The standard solution

Consider a rotationally symmetric metric on $\mathbb{R}^3$ with nonnegative sectional curvature, which splits at infinity as the metric product of a ray and the round 2-sphere of scalar curvature one. At this point we make some choice for the metric on the cap, and will refer to it as the standard cap; unfortunately, the most obvious choice, the round hemisphere, does not fit, because the metric on $\mathbb{R}^3$ would not be smooth enough, however we can make our choice as close to it as we like. Take such a metric on $\mathbb{R}^3$ as the initial data for a solution $g_{ij}(t)$ to the Ricci flow on some time interval $[0,T),$ which has bounded curvature for each $t\in [0,T).$

Claim 1. The solution is rotationally symmetric for all $t.$

Indeed, if $u^i$ is a vector field evolving by $u^i_t=\triangle u^i+R^i_ju^j,$ then $v_{ij}=\nabla_i u_j$ evolves by $(v_{ij})_t=\triangle v_{ij}+2R_{ikjl}v_{kl}-R_{ik}v_{kj}-R_{kj}v_{ik}.$ Therefore, if $u^i$ was a Killing field at time zero, it would stay Killing by the maximum principle. It is also clear that the center of the cap, that is the unique maximum point for the Busemann function, and the unique point, where all the Killing fields vanish, retains these properties, and the gradient of the distance function from this point stays orthogonal to all the Killing fields. Thus, the rotational symmetry is preserved.

Claim 2. The solution converges at infinity to the standard solution on the round infinite cylinder of scalar curvature one. In particular, $T\le 1.$

Claim 3. The solution is unique.

Indeed, using Claim 1, we can reduce the linearized Ricci flow equation to the system of two equations on $(-\infty,+\infty)$ of the following type

$$f_t=f''+a_1 f'+b_1 g'+c_1 f+d_1 g,\ \ g_t=a_2f'+b_2 g'+c_2 f+d_2 g,$$

where the coefficients and their derivatives are bounded, and the unknowns $f,g$ and their derivatives tend to zero at infinity by Claim 2. So we get uniqueness by looking at the integrals $\int_{-A}^A{(f^2+g^2)}$ as $A\to \infty.$

Claim 4. The solution can be extended to the time interval $[0,1).$

Indeed, we can obtain our solution as a limit of the solutions on $\mathbb{S}^3,$ starting from the round cylinder $\mathbb{S}^2\times\mathbb{I}$ of length $L$ and scalar curvature one, with two caps attached; the limit is taken about the center $p$ of one of the caps, $L\to\infty.$ Assume that our solution goes singular at some time $T<1.$ Take $T_1<T$ very close to $T, \ \ T-T_1<<1-T.$ By Claim 2, given $\delta >0,$ we can find $\bar{L},\bar{D}<\infty,$ depending on $\delta$ and $T_1,$ such that for any point $x$ at distance $\bar{D}$ from $p$ at time zero, in the solution with $L\ge\bar{L},$ the ball $B(x,T_1,1)$ is $\delta$-close to the corresponding ball in the round cylinder of scalar curvature $(1-T_1)^{-1}.$ We can also find $r=r(\delta ,T),$ independent of $T_1,$ such that the ball $B(x,T_1,r)$ is $\delta$-close to the corresponding euclidean ball. Now we can apply Theorem I.10.1 and get a uniform estimate on the curvature at $x$ as $t\to T$, provided that $T-T_1<\epsilon ^2 r(\delta ,T)^2.$ Therefore, the $t\to T$ limit of our limit solution on the capped infinite cylinder will be smooth near $x.$ Thus, this limit will be a positively curved space with a conical point. However, this leads to a contradiction via a blow-up argument; see the end of the proof of the Claim 2 in I.12.1.

The solution constructed above will be called the standard solution.

Claim 5. The standard solution satisfies the conclusions of 1.5 , for an appropriate choice of $\epsilon , \ \eta , C_1(\epsilon ),C_2(\epsilon ),$ except that the $\epsilon$-neck neighborhood need not be strong; more precisely, we claim that if $(x,t)$ has neither an $\epsilon$-cap neighborhood as in 1.5(b), nor a strong $\epsilon$-neck neighborhood as in 1.5(a), then $x$ is not in $B(p,0,\epsilon ^{-1}),\ \ t<3/4,$ and there is an $\epsilon$-neck $B(x,t,\epsilon ^{-1}r),$ such that the solution in $P(x,t,\epsilon ^{-1}r,-t)$ is, after scaling with factor $r^{-2}, \ \ \epsilon$-close to the appropriate piece of the evolving round infinite cylinder.

Moreover, we have an estimate $R_{\mathrm{min}}(t)\ge \mathrm{const}\cdot (1-t)^{-1}.$

Indeed, the statements follow from compactness and Claim 2 on compact subintervals of $[0,1),$ and from the same arguments as for ancient solutions, when $t$ is close to one.