In [3] the authors proved the long-time existence of Ricci flow starting from complete bounded curvature Riemannian manifolds with scale-invariant integral curvature bounded by a dimensional constant times the inverse of the Sobolev constant. We generalize this result by replacing the bounded curvature assumption with the assumption that g is only equivalent to a complete bounded curvature metric h while satisfying a Morrey-type condition on the gradient of g relative to h: a local integral condition on the covariant derivative \(\nabla _h g\) . The Morrey-type condition was first considered in [10] in the context of Ricci flow on non-compact manifolds, and in particular allows the possibility for g to have unbounded curvature on M. As in [3], our long-time solution enjoys curvature decay estimates implying in particular that M is diffeomorphic to \(\mathbb {R}^n\) .
