%%%%%%%%% \noindent{\bf To prove \eqref{claim is spherical graph}$_{j=\ell+1}$}: Notice that \eqref{ext u ell plus one 1 lambda} and \eqref{claim is spherical graph}$_{j=\ell}$ do not imply \eqref{claim is spherical graph}$_{j=\ell+1}$, since, in \eqref{claim is spherical graph}$_{j=\ell+1}$, we are claiming the graphicality of $V$ inside $A_{s/32}^{4\,s_{\ell+1}}$ (which is strictly larger than $A_{s/32}^{4\,s_\ell}$), and we are claiming that $u_{\ell+1}$ has $C^1$-norm bounded by $\s$, and not just by $C_0\,\s_0$ (with $C_0$ as in Lemma \ref{lemma step one}-(ii)). We want to apply Lemma \ref{lemma graphicality lambda} with
%%%%%%%%%  \begin{equation}
%%%%%%%%%    \label{ext parameters for step two lambda}
%%%%%%%%%      r=8\,s_{\ell+1}\,,\,\,(\l_1,\l_2)=\Big(\frac1{16},\frac18\Big)\,,\,\,(\l_3,\l_4)=\Big(\frac{3}{32},\frac{7}{64}\Big)\,,\,\, K=H_{\ell+1}\,.
%%%%%%%%%  \end{equation}
%%%%%%%%%  We check the validity of \eqref{conditions on r lambda}, \eqref{ext hp step two 1 lambda}, \eqref{ext hp step two 2 lambda} and \eqref{ext hp step two 3 lambda} for these choices of $r$, $\l_1$, $\l_2$, $\l_3$, $\l_4$ and $K$. Since $r=8\,s_{\ell+1}\ge s\ge\max\{M_0,64\,R\}$, and since \eqref{def of N} and $\ell+1\le N$ give $r=8\,s_{\ell+1}\le \e_0/\Lambda$, we deduce the validity of \eqref{conditions on r lambda} with $r=8\,s_{\ell+1}$. The validity of \eqref{ext hp step two 1 lambda} with $r=8\,s_{\ell+1}$ is immediate from \eqref{delta below eps0 in absolute value} by our choice \eqref{choice of eps0} of $\e_0$. Next we notice that
%%%%%%%%%  \[
%%%%%%%%%  \|V\|(A_{\l_3\,r}^{\l_4\,r})=\|V\|(A_{3\,[8\,s_{\ell+1}]/32}^{7\,[8\,s_{\ell+1}]/64})=\|V\|(A_{3\,s_{\ell}/2}^{7\,s_\ell/4})>0
%%%%%%%%%  \]
%%%%%%%%%  thanks to \eqref{claim is spherical graph}$_{j=\ell}$, so that \eqref{ext hp step two 3 lambda} holds for $r$, $\l_3$ and $\l_4$ as in \eqref{ext parameters for step two lambda}. Finally, by \eqref{equuivalence between u square and omega square} (which applies to $u_{\ell+1}$ by \eqref{choice of sigma0}), \eqref{ext u ell plus one 1 lambda}, \eqref{claim is spherical graph}$_{j=\ell}$, and \eqref{ext u ell plus one 4 lambda},
%%%%%%%%%  \begin{eqnarray*}
%%%%%%%%%      &&\frac1{r^n}\,\int_{A_{\l_1\,r}^{\l_2\,r}}\,\om_{H_{\ell+1}}^2\,d\|V\|\le
%%%%%%%%%      \frac{C(n)}{s_{\ell+1}^n}\,\int_{s_\ell}^{s_{\ell+1}}\,r^{n-1}\,dr\,\int_{\Sigma_{H_{\ell+1}}}[u_{\ell+1}]_r^2
%%%%%%%%%      \\
%%%%%%%%%      &&\le C(n)\,T_\ell+
%%%%%%%%%      \frac{C(n)}{s_{\ell+1}^n}\,\int_{s_\ell}^{s_{\ell+1}}\,r^{n-1}\,dr\,\int_{\Sigma_{H_{\ell}}}[u_{\ell}]_r^2
%%%%%%%%%      \le C(n)\,T_\ell\le  C(n)\,\e_0\,,
%%%%%%%%%  \end{eqnarray*}
%%%%%%%%%  where in the last inequality we have used \eqref{claim Tj smaller eps0}$_{j=\ell}$. Again by our choice \eqref{choice of eps0} of $\e_0$, we deduce that \eqref{ext hp step two 2 lambda} holds with $r$, $\l_1$ and $\l_2$ as in \eqref{ext parameters for step two lambda}. We can thus apply Lemma \ref{lemma graphicality lambda}, and find $v\in\X_\s(\Sigma_{H_{\ell+1}},s_{\ell+1}/4,4\,s_{\ell+1})$ such that
%%%%%%%%%  \begin{equation}
%%%%%%%%%    \label{ext jelly lambda}
%%%%%%%%%      \mbox{$V$ corresponds to $\Sigma_{H_{\ell+1}}(v,s_{\ell+1}/4,4\,s_{\ell+1})$ on $A_{s_{\ell+1}/4}^{4\,s_{\ell+1}}$}\,.
%%%%%%%%%  \end{equation}
%%%%%%%%%  By \eqref{ext u ell plus one 1 lambda}, \eqref{claim is spherical graph}$_{j=\ell}$, and \eqref{ext jelly lambda}, $v=u_{\ell+1}$ on $\S_{H_{\ell+1}}\times(s_{\ell+1}/4, 4\,s_\ell)$. We can thus use $v$ to extend $u_{\ell+1}$ from $\S_{H_{\ell+1}}\times(s/32, 4\,s_\ell)$ to $\S_{H_{\ell+1}}\times(s/32,4\,s_{\ell+1})$, and, thanks to \eqref{ext u ell plus one 1 lambda}, \eqref{claim is spherical graph}$_{j=\ell}$ and \eqref{ext jelly lambda}, the resulting extension is such that \eqref{claim is spherical graph}$_{j=\ell+1}$ holds. {\bf To prove \eqref{claim deltaj smaller deltaj-1}$_{j=\ell+1}$}: We set $r_0=(s_\ell+s_{\ell+1})/2$ and notice that for $\eta_0=1/3$ we have
%%%%%%%%%  \begin{equation}
%%%%%%%%%    \label{ext choices 1 lambda}
%%%%%%%%%      r_1=r_0\,(1-\eta_0)=s_\ell\,,\qquad r_2=r_0\,(1+\eta_0)=s_{\ell+1}\,.
%%%%%%%%%  \end{equation}
%%%%%%%%%  For $\eta=1/6$ we correspondingly set
%%%%%%%%%  \begin{equation}
%%%%%%%%%    \label{ext choices 2 lambda}
%%%%%%%%%    r_3=r_0\,(1-\eta)=:s_\ell^-\,,\qquad r_4=r_0\,(1+\eta)=:s_\ell^+\,,
%%%%%%%%%  \end{equation}
%%%%%%%%%  and notice that $(\eta_0,\eta)\in J$, see \eqref{def of J lambda}. With the aim of applying Theorem \ref{theorem 7.17AA exterior lambda}-(iii) to these radii, we notice that \eqref{claim is spherical graph}$_{j=\ell+1}$ implies that assumption \eqref{V corresponds to u monoto lambda} holds with $H=H_{\ell+1}$ and $u=u_{\ell+1}$, while, by \eqref{ext u ell plus one 2 lambda}, $r=s_\ell^*\in(s_\ell,s_{\ell+1})$ is such that \eqref{lambdaAA 7.15(4) for 7.17 hole} holds. By $\Lambda\,s_{\ell+1}\le\e_0\le 1$, \eqref{def of N}, and \eqref{lambdaAA tesi 7.17 hole}, with $C(n)=C_0(n,1/6,1/3)$ for $C_0$ as in Theorem \ref{theorem 7.17AA exterior lambda}-(iii), we have
%%%%%%%%%  \begin{eqnarray}
%%%%%%%%%    \nonumber
%%%%%%%%%    &&s_{\ell+1}^{-n}\,\,\big|\|V\|\big(B_{s_\ell^+}\setminus B_{s_\ell^-}\big)-\om_n\,\big((s_\ell^+)^n-(s_\ell^-)^n\big)\big|
%%%%%%%%%    \\\nonumber
%%%%%%%%%    &&= s_{\ell+1}^{-n}\big|\H^n(\Sigma_{H_{\ell+1}}(u_{\ell+1},s_\ell^-,s_\ell^+))-\H^n(\Sigma_{H_{\ell+1}}(0,s_\ell^-,s_\ell^+))\big|
%%%%%%%%%   \\\label{compare to lambda}
%%%%%%%%%&&\le C(n)\,\big\{(\Lambda\,s_{\ell+1})^2\,\,
%%%%%%%%%  \end{eqnarray}
%%%%%%%%%Setting for brevity $\de=\de_{V,R,\Lambda}$ and $\Theta=\Theta_{V,R,\Lambda}$, and recalling that
%%%%%%%%%&&=\om_n\,r^n-\|V\|(B_r\setminus B_R)
%%%%%%%%%-\Lambda\,r^n\,\int_R^r\,\frac{\|V\|(B_\rho\setminus B_R)}{\rho^n}\,d\rho+\frac{R\,\|\de V\|(\pa B_R)}{n}
%%%%%%%%%we have
%%%%%%%%%  -(s_{\ell}^+)^n\,\de(s_{\ell}^+)\big|
%%%%%%%%%  \le C(n)\,\big\{(\Lambda\,s_\ell)^2+\Theta(s_{\ell+1})-\Theta(s_{\ell})\big\}
%%%%%%%%%&&\!\!\!\!\!\!\!\!\!\!+C(n)\,\Lambda\,s_\ell^{-n}\Big\{(s_\ell^+)^n\int_{R}^{s_{\ell}^+}\frac{\|V\|(B_\rho\setminus B_R)}{\rho^n}\,d\rho
%%%%%%%%%-(s_\ell^-)^n\int_{R}^{s_{\ell}^-}\frac{\|V\|(B_\rho\setminus B_R)}{\rho^n}\,d\rho\Big\}
%%%%%%%%%&&\!\!\le C(n)\,\Big\{(\Lambda\,s_\ell)^2 +\Theta(s_{\ell+1})-\Theta(s_{\ell})\Big\}
%%%%%%%%%+C(n)\,\Lambda\, \int_{R}^{s_{\ell}^+}\frac{\|V\|(B_\rho\setminus B_R)}{\rho^n}\,d\rho\,.
%%%%%%%%%By $\Lambda\,s_\ell\le 1$ and since $s_{\ell}^+\le s_\ell\le \e_0/8\,\Lambda$ thanks to $\ell<N$, we can use the upper bound  $\|V\|(B_\rho\setminus B_R)\le \Gamma\,\rho^n$ with $\rho\in(R,s_\ell^+)\subset(R,1/\Lambda)$, to find that
%%%%%%%%%  -\frac{(s_{\ell}^+)^n}{s_\ell^n}\,\de(s_{\ell}^+)\Big|
%%%%%%%%%\le\! C_*(n)\,\big\{\de(s_{\ell})-\de(s_{\ell+1})\big\}+C_*(n)\,(\Gamma+1)\,\Lambda\,s_\ell\,,
%%%%%%%%%for a constant $C_*(n)$. By rearranging terms, and using the monotonicity of $\de$ on $(R,\infty)$ and $(s_\ell^-,s_\ell^+)\subset(s_\ell,s_{\ell+1})$ we find that
%%%%%%%%%  \begin{eqnarray}\nonumber
%%%%%%%%%  &&\big(C_*(n)\,+(s_{\ell}^+)^n/(s_\ell^n)\big)\,\de(s_{\ell+1})\le C_*(n)\,\de(s_{\ell+1})+\big((s_{\ell}^+)^n/(s_\ell^n)\big)\,\de(s_{\ell}^+)
%%%%%%%%%  \\\nonumber
%%%%%%%%%  &\le&
%%%%%%%%%  C_*(n)\,\de(s_\ell)+\big((s_{\ell}^-)^n/(s_\ell^n)\big)\,\de(s_\ell^-)+C_*(n)\,(1+\Gamma)\,\Lambda \,s_\ell
%%%%%%%%%  \\\label{ext grandi aa lambda}
%%%%%%%%%  &\le&
%%%%%%%%%  \big(C_*(n)\,+(s_{\ell}^-)^n/(s_\ell^n)\big)\,\de(s_{\ell})+C_*(n)\,(1+\Gamma)\,\Lambda \,s_\ell\,.
%%%%%%%%%  \end{eqnarray}
%%%%%%%%%  We finally notice that by \eqref{ext choices 1 lambda}, \eqref{ext choices 2 lambda}, $\eta_0=1/3$, and $\eta=1/6$, we have
%%%%%%%%%  \[
%%%%%%%%%  \frac{s_\ell^-}{s_\ell}=\frac{r_0\,(1-\eta)}{r_0\,(1-\eta_0)}=\frac{5}4\,,\qquad
%%%%%%%%%  \frac{s_\ell^+}{s_\ell}=2\,\frac{s_\ell^+}{s_{\ell+1}}=2\,\frac{1+\eta}{1+\eta_0}=\frac{7}4\,,
%%%%%%%%%  \]
%%%%%%%%%  so that, we find $\de(s_{\ell+1})\le\tau\{\de(s_\ell)+(1+\Gamma)\,\Lambda\,s_\ell\}$ (i.e. \eqref{claim deltaj smaller deltaj-1}$_{j=\ell+1}$) with
%%%%%%%%%  \[
%%%%%%%%%  \tau=\tau(n)=\frac{C_*(n)+(5/4)^n}{C_*(n)+(7/4)^n}\,,\qquad \tau_*=\tau_*(n)=\frac{C_*(n)}{C_*(n)+(7/4)^n}<\tau\,.
%%%%%%%%%  \]
%%%%%%%%%    %%%Finally, by $\Lambda\,s_\ell\le \e_1\le \e_0/[(1+\Gamma)\,C_*]$ (see \eqref{def of N} and \eqref{choice of eps1}), and since $\de(s_\ell)\le\e_0$ (see \eqref{delta below eps0 in absolute value}), we deduce from \eqref{ext itera lambda} that
%%%%%%%%%%%%  \[
%%%%%%%%%%%%  \de(s_{\ell+1})\le \Big(\tau+\frac1{C_*}\Big)\,\e_0\le\e_0\,,
%%%%%%%%%%%%  \]
%%%%%%%%%%%%  provided $C_*=C_*(n)$ in \eqref{choice of eps1} is small enough in terms of $\tau(n)$. This proves \eqref{claim delta smaller eps0}$_{j=\ell+1}$.
%%%%%%%%%{\bf We finally prove  \eqref{claim Tj smaller deltaj-1}$_{j=\ell+1}$}, i.e.
%%%%%%%%%  \begin{equation}
%%%%%%%%%    \label{ext bd decay Tj j ell plus one lambda}  \frac1{s_{\ell+1}^n}\int_{s_{\ell+1}}^{2\,s_{\ell+1}}\!\!\!r^{n-1}\!\!\int_{\Sigma_{H_{\ell+1}}}\!\!\!\!\![u_{\ell+1}]_r^2\le
%%%%%%%%%    C(n)\big\{\de_{V,R,\Lambda}(s_{\ell})-\de_{V,R,\Lambda}(s_{\ell+3})+\Lambda\,s_{\ell}\big\}.
%%%%%%%%%  \end{equation}
%%%%%%%%%  By \eqref{claim is spherical graph}$_{j=\ell+1}$ we know that
%%%%%%%%%  \begin{equation}
%%%%%%%%%    \label{ext yes we know lambda}
%%%%%%%%%    \mbox{$V$ corresponds to $\S_{H_{\ell+1}}(u_{\ell+1},s/32,4\,s_{\ell+1})$ on $A_{s/32}^{4\,s_{\ell+1}}$}\,.
%%%%%%%%%  \end{equation}
%%%%%%%%%  Now, \eqref{r1r2r3r4} holds with $r_0=3\,s_\ell$ and $(\eta_0,\eta)=(2/3,1/3)\in J$, see \eqref{def of J lambda}, if
%%%%%%%%%  \begin{eqnarray*}
%%%%%%%%%  &&r_1=s_\ell=3\,s_\ell-2\,s_\ell\,,\qquad r_2=5\,s_{\ell}=3\,s_\ell+2\,s_\ell\,,
%%%%%%%%%  \\
%%%%%%%%%  &&r_3=s_{\ell+1}=3\,s_\ell-s_\ell\,,\qquad\,\,\,\, \,r_4=2\,s_{\ell+1}=3\,s_\ell+s_\ell\,.
%%%%%%%%%  \end{eqnarray*}
%%%%%%%%%  Since $s_\ell^*\in(s_\ell,s_{\ell+1})\subset (r_1,r_2)$,
%%%%%%%%%  by \eqref{ext yes we know lambda}, \eqref{ext u ell plus one 2 lambda} and $(r_1,r_2)\subset (s/32,4\,s_{\ell+1})$ we can apply  Theorem \ref{theorem 7.15AA main estimate lambda}  to deduce that
%%%%%%%%%  \[
