Mixed initial boundary value problem for hyperbolic geometric flow on Riemann surfaces

Abstract

The hyperbolic geometric flow equations is introduced recently by Kong and Liu motivated by Einstein equation and Hamilton Ricci flow. In this paper, we consider the mixed initial boundary value problem for hyperbolic geometric flow, and prove the global existence of classical solutions. The results show that, for any given initial metric on R 2 in certain class of metric, one can always choose suitable initial velocity symmetric tensor such that the solutions exist, and the scalar curvature corresponding to the solution metric g i j keeps bounded. If the initial velocity tensor does not satisfy the certain conditions, the solutions will blow up at a finite time. Some special explicit solutions to the reduced equation are given.