Global classical solutions to hyperbolic geometric flow on Riemann surfaces

Fagui Liu and Yuanzhang Zhang

Vol. 246 (2010), No. 2, 333–343

DOI: 10.2140/pjm.2010.246.333

Abstract

We consider the Cauchy problem for hyperbolic geometric flow equations introduced recently by Kong and Liu motivated by the Einstein equation and Hamilton Ricci flow, and obtain a necessary and sufficient condition for the global existence of classical solutions to this kind of flow on Riemann surfaces. The results show that the scalar curvature of the solution metric g_ij converges to one of flat curvature, and the hyperbolic geometric flow has the advantage that the surgery technique may be replaced by choosing a suitable initial velocity tensor.

Keywords

hyperbolic geometric flow, Riemann surface, quasilinear hyperbolic system, classical solution

Mathematical Subject Classification 2000

Primary: 30F45, 58J45

Secondary: 35L65, 35L45

Milestones

Received: 15 December 2008

Accepted: 13 January 2009

Published: 1 June 2010