%%%%%%%%%  \int_{s_{\ell+1}}^{2\,s_{\ell+1}}\!\!\!r^{n-1}\!\int_{\Sigma_{H_{\ell+1}}}\,[u_{\ell+1}]_r^2\le C(n)\,\int_{s_\ell}^{5\,s_\ell}\!\!\!\!r^{n+1}\!\int_{\Sigma_{H_{\ell+1}}}\!\!\!\!\!\!(\pa_ru_{\ell+1})_r^2
%%%%%%%%%  +C(n)\,\Lambda\,(s_\ell)^{n+1}\,.
%%%%%%%%%  \]
%%%%%%%%%  Again by \eqref{ext yes we know lambda}, Theorem \ref{theorem 7.17AA exterior lambda}-(ii) with $(r_1,r_2)=(s_\ell,8\,s_\ell)$ gives
%%%%%%%%%  \begin{eqnarray*}
%%%%%%%%%  &&s_\ell^{-n}\,\int_{s_\ell}^{5\,s_\ell}\,r^{n+1}\,\int_{\Sigma_{H_{\ell+1}}}\,(\pa_r[u_{\ell+1}])_r^2
%%%%%%%%%  \le
%%%%%%%%%  s_\ell^{-n}\,\int_{s_\ell}^{8\,s_\ell}\,r^{n+1}\,\int_{\Sigma_{H_{\ell+1}}}\,(\pa_r[u_{\ell+1}])_r^2
%%%%%%%%%  \\
%%%%%%%%%  &&\le C(n)\,\big\{\Theta_{V,R,\Lambda}(8\,s_\ell)-\Theta_{V,R,\Lambda}(s_{\ell})\big\}
%%%%%%%%%  \le C(n)\,\big\{\de_{V,R,\Lambda}(s_{\ell})-\de_{V,R,\Lambda}(s_{\ell+3})\big\}\,.
%%%%%%%%%  \end{eqnarray*}
%%%%%%%%%  The last two estimates combined give \eqref{ext bd decay Tj j ell plus one lambda}. This completes the proof of the {\bf claim}. {\bf Proof of statement (i)}: We assume $S_*<\infty$ (that is either $\Lambda>0$ or $R_*<\infty$). In this case $N$ (as defined in \eqref{def of N}) is finite, with $2^N\le S_*/s<2^{N+1}$. By \eqref{claim is spherical graph}$_{j=N-2}$, $V$ corresponds to $\Sigma_{H_{N-2}}(u_{N-2},s/32,4\,s_{N-2})$ on $A_{s/32}^{4\,s_{N-2}}$, and since $4\,s_{N-2}=4\,2^{N-2-3}\,s=2^{N+1}\,s/16>S_*/16$,
%%%%%%%%%  we deduce \eqref{mesoscale thesis graphicality} with $K=H_{N-2}$ and $u=u_{N-2}$. {\bf Proof of statement (ii)}: We assume  $\Lambda=0$ and $R_*=+\infty$ (that is, $\de_{V,R,0}\ge-\e_0$ on $(s/8,\infty)$). In this case,
%%%%%%%%%  we have $N=+\infty$, and the iteration procedure set up in the claim actually defines a sequence $\{(H_j,u_j)\}_{j=0}^\infty$
%%%%%%%%%  with $u_j\in\X_\s(\S_{H_j},s/32,4\,s_j)$ and
%%%%%%%%%  \begin{equation}
%%%%%%%%%      \label{claim is spherical graph jj}
%%%%%%%%%  \mbox{$V$ corresponds to $\Sigma_{H_j}(u_j,s/32,4\,s_j)$ on $A_{s/32}^{4\,s_j}$}
%%%%%%%%%  \end{equation}
%%%%%%%%%  for every $j\ge0$. By compactness of $\SS^n$, we can find $K\in\H$ and a subsequence $j(k)$ such that $\e_k=|\nu_K-\nu_{H_{j(k)}}|\to 0$ as $k\to\infty$. In particular, for $k$ large enough, we have $\e_k<\e_0$, and thus, by Lemma \ref{lemma step one}-(i) and by \eqref{claim is spherical graph jj} we can find $v_k\in\X_{C(n)\,(\s+\e_k)}(\S_K;s/32,4\,s_{j(k)})$ such that
%%%%%%%%%  \begin{equation}
%%%%%%%%%      \label{claim is spherical graph jjj}
%%%%%%%%%  \mbox{$V$ corresponds to $\Sigma_K(v_{j(k)},s/32,4\,s_{j(k)})$ on $A_{s/32}^{4\,s_{j(k)}}$}\,.
%%%%%%%%%  \end{equation}
%%%%%%%%%  By \eqref{claim is spherical graph jjj}, $v_{j(k)+1}=v_{j(k)}$ on $\Sigma_K\times(s/32,4\,s_{j(k)})$. Since $s_{j(k)}\to\infty$, we have found $u\in\X_{C(n)\,\s}(\S_K;s/32,\infty)$ such that $V$ corresponds to $\Sigma_K(u,s/32,\infty)$ on $A_{s/32}^\infty$. This proves statement (ii). {\bf Proof of statement (iii)}: We finally assume that $\Lambda=0$ and, setting for brevity $\de=\de_{V,R,0}$,
%%%%%%%%%  \begin{equation}
%%%%%%%%%    \label{positive}
%%%%%%%%%    \de(r)\ge0\,,\qquad\forall r\ge s/8\,.
%%%%%%%%%  \end{equation}
%%%%%%%%%  As for statement (ii), $N=+\infty$, and there is $\{(H_j,u_j)\}_{j=0}^\infty$ satisfying
%%%%%%%%%  \begin{eqnarray}
%%%%%%%%%  \label{claim is spherical graph fine}
%%%%%%%%%  &&\mbox{$V$ corresponds to $\Sigma_{H_j}(u_j,s/32,4\,s_j)$ on $A_{s/32}^{4\,s_j}$}\,,\qquad\forall j\ge0\,,
%%%%%%%%%  \end{eqnarray}
%%%%%%%%%  \vspace{-0.5cm}
%%%%%%%%%  \begin{eqnarray}
%%%%%%%%%  \label{claim decay normal vectors fine}
%%%%%%%%%  |\nu_{H_j}-\nu_{H_{j-1}}|^2&\le& C(n)\,T_{j-1}\,,\qquad\hspace{0.8cm}\mbox{if $j\ge 1$}\,,
%%%%%%%%%  \\
%%%%%%%%%  \label{claim delta fine}
%%%%%%%%%  \de(s_j)&\le&
%%%%%%%%%  \left\{
%%%%%%%%%  \begin{split}
%%%%%%%%%  &\e_0\,,\hspace{1.75cm}\qquad\mbox{if $j=0$}\,,
%%%%%%%%%  \\
%%%%%%%%%  &\tau\,\de(s_{j-1})\,,\hspace{0.63cm}\qquad\mbox{if $j\ge1$}\,,
%%%%%%%%%  \end{split}
%%%%%%%%%  \right .
%%%%%%%%%  \\
%%%%%%%%%  \label{claim Tj fine}
%%%%%%%%%  T_j&\le&\left\{
%%%%%%%%%  \begin{split}
%%%%%%%%%  &C(n)\,\e_0\,,\hspace{0.85cm}\qquad\mbox{if $j=0$}\,,
%%%%%%%%%  \\
%%%%%%%%%  &C(n)\,\de(s_{j-1})\,,\qquad\mbox{if $j\ge1$}\,.
%%%%%%%%%  \end{split}
%%%%%%%%%  \right .
%%%%%%%%%  \end{eqnarray}
%%%%%%%%%  Notice that, in asserting the validity of \eqref{claim Tj fine} with $j\ge 1$, we have used \eqref{positive} to estimate $-\de(s_{j+2})\le0$ in \eqref{claim Tj smaller deltaj-1}$_{j}$. By iterating \eqref{claim delta fine} we find
%%%%%%%%%  \begin{equation}
%%%%%%%%%    \label{fine deltaj}
%%%%%%%%%      \de(s_j)\le \tau^j\,\de(s/8)\le \tau^j\,\e_0\,,\qquad\forall j\ge 1\,,
%%%%%%%%%  \end{equation}
%%%%%%%%%  which, combined with \eqref{claim Tj fine} and \eqref{claim decay normal vectors fine}, gives, for every $j\ge 1$,
%%%%%%%%%  \begin{eqnarray}\label{fine Tj}
%%%%%%%%%  T_j\le C(n)\,\min\{1,\tau^{j-1}\}\,\de(s/8)\le C(n)\,\tau^j\,\de(s/8)\,,
%%%%%%%%%  \\\label{fine nuj}
%%%%%%%%%  |\nu_{H_j}-\nu_{H_{j-1}}|^2\le C(n)\,\min\{1,\tau^{j-2}\}\,\de(s/8)\le C(n)\,\tau^j\,\de(s/8)\,,
%%%%%%%%%  \end{eqnarray}
%%%%%%%%%  thanks also to $\tau=\tau(n)$ and, again, to \eqref{positive}. By \eqref{fine nuj}, for every $j\ge 0$, $k\ge 1$, we have
%%%%%%%%%  $|\nu_{H_{j+k}}-\nu_{H_j}|\le C(n)\,\sqrt{\de(s/8)}\,\sum_{h=1}^{k+1}\big(\sqrt\tau\big)^{j-1+h}$,
%%%%%%%%%  so that $|\nu_{H_j}-\nu_K|\to 0$ for some $K\in\H$ thanks to
%%%%%%%%%  \begin{equation}
%%%%%%%%%    \label{nuK convergence}
%%%%%%%%%    |\nu_K-\nu_{H_j}|^2\le C(n)\,\tau^j\,\de(s/8)\,,\qquad\forall j\ge 1\,.
%%%%%%%%%  \end{equation}
%%%%%%%%%  By arguing as in the proof of statement (ii) we find $u\in\X_{\s'}(\Sigma_K,s/32,\infty)$ for every $\s'>\s$ such that, for every $j$ large enough,
%%%%%%%%%  $\Sigma_K(u,s/32,4\,s_j)=\Sigma_{H_j}(u_j,s/32,4\,s_j)$,  and hence, by \eqref{claim is spherical graph},
%%%%%%%%%  \begin{equation}
%%%%%%%%%  \label{lambdaext bd easy corresponds final}
%%%%%%%%%  \mbox{$V$ corresponds to $\Sigma_K(u,s/32,\infty)$ on $\R^{n+1}\setminus B_{s/32}$}\,,
%%%%%%%%%  \end{equation}
%%%%%%%%%%%%%  Moreover, by construction (see \eqref{tr} and \eqref{vrtr}),
%%%%%%%%%%%%%  \[
%%%%%%%%%%%%%  u\Big(\frac{\p_K[\om+u_j\,\nu_{H_j}]}{|\p_K[\om+u_j\,\nu_{H_j}]|}\Big)
%%%%%%%%%%%%%  =\frac{\nu_K\cdot(\om+u_j\,\nu_{H_j})}{|\p_K[\om+u_j\,\nu_{H_j}]|}\,,\qquad\forall (\om,r)\in\S_{H_j}\times\big(s/32,4\,s_j\big)\,,
%%%%%%%%%%%%%  \]
%%%%%%%%%%%%%  so that $u_j\in\X_\s(\Sigma_{H_j},s/32,4\,s_j)$ and $|\nu_K-\nu_{H_j}|\to 0$ as $j\to\infty$ imply $u\in\X_\s(\Sigma_K,s/32,4\,s_j)$ for every $j$ large enough; in particular, $u\in\X_\s(\Sigma_K,s/32,\infty)$, and thus
%%%%%%%%%  which is \eqref{mesoscale thesis graphicality} with $S_*=+\infty$. To prove \eqref{decay deficit for exterior minimal surfaces}, we notice that if $r\in(s_j,s_{j+1})$ for some $j\ge1$, then, setting $\tau=(1/2)^\a$ (i.e., $\a=\log_{1/2}(\tau)\in(0,1)$) and noticing that $r/s\le 2^{j+1-3}$, by \eqref{is decreasing} and \eqref{fine deltaj} we have
%%%%%%%%%  \begin{eqnarray}\nonumber
%%%%%%%%%  \de(r)\!\!\!&\le&\!\!\!\de(s_j)\le \tau^j\,\de(s/8)=2^{-j\,a}\,\de(s/8)=
%%%%%%%%%  4^{-\a}\,2^{-(j-2)\a}\,\de(s/8)
%%%%%%%%%  \\\label{ex cont}
%%%%%%%%%  \!\!\!&\le&\!\!\! C(n)\,(s/r)^\a\,\de(s/8)\,,\,\,\,\,\,
%%%%%%%%%  \end{eqnarray}
%%%%%%%%%  where in the last inequality \eqref{positive} was used again; this proves \eqref{decay deficit for exterior minimal surfaces}. To prove \eqref{decay flatness for exterior minimal surfaces}, we recall that $\om_K(y)=\atan(|\nu_K\cdot\hat{y}|/|\p_K\,\hat{y}|)$, provided $\atan$ is defined on $\R\cup\{\pm\infty\}$, and where $\hat{y}=y/|y|$, $y\ne 0$. Now, by \eqref{lambdaext bd easy corresponds final},
%%%%%%%%%  \[
%%%%%%%%%  y=|y|\,\frac{\p_K\,\hat{y}+u(\p_K\,\hat{y},|y|)\,\nu_K}{\sqrt{1+u(\p_K\,\hat{y},|y|)^2}}\,,\qquad\forall y\in(\spt\,V)\setminus B_{s/32}\,,
%%%%%%%%%  \]
%%%%%%%%%  so that $|\p_K\,\hat{y}|\ge 1/2$ for $y\in(\spt V)\setminus B_{s/32}$; therefore, by \eqref{nuK convergence}, up to further decrease the value of $\e_0$, and recalling $\de(s/8)\le\e_0$, we conclude that
%%%%%%%%%  \begin{equation}
%%%%%%%%%    \label{nuK uniform projections}
%%%%%%%%%      |\p_{H_j}\,\hat{y}|\ge 1/3\,,\qquad\forall y\in(\spt V)\setminus B_{s/32}\,,
%%%%%%%%%  \end{equation}
%%%%%%%%%  for every $j\in\N\cup\{+\infty\}$ (if we set $H_\infty=K$). By \eqref{nuK uniform projections} we easily find
%%%%%%%%%  \[
%%%%%%%%%  |\om_K(y)-\om_{H_j}(y)|\le C\,|\nu_{H_j}-\nu_K|\,,\qquad\forall y\in(\spt V)\setminus B_{s/32}\,,\forall j\ge 1\,,
%%%%%%%%%  \]
%%%%%%%%%  from which we deduce that, if $j\ge 1$ and $r\in(s_j,s_{j+1})$, then
%%%%%%%%%  \begin{eqnarray*}
%%%%%%%%%    \frac1{r^n}\,\int_{A_r^{2\,r}}\om_K^2\,d\|V\|\!\!\!&\le&\!\!\!C(n)\,\Big\{\frac1{s_j^n}\,\int_{A_{s_j}^{s_{j+1}}}\om_K^2\,d\|V\|+
%%%%%%%%%    \frac1{s_{j+1}^n}\,\int_{A_{s_{j+1}}^{s_{j+2}}}\om_K^2\,d\|V\|\Big\}
%%%%%%%%%    \\
%%%%%%%%%    \!\!\!&\le&\!\!\!C(n)\,\Big\{\frac1{s_j^n}\,\int_{A_{s_j}^{s_{j+1}}}\om_{H_j}^2\,d\|V\|+
%%%%%%%%%    \frac1{s_{j+1}^n}\,\int_{A_{s_{j+1}}^{s_{j+2}}}\om_{H_{j+1}}^2\,d\|V\|\Big\}
%%%%%%%%%    \\
%%%%%%%%%    \!\!\!&&\!\!\!+C(n)\,\Gamma\,\big\{|\nu_K-\nu_{H_j}|^2+|\nu_K-\nu_{H_{j+1}}|^2\big\}\,,
%%%%%%%%%  \end{eqnarray*}
%%%%%%%%%  where \eqref{bounds needed for proof meso} was used to bound $\|V\|(A_\rho^{2\,\rho})\le \Gamma\,(2\,\rho)^n$ with $\rho=s_j,s_{j+1}\in(R,1/\Lambda)$.
%%%%%%%%%  By \eqref{claim is spherical graph fine} we can exploit \eqref{equuivalence between u square and omega square} on the first two integrals, so that taking \eqref{nuK convergence} into account we find that, if $j\ge 1$ and $r\in(s_j,s_{j+1})$, then
%%%%%%%%%  $r^{-n}\,\int_{A_r^{2\,r}}\om_K^2\,d\|V\|\le C(n)\{T_j+T_{j+1}\big\}+C(n)\,\Gamma\,\tau^j\,\de(s/8)
%%%%%%%%%        \le C(n)\,(1+\Gamma)\,\tau^j\,\de(s/8)$
%%%%%%%%%  where in the last inequality we have used \eqref{fine Tj}. Since $\tau^j\le C(n)\,(s/r)^\a$, we conclude the proof of \eqref{decay flatness for exterior minimal surfaces}, and thus, of the theorem.
\end{proof}

