Abstract

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 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 gij 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.