Yamabe Flow on Non-compact Manifolds with Unbounded Initial Curvature

We prove global existence of Yamabe flows on non-compact manifolds M of dimension m≥ 3 under the assumption that the initial metric g= ugM is conformally equivalent to a complete background metric gM of bounded, non-positive scalar curvature and positive Yamabe invariant with conformal factor u bounded from above and below. We do not require initial curvature bounds. In particular, the scalar curvature of (M, g) can be unbounded from above and below without growth condition.