\section{Application of quantitative isoperimetry}\label{section existence and quantitative isoperimetry} Here we apply quantitative isoperimetry to prove Theorem \ref{thm main psi}-(i) and parts of Theorem \ref{thm main psi}-(iv).

\begin{theorem}\label{thm existence and uniform min}
If $W\subset\R^{n+1}$ is compact, $v>0$, then ${\rm Min}[\psi_W(v)]\ne\emptyset$.  Moreover, depending on $n$ and $W$ only, there are $v_0$, $C_0$, $\Lambda_0$ positive, $s_0\in(0,1)$, and $R_0(v)$  with $R_0(v)\to 0^+$ and $R_0(v)\,v^{1/(n+1)}\to\infty$ as $v\to\infty$, such that, if $v> v_0$ and $E_v$ is a minimizer of $\psi_W(v)$, then:

\noindent {\bf (i):} $E_v$ is a {\bf $(\Lambda_0/v^{1/(n+1)},s_0\,v^{1/(n+1)})$-perimeter minimizer with free boundary in $\Om$}, that is
\begin{equation}
\label{uniform lambda minimality}
P(E_v; \Om\cap B_r(x))\le P(F;\Om\cap B_r(x))+\frac{\Lambda_0}{v^{1/(n+1)}}\,\big|E_v\Delta F\big|\,,
\end{equation}
for every $F\subset\Om=\R^{n+1}\setminus W$ with $E_v\Delta F\cc B_r(x)$ and $r<s_0\,v^{1/(n+1)}$;
\noindent {\bf (ii):} $E_v$ determines $x\in\R^{n+1}$ such that
\begin{equation}\label{isop estimate 2}
|E_v\Delta B^{(v)}(x)| \le C_0\,v^{-1+1/[2(n+1)]}\,;
\end{equation}
if $\Rr(W)>0$, then $E_v$ also determines $u\in C^\infty(\pa B^{(1)})$ such that
\begin{eqnarray}
\label{x and u of Ev take 2}
&&(\pa E_v)\setminus B_{R_0\,v^{1/(n+1)}}
\\\nonumber
&&=\Big\{y+ v^{1/(n+1)}u\Big(\frac{y-x}{v^{1/(n+1)}}\Big)\,\nu_{B^{(v)}(x)}(y):y\in\pa B^{(v)}(x)\Big\}\setminus B_{R_0\,v^{1/(n+1)}}\,;
\end{eqnarray}
\noindent {\bf (iii):} if $\Rr(W)>0$ and $x$ and $u$ depend on $E_v$ as in \eqref{isop estimate 2} and \eqref{x and u of Ev take 2}, then
\begin{equation}\label{limsupmax goes to zero take 2}
\lim_{v\to\infty}\sup_{E_v\in{\rm Min}[\psi_W(v)]}\!\!\!\max\big\{\big||x|\,v^{-1/(n+1)}-\omega_{n+1}^{-1/(n+1)}\big|\,,
\|u\|_{C^1(\pa B^{(1)})}\big\}=0\,.
\end{equation}
\end{theorem}

\begin{remark}[Improved convergence]\label{remark improved convergence}
{\rm We will repeatedly use the following fact (see, e.g. [19, 17, 6, 7]): {\it If $\Om$ is an open set, $\Lambda\ge0$, $s>0$, if $\{F_j\}_j$ are {\bf $(\Lambda,s)$-perimeter minimizers in $\Om$}, i.e. if it holds that
\begin{equation}
\label{in comparing}
P(F_j;B_r(x))\le P(G_j; B_r(x))+\Lambda\,|F_j\Delta G_j|\,,
\end{equation}
whenever $G_j\Delta F_j\cc B_r(x)\cc \Om$ and $r<s$, and if $F$ is an open set with smooth boundary in $\Om$ such that $F_j\to F$ in $L^1_{{\rm loc}}(\Om)$ as $j\to\infty$, then for every $\Om'\cc \Om$ there is $j(\Om')$ such that
\[
(\pa F_j)\cap\Om'= \Big\{y+u_j(y)\,\nu_F(y):y\in \Om\cap\pa F\Big\}\cap\Om'\,,\qquad\forall j\ge j(\Om')\,,
\]
for a sequence $\{u_j\}_j\subset C^1(\Om\cap\pa F)$ with $\|u_j\|_{C^1(\Om\cap\pa F)}\to 0$.}
%(A proof is quickly obtained by combining \cite[Lemma 4.4]{} with a covering argument.)
Compare the terminology used in \eqref{uniform lambda minimality} and \eqref{in comparing}: when we add ``with free boundary'', the ``localizing balls'' $B_r(x)$ are not required to be compactly contained in $\Om$, and the perimeters are computed in $B_r(x)\cap\Om$.
%This allows competitors in \eqref{uniform lambda minimality} to differ from the minimizer near $\pa\Om$, while being constrained to be subsets of $\Om$. Hence, $(\Lambda,s)$-perimeter minimality as defined in \eqref{in comparing} is weaker than its ``with free boundary'' variant defined in \eqref{uniform lambda minimality}.
\end{remark}

\begin{proof}[Proof of Theorem \ref{thm existence and uniform min}] {\bf Step one:} We prove ${\rm Min}[\psi_W(v)]\ne\emptyset$ for all $v>0$. Since $W$ is compact, $B^{(v)}(x)\cc\Om$ for $|x|$ large. Hence there is $\{E_j\}_j$ with
\begin{equation}
\label{minimizing sequence}
E_j\subset\Om\,,\,\,\, |E_j|=v\,,\,\,\,P(E_j;\Om)\le \min\Big\{P(B^{(v)}),P(F;\Om)\Big\}+(1/j)\,,
\end{equation}
for every $F\subset\Om$ with $|F|=v$. Hence, up to extracting subsequences, $E_j\to E$ in $L^1_{\rm loc}(\R^{n+1})$ with $P(E;\Omega) \leq \varliminf_{j\to \infty}P(E_j;\Omega)$, where $E\subset\Om$ and $|E|\le v$. We now make three remarks concerning $E$:

\noindent {\bf (a):} {\it If $\{\Omega_i\}_{i\in I}$ are the connected components of $\Om$, then $\Om\cap\pa^*E=\emptyset$ if and only  $E = \bigcup_{i\in I_0} \Omega_i$} ($I_0\subset I$). Indeed, $\Omega \cap \partial^\ast E=\emptyset$ implies $\cl(\pa^*E)\cap \Omega =\partial E\cap \Omega$, hence $\pa E\subset\pa\Om$ and $E = \bigcup_{i\in I_0} \Omega_i$. The converse is immediate.

\noindent {\bf (b):} {\it If $\Om\cap\pa^*E\ne\emptyset$, then we can construct a system of ``volume--fixing variations'' for $\{E_j\}_j$}. Indeed, if $\Om\cap\pa^*E\ne\emptyset$, then there are $B_{S_0}(x_0)\cc\Om$ with $P(E;\pa B_{S_0}(x_0))=0$ and a vector field $X\in C^\infty_c(B_{S_0}(x_0);\R^{n+1})$ such that $\int_E\Div\,X=1$. By \cite[Theorem 29.14]{maggiBOOK}, there are constants $C_0,c_0>0$, depending on $E$ itself, with the following property: whenever $|(F\Delta E)\cap B_{S_0}(x_0)|<c_0$, then there is a smooth function $\Phi^F:\R^n\times(-c_0,c_0)\to\R^n$ such that, for each $|t|<c_0$, the map $\Phi_t^F=\Phi^F(\cdot,t)$ is a smooth diffeomorphism with $\{\Phi_t^F\ne\id\}\cc B_{S_0}(x_0)$, $|\Phi_t^F(F)|=|F|+t$, and $P(\Phi_t^F(F);B_{S_0}(x_0))\le(1+C_0\,|t|)\,P(F;B_{S_0}(x_0))$. For $j$ large enough, we evidently have $|(E_j\Delta E)\cap B_{S_0}(x_0)|<c_0$, and thus we can construct smooth functions $\Phi^j:\R^n\times(-c_0,c_0)\to\R^n$ such that, for each $|t|<c_0$, the map $\Phi_t^j=\Phi^j(\cdot,t)$ is a smooth diffeomorphism with $\{\Phi_t^j\ne\id\}\cc B_{S_0}(x_0)$, $|\Phi_t^j(E_j)|=|E_j|+t$, and $P(\Phi_t^j(E_j);B_{S_0}(x_0))\le(1+C_0\,|t|)\, P(E_j;B_{S_0}(x_0))$.

\noindent {\bf (c):} {\it If $\Om\cap\pa^*E\ne\emptyset$, then $E$ is bounded}. Since $|E|\le v<\infty$, it is enough to prove that $\Om\cap\pa^*E$ is bounded. In turn, taking $x_0\in\Om\cap\pa^*E$, and since $W$ is bounded and $|E|<\infty$, the boundedness of $\Om\cap\pa^*E$ descends immediately by the following density estimate: there is $r_1>0$ such that
\begin{equation}
\label{lower bound volume of E}
\begin{split}
&|E\cap B_r(x)|\ge c(n)\,r^{n+1}
\\
&\forall\,\,x\in\Om\cap\pa^*E\,,\,\, r<r_1\,,\,\, B_r(x)\cc \R^{n+1}\setminus \big(I_{r_1}(W)\cup B_{S_0}(x_0)\big)\,.
\end{split}
\end{equation}
To prove \eqref{lower bound volume of E}, let $r_1>0$ be such that $|B_{r_1}|<c_0$, let $x$ and $r$ be as in \eqref{lower bound volume of E}, and set $F_j=(\Phi_t^j(E_j)\cap B_{S_0}(x_0))\cup[E_j\setminus(B_r(x)\cup B_{S_0}(x_0))]$ for $t=|E_j\cap B_r(x)|$ (which is an admissible value of $t$ by $|B_{r_1}|<c_0$). In this way, $|F_j|=|E_j|=v$, and thus we can exploit \eqref{minimizing sequence} with $F=F_j$. A standard argument (see, e.g. \cite[Theorem 21.11]{maggiBOOK}) leads then to \eqref{lower bound volume of E}.
%  find that, for a.e. $r<r_1$,
%  \begin{eqnarray*}
%    &&P(E_j;\Om)-(1/j)\le P\big(E_j;\Om\setminus(B_r(x)\cup B_{S_0}(x_0))\big)+P(E_j;B_{S_0}(x_0))
%    \\
%    &&\hspace{4.5cm}+C_0\,\big(\psi_W(v)+1/j\big)\,u_j(r)+u_j'(r)
%  \end{eqnarray*}
%  where $u_j(r)=|E_j\cap B_r(x)|$, and thus $u_j'(r)=\H^n(E_j\cap\pa B_r(x))$ for a.e. $r>0$. Thus
%  $P(E_j;B_r(x))\le u'_j(r)+C_0\,(\psi_W(v)+1)\,u_j(r)+1/j$ for a.e. $r<r_1$, and adding $u_j'(r)$ on both sides and applying the Euclidean isoperimetric inequality to $E_j\cap B_r(x)$, we find
%  \[
%  c(n)\,u_j(r)^{n/(n+1)}\le 2\,u'_j(r)+C_0\,(\psi_W(v)+1)\,u_j(r)+\frac1j\,,\qquad\mbox{for a.e. $r<r_1$}\,.
%  \]
%  Since $C_0\,(\psi_W(v)+1)\,u_j(r)\le [c(n)/2] u_j(r)^{n/(n+1)}$ is equivalent to
%u_j(r)\le \Big(\frac{c(n)}{2\,C_0\,(\psi_W(v)+1)}\Big)^{n+1}\,,
%and since $u_j(r)\le \om_{n+1}\,r_1^{n+1}$, up to further decreasing the value of $r_1$ we can assume that $C_0\,(\psi_W(v)+1)\,u_j\le [c(n)/2] u_j^{n/(n+1)}$ on $(0,r_1)$, and thus deduce
%  \[
%  \frac{c(n)}2\,u_j(r)^{n/(n+1)}\le u'_j(r)+\frac1j\,,\qquad\mbox{for a.e. $r\in(0,r_1)$}\,.
%  \]
%  Letting $j\to\infty$ and setting $u(r)=|E\cap B_r(x)|$, we find $u^{n/(n+1)}\le C(n)\, u'$ a.e. on $(0,r_1)$, and thus (by $x\in\pa^*E \cap \Omega$) that $u(r)\ge c_0\,r^{n+1}$ for every $r\in(0,r_1)$, which is \eqref{lower bound volume of E}.

Now, since $\pa\Om\subset W$ is bounded, every connected component of $\Om$ with finite volume is bounded. Thus, by (a), (b) and (c) above, there is $R>0$ such that $W\cup E\,\cc \,B_R$. Since $|E\cap [B_{R+1}\setminus B_R]|=0$, we can pick $T\in(R,R+1)$ such that $\H^n(E_j\cap\pa B_T)\to 0$ and $P(E_j\setminus B_T)=\H^n(E_j\cap\pa B_T)+P(E_j;\Om\setminus B_T)$,  and consider the sets $F_j=(E_j\cap B_T)\cup B_{\rho_j}(y)$ corresponding to $\rho_j=(|E_j\setminus B_T|/\om_{n+1})^{1/(n+1)}$ and to $y\in\R^{n+1}$ which is independent from $j$ and such that $|y|>\rho_j+T$ (notice that $\sup_j\rho_j\le C(n)\,v^{1/(n+1)}$). Since $|F_j|=|E_j|=v$, \eqref{minimizing sequence} with $F=F_j$ and $P(B_{\rho_j})\le P(E_j\setminus B_T)$ give
\begin{eqnarray*}
P(E_j;\Om)-(1/j)\!\!&\le& \!\!\!\!P(F_j;\Om)\le P(E_j;\Om\cap B_T)+\H^n(E_j\cap\pa B_T)+P(B_{\rho_j})
\\
&\le&\!\!\!\!P(E_j;\Om)+2\,\H^n(E_j\cap\pa B_T)\,,
\end{eqnarray*}
so that, by the choice of $T$, $\{F_j\}_j$ is a minimizing sequence for $\psi_W(v)$, with $F_j\subset B_{T^*}$ and $T^*$ independent of $j$. We conclude by the Direct Method.

