Notation and terminology

$\ \ \ \ \ \ B(x,t,r)$ denotes the open metric ball of radius $r,$ with respect to the metric at time $t,$ centered at $x.$

$P(x,t,r,\triangle t)$ denotes a parabolic neighborhood, that is the set of all points $(x',t')$ with $x'\in B(x,t,r)$ and $t'\in [t,t+\triangle t]$ or $t'\in [t+\triangle t,t],$ depending on the sign of $\triangle t.$

A ball $B(x,t,\epsilon ^{-1}r)$ is called an $\epsilon$-neck, if, after scaling the metric with factor $r^{-2},$ it is $\epsilon$-close to the standard neck $\mathbb{S}^2\times\mathbb{I},$ with the product metric, where $\mathbb{S}^2$ has constant scalar curvature one, and $\mathbb{I}$ has length $2\epsilon ^{-1};$ here $\epsilon$-close refers to $C^N$ topology, with $N>\epsilon ^{-1}.$

A parabolic neighborhood $P(x,t,\epsilon ^{-1}r,r^2)$ is called a strong $\epsilon$-neck, if, after scaling with factor $r^{-2},$ it is $\epsilon$-close to the evolving standard neck, which at each time $t'\in [-1,0]$ has length $2\epsilon ^{-1}$ and scalar curvature $(1-t')^{-1}.$

A metric on $\mathbb{S}^2\times\mathbb{I},$ such that each point is contained in some $\epsilon$-neck, is called an $\epsilon$-tube, or an $\epsilon$-horn, or a double $\epsilon$-horn, if the scalar curvature stays bounded on both ends, stays bounded on one end and tends to infinity on the other, and tends to infinity on both ends, respectively.

A metric on $\mathbb{B}^3$ or $\mathbb{RP}^3\setminus\bar{\mathbb{B}}^3,$ such that each point outside some compact subset is contained in an $\epsilon$-neck, is called an $\epsilon$-cap or a capped $\epsilon$-horn, if the scalar curvature stays bounded or tends to infinity on the end, respectively.

We denote by $\epsilon$ a fixed small positive constant. In contrast, $\delta$ denotes a positive quantity, which is supposed to be as small as needed in each particular argument.