of knots into the plane) in plane, i suspect there might be interesting connections of this remark to his other activities. Anyway, my last statement is entirely speculation, and and i'd rather prefer to concentrate at giving an explanation to Gauss's remark in the context of conformal mappings.

Any explanation\usefull comment will be blessed!

Answer 1

First of all, one would need to define the meaning of conformality of a map $f$ from a nonregular surface $S$ to the plane at points $p$ of where $S$ is non-regular. This is actually a nontrivial issue. One possibility is to use the "tangent cone" (if it exists, which is not always the case!) $C_p(S)$, and then require for the map $f$ to preserve angles between the directions. In the example of the cone that I gave you in my comment and , $C_p(S)$ is $S$ itself. Hence, there is no chance for a conformal map to the plane because the total angle of the cone at $p$ is $<2\pi$. There are other examples, where $C_p(S)$ is actually 1-dimensional (cuspidal singularities). One can give examples of non-regular surfaces with tangent cones isometric to the plane. Could Gauss handle these? No: His achievement was actually quite spectacular, he proved that for real-analytic regular surfaces, the metric is locally conformally flat. However, he could not even handle the case of infinitely-differentiable surfaces (or metrics).

Gauss' work set off a long chain of developments, documented for instance in this Wikipedia article, which culminated in the Measurable uniformization theorem. The modern viewpoint on this issue is to work within a suitable Sobolev space ($W^{2,1}$). But Gauss simply did not have the tools needed to overcome the difficulties that one has to deal with in the singular case.