\noindent{\bf Step two:} We prove \eqref{isop estimate 2}. If $E_v$ a minimizer of $\psi_W(v)$ and $R>0$ is such that $W \cc B_R$, then by $P(E_v;\Om)\le P(B^{(v)})$ we have, for $v>v_0$, and $v_0$ and $C_0$ depending on $n$ and $W$,
\begin{eqnarray}
\label{cookie 1}
P(E_v\setminus B_{R})&\le&P(E_v;\Omega) + n\,\om_n\,R^n\le P(B^{(v)})+C_0
\\\nonumber
&\le&(1+(C_0/v))\,P(B^{(|E_v\setminus B_{R}|)})  + C_0\,,
\end{eqnarray}
where we have used that, if $v>2\,b>0$ and $\a=n/(n+1)$, then
\[
P(B^{(v)})\,P(B^{(v-b)})^{-1}-1=(v/(v-b))^\a-1\le \a\,b/(v-b)\le 2\,\a\,b\,v^{-1}\,.
\]
By combining \eqref{quantitative euclidean isop} and \eqref{cookie 1} we conclude that, for some $x\in\R^{n+1}$,
\[
c(n)\,\Big(\frac{|(E_v\setminus B_{R})\Delta B^{(|E_v\setminus B_{R}|)}(x)|}{|E_v \setminus B_{R}|}\Big)^2
\le \frac{P(E_v\setminus B_{R})}{P(B^{(|E_v\setminus B_{R}|)})}-1\le \frac{C_0}{v^{n/(n+1)}}\,,
\]
provided $v>v_0$. Hence we deduce \eqref{isop estimate 2} from
\begin{eqnarray*}
&&|E_v \Delta B^{(v)}(x)|=2\,|E_v \setminus B^{(v)}(x)|\le C_0+2\,\big|(E_v\setminus B_R) \setminus B^{(v)}(x)\big|
\\
&&\le C_0+2\,\big|(E_v\setminus B_R) \setminus B^{(|E_v\setminus B_{R}|)}(x)\big|\le C_0+|E_v\setminus B_R|\,C_0\,v^{-n/2\,(n+1)}\,.
\end{eqnarray*}

\noindent {\bf Step three:} We prove the existence of $v_0$, $\Lambda_0$, and $s_0$ such that every $E_v\in{\rm Min}[\psi_W(v)]$ with $v>v_0$ satisfies \eqref{uniform lambda minimality}. Arguing by contradiction, we assume the existence of $v_j\to\infty$, $E_j\in{\rm Min}[\psi_W(v_j)]$, $F_j\subset\Om$ with $|F_j\Delta E_j|>0$ and $F_j\Delta E_j\cc B_{r_j}(x_j)$ for some $x_j\in\R^{n+1}$ and $r_j=v_j^{1/(n+1)}/j$, such that
\[
P(E_j;\Om\cap B_{r_j}(x_j))\ge  P(F_j;\Om\cap B_{r_j}(x_j))+j\,v_j^{-1/(n+1)}\,\big|E_j\Delta F_j\big|\,.
\]
Denoting by $E_j^*$, $F_j^*$ and $\Om_j$ the sets obtained by scaling $E_j$, $F_j$ and $\Om$ by a factor $v_j^{-1/(n+1)}$, we find that $F_j^*\Delta E_j^*\cc B_{1/j}(y_j)$ for some $y_j\in\R^{n+1}$, and
\begin{equation}
\label{unifor min contra scaled}
P(E_j^*;\Om_j\cap B_{1/j}(y_j))\ge  P(F_j^*;\Om_j\cap B_{1/j}(y_j))+j\,\big|E_j^*\Delta F_j^*\big|\,.
\end{equation}
By \eqref{isop estimate 2} there are $z_j\in\R^{n+1}$ such that $|E_j^*\Delta B^{(1)}(z_j)|\to 0$. We can therefore use the volume-fixing variations of $B^{(1)}$ to find diffeomorphisms $\Phi^j_t:\R^n\to\R^n$ and constants $c(n)$ and $C(n)$ such that, for every $|t|<c(n)$, one has $\{\Phi^j_t\ne\id\}\cc U_j$ for some open ball $U_j$ with $U_j\cc \Om_j\setminus B_{1/j}(y_j)$,
$|\Phi^j_t(E_j^*)\cap U_j|=|E_j^*\cap U_j|+t$, and $P(\Phi^j_t(E_j^*);U_j)\le (1+C(n)\,|t|)\,P(E_j^*;U_j)$.  Since $F_j^*\Delta E_j^*\cc B_{1/j}(y_j)$ implies $||F_j^*|-|E_j^*||<c(n)$ for $j$ large, if $t=|E_j^*|-|F_j^*|$, then $G_j^*=\Phi^j_t(F_j^*)$ is such that $|G_j^*|=|E_j^*|$, and by $E_j\in{\rm Min}[\psi_W(v_j)]$,
\begin{eqnarray*}
&&\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!P(E_j^*;\Om_j)\le P(G_j^*;\Om_j)
\le P\big(E_j^*;\Om_j\setminus(U_j\cup B_{1/j}(y_j))\big)
\\
&&+P(F_j^*;\Om_j\cap B_{1/j}(y_j))+P(E_j^*;U_j)+C(n)\,P(E_j^*;U_j)\,\big|E_j^*\Delta F_j^*\big|\,.
\end{eqnarray*}
Taking into account $P(E^*_j;U_j)\le \psi_W(v_j)/v_j^{n/(n+1)}\le C(n)$, we thus find
\begin{eqnarray*}
P(E_j^*;\Om_j\cap B_{1/j}(y_j))\le P(F_j^*;\Om_j\cap  B_{1/j}(y_j))+C(n)\,\big|E_j^*\Delta F_j^*\big|\,,
\end{eqnarray*}
which, by \eqref{unifor min contra scaled}, gives $j\,\big|E_j^*\Delta F_j^*\big|\le C(n)\,\big|E_j^*\Delta F_j^*\big|$. Since $|E_j^*\Delta F_j^*|>0$, this is a contradiction for $j$ large enough.

\noindent {\bf Step four:} We now prove that, if $\Rr(W)>0$, then
\begin{equation}\label{xv equation}
\lim_{v\to\infty}\,\sup_{E_v\in{\rm Min}[\psi_W(v)]}\,\big||x|\,v^{-1/(n+1)}-\omega_{n+1}^{-1/(n+1)}\big|=0\,,
\end{equation}
where $x$ is related to  $E_v$  by \eqref{isop estimate 2}. In proving \eqref{xv equation} we will use the assumption $\Rr(W)>0$ and the energy upper bound
\begin{equation}
\label{bello}
\varlimsup_{v\to \infty} \psi_W(v) - P(B^{(v)})\le-\Rr(W)\,.
\end{equation}
A proof of \eqref{bello} is given in step one of the proof of Theorem \ref{thm main psi}, see section \ref{section resolution for exterior}; in turn, that proof is solely based on the results from section \ref{section isoperimetric residues}, where no part of Theorem \ref{thm existence and uniform min} (not even the existence of minimizers in $\psi_W(v)$) is ever used. This said, when $|W|>0$, and thus $\Ss(W)>0$, one can replace \eqref{bello} in the proof of \eqref{xv equation} by the simpler upper bound
\begin{equation}\label{strict}
\varlimsup_{v\to \infty} \psi_W(v) - P(B^{(v)})\le-\Ss(W)\,,
\end{equation}
where, we recall, $\Ss(W)=\sup\{\H^n(W\cap\Pi):\mbox{$\Pi$ is a hyperplane in $\R^{n+1}$}\}$. To prove \eqref{strict}, given $\Pi$, we construct competitors for $\psi_W(v)$ by intersecting $\Om$ with balls $B^{(v')}(x_v)$ with $v'>v$ and $x_v$ such that
$|B^{(v')}(x_v)\setminus W|=v$ and $\H^n(W\cap\pa B^{(v')}(x_v))\to\H^n(W\cap\Pi)$ as $v\to\infty$. Hence,
thus giving \eqref{strict}. The proof of \eqref{bello} is identical in spirit to that of \eqref{strict}, with the difference that to glue a large ball to $(F,\nu)\in{\rm Max}[\Rr(W)]$ we will need to establish the decay of $\pa F$ towards a hyperplane parallel to $\nu^\perp$ to the high degree of precision expressed in \eqref{asymptotics of F}. Now to prove \eqref{xv equation}: by contradiction, consider $v_j\to\infty$, $E_j\in{\rm Min}[\psi_W(v_j)]$, and $x_j\in\R^{n+1}$ with $\inf_{x\in\R^{n+1}}|E_j\Delta B^{(v_j)}(x)|=|E_j\Delta B^{(v_j)}(x_j)|$, such that
\begin{equation}\label{bad assumption on xv}
\varliminf_{j\to\infty}\big||x_j|\,v_j^{-1/(n+1)}-\omega_{n+1}^{-1/(n+1)}\big|>0\,,
\end{equation}
and set $\l_j=v_j^{-1/(n+1)}$, $E_j^*=\l_j\,(E_j-x_j)$, $W_j^*=\l_j\,(W-x_j)$, and $\Om_j^*=\l_j\,(\Om-x_j)$. By \eqref{uniform lambda minimality}, each $E_j^*$ is a  $(\Lambda_0,s_0)$-perimeter minimizer with free boundary in $\Om_j^*$. By \eqref{isop estimate 2} and the defining property of $x_j$, $E_j^*\to B^{(1)}$ in $L^1(\R^{n+1})$. Moreover, $\diam(W_j^*)\to 0$ and, by \eqref{bad assumption on xv},
\begin{equation}
\label{because}
\varliminf_{j\to\infty}\dist\big(W_j^*,\pa B^{(1)}\big)>0\,.
\end{equation}
Thus there is $z_0\not\in\pa B^{(1)}$ such that, for every $\rho<\dist(z_0,\pa B^{(1)})$, there is $j(\rho)$ such that $\{E_j^*\}_{j\ge j(\rho)}$ is a sequence of $(\Lambda_0,s_0)$-perimeter minimizers in $\R^{n+1}\setminus B_{\rho/2}(z_0)$. By Remark \ref{remark improved convergence}, up to increasing $j(\rho)$, $(\pa E_j^*)\setminus B_{\rho}(z_0)$ is contained in the normal graph over $\pa B^{(1)}$ of $u_j$ with $\|u_j\|_{C^1(\pa B^{(1)})}\to 0$; in particular, by \eqref{because}, $(\pa E_j^*)\setminus B_{\rho}(z_0)$ is disjoint from $W_j^*$. By the constant mean curvature condition satisfied by $\Om\cap\pa E_j^*$, and by Alexandrov's theorem [2], $(\pa E_j^*)\setminus B_{\rho}(z_0)$ is a sphere $M_j^*$ for $j\ge j(\rho)$. Let $B_j^*$ be the ball bounded by $M_j^*$. Since $M_j^*\cap W_j^*=\emptyset$, we have either one of the following:

\noindent {\bf Case one:} $W_j^*\subset B_j^*$. We have $\pa[B_j^*\cup E_j^*]\subset M_j^*\cup[(\pa E_j^*)\setminus \cl(B_j^*)]\subset (\pa E_j^*)\setminus W_j^*$, so that, by $|B_j^*\cup E_j^*|\ge |E_j^*|+|W_j^*|\ge 1$, we find $P(E_j^*;\Om_j^*)\ge P(B_j^*\cup E_j^*)\ge P(B^{(1)})$, that is, $\psi_W(v_j)\ge P(B^{(1)})$, against \eqref{bello}.

\noindent {\bf Case two:}  $W_j^*\cap B_j^*=\emptyset$. In this case, $E_j^*=B_j^*\cup G_j^*$, where $G_j^*$ is the union of the connected components of $E_j^*$ whose boundaries have non-empty intersection with $W_j^*$: in other words, we are claiming that $B_j^*$ is the only connected component of $E_j^*$ whose closure is disjoint from $W_j^*$. Indeed, if this were not the case, we could recombine all the connected components of $E_j^*$ with closure disjoint from $W_j^*$ into a single ball of same total volume, centered far away from $W_j^*$, in such a way to strictly decrease $P(E_j^*;\Om_j^*)$, against $E_j\in{\rm Min}[\psi_W(v_j)]$. Let us now set
$G_j=x_j+v_j^{1/(n+1)}\,G_j^*$ and $U_j=x_j+v_j^{1/(n+1)}\,B_j^*$, so that $E_j=G_j\cup U_j$ and $\dist(G_j,U_j)>0$.

If we start sliding $U_j$ from infinity towards $G_j\cup W$ along arbitrary directions, then at least one of the resulting ``contact points'' $z_j$ belongs to $\Om\cap\pa G_j$: if this were not the case, then $G_j$ would be contained in the convex envelope of $W$, so that $|B_j|=|E_j|-|G_j|\ge v_j-C(W)$, and thus, by $\psi_W(v_j)=P(E_j;\Om)\ge P(B_j;W)=P(B_j)$, and by $P(B_j)\ge P(B^{(v_j-C(W))})\ge P(B^{(v_j)})-C(W)\,v_j^{-1/(n+1)}$, against with \eqref{bello} for $j$ large.

By construction, there is a half-space $H_j$ such that $G_j\subset H_j$, $z_j\in(\pa G_j)\cap(\pa H_j)$, and $G_j$ is a perimeter minimizer in $B_r(z_j)$ for some small $r>0$. By the strong maximum principle, see, e.g. \cite[Lemma 2.13]{dephilippismaggiCAP-ARMA}, $G_j$ has $H_j-z_j$ as its unique blowup at $z_j$. By De Giorgi's regularity theorem, see e.g. \cite[Part III]{maggiBOOK}, $G_j$ is an open set with smooth boundary in a neighborhood of $z_j$. Therefore, if we denote by $U_j'$ the translation of $U_j$ constructed in the sliding argument, then, $E_j'=G_j\cup U_j'\in{\rm Min}[\psi_W(v)]$ and, in a neighborhood of $z_j$, $E_j'$ is the union of two disjoint sets with smooth boundary which touch tangentially at $z_j$. In particular, $|E_j'\cap B_r(z_j)|/|B_r|\to 1$ as $r\to 0^+$, against volume density estimates implied by \eqref{uniform lambda minimality}, see, e.g. \cite[Theorem 21.11]{maggiBOOK}.

\noindent {\bf Step five:} We finally show the existence of $v_0$ and $R_0(v)$ with $R_0(v)\to 0^+$ and $R_0(v)\,v^{1/(n+1)}\to \infty$, such that each $E_v\in{\rm Min}[\psi_W(v)]$ with $v>v_0$ determines $x$ and $u\in C^{\infty}(\partial B^{(1)})$ such that \eqref{x and u of Ev take 2} holds and $\sup_{E_v} \|u \|_{C^1(\partial B^{(1)})}\to 0$ as $v\to\infty$. To this end, let us consider $v_j\to\infty$, $E_j\in{\rm Min}[\psi_W(v_j)]$, and define $x_j$, $E_j^*$ and $W_j^*$ as in step four. Thanks to \eqref{xv equation}, there is $z_0\in\pa B^{(1)}$ s.t. $\dist(z_0,W_j^*)\to 0$. In particular, for every $\rho>0$, we can find $j(\rho)\in\N$ such that if $j\ge j(\rho)$, then $E_j^*$ is a $(\Lambda_0,s_0)$-perimeter minimizer in $\R^{n+1}\setminus B_\rho(z_0)$, with $E_j^*\to B^{(1)}$. By Remark \ref{remark improved convergence}, there are $u_j\in C^1(\pa B^{(1)})$ such that
\[
(\pa E_j^*)\setminus B_{2\,\rho}(z_0)=\big\{y+u_j(y)\,\nu_{B^{(1)}}(y):y\in\pa B^{(1)}\big\}\setminus B_{2\,\rho}(z_0)\,,\,\,\forall j\ge j(\rho)\,,
\]
and $\|u_j\|_{C^1(\pa B^{(1)})}\to 0$. By the arbitrariness of $\rho$ and by a contradiction argument, \eqref{x and u of Ev take 2} holds with $R_0(v)\to 0^+$ such that $R_0(v)\,v^{1/(n+1)}\to\infty$ as $v\to\infty$, and with the uniform decay of $\|u \|_{C^1(\partial B^{(1)})}$.
\end{proof}

