On Stability of the Hyperbolic Space Form under the Normalized Ricci Flow

Abstract

Abstract This paper studies the normalized Ricci flow from a slight per turbation of the hy-perbolic metric on H n . It’s proved that if the perturbation is small and decays sufficiently fastat the infinity, then the flow will converge exponentially fas t to the hyperbolic metric when thedimension n > 5. 1 Introduction The Ricci flow of Hamilton evolves the metric of a Riemannian m anifold in the direction ofan Einstein metric. There is a natural question that if one starts from a small perturbation ofan Einstein metric, or without the a priori knowledge of the existence of an Einstein metric,from a sufficiently (Ricci) pinched metric, can we show the flo w converges to (the) Einsteinmetric? The problem is addressed by R. Ye in [7]. Ye proved several theorems in this directionin the case of closed Riemannian manifolds. Central to his proof is a concept of stability. Onone hand, if the solution to the Ricci flow remains stable, the L 2 -norm of traceless Ricci tensordecays exponentially. On the other hand, if the solution remains pinched, it is stable. Since thetraceless Ricci tensor is almost the right hand side of the Ricci flow equation, Ye was able tocombine the above two observations to conclude if the initial metric is sufficiently pinched thenthe solution will remain so for any later time and converge to some Einstein metric.In this paper, we try to study a similar problem for complete noncompact manifolds. Moreprecisely, we are concerned with the normalized Ricci flow∂g