Horizon smallness is the induced norm of the tangential stationary field

The decomposition in [F2] gives $g(T,T)=n^{2}g(\nu ,\nu )+2ng(\nu ,X)+g(X,X)=-n^2+h(X,X)$. At a point $p\in S_0$, [F1] supplies $T_p\in T_p{H}^{+}\cap T_p{H}^{-}$, and transversality allows [F3] to identify this intersection with $T_p{S}_0$. Since $S_0\subset {\mathrm{\Sigma }}_0$, it follows that $g(T,\nu )=0$, hence $n=0$ and $X=T$.

The package identifies $S_0$ as the image of the nonempty unit sphere $\{x\in {\mathbb{R}}^3:\mathrm{\mid }x\mathrm{\mid }=1\}$ under the smooth map ${\mathrm{\Phi }}_0$. The unit sphere is compact by [F4], so its continuous image $S_0$ is nonempty and compact by [F5]. The induced metric is positive definite, and the function $f(p)=\mathrm{\mid }{T}_p{\mathrm{\mid }}_h=\sqrt{h_p(T_p,T_p)}$ is continuous: in local coordinates its square is a finite sum of products of smooth metric and field components, and the nonnegative square root is continuous. Thus [F5] gives a maximum $m={\max }_{S_0}f\ge 0$.

By step 1.1, $g(T,T)=h(T,T)=f^2\ge 0$ on $S_0$. Since $0\le f\le m$ and $f$ attains $m$, the increasing square function gives ${\sup }_{S_0}\mathrm{\mid }g(T,T)\mathrm{\mid }=m^2$ and ${\sup }_{S_0}\mathrm{\mid }T{\mathrm{\mid }}_h=m$. For $ϵ>0$, $m^2<{ϵ}^2$ holds exactly when $m<ϵ$, proving the two strict smallness conditions equivalent. It does not identify either with ADM angular-momentum smallness.