\section{Properties of isoperimetric residues}\label{section isoperimetric residues} Here we prove Theorem \ref{thm main of residue}. It will be convenient to introduce some notation for cylinders and slabs in $\R^{n+1}$: precisely, given $r>0$, $\nu\in\SS^n$ and $I\subset\R$, and setting $\pp_{\nu^\perp}(x)=x-(x\cdot\nu)\,\nu$ ($x\in\R^{n+1}$), we let
\begin{eqnarray}\nonumber
\textbf{D}_r^\nu&=&\big\{x\in\R^{n+1}:|\pp_{\nu^\perp}x|<r\,,x\cdot\nu=0\big\}\,,
\\\nonumber
\textbf{C}_r^\nu&=&\big\{x\in\R^{n+1}:|\pp_{\nu^\perp}x|<r\big\}\,,
\\\label{cylinders and slabs}
\textbf{C}_{r,I}^\nu&=&\big\{x\in\R^{n+1}:|\pp_{\nu^\perp}x|<r\,,x\cdot\nu\in I\big\}\,,
\\\nonumber
\partial_\ell \textbf{C}_{r,I}^\nu&=&\big\{x\in\R^{n+1}:|\pp_{\nu^\perp}x|=r\,,x\cdot\nu\in I\big\}\,,
\\\nonumber
\textbf{S}_{I}^\nu &=& \big\{x\in\R^{n+1}: x \cdot \nu \in I \big\}\,.
\end{eqnarray}
In each case, given $x\in\R^{n+1}$, we also set $\textbf{D}_r^\nu(x)=x+\textbf{D}_r^\nu$, $\textbf{C}_r^\nu(x)=x+\textbf{C}_r^\nu$, etc. We premise the following proposition, used in the proof of Theorem \ref{thm main of residue} and Theorem \ref{thm main psi}, and based on \cite[Proposition 1 and Proposition 3]{Scho83}.

\begin{proposition}\label{prop schoen}
Let $n\ge 2$, $\nu\in\SS^n$, and let $f$ be a Lipschitz solution to the minimal surface equation on $\nu^\perp\setminus\cl(\DD_R^\nu)$.
If $n=2$, assume in addition that $M=\{x+f(x)\,\nu:|x|>R\}$ is stable and has natural area growth, i.e.
\begin{eqnarray}
\label{schoen 1}
\int_M\,|\nabla^M\vphi|^2-|A|^2\,\vphi^2\ge0\,,&&\qquad\forall\vphi\in C^1_c(\R^3\setminus B_R)\,,
\\
\label{schoen 2}
\H^2(M\cap B_r)\le C\,r^2\,,&&\qquad\forall r>R\,.
\end{eqnarray}
Then there are $a,b\in\R$ and $c\in\nu^\perp$ such that, for every $|x|>R$,
\begin{eqnarray}
\label{schoen conclusion 1}
&&\big|f(x)-\big(a+b\,|x|^{2-n}+(c\cdot x)\,|x|^{-n}\big)\big|\le C\,|x|^{-n}\,,\,\,(n\ge 3)
\\
\label{schoen conclusion 2}
&&\big|f(x)-\big(a+b\,\log|x|+(c\cdot x)\,|x|^{-2}\big)\big|\le C\,|x|^{-2}\,,\,\, (n=2)
\\
\label{schoen derivatives}
&&\max\Big\{|x|^{n-1}\,|\nabla f(x)|,|x|^n\,|\nabla^2f(x)|:|x|>R\Big\}\le C\,,\,\,(\mbox{every $n$})\,.
\end{eqnarray}
\end{proposition}

\begin{proof} If $n\ge 3$, the fact that $\nabla f$ is bounded allows one to represent $f$ as the convolution with a singular kernel which, by a classical result of Littman,  Stampacchia, and Weinberger [25], is comparable to the Green's function of $\R^n$; \eqref{schoen conclusion 1} is then deduced starting from that representation formula. For more details, see \cite[Proposition 3]{Scho83}. In the case $n=2$, by \eqref{schoen 1} and \eqref{schoen 2}, we can exploit a classical ``logarithmic cut-off argument'' to see that $M$ has finite total curvature, i.e. $\int_M |K| \,d\H^2< \infty$, where $K$ is the Gaussian curvature of $M$. As a consequence, see, e.g. \cite[Section 1.2]{PerezRos}, the compactification $\ov{M}$ of $M$ is a Riemann surface with boundary, and $M$ is conformally equivalent to $\ov{M}\setminus\{p_1,...,p_m\}$, where $p_i$ are interior points of $\ov{M}$. One can thus conclude by the argument in \cite[Proposition 1]{Scho83} that $M$ has $m$-many ends satisfying the decay \eqref{schoen conclusion 2}, and then that $m=1$ thanks to the fact that $M=\{x+f(x)\,\nu:|x|>R\}$.
%to see that, for each $i=1,...,m$, there exist a plane $\Pi_i$ and a solution of the minimal surfaces equation $f_i$ over an exterior domain in $\Pi_i$ such that $M$ contains the graph of $f_i$, and such that $f_i$ satisfies
%  \label{schoen conclusion 2 proof}
%\Big|f_i(x) -\Big( a_i + b_i\,\log |x| + \frac{c_i\cdot x}{|x|^2}\Big)\Big|\le \frac{C}{|x|^2}\,,\qquad\forall x\in\Pi_i\,,|x|\ge R_i\,,
%for suitable $a_i,b_i\in\R$ and $c_i\in\Pi$. Evidently, the fact that $M=\{x+f(x)\,\nu:|x|>R\}$ implies $m=1$, and \eqref{schoen conclusion 2 proof} implies \eqref{schoen conclusion 2}.
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm main of residue}] {\bf Step one:} Given a hyperplane $\Pi$ in $\R^{n+1}$, if $F$ is a half-space with $\pa F=\Pi$ and $\nu$ is a unit normal to $\Pi$, then ${\rm res}_W(F,\nu)=\H^n(W\cap\Pi)$. Therefore the lower bound in \eqref{R larger than S} follows by
\begin{equation}
\label{famous lb}
\Rr(W)\ge\Ss(W)=\sup\big\{\H^n(\Pi\cap W):\mbox{$\Pi$ an hyperplane in $\R^{n+1}$}\big\}\,.
\end{equation}

\noindent {\bf Step two:} We notice that, if  $(F,\nu)\in\F$, then by \eqref{def Sigma nu 1}, \eqref{def Sigma nu 2}, and the divergence theorem (see, e.g., \cite[Lemma 22.11]{maggiBOOK}), we can define a Radon measure on the open set $\nu^\perp\setminus\pp_{\nu^\perp}(W)$ by setting
\begin{equation}
\label{is a radon measure}
\mu(U)=P\big(F;(\pp_{\nu^\perp})^{-1}(U)\big)-\H^n(U)\,,\qquad U\subset\nu^\perp\setminus\pp_{\nu^\perp}(W)\,.
\end{equation}
In particular, setting $R'=\inf\{\rho: W\subset\CC_\rho^\nu\}$, $\mu(\DD_R^\nu\setminus\pp_{\nu^\perp}(W))\ge0$  gives
\[
P(F;\CC_R^\nu\setminus W)\ge\om_n\,R^n-\H^n(\pp_{\nu^\perp}(W))\,,\qquad\forall R>R'\,,
\]
while the identity
\begin{eqnarray*}
\om_n\,R^n-P(F;\CC_R^\nu\setminus W)=-\mu(\DD_R^\nu\setminus\DD_{R'}^\nu)+\om_n\,(R')^n-P(F;\CC_{R'}^\nu\setminus W)
\end{eqnarray*}
(which possibly holds as $-\infty=-\infty$ if $P(F;\CC_{R'}^\nu\setminus W)=+\infty$) gives that
\begin{equation}
\label{perimeter is decreasing}
R\in(R',\infty)\mapsto \om_n\,R^n-P(F;\CC_R^\nu\setminus W)\,\,\,
\mbox{is {\it decreasing} on $(R',\infty)$}\,.
\end{equation}
In particular, the limsup defining ${\rm res}_W$ always exists as a limit.

\noindent {\bf Step three:} We prove the existence of $(F,\nu)\in{\rm Max}[\Rr(W)]$ and \eqref{local perimeter minimizer}. We first claim that if $\{(F_j,\nu_j)\}_j$ is a maximizing sequence for $\Rr(W)$, then, in addition to $\pp_{\nu_j^\perp}(\pa F_j)=\nu_j^\perp$, one can modify $(F_j,\nu_j)$, preserving the optimality in the limit $j\to \infty$, so that (setting $X\subset^{\L^{n+1}} Y$ for $|X\setminus Y|=0$)
\begin{eqnarray}
\label{properties of Fj updated}
&&\pa F_j\subset \textbf{S}_{[A_j,B_j]}^{\nu_j}\,,\,\,\, \textbf{S}_{(-\infty, A_j)}^{\nu_j}\stackrel{\L^{n+1}}{\subset}F_j\,,
\,\,\, \textbf{S}_{(B_j,\infty)}^{\nu_j} \stackrel{\L^{n+1}}{\subset}\R^{n+1}\setminus F_j\,,\hspace{0.5cm}
\\
\label{E containment}
&&\mbox{where}\,\,[A_j,B_j]=\bigcap\big\{(\a,\b):W\subset \textbf{S}_{(\a,\b)}^{\nu_j}\big\}\,.
\end{eqnarray}
Indeed, since $(F_j,\nu_j)\in\F$, for some $\a_j<\b_j\in\R$ we have
\begin{equation}
\label{properties of Fj}
\pa F_j\subset \textbf{S}_{[\a_j,\beta_j]}^{\nu_j}\,,\qquad \pp_{\nu_j^\perp}(\pa F_j)=\nu_j^\perp\,.
\end{equation}
Would it be that either $\textbf{S}_{(-\infty, \a_j)\cup(\b_j,\infty)}^{\nu_j} \subset_{\L^{n+1}} F_j$ or $\textbf{S}_{(-\infty, \a_j)\cup(\b_j,\infty)}^{\nu_j} \subset_{\L^{n+1}} \R^{n+1}\setminus F_j$, then, by the divergence theorem and by $\pp_{\nu_j^\perp}(\pa F_j)=\nu_j^\perp$,
\[
P(F_j;\CC_R^{\nu_j}\cap\Om)\ge 2\,\big(\om_n\,R^n-\H^n(\pp_{\nu_j^\perp}(W))\big)\,,\qquad\forall R>0\,,
\]
and thus ${\rm res}_{W}(F_j,\nu_j)=-\infty$; in particular, $(F_j,\nu_j)\in\F$ being a maximizing sequence, we would have $\Rr(W)=-\infty$, against \eqref{famous lb}. This proves the validity (up to switching $F_j$ with $\R^{n+1}\setminus F_j$), of the inclusions
\begin{equation}
\label{properties of Fj updated new}
\textbf{S}_{(-\infty, \a_j)}^{\nu_j} \subset_{\L^{n+1}} F_j\,,
\qquad \textbf{S}_{(\b_j,\infty)}^{\nu_j} \subset_{\L^{n+1}} \R^{n+1}\setminus F_j\,.
\end{equation}
Thanks to \eqref{properties of Fj updated new} (and by exploiting basic set operations on sets of finite perimeter, see, e.g., \cite[Theorem 16.3]{maggiBOOK}),  we see that
\begin{eqnarray}\label{truncate up to the obstacle}
&&\mbox{$F_j^*=\big(F_j\cup \textbf{S}_{(-\infty,A_j-1/j)}^{\nu_j}\big)\cap \textbf{S}_{(-\infty,B_j+1/j)}^{\nu_j}$ satisfies}
\\\nonumber
&&(F_j^*,\nu_j)\in\F\,,\qquad P\big(F_j^*;\CC_R^{\nu_j}\setminus W\big)\le P\big(F_j;\CC_R^{\nu_j}\setminus W\big)\,,\qquad\forall R>0\,;
\end{eqnarray}
in particular, $\{(F_j^*,\nu_j)\}_j$ is also a maximizing sequence for $\Rr(W)$. By standard compactness theorems there are $F$ of locally finite perimeter in $\R^{n+1}$ and $\nu\in\SS^n$ such that $F_j\to F$ in $L^1_{{\rm loc}}(\R^{n+1})$ and $\nu_j\to\nu$. If $A\cc\CC_R^\nu\setminus W$ is open, then, for $j$ large enough, $A\cc \CC_R^{\nu_j}\setminus W$, and thus
\begin{equation}
\label{liminf on cylinders}
P(F;\CC_R^\nu\setminus W)=\sup_{A\cc\CC_R^\nu\setminus W}\,P(F;A)\le\varliminf_{j\to\infty}P(F_j;\CC_R^{\nu_j}\setminus W)\,.
\end{equation}
By \eqref{perimeter is decreasing}, $R\mapsto \om_n\,R^n-P(F_j;\CC_R^{\nu_j}\setminus W)$ is decreasing on $R>R_j=\inf\{\rho:W\subset \CC_\rho^{\nu_j}\}$. By $\sup_jR_j\le C(W)<\infty$ and \eqref{liminf on cylinders} we have
\[
\om_n\,R^n-P(F;\CC_R^\nu\setminus W)\ge\varlimsup_{j\to\infty}\om_n\,R^n-P(F_j;\CC_R^{\nu_j}\setminus W)\ge\varlimsup_{j\to\infty}{\rm res}_W(F_j,\nu_j)\,,
\]
for every $R>C(W)$; in particular, letting $R\to\infty$,
\begin{equation}
\label{max inq}
{\rm res}_W(F,\nu)\ge\varlimsup_{j\to\infty}{\rm res}_W(F_j,\nu_j)=\Rr(W)\,.
\end{equation}
By $F_j\to F$ in $L^1_{{\rm loc}}(\R^{n+1})$, $\pa F=\cl(\pa^*F)$ is contained in the set of accumulation points of sequences $\{x_j\}_j$ with $x_j\in\pa F_j$, so that \eqref{properties of Fj updated} gives
\begin{equation}
\label{properties of F updated}
\pa F\subset \textbf{S}_{[A,B]}^{\nu}\,,\qquad \textbf{S}_{(-\infty, A)}^{\nu} \subset_{\L^{n+1}} F\,,
\qquad \textbf{S}_{(B,\infty)}^{\nu} \subset_{\L^{n+1}}\R^{n+1}\setminus F\,,
\end{equation}
if $[A,B]=\bigcap\{(\a,\b):W\subset\textbf{S}^\nu_{(\a,\b)}\}$. Therefore $(F,\nu)\in\F$, and thus, by \eqref{max inq}, $(F,\nu)\in{\rm Max}[\Rr(W)]$. We now show that \eqref{max inq} implies \eqref{local perimeter minimizer}, i.e.
\begin{equation}\label{local perimeter minimizer proof}
P(F;\Omega \cap B) \leq P(G;\Omega \cap B)\,,\qquad\mbox{$\forall F\Delta G\cc B$, $B$ a ball}\,.
\end{equation}
Indeed, should \eqref{local perimeter minimizer proof} fail, we could find $\de>0$ and $G\subset\R^{n+1}$ with $F\Delta G\cc B$ for some ball $B$, such that $P(G;B\setminus W)+\de\le P(F;B\setminus W)$. For $R$ large enough to entail $B\cc \CC_R^\nu$ we would then find
\[
{\rm res}_W(F,\nu)+\de\le\om_n\,R^n-P(F;\CC_R^\nu\setminus W)+\de\le\om_n\,R^n-P(G;\CC_R^\nu\setminus W)\,,
\]
which, letting $R\to\infty$, would violate the maximality of $(F,\nu)$ in $\Rr(W)$.

\noindent {\bf Step four:} We show that if $\Rr(W)>0$ and $(F,\nu)\in{\rm Max}[\Rr(W)]$, then $\partial F\subset\mathbf{S}_{[A,B]}^\nu$ for $A,B$ as in \eqref{properties of F updated}. Otherwise, by the same truncation procedure leading to \eqref{truncate up to the obstacle} and by $(F,\nu)\in{\rm Max}[\Rr(W)]$, we would find
\begin{equation*}
\omega_n R^n - P\big(F^*;\CC_R^{\nu_j}\setminus W\big)\geq \omega_n R^n -P\big(F;\CC_R^{\nu_j}\setminus W\big)\geq \mathcal{R}(W) \qquad\forall R>0\,,
\end{equation*}
so that $(F^\ast,\nu)\in{\rm Max}[\Rr(W)]$ too. Now $P\big(F;\CC_R^{\nu_j}\setminus W\big) - P\big(F^*;\CC_R^{\nu_j}\setminus W\big)  $ is increasing in $R$, and since ${\rm res}_W(F,\nu) = {\rm res}_W(F^\ast,\nu)$, it follows that $P\big(F;\CC_R^{\nu_j}\setminus W\big) = P\big(F^*;\CC_R^{\nu_j}\setminus W\big) $ for large $R$. But this can hold only if $\partial F \cap \Omega$ is an hyperplane disjoint from $W$, in which case $\mathcal{R}(W)={\rm res}_W(F,\nu)=0$.

