  In this paper the authors study the hyperbolic geometric flow on Riemann
surfaces. This new nonlinear geometric evolution equation was recently
introduced by the first two authors motivated by Einstein equation and
Hamilton's Ricci flow. We prove that, for any given initial metric on
${\mathbb{R}}^{2}$ in certain class of metrics, one can always choose suitable
initial velocity symmetric tensor such that the solution exists for all time,
and the scalar curvature corresponding to the solution metric $g_{ij}$ keeps
uniformly bounded for all time. If the initial velocity tensor does not satisfy
the condition, then the solution blows up at a finite time, and the scalar
curvature $R(t,x)$ goes to positive infinity as $(t,x)$ tends to the blowup
points, and a flow with surgery has to be considered. The authors attempt to
show that, comparing to Ricci flow, the hyperbolic geometric flow has the
following advantage: the surgery technique may be replaced by choosing suitable
initial velocity tensor. Some geometric properties of hyperbolic geometric flow
on general open and closed Riemann surfaces are also discussed.
