Ancient solutions to the Ricci flow in dimension $3$

brendle in time, starting from time tk . As long as the solution is nearly rotationally symmetric, it will remain close to the Bryant soliton by Theorem 1.1. On the other hand, as long as the cap is close to the Bryant soliton, we are able to show that the symmetry improves under the evolution (see Proposition 9.19). Using a continuity argument, we are able to show that there exists a sequence k 2 k such that k !0 and the flow is k -symmetric at time t for all t[ tk , 0]. Passing to the limit as k!, it follows that (M, g(t)) is rotationally symmetric for all t.