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?
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Sign up to join this communityConsider 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.
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.