Symmetries and solutions to geometrical flows

In this paper, we investigate group-invariant solutions to the hyperbolic geometric flow on Riemann surfaces, which include solutions of separation variables, traveling wave solutions, self-similar solutions and radial solutions. In the proceeding of reduction, there are elliptic, hyperbolic and mixed types of equations. For the first kind of equation, some exact solutions are found; while for the last two kinds, with implicit solutions found, we furthermore investigate whether there will be a global solution or blowing up. Referring to the work of Kong et al. (2009), the results come out perfectly.