Extending Yamabe flow on complete Riemannian manifolds

In this paper, we study the extending problem of the Yamabe flow on complete Riemannian manifolds. Suppose that (Mn,g(t)) is a solution to the Yamabe flow on a complete Riemannian manifold on time interval [0,T), where n⩾3 and T<+∞. We first prove that the Yamabe flow can be extended over T provided the scalar curvature function stays uniformly bounded on [0,T). Next, we show that the Yamabe flow with positive Yamabe invariant can be extended beyond T provided the (local) norm of the scalar curvature function is uniformly bounded on [0,T). This latter result is obtained by the Moser iteration method.