We study the Yamabe flow on a Riemannian manifold of dimension $mgeq3$ minus a closed submanifold of dimension $n$ and prove that there exists an instantaneously complete solution if and only if $n>frac{m-2}{2}$. In the remaining cases $0leq nleqfrac{m-2}{2}$ 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.
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