  We define a formal Riemannian metric on a given conformal class of metrics on
a closed Riemann surface. We show interesting formal properties for this
metric, in particular the curvature is nonpositive and the Liouville energy is
geodesically convex. The geodesic equation for this metric corresponds to a
degenerate elliptic fully nonlinear PDE, and we prove that any two points are
connected by a $C^{1,1}$ geodesic. Using this we can define a length space
structure on the given conformal class. We present a different approach to the
uniformization theorem by studying the negative gradient flow of the normalized
Liouville energy, a new geometric flow whose principal term is the inverse of
the Gauss curvature. We prove long time existence of solutions with arbitrary
initial data and weak convergence to constant scalar curvature metrics. This is
all a special case of a more general construction on even dimensional manifolds
related to the $\sigma_{\frac{n}{2}}$-Yamabe problem, which will appear in a
forthcoming article.