\noindent {\bf Step five:} Still assuming $\Rr(W)>0$, we complete the proof of statement (ii) by proving \eqref{asymptotics of F}. By \eqref{properties of F updated}, if $(F,\nu)\in{\rm Max}[\Rr(W)]$, then $F/R\to H^-=\{x\in\R^{n+1}:x\cdot\nu<0\}$ in $L^1_{{\rm loc}}(\R^{n+1})$ as $R\to\infty$. By \eqref{local perimeter minimizer proof} and by improved convergence (i.e., Remark \ref{remark improved convergence} -- notice carefully that $\pa F$ is bounded in the direction $\nu$ thanks to step four), we find $R_F>0$ and functions $\{f_R\}_{R>R_F}\subset C^1(\DD_2^\nu\setminus\DD_1^\nu)$ such that
\[
\big(\CC_2^\nu\setminus\CC_1^\nu\big)\cap\pa (F/R)=\big\{x+f_R(x)\,\nu:x\in \DD_2^\nu\setminus\DD_1^\nu\big\}\,,\qquad\forall R>R_F\,.
\]
with $\|f_R\|_{C^1(\DD_2^\nu\setminus\DD_1^\nu)}\to 0$ as $R\to\infty$. Scaling back to $F$ we deduce that
\begin{equation}
\label{F represented by u}
(\pa F)\setminus\CC_{R_F}^\nu=\big\{x+f(x)\,\nu:x\in\nu^\perp\setminus\DD_{R_F}^\nu\big\}\,,
\end{equation}
for a (necessarily smooth) solution $f$ to the minimal surfaces equation with
\begin{equation}
\label{u estimates}
\|f\|_{C^0(\nu^\perp\setminus\DD_{R_F}^\nu)}\le B-A\,,\qquad
\lim_{R\to\infty}\|\nabla f\|_{C^0(\DD_{2\,R}^\nu\setminus \DD_{R}^\nu)}=0\,,
\end{equation}
thanks to the fact that $f(x)=R\,f_R(x/R)$ if $x\in\DD_{2\,R}^\nu\setminus \DD_{R}^\nu$. {\bf When $n\ge 3$}, \eqref{asymptotics of F} follows by \eqref{F represented by u} and Proposition \ref{prop schoen}. {\bf When $n=2$}, \eqref{schoen 1} holds by \eqref{local perimeter minimizer proof}. To check \eqref{schoen 2}, we deduce by ${\rm res}_W(F,\nu)\ge0$ the existence of $R'>R_F$ such that $\om_n\,R^n\ge P(F;\CC_R^\nu\setminus W)-1$ if $R>R'$. In particular, setting $M=(\pa F)\setminus B_{R_F}$, for $R>R'$ we have
\[
\H^2(M\cap B_R)\le \H^2(M\cap W)+P(F;\CC_R^\nu\setminus W)\le \om_n\,R^n+1+\H^2(M\cap W)\le C\,R^n\,,
\]
provided $C=\om_n+[(1+\H^2(M\cap W))/(R')^n]$; while if $R\in(R_F,R')$, then $\H^2(M\cap B_R)\le C\,R^n$ with $C=\H^2(M\cap B_{R'})/R_F^n$. This said, we can apply Proposition \ref{prop schoen} to deduce \eqref{schoen conclusion 2}. Since $\pa F$ is bounded in a slab, the logarithmic term in \eqref{schoen conclusion 2} must vanish (i.e. \eqref{schoen conclusion 2} holds with $b=0$), and thus \eqref{asymptotics of F} is proved. {\bf Finally, when $n=1$}, by \eqref{F represented by u} and \eqref{u estimates} there are $a_1,a_2\in\R$, $x_1<x_2$, $x_1,x_2\in\nu^\perp\equiv\R$ such that $f(x)=a_1$ for $x\in\nu^\perp$, $x<x_1$, and $f(x)=a_2$ for $x\in\nu^\perp$, $x>x_2$. Now, setting $M_1=\{x+a_1\,\nu:x\in\nu^\perp,x<x_1\}$ and $M_2=\{x+a_2\,\nu:x\in\nu^\perp,x>x_2\}$, we have that
\[
P(F;\CC_R^\nu\setminus W)=\H^n\big(\CC_R^\nu\cap(\pa F)\setminus(W\cup M_1\cup M_2)\big)+2\,R-|x_2-x_1|\,;
\]
while, if $L$ denotes the line through $x_1+a_1\,\nu$ and $x_2+a_2\,\nu$, then we can find $\nu_L\in\SS^1$ and a set $F_L$ such that $(F_L,\nu_L)\in\F$ with $\pa F_L=\big[\big((\pa F)\setminus(M_1\cup M_2)\big)\cup (L_1\cup L_2)\big]$, where $L_1$ and $L_2$ are the two half-lines obtained by removing from $L$ the segment joining $x_1+a_1\,\nu$ and $x_2+a_2\,\nu$. In this way, $P(F_L;\CC_R^{\nu_L}\setminus W)=\H^n\big(\CC_R^\nu\cap(\pa F)\setminus(W\cup M_1\cup M_2)\big)+2\,R-\big|(x_1+a_1\,\nu)-(x_2+a_2\,\nu)\big|$, so that ${\rm res}_W(F_L,\nu_L)-{\rm res}_W(F,\nu)=\big|(x_1+a_1\,\nu)-(x_2+a_2\,\nu)\big|-|x_2-x_1|>0$, against $(F,\nu)\in{\rm Max}[\Rr(W)]$ if $a_1\ne a_2$. Hence, $a_1=a_2$.

We are left to prove that \eqref{F represented by u} holds with $R_2=R_2(W)$ in place of $R_F$, and the constants $a$, $b$, $c$ and $C_0$ appearing in \eqref{asymptotics of F} can be bounded in terms of $W$ only. To this end, we notice that the argument presented in step one shows that ${\rm Max}[\Rr(W)]$ is pre-compact in $L^1_{\rm loc}(\R^{n+1})$. Using this fact and a contradiction argument based on improved convergence (Remark \ref{remark improved convergence}), we conclude the proof of statement (ii).

\noindent {\bf Step six:} We complete the proof of statement (i) and begin the proof of statement (iii) by showing that, setting for brevity $d=\diam(W)$, it holds
\begin{equation}
\label{RW upper}
\H^n(W\cap\Pi)\le\Rr(W) \leq \sup_{\nu\in\SS^n}\hn(\pp_{\nu^\perp} (W))\leq\om_n\,(d/2)^n\,,
\end{equation}
whenever $\Pi$ is a hyperplane in $\R^{n+1}$. We have already proved the first inequality in step one. To prove the others, we notice that, if $(F,\nu)\in\F$, then $\pp_{\nu^\perp}(\pa F)=\nu^\perp$ and \eqref{perimeter is decreasing}
%(that is, the monotone increasing character of $R\mapsto  P(F;\CC_R^\nu\setminus W)-\om_n\,R^n$ over $R>R'=\inf\{\rho:W\subset\CC_\rho^\nu\}$),
give, for every $R>R'$,
\begin{eqnarray}
\nonumber
&&\!\!\!\!\!\!\!\!\!\!-{\rm res}_W(F,\nu)
\ge
P(F;\CC_R^\nu\setminus W)-\om_n\,R^n
\ge\H^n\big(\pp_{\nu^\perp}(\pa F\setminus W)\cap\DD_R^\nu\big)-\om_n\,R^n
\\
\label{projection type inequality}
&&\!\!\!\!=-\H^n\big(\DD_R^\nu\setminus\pp_{\nu^\perp}(\pa F\setminus W)\big)
\ge-\H^n(\pp_{\nu^\perp}(W))\ge-\om_n\,(d/2)^n\,,
\end{eqnarray}
where in the last step we have used the isodiametric inequality. Maximizing over $(F,\nu)$ in \eqref{projection type inequality} we complete the proof of \eqref{RW upper}. Moreover, if $W=\cl(B_{d/2})$, then, since $\Ss(\cl(B_{d/2}))=\H^n(\cl(B_{d/2})\cap\Pi)=\om_n\,(d/2)^n$ for any hyperplane $\Pi$ through the origin, we find that $\Rr(\cl(B_{d/2}))=\om_n\,(d/2)^n$; in particular, \eqref{RW upper} implies \eqref{optimal RW}.

\noindent {\bf Step seven:} We continue the proof of statement (iii) by showing \eqref{characterization 2}. Let $\Rr(W)=\om_n\,(d/2)^n$ and let $(F,\nu)\in{\rm Max}[\Rr(W)]$. Since every inequality in \eqref{projection type inequality} holds as an equality, we find in particular that
\begin{eqnarray}\label{dai 1}
&&\sup_{R>R'}P(F;\CC_R^\nu\setminus W)-\H^n\big(\pp_{\nu^\perp}(\pa F\setminus W)\cap\DD_R^\nu\big)=0\,,
%  \\\label{two conditions ii}
%  &&\H^n\big(\pp_{\nu^\perp}(W)\cap\pp_{\nu^\perp}((\pa F)\setminus W)\big)=0\,,
\\\label{dai 3}
\end{eqnarray}
By \eqref{dai 3} and the discussion of the equality cases for the isodiametric inequality (see, e.g. [28]), we see that, for some $x_0\in\nu^\perp$,
\begin{equation}
\label{third condition}
\pp_{\nu^\perp}(W)=\cl(\DD_{d/2}^\nu(x_0))\,,\qquad\mbox{so that $W\subset\CC_{d/2}^\nu(x_0)$}\,.
\end{equation}
Condition \eqref{dai 1} implies that \eqref{asymptotics of F} holds with $u\equiv a$ for some $a\in[A,B]=\bigcap\{(\a,\b):W\subset\mathbf{S}^\nu_{(\a,\b)}\}$; in particular, since $(\pa F)\setminus W$ is a minimal surface and $W\subset\CC_{d/2}^\nu(x_0)$, by analytic continuation we find that
\begin{equation}
\label{two conditions i}
(\partial F)\setminus\CC_{d/2}^\nu(x_0) =\Pi\setminus\CC_{d/2}^\nu(x_0)\,,\qquad\Pi=\big\{x:x\cdot\nu=a\big\}\,.
\end{equation}
By \eqref{two conditions i}, we have that for $R>R'$,
\[
P(F;\CC_R^\nu\setminus W)-\om_n\,R^n=P(F;\CC_{d/2}^\nu(x_0)\setminus W)-\om_n\,(d/2)^n\,.
\]
Going back to \eqref{projection type inequality}, this implies $P(F;\CC_{d/2}^\nu(x_0)\setminus W)=0$. However, since $(\pa F)\setminus W$ is (distributionally) a minimal surface, $P(F;B_\rho(x)\setminus W)\ge\om_n\,\rho^n$ whenever $x\in (\pa F)\setminus W$ and $\rho<\dist(x,W)$, so that
$P(F;\CC_{d/2}^\nu(x_0)\setminus W)=0$ gives $((\pa F)\setminus W)\cap\CC_{d/2}^\nu(x_0)=\emptyset$. Hence, using also \eqref{two conditions i}, we find $(\partial F)\setminus W=\Pi\setminus\cl\big(B_{d/2}(x)\big)$ for some $x\in\Pi$, that is \eqref{characterization 2}.

\noindent {\bf Step eight:} We finally prove that $\Rr(W)=\om_n\,(d/2)^n$ if and only if there are a hyperplane $\Pi$ and a point $x\in\Pi$ such that
\begin{eqnarray}
\label{condo 1}
&&\Pi\cap\pa B_{d/2}(x)\subset W\,,
\\
\label{condo 2}
&&\mbox{$\Om\setminus(\Pi\setminus B_{d/2}(x))$ has two unbounded connected components}\,.
\end{eqnarray}
We first prove that the two conditions are sufficient. Let $\nu$ be a unit normal to $\Pi$ and let $\Pi^+$ and $\Pi^-$ be the two open half-spaces bounded by $\Pi$. The condition $\Pi\cup\pa B_{d/2}(x)\subset W$ implies $W\subset\CC_{d/2}^\nu(x)$, and thus
\[
\Om\setminus\cl\big[ \CC^\nu_{d/2,(-d,d)}(x)\big]=(\Pi^+\cup\Pi^-)\setminus\cl\big[\CC^\nu_{d/2,(-d,d)}(x)\big]\,.
\]
In particular, $\Omega \setminus(\Pi\setminus B_{d/2}(x))$ has a connected component $F$ which contains
\[
\Pi^+\setminus\cl\big[\CC^\nu_{d/2,(-d,d)}(x)\big]\,;
\]
and since $\Om\setminus(\Pi\setminus B_{d/2}(x))$ contains exactly two unbounded connected components, it cannot be that $F$ contains also $\Pi^-\setminus\cl[\CC^\nu_{d/2,(-d,d)}(x)]$, therefore
\begin{equation}
\label{pi plus pi minus}
\Pi^+\setminus\cl\big[\CC^\nu_{d/2,(-d,d)}(x)\big]\subset F\,,\qquad \Pi^-\setminus\cl\big[\CC^\nu_{d/2,(-d,d)}(x)\big]\subset\R^{n+1}\setminus\cl(F)\,.
\end{equation}
As a consequence $\pa F$ is contained in the slab $\{y:|(y-x)\cdot\nu|<d\}$, and is such that $\pp_{\nu^\perp}(\pa F)=\nu^\perp$, that is, $(F,\nu)\in\F$. Moreover, \eqref{pi plus pi minus} implies
\[
\Pi\setminus \cl(B_{d/2}(x))\subset\Om\cap\pa F\,,
\]
while the fact that $F$ is a connected component of $\Omega \setminus (\Pi\setminus B_{d/2}(x))$ implies $\Om\cap\pa F\subset \Pi\setminus \cl(B_{d/2}(x))$. In conclusion, $\Om\cap\pa F=\Pi\setminus\cl(B_{d/2}(x))$, hence
\begin{equation}
\om_n\,(d/2)^n=
\lim_{r\to \infty}\omega_n r^n - P(F; \textbf{C}_r^{\nu}\setminus W) \leq \Rr(W) \leq \om_n\,(d/2)^n\,,
\end{equation}
and $\Rr(W)=\om_n\,(d/2)^n$, as claimed. We prove that the two conditions are necessary. Let $(F,\nu)\in{\rm Max}[\Rr(W)]$. As proved in step seven, there is a hyperplane $\Pi$ and $x\in\Pi$ such that $\Om\cap\pa F=\Pi\setminus \cl(B_{d/2}(x))$. If $z\in\Pi\cap\pa B_{d/2}(x)$ but $z\in\Om$, then there is $\rho>0$ such that $B_\rho(z)\subset\Om$, and since $\pa F$ is a minimal surface in $\Om$, we would obtain that $\Pi\cap B_\rho(z)\subset\Om\cap\pa F$, against $\Om\cap\pa F=\Pi\setminus\cl(B_{d/2}(x))$. So it must be $\Pi\cap\pa B_{d/2}(x)\subset W$, and the necessity of \eqref{condo 1} is proved. To prove the necessity of \eqref{condo 2}, we notice that since $\Pi^+\setminus\cl[\CC^\nu_{d/2,(-d,d)}(x)]$ and $\Pi^-\setminus\cl[\CC^\nu_{d/2,(-d,d)}(x)]$ are both open, connected, and unbounded subsets of $\Omega \setminus (\Pi\setminus B_{d/2}(x))$, and since the complement in $\Omega \setminus (\Pi\setminus B_{d/2}(x))$ of their union is bounded, it must be that $\Omega \setminus (\Pi\setminus B_{d/2}(x))$ has {\it at most} two unbounded connected components: therefore we just need to exclude that {\it it has only one}. Assuming by contradiction that this is the case, we could then connect any point $x^+\in\Pi^+\setminus\cl[\CC^\nu_{d/2,(-d,d)}(x)]$  to any point $x^-\in\Pi^-\setminus\cl[\CC^\nu_{d/2,(-d,d)}(x)]$ with a continuous path $\g$ entirely contained in $\Omega \setminus (\Pi\setminus  B_{d/2}(x))$. Now, recalling that $\Om\cap\pa F=\Pi\setminus\cl(B_{d/2}(x))$, we can pick $x_0\in\Pi\setminus\cl(B_{d/2}(x))$ and $r>0$ so that
\begin{equation}
\label{separating F}
B_r(x_0)\cap\Pi^+\subset F\,,\qquad B_r(x_0)\cap\Pi^-\subset\R^{n+1}\setminus\cl(F)\,,
\end{equation}
and $B_r(x_0)\cap\cl[\CC^\nu_{d/2,(-d,d)}(x)]=\emptyset$. We can then pick $x^+\in B_r(x_0)\cap\Pi^+$, $x^-\in B_r(x_0)\cap\Pi^-$, and then connect them by a path $\g$ entirely contained in $\Omega \setminus (\Pi\setminus  B_{d/2}(x))$. By \eqref{separating F}, $\g$ must intersect $\pa F$, and since $\g$ is contained in $\Om$, we see that $\g$ must intersect $\Om\cap\pa F=\Pi\setminus\cl(B_{d/2}(x))$, which of course contradicts the containment of $\g$ in $\Omega \setminus (\Pi\setminus  B_{d/2}(x))$. We have thus proved that $\Omega \setminus (\Pi\setminus  B_{d/2}(x))$ has exactly two unbounded connected components.
\end{proof}

\section{Resolution theorem for exterior isoperimetric sets}\label{section resolution for exterior} The notation set in \eqref{cylinders and slabs} is in use. Given $v_j\to\infty$, we set $\l_j=v_j^{1/(n+1)}$.

\begin{proof}[Proof of Theorem \ref{thm main psi}] We recall that, throughout the proof, $\Rr(W)>0$. Theorem \ref{thm main psi}-(i) and the estimate for $|v^{-1/(n+1)}\,|x|-\om_{n+1}^{-1/(n+1)}|$ in Theorem \ref{thm main psi}-(iv), have already been proved in Theorem \ref{thm existence and uniform min}-(ii, iii).

