# flat metrics on the 2-sphere with conical singularities

Consider some flat metric on $S^2$ with a fixed finite number of conical singularities $p_1,\ldots,p_n$.

What is the moduli space of such metrics up to isometry? In particular what is its dimension?

There is a necessary and sufficient condition that such a metric exist (at least when the cone angles are all less than $2\pi$), which is the simple linear condition on the cone angles imposed by Gauss-Bonnet.

This allows for $n-1$ degrees of freedom. There is also an extra $2n$-dimensional freedom from moving the points independently of one another; however, one may use a conformal position to fix the positions of exactly three points, so there is really only $2n-6$ degrees of freedom obtained this way. Altogether the moduli space of flat metrics with cone angles less than $2\pi$ is $3n-7$ dimensional once one has modded out by the Möbius group.

All of this (in the flat case particularly) follows immediately from work of Marc Troyanov, though certain global aspects of the moduli space require further arguments.

• Thank you very much Rafe! Let me think about what you wrote. Couple more questions: 1) $3n-7=-1$ if $n=2$ so this means there is no flat metric with $2$ singularities? Is it important that the angles are less than $2\pi$? Can it be possible if the angles more than $2\pi$ are allowed? 2) Could you please provide the title of Marc Troyanov's paper. – Axel Nov 10 '12 at 7:50
• Indeed, there is no flat metric on the 2-sphere with only two singularities. Another way to see it is that if $D=\sum a_i (p_i)$ with $a_i$ the cone angles, then we must have $K_{\mathbb P^1}+D \simeq 0$. This is equivalent to $\sum a_i=2$. – Henri Nov 10 '12 at 9:14

Another example: any meromorphic quadratic differential on the Riemann sphere (with at most simple poles) gives such a metric. There the cone angles are always positive integer multiples of $\pi$ .

These moduli spaces have been extensively studied from many different points of view.

• Thank you, these moduli spaces are a little rigid from this point of view as the cone angles can not change. I understand that nevertheless the theory behind is extremely rich. I am mostly interested in the deformations which CAN change the cone angles. – Axel Nov 11 '12 at 8:31