Incomplete Yamabe flows and removable singularities

We study the Yamabe flow on a Riemannian manifold of dimension minus a closed submanifold of dimension n and prove that there exists an instantaneously complete solution if and only if . In the remaining cases including the borderline case, we show that the removability of the n-dimensional singularity is necessarily preserved along the Yamabe flow. In particular, the flow must remain geodesically incomplete as long as it exists. This is contrasted with the two-dimensional case, where instantaneously complete solutions always exist.