Let $M$ be a closed orientable smooth three-dimensional manifold, and let $g_0$ be a smooth Riemannian metric on $M$. Then there exists a choice of admissible surgery parameters, including canonical-neighbourhood scales, noncollapsing constants, curvature cutoff thresholds, and surgery radii, for which there is a Ricci flow with surgery starting from $(M,g_0)$.

More precisely, there exists a maximal surgery time interval $[0,T_{\mathrm{ext}})$, where $0<T_{\mathrm{ext}}\leq \infty$, and a piecewise smooth family of three-dimensional Riemannian manifolds with metrics $(M(t),g(t))$ such that $M(0)=M$, $g(0)=g_0$, and on every nonsurgery time interval the metric satisfies

\begin{align*} \frac{\partial g}{\partial t}=-2\operatorname{Ric}_{g(t)}. \end{align*}

If $T_{\mathrm{ext}}<\infty$, then the flow becomes extinct at time $T_{\mathrm{ext}}$, meaning that no components remain after the final surgery or limiting extinction time. If extinction does not occur, then $T_{\mathrm{ext}}=\infty$.

For every compact interval $[0,T]\subset [0,T_{\mathrm{ext}})$, the set of surgery times in $[0,T]$ is finite. On each such finite time interval, the flow satisfies the [Hamilton-Ivey pinching estimate](/theorems/6022), the canonical-neighbourhood assumption at the prescribed canonical scales, and Perelman $\kappa$-noncollapsing on all controlled metric balls whose radii are bounded below by a fixed multiple of the corresponding surgery radius and bounded above by the canonical or macroscopic scale assigned to that interval.