Riemann sphere as plane with delta-function curvature

Question

Standard metric on the round Riemann sphere is $ds^2=\frac{dzd\bar{z}}{(1+|z|^2)^2}$. It has constant scalar curvature. If the sphere is not round one gets another metric. A somewhat degenerate case that I have encountered in the literature is when the metric is everywhere flat $ds^2=dzd\bar{z}$ except singularity at infinity. Near infinity $dzd\bar{z}\to\frac{dzd\bar{z}}{|z|^4}$ and the Riemann curvature is proportional to delta-function $\sqrt{g}R=8\pi \delta^2(z)$. This agrees with the fact that integral of the curvature gives the Euler characteristic.

I would like to understand this construction better.

1. Is it true that this sphere is necessarily of infinite area? Integral $\int \sqrt{g}d^2z$ does not seem bounded.
2. Can one visualize this sphere as embedded into $\mathbb{R}^3$? I fail to see how can one bend a disc so that it becomes $S^2$ without curving it (except at one point, or a small neighborhood).
3. Is this metric singularity called conic? If so, what is a good quick introduction/reference on these?

Answer 1

I have to say, I still find the question unclear. However:

1. The metric that you are working with is the standard Euclidean metric on ${\mathbb C}$, hence, it clearly has infinite area. This provides a trivial answer to 1.

2. Regarding 2, I think the answer depends on the degree of smoothness. To the extent I understand the question, you are looking for a topological embedding $$ f: S^2= {\mathbb C}\cup \{\infty\}\to E^3 $$ which is a $C^k$-smooth isometric embedding when restricted to ${\mathbb C}$ (where ${\mathbb C}$ is equipped with the standard flat metric). The answer probably depends on $k$.

a. Assuming that $k=2$, the answer is negative. The reason is that ${\mathbb C}$ with its standard metric is complete, but every complete $C^2$-smooth Riemannian surface of zero curvature embedded in $E^3$ has to be a (not necessarily round) cylinder. See

W.Massey, Surfaces of Gaussian curvature zero in Euclidean 3-space. Tohoku Math. J. (2) 14 (1962), no. 1, 73--79.

for a simple proof in the case of $k=4$ and a reference to Hartman and Nirenberg for the case $k=2$.

b. I am not sure what happens when $k=1$. I suspect that one can use the technique of the Nash-Kuiper isometric embedding theorem in order to prove the existence of $f$ as above. But that would be a serious research paper.

3. I am not sure such metrics have a name, I do not think people in the area refer to them as metrics with conical singularities; that is reserved for metric completions of the flat metrics obtained from ${\mathbb C}^*$ by pullback via a multivalived map $$ z\mapsto z^{\alpha}, \alpha>0. $$ (I once heard somebody referring to a flat metric on a cone of revolution in a neighborhood of infinity as an "anticonical singularity", or a "party-hat" metric.) If you want to find out, I suggest asking Rafe Mazzeo (in Stanford) or Alexandre Eremenko (in Purdue).