Global existence of classical solutions to the hyperbolic geometry flow with time-dependent dissipation

Abstract

In this article, we investigate the hyperbolic geometry flow with time-dependent dissipation ∂2gij∂t2+μ(1+t)λ∂gij∂t=-2Rij,on Riemann surface. On the basis of the energy method, for 0 < λ ≤ 1, μ > λ + 1, we show that there exists a global solution gij to the hyperbolic geometry flow with time-dependent dissipation with asymptotic flat initial Riemann surfaces. Moreover, we prove that the scalar curvature R(t,x) of the solution metric gij remains uniformly bounded.