\noindent {\bf Step one:} We prove that
\begin{equation}\label{energy upper bound}
\varlimsup_{v \to \infty} \psi_W(v)- P(B^{(v)})\leq -\Rr(W)\,.
\end{equation}
To this end, let $(F,\nu)\in{\rm Max}[\Rr(W)]$, so that by \eqref{main residue graphicality of F} and \eqref{asymptotics of F}, we have
\begin{equation}
\label{order of F0}
\end{equation}
for a function $f\in C^1(\nu^\perp)$ satisfying
\begin{eqnarray}
\label{order of F}
&&\big|f(x)-\big(a+b\,|x|^{2-n}+(c\cdot x)\,|x|^{-n}\big)\big|\le C_0\,|x|^{-n}\,,
\\
\nonumber
&&\max\big\{|x|^{n-1}\,|\nabla f(x)|,|x|^{n}\,|\nabla^2 f(x)|\big\}\le C_0\,,\qquad\forall x\in\nu^\perp\,,|x|>R_2\,,
\end{eqnarray}
and for some $a,b\in\R$ and $c\in\nu^\perp$ such that $\max\{|a|,|b|,|c|\}\le C(W)<\infty$ (moreover, we can take $b=0$, $c=0$ and $C_0=0$ if $n=1$). We are going to construct competitors for $\psi_W(v)$ with $v$ large by gluing a large sphere $S$ to $\pa F$ along $\pa\CC_r^\nu$ for $r>R_2$. This operation comes at the price of an area error located on the cylinder $\pa\CC_r^\nu$. We can make this error negligible thanks to the fact that \eqref{order of F} determines the distance (inside of $\pa\CC_r^\nu$) of $\pa F$ from a hyperplane (namely, $\pa G_r$ for the half-space $G_r$ defined below) up to ${\rm o}(r^{1-n})$ as $r\to\infty$. Thus, the asymptotic expansion \eqref{asymptotics of F} is just as precise as needed in order to perform this construction, i.e. our construction would not be possible with a less precise information.

We now discuss the construction in detail. Given $r>R_2$, we consider the half-space $G_r\subset\R^{n+1}$ defined by the condition that
\begin{equation}
\label{def of Gr}
G_r \cap \partial\CC_r^\nu =\big\{x+t\,\nu:x\in\nu^\perp\,,|x|=r\,, t< a + b\,r^{2-n}+(c\cdot x)\,r^{-n}\big\}\,,
\end{equation}
so that $G_r$ is the ``best half-space approximation'' of $F$ on $\pa\CC_r^\nu$ according to \eqref{order of F}. Denoting by $\hd(X,Y)$ the Hausdorff distance between $X,Y\subset\R^{n+1}$, for every $r>R_2$ and $v>0$ we can define $x_{r,v}\in\R^{n+1}$ in such a way that $v\mapsto x_{r,v}$ is continuous and
\begin{eqnarray}
\label{hausdorff convergence to halfspace}
\lim_{v\to\infty}\hd(B^{(v)}(x_{r,v})\cap K,G_r\cap K)=0\qquad\forall\,K\cc\R^{n+1}\,.
\end{eqnarray}
Thus, the balls $B^{(v)}(x_{r,v})$ have volume $v$ and are locally converging in Hausdorff distance, as $v\to\infty$, to the optimal half-space $G_r$. Finally, we notice that by \eqref{order of F} we can find $\a<\b$ such that
\begin{align}\label{containment in finite cylinder}
\big((\partial F) \cup (\partial G_r) \cup (G_r \Delta F)\big) \cap \CC_r^\nu \,\,\subset\,\, \CC_{r,(\a+1,\b-1)}^\nu\,,
\end{align}
and then define $F_{r,v}$ by setting
\begin{equation}\label{the competitors}
\end{equation}
\begin{figure}\input{upperbound2.pstex_t}\caption{\small{The competitors $F_{r,v}$ constructed in \eqref{the competitors}. A maximizer $F$ in the isoperimetric residue $\Rr(W)$ is joined to a ball of volume $v$, whose center $x_{r,v}$ is determined by looking at best hyperplane $\pa G_r$ approximating $\pa F$ on the ``lateral'' cylinder $\pa\CC_r^\nu$. To ensure the area error made in joining this large sphere to $\pa F$ is negligible, the distance between $\pa F$ and the sphere inside $\pa\CC_r^\nu$ must be ${\rm o}(r^{1-n})$ as $r\to\infty$. The asymptotic expansion \eqref{order of F} gives a hyperplane $\pa G_r$ which is close to $\pa F$ up to ${\rm O}(r^{-n})$, and is thus just as precise as needed to perform the construction.}}\label{fig upperbound}\end{figure}
Figure \ref{fig upperbound}. We claim that, by using $F_{r,v}$ as comparisons for $\psi_W(|F_{r,v}|)$, and then sending first $v\to\infty$ and then $r\to\infty$, one obtains \eqref{energy upper bound}. We first notice that by \eqref{hausdorff convergence to halfspace} and \eqref{containment in finite cylinder} (see, e.g. \cite[Theorem 16.16]{maggiBOOK}), we have
\begin{eqnarray}\nonumber
P(F_{r,v};\Om)&=&P(F;\CC_{r,(\a,\b)}^\nu\setminus W)+P\big(B^{(v)}(x_{r,v});\R^{n+1}\setminus\cl\big[\CC_{r,(\a,\b)}^\nu\big]\big)
\\\label{perimeter Frv}
&&+\H^n\big((F\Delta B^{(v)}(x_{r,v}))\cap\pa_\ell\CC_{r,(\a,\b)}^\nu\big)\,,
\end{eqnarray}
where the last term is the ``gluing error'' generated by the mismatch between the boundaries of $\pa F$ and $\pa B^{(v)}(x_{r,v})$ along $\pa_\ell\CC_{r,(\a,\b)}^\nu$. Now, thanks to \eqref{order of F} we have
$\hd(G_r\cap\pa\CC_r^\nu,F\cap\pa\CC_r^\nu)\le C_0\,r^{-n}$, so that
\begin{equation}
\label{small error on lateral boundary}
\H^n\big((F\Delta G_r)\cap\pa\CC_r^\nu\big)\le n\,\om_n\,r^{n-1}\,  \hd(G_r\cap\pa\CC_r^\nu,F\cap\pa\CC_r^\nu)\le C(n,W)/r\,.
\end{equation}
At the same time, by \eqref{hausdorff convergence to halfspace},
\begin{equation}
\label{L1 convergence on cylinder boundaries}
\lim_{v\to\infty}\H^n\big((G_r\Delta B^{(v)}(x_{r,v}))\cap\pa_\ell\CC_{r,(\a,\b)}^\nu\big)=0\,,
\end{equation}
and thus we have the following estimate for the gluing error,
\begin{equation}
\label{gin 1}
\varlimsup_{v\to\infty}\H^n\big((F\Delta B^{(v)}(x_{r,v}))\cap\pa_\ell\CC_{r,(\a,\b)}^\nu\big)\le\frac{C(n,W)}r\,,\qquad\forall r>R_2\,.
\end{equation}
Again by \eqref{hausdorff convergence to halfspace}, we find
\begin{eqnarray}
\label{ball perimeter converges to halfspace perimeter}
\\
\label{Gr perimeter}
&&\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!1\le(\om_n\,r^n)^{-1}\,P\big(G_r;\CC_{r,(\a,\b)}^\nu\big)=\fint_{\DD_r^\nu}\sqrt{1+(c/r^n)^2}\le 1+C_0\,r^{-2\,n}\,,
\end{eqnarray}
so that, by \eqref{ball perimeter converges to halfspace perimeter} and by the lower bound in \eqref{Gr perimeter}, for every $r>R_2$,
\begin{equation}\label{gin 2}
\varlimsup_{v\to\infty}P\big(B^{(v)}(x_{r,v});\R^{n+1}\setminus\cl\big[\CC_{r,(\a,\b)}^\nu\big]\big)-P(B^{(v)})
\le -\om_n\,r^n\,.
\end{equation}
Combining \eqref{gin 1} and \eqref{gin 2} with \eqref{perimeter Frv} and the fact that $\CC_{r,(\a,\b)}^\nu\cap\pa F=\CC_r^\nu\cap\pa F$ (see \eqref{containment in finite cylinder}), we find that for every $r>R_2$,
\begin{eqnarray}\nonumber
&&\varlimsup_{v\to\infty}P(F_{r,v};\Om)-P(B^{(v)})\le P(F;\CC_r^\nu\setminus W)-\om_n\,r^n+ C(n,W)/r
\\
\label{perimeter Frv 2}
&&\le-{\rm res}_W(F,\nu)+C(n,W)/r=-\Rr(W)+C(n,W)/r\,.
\end{eqnarray}
where \eqref{perimeter is decreasing} has been used. Now, combining the elementary estimates
\begin{equation}
\label{volum of Frv}
\max\big\{\big||F_{r,v}|-v\big|\,,v^{-1/(n+1)}\,|P(B^{(v)})-P(B^{(|F_{r,v}|)})|\big\}\le C(n)\,r^{n+1}
\end{equation}
with \eqref{perimeter Frv 2}, we see that
\begin{eqnarray}\label{perimeter Frv 3}
\varlimsup_{v\to\infty}\psi_W(|F_{r,v}|)-P(B^{(|F_{r,v}|)})\le-\Rr(W)+ C(n,W)/r\,,\,\,\forall r>R_2\,.
\end{eqnarray}
Again by \eqref{volum of Frv} and since $v\mapsto|F_{r,v}|$ is a continuous function, we see that
$\varlimsup_{v\to\infty}\psi_W(|F_{r,v}|)-P(B^{(|F_{r,v}|)})=\varlimsup_{v\to\infty}\psi_W(v)-P(B^{(v)})$. This last identity combined with \eqref{perimeter Frv 3} implies \eqref{energy upper bound} in the limit $r\to\infty$.

\noindent {\bf Step two:} Now let $E_j\in{\rm Min}[\psi_W(v_j)]$ for $v_j \to \infty$. By \eqref{uniform lambda minimality} and a standard argument (see, e.g. \cite[Theorem 21.14]{maggiBOOK}), there is a local perimeter minimizer with free boundary $F$ in $\Om$ such that, up to extracting subsequences,
\begin{eqnarray}\nonumber
&&\mbox{$E_j\to F$ in $L^1_{{\rm loc}}(\R^{n+1})$, $\H^n\llcorner\pa E_j\weak\H^n\llcorner\pa F$ as Radon measures in $\Om$}\,,
\\
&&\hd(K\cap\pa E_j;K\cap\pa F)\to0\qquad\mbox{for every $K\cc\Om$}\,.\label{convergence of Ej to F last proof}
\end{eqnarray}
Notice that it is not immediate to conclude from $E_j\in{\rm Min}[\psi_W(v_j)]$ that (for some $\nu\in\SS^n$) $(F,\nu)\in{\rm Max}[\Rr(W)]$ (or even that $(\nu,F)\in\F$), nor that $P(E_j;\Om)-P(B^{(v_j)})$ is asymptotically bounded from below by $-{\rm res}_W(F,\nu)$. In this step we prove some preliminary properties of $F$, and in particular we exploit the blowdown result for exterior minimal surfaces contained in Theorem \ref{theorem mesoscale criterion}-(ii) to prove that $F$ satisfies \eqref{order of F0} and \eqref{order of F} (see statement (c) below). Then, in step three, we use the decay rates \eqref{order of F} to show that $E_j$ can be ``glued'' to $F$, similarly to the construction of step one, and then derive from the corresponding energy estimates the lower bound matching \eqref{energy upper bound} and the optimality of $F$ in $\Rr(W)$.

\noindent {\bf (a)} {\it $\Om\cap\pa F\cap\pa B_\rho\ne\emptyset$ for every $\rho$ such that $W\cc B_\rho$}: If not there would be $\e>0$ such that $W\cc B_{\rho-\e}$ and $\Om\cap\pa F\cap A_{\rho-\e}^{\rho+\e}=\emptyset$ (recall that $A_r^s=\{x:s>|x|>r\}$). By \eqref{convergence of Ej to F last proof} and the constant mean curvature condition satisfied by $\Om\cap\pa E_j$, we would then find that each $E_j$ (with $j$ large enough) has a connected component of the form $B^{(w_j)}(x_j)$, with $B^{(w_j)}(x_j)\cc \R^{n+1}\setminus B_{\rho+\e}$ and $w_j\ge v_j-C(n)\,(\rho+\e)^{n+1}$. In particular, against $\Rr(W)>0$,
\[
\psi_W(v_j)=P(E_j;\Om)\ge P(B^{(v_j-C\,(\rho+\e)^{n+1})})\ge P(B^{(v_j)})-C\l_j^{-1}(\rho+\e)^{n+1}\,.
\]

\noindent {\bf (b)} {\it Sharp area bound}: We combine the upper energy bound \eqref{energy upper bound} with the perimeter inequality for spherical symmetrization, to prove
\begin{equation}\label{growth bound for F}
P(F;\Omega \cap B_r) \leq \omega_n r^n - \Rr(W)\,,\qquad\mbox{for every $r$ s.t. $W\cc B_r$}\,.
\end{equation}
(Notice that \eqref{growth bound for F} does not immediately imply the bound for $P(F;\Omega\cap\CC_r^\nu)$ which would be needed to compare $\Rr(W)$ and ${\rm res}_W(F,\nu)$.) To prove \eqref{growth bound for F} we argue by contradiction, and consider the existence of $\de>0$ and $r$ with $W\cc B_r$ such that
$P(F;\Om\cap\ B_r)\ge \om_n\,r^n-\Rr(W)+\de$. In particular, for $j$ large enough, we would then have
\begin{equation}\label{bad one}
P(E_j;\Om\cap B_r)\ge\omega_nr^n - \Rr(W)+\de\,.
\end{equation}
Again for $j$ large, it must be $\H^n(\pa E_j\cap\pa B_r)=0$: indeed, by \eqref{uniform lambda minimality}, $\Om\cap\pa E_j$ has mean curvature of order ${\rm O}(\l_{j}^{-1})$, while of course $\pa B_r$ has constant mean curvature equal to $n/r$. Thanks to $\H^n(\pa E_j\cap\pa B_r)=0$,
\begin{equation}
\label{bad one 1}
P(E_j;\Om)=P(E_j;\Om\cap B_r)+P\big(E_j;\R^{n+1}\setminus\cl(B_r)\big)\,.
\end{equation}
If $E_j^s$ denotes the spherical symmetral of $E_j$ such that $E_j^s\cap\pa B_\rho$ is a spherical cap in $\pa B_\rho$, centered at $\rho\,e_{n+1}$, with area equal to $\H^n(E_j\cap\pa B_\rho)$, then we have the perimeter inequality
\begin{equation}
\label{bad one 2}
P\big(E_j;\R^{n+1}\setminus\cl(B_r)\big)\ge P\big(E_j^s;\R^{n+1}\setminus\cl(B_r)\big)\,;
\end{equation}
see [9]. Now, we can find a half-space $J$ orthogonal to $e_{n+1}$ and such that $\H^n(J\cap\pa B_r)=\H^n(E_j\cap\pa B_r)$. In this way, using that $|E_j^s\setminus B_r|=|E_j\setminus B_r|$ (by Fubini's theorem in spherical coordinates), and that $\H^n(B_r\cap\pa J)\le\om_n\,r^n$ (by the fact that $\pa J$ is a hyperplane), we find
\begin{eqnarray*}
P\big(E_j^s;\R^{n+1}\setminus\cl(B_r)\big)&=&P\big((E_j^s\setminus\cl(B_r))\cup (J\cap B_r)\big)-\H^n(B_r\cap\pa J)
\\
&\ge&P\big(B^{(|E_j|-|E_j\cap B_r|+|J\cap B_r|)}\big)-\om_n\,r^n
\\
\end{eqnarray*}
which, with \eqref{bad one}, \eqref{bad one 1} and \eqref{bad one 2}, finally gives
$P(E_j;\Om)-P(B^{(v_j)})> -\Rr(W)+\de-C(n)\,r^{n+1}\,\l_{j}^{-1}$
for $j$ large, against \eqref{energy upper bound}.

