How many metrics of constant curvature exist on a Riemannian surface?

Question

I have been trying to determine the number of metrics of constant curvature on a surface of genus $n$, say $\Sigma$. For low values, the answer is clear, the moduli space is a point for the sphere, and is two dimensional for the torus, but the higher dimensional cases stump me, and I am unable to find the result. Any help or a reference would be appreciated.

Answer 1

First on terminology. "Riemannian surface" is a surface already equipped with a Riemannian metric. So the question "how many metrics of constant curvature exist on a Riemannian surface" makes sense only if you state what is the relation between the metric of constant curvature and the original metric on the Riemannian surface.

"Riemann surface" is a surface equipped with a conformal structure.

You probably mean "on a RIEMANN surface", this means that the conformal structure is fixed. Then the answer is: one parametric family. Parameter is just the scaling factor.

For the sphere, these metrics are of positive curvature, for the plane, punctured plane and tori of zero curvature, and for the rest of Riemann surfaces negative curvature.

If you change your question and ask about "a topological orientable surface", so that the conformal structure is not fixed then there are usually many conformal structures (and for each there is a metric as above). For example, in the case of compact surfaces of genus $g>1$ the number of conformal structures depends on $6g-6$ parameters. For a torus ($g=1$) there is a 2-parametric family, for the sphere, the structure is unique. For the disk, there are exactly $2$ different structures, for the annulus, a $1$-parametric family (of non-degenerate annuli) and two other structures (degenerate annuli), etc. (I always mean real parameters when I count, though in many cases there exists also complex analytic structure on the moduli spaces of conformal structures).

All these statements are consequences of the general Uniformization theorem. Some references where a complete proof is given are:

Ahlfors, Conformal invariants

Hubbard, Teichmuller theory

H. P. de Saint-Gervais, Uniformisation des surfaces de Riemann.

(The last book is completely devoted to the discussion of the history, meaning and various proofs of this theorem).

Answer 2

This is the geometric version of Riemann's "problem of moduli". A connected, closed, oriented surface of genus $g$ has a moduli space of dimension $6g - 6$.

One way to count the dimension of hyperbolic structures is via Fenchel-Nielsen coordinates. See also Section 10.6 of Farb and Margalit's book "Primer on mapping class groups".