This result follows from i) Uniformization theorem and ii) Gauss-Bonnet theorem in 2d.

According to the statement of uniformization theorem from this wiki page :

every connected Riemann surface X admits a unique complete 2-dimensional real Riemann metric with constant curvature −1, 0 or 1 inducing the same conformal structure

On the other hand, according to Gauss-Bonnet theorem, integral of the scalar curvature on a 2d surface is a positive multiple of the Euler characteristic ($\chi=2-2g$). Since the Euler characteristic is negative for $g>1$ so from the uniformization theorem it follows that -

any Riemann surface with genus $g>1$ admits a unique complete 2-dimensional real Riemann metric with constant curvature −1 inducing the same conformal structure

Note : I am not aware of any good reference where uniformization thereom is proved in the form stated above. However, I hope you can find a proof in some of the references mentioned in the corresponding wiki article.

This post imported from StackExchange Physics at 2015-03-30 13:49 (UTC), posted by SE-user user10001