Note that two sphere with nonnegative curvature converges to canonical sphere along normalized Ricci flow. So assume that if curvature is bounded below by $-\epsilon$ then the sphere meets a singularity or converges to a canonical sphere. Fix a metric $g$ with nonnegative curvature. Then is there $g_0$ such that
(1) curvature is not nonnegative
(2) if $g(t)$ is a solution converging to canonical sphere with $g(0)=g_0$ then $g(\epsilon_2)=g$ where $\epsilon_2>0$.
Is there such $g_0$ ?