\noindent {\bf (c)} {\it Asymptotic behavior of $\pa F$}: We prove that there are $\nu\in\SS^n$, $f\in C^\infty(\nu^\perp)$, $a,b\in\R$, $c\in\nu^\perp$, $R'>\sup\{\rho:W\subset\CC_\rho^\nu\}$ and $C$ positive, with
\begin{eqnarray}
\label{main residue graphicality of F end}
&&\partial F \setminus \CC^\nu_{R'}=\big\{x+f(x)\,\nu:x\in\nu^\perp\,,|x|>R'\big\}\,,
\\\nonumber
&&f(x)=a\,,\hspace{6.6cm} (n=1)
\\\label{asymptotics of F end}
&&\big|f(x)-\big(a+b\,|x|^{2-n}+(c\cdot x)\,|x|^{-n}\big)\big|\le C\,|x|^{-n}\,,\,\,\, (n\ge 2)\,,
\\\nonumber
\max\big\{|x|^{n-1}\,|\nabla f(x)|,|x|^n\,|\nabla^2f(x)|\big\}\le C_0\,,\qquad\forall x\in\nu^\perp\,,|x|>R'\,.
\end{eqnarray}
To this end, by a standard argument exploiting the local perimeter minimality of $F$ in $\Om$, given $r_j\to\infty$, then, up to extracting subsequences, $F/r_j\toloc J$ in $L^1_{\rm loc}(\R^{n+1})$, where $J$ is a perimeter minimizer in $\R^{n+1}\setminus\{0\}$, $0\in\pa J$ (thanks to property (a)), $J$ is a cone with vertex at $0$
%, i.e., $\l\,J=J$ for every $\l>0$
(thanks to Theorem \ref{theorem 7.17AA exterior lambda} and, in particular to \eqref{conelimit}), and $P(J;B_1)\le\om_n$ (by \eqref{growth bound for F}). {\bf If $n\ge 2$}, then $\pa J$ has vanishing distributional mean curvature in $\R^{n+1}$ (as points are removable singularities for the mean curvature operator when $n\ge 2$), thus $P(J;B_1)\ge\om_n$ by upper semicontinuity of area densities, and, finally, by $P(J;B_1)=\om_n$ and Allard's regularity theorem, $J$ is a half-space. {\bf If $n=1$}, then $\pa J$ is the union of two half-lines $\ell_1$ and $\ell_2$ meeting at $\{0\}$. If $\ell_1$ and $\ell_2$ are not opposite (i.e., if $J$ is not a half-space), then we can find a half-space $J^*$ such that $(J\cap J^*)\Delta J\cc B\cc \R^2\setminus\{0\}$ for some ball $B$, and $P(J\cap J^*;B)<P(J;B)$, thus violating the fact that $J$ is a perimeter minimizer in $\R^{n+1}\setminus\{0\}$.

If $n=1$ it is immediate from the above information that, for some $R'>0$, $F\setminus B_{R'}=J\setminus B_{R'}$; this proves \eqref{main residue graphicality of F end} and \eqref{asymptotics of F end} in the case $n=1$. To prove \eqref{main residue graphicality of F end} and \eqref{asymptotics of F end} when $n\ge 2$, we let $M_0$ and $\e_0$ be as in Theorem \ref{theorem mesoscale criterion}-(ii) with $(n,\Gamma=2\,n\,\om_n,\s=1)$. Since $J$ is a half-space, by using Remark \ref{remark improved convergence} and $F/r_j\toloc J$ on the annulus $A_{1/2}^{2\,L}$, for some $L>\max\{M_0,64\}$ to be chosen later on depending also on $\e_0$, we find that
\begin{equation}\label{between one half and L}
(\pa F)\cap A^{4\,L\,r_j}_{r_j/2}
=\big\{x+r_j\,f_j\big(x/r_j\big)\,\nu:x\in\nu^\perp\big\}\cap A^{4\,L\,r_j}_{r_j/2}\,,\qquad \nu^\perp=\pa J\,,
\end{equation}
for  $f_j\in C^1(\nu^\perp)$ with $\|f_j\|_{C^1(\nu^\perp)}\to 0$. By \eqref{between one half and L},
$V_j=\var\big((\pa F)\setminus B_{r_j},1)\in\V_n(0,r_j,\infty)$, with (for ${\rm o}(1)\to 0$ as $j\to\infty$)
\begin{eqnarray}\label{FF1}
&&r_j^{-n}\,\int x\cdot\nu^{\rm co}_{V_j}\,d{\rm bd}_{V_j}=-n\,\om_n+{\rm o}(1)
\\\label{FF2}
&&r_j^{1-n}\|{\rm bd}_{V_j}\|(\pa B_{r_j})=n\,\om_n+{\rm o}(1)\,,
\\\label{FF3}
&&\sup_{r\in(r_j,3\,L\,r_j)}\big|(r^n-r_j^n)^{-1}\,\|V_j\|(B_r\setminus B_{r_j})-\om_n\big|={\rm o}(1)\,.
\end{eqnarray}
By our choice of $\Gamma$, by \eqref{growth bound for F} and \eqref{FF1} we see that, for $j$ large, we have
\begin{equation}
\label{blowdown hp check 1}
\|{\rm bd}_{V_j}\|(\pa B_{r_j})\le \Gamma\,r_j^{n-1}\,,\qquad \|V_j\|(B_\rho\setminus B_{r_j})\le\Gamma\rho^n\,,\,\,\forall\rho>r_j\,.
\end{equation}
Moreover, we claim that setting
\[
\]
(so that, in particular, $s_j>\max\{M_0,64\}\,r_j$), then
\begin{eqnarray}
\label{blowdown hp check 2}
|\de_{V_j,r_j,0}(s_j/8)|\le\e_0\,,\qquad \inf_{r>s_j/8}\de_{V_j,r_j,0}(r)\ge-\e_0\,,
\end{eqnarray}
provided $j$ and $L$ are taken large enough depending on $\e_0$. To check the first inequality in \eqref{blowdown hp check 2} we notice that, by \eqref{FF1} and \eqref{FF3},
\begin{eqnarray*}
\de_{V_j,r_j,0}(s_j/8)\!\!&=&\!\!\om_n-\frac{\|V_j\|(B_{s_j/8}\setminus B_{r_j})}{(s_j/8)^n}+\frac1{n\,(s_j/8)^n}\,\int x\cdot\nu^{\rm co}_{V_j}\,d\,{\rm bd}_{V_j}
\\
&=&\!\!\om_n-\big(\om_n+{\rm o}(1)\big)\,\frac{(s_j/8)^n-r_j^n}{(s_j/8)^n}-\frac{\om_n\,r_j^n}{(s_j/8)^n}\,\big(1+{\rm o}(1)\big)
\\
&=&\!\!{\rm o}(1)\,(1+(r_j/s_j)^n)={\rm o}(1)\,,
\end{eqnarray*}
so that $|\de_{V_j,r_j,0}(s_j/8)|\le\e_0$ as soon as $j$ is large with respect to $\e_0$. Similarly, if $r>s_j/8=(L\,r_j)/4$, then by \eqref{FF1}, \eqref{FF3}, \eqref{growth bound for F}, and $r_j/r\le 4/L$,
\begin{eqnarray*}
&&\!\!\!\!\!\!\!\de_{V_j,r_j,0}(r)=\om_n-\frac{\|V_j\|(B_r\setminus B_{2\,r_j})}{r^n}-\frac{\|V_j\|(B_{2\,r_j}\setminus B_{r_j})}{r^n}
-\frac{\om_n\,r_j^n}{r^n}\,\big(1+{\rm o}(1)\big)
\\
&&\ge\om_n-\frac{\om_n\,r^n-\Rr(W)}{r^n}-\big(\om_n+{\rm o}(1)\big)\,\frac{(2\,r_j)^n-r_j^n}{r^n}-\frac{\om_n\,r_j^n}{r^n}\,\big(1+{\rm o}(1)\big)
\\
&&\ge r^{-n}\, \Rr(W) -2\,(4/L)^n\,\big(\om_n+{\rm o}(1)\big)-(4/L)^n\,{\rm o}(1)\ge-3\,(4/L)^n\,\om_n\,,
\end{eqnarray*}
provided $j$ is large; hence the second inequality in \eqref{blowdown hp check 2} holds if $L$ is large in terms of $\e_0$.
By \eqref{blowdown hp check 1} and \eqref{blowdown hp check 2}, Theorem \ref{theorem mesoscale criterion}-(ii) can be applied to $(V,R,\Lambda,s)=(V_j,r_j,0,s_j)$ with $j$ large. As a consequence, passing from spherical graphs to cylindrical graphs with the aid of Lemma \ref{lemma D1}, we find that, for some large $j$,
\begin{equation}\label{between one half and L fine}
(\pa F)\setminus B_{s_j/16}
=\big\{x+f(x)\,\nu:x\in\nu^\perp\big\}\setminus B_{s_j/16}\,,
\end{equation}
where $f:\nu^\perp\to\R$ is a smooth function which solves the minimal surfaces equation on $\nu^\perp\setminus B_{s_j/16}$. Since $\pa F$ admits at least one sequential blowdown limit hyperplane (namely, $\nu^\perp=\pa J$), by a theorem of Simon \cite[Theorem 2]{SimonAIHP} we find that $\nabla f$ has a limit as $|x|\to\infty$; in particular, $|\nabla f|$ is bounded. Moreover, by \eqref{between one half and L fine} (or by the fact that $F$ is a local perimeter minimizer in $\Om$), $\pa F$ is a stable minimal surface in $\R^{n+1}\setminus B_{s_j/16}$, which, thanks to \eqref{growth bound for F}, satisfies an area growth bound like \eqref{schoen 2}. We can thus apply Proposition \ref{prop schoen} to deduce the validity of \eqref{asymptotics of F end} when $n\ge 3$, and of $|f(x)-[a+b\,\log\,|x|+(c\cdot x)\,|x|^{-2}]|\le C\,|x|^{-2}$ for all $|x|>R'$ when $n=2$ (with $R'>s_j$). Recalling that $F$ is a local perimeter minimizer with free boundary in $\Om$ (that is, $P(F;\Om\cap B)\le P(F';\Om\cap B)$ whenever $F\Delta F'\cc B\cc\R^3$) it must be that $b=0$, as it can be seen by comparing $F$ with the set $F'$ obtained by changing $F$ inside $\CC_r^\nu$ ($r>>R'$) with the half-space $G_r$ bounded by the plane $\{x+t\,\nu:x\in\nu^\perp,t=a+b\,\log(r)+c\cdot x/r^2\}$ and such that $\H^2((F\Delta G_r)\cap\pa\CC_r^\nu)\le C/r^2$ (we omit the details of this standard comparison argument). Having shown that $b=0$, the proof is complete.

\noindent {\bf (d)} {\it $F\cup W$ defines an element of $\F$}: With $R>R'$ as in \eqref{main residue graphicality of F end} and \eqref{asymptotics of F end}, $V_R=\var((\pa F)\cap(B_R\setminus W))$ is a stationary varifold in $\R^{n+1}\setminus K_R$ for
$K_R=W\cup\big\{x+f(x)\,\nu:x\in\nu^\perp\,,|x|=R\}$, and has bounded support. By the convex hull property \cite[Theorem 19.2]{SimonLN}, we deduce that, for every $R>R'$, $\spt V_R$ is contained in the convex hull of $K_R$, for every $R>R'$. Taking into account that $f(x)\to a$ as $|x|\to\infty$ we conclude that $\Om\cap\pa F$ is contained in the smallest slab $\mathbf{S}_{[\a,\b]}^\nu$ containing both $W$ and $\{x:x\cdot\nu=a\}$. Now set
$F'=F\cup W$. Clearly $F'$ is a set of locally finite perimeter in $\Om$ (since $P(F';\Om')=P(F;\Om')$ for every $\Om'\cc\Om$). Second, $\pa F'$ is contained in $\mathbf{S}_{[\a,\b]}^\nu$ (since $\pa F'\subset [(\pa F)\cap\Om]\cup W$). Third, by \eqref{main residue graphicality of F end} and \eqref{asymptotics of F end},
\begin{eqnarray}
\label{coo1}
&&\big\{x+t\,\nu:x\in\nu^\perp\,,|x|>R'\,,t<\a\big\}\subset F'\,,
\\
\label{coo2}
&&\big\{x+t\,\nu:x\in\nu^\perp\,,|x|>R'\,,t>\b\big\}\subset \R^{n+1}\setminus F'\,,
\\
\label{coo3}
&&\big\{x+t\,\nu:x\in\nu^\perp\,,|x|<R'\,,t\in\R\setminus[\a,\b]\big\}\cap(\pa F')=\emptyset\,.
\end{eqnarray}
By combining \eqref{coo1} and \eqref{coo3} we see that $\{x+t\,\nu:x\in\nu^\perp\,,t<\a\}\subset F'$, and by combining \eqref{coo2} and \eqref{coo3} we see that $\{x+t\,\nu:x\in\nu^\perp\,,t>\b\}\subset \R^{n+1}\setminus F'$: in particular, $\pp_{\nu^\perp}(\pa F')=\nu^\perp$, and thus $(F',\nu)\in\F$.

\noindent {\bf Step three:} We prove that
\begin{equation}\label{lower bound equation}
\varliminf_{v\to \infty} \psi_W(v) - P(B^{(v)}) \geq -\Rr(W)\,.
\end{equation}
For  $v_j\to\infty$ achieving the liminf in \eqref{lower bound equation}, let $E_j\in{\rm Min}[\psi_W(v_j)]$ and let $F$ be a (sub-sequential) limit of $E_j$, so that properties (a), (b), (c) and (d) in step two hold for $F$. In particular, properties \eqref{main residue graphicality of F end} and \eqref{asymptotics of F end} from (c) are entirely analogous to properties \eqref{order of F0} and \eqref{order of F} exploited in step one: therefore, the family of half-spaces $\{G_r\}_{r>R'}$ defined by \eqref{def of Gr} is such that
\begin{eqnarray}
\label{b1}
\big((\partial F) \cup (\partial G_r) \cup (G_r \Delta F)\big) \cap \CC_r^\nu \,\,\subset\,\, \CC_{r,(\a+1,\b-1)}^\nu\,,
\\
\label{b2}
\H^n\big((F\Delta G_r)\cap\pa\CC_r^\nu\big)\le r^{-1}\,C(n,W)\,,
\\
\label{b3}
\big|P\big(G_r;\CC_{r,(\a,\b)}^\nu\big)-\om_n\,r^n\big|\le r^{-n}\,C(n,W)\,,
\end{eqnarray}
(compare with \eqref{containment in finite cylinder}, \eqref{small error on lateral boundary}, and \eqref{Gr perimeter} in step one). By \eqref{b3} we find
\begin{equation}
\label{coo4}
-{\rm res}_W(F',\nu)
%=\lim_{r\to\infty}\,P(F;\CC_r^\nu\setminus W)-\om_n\,r^n
=\lim_{r\to\infty}P(F;\CC_r^\nu\setminus W)-P(G_r;\CC_{r,(\a,\b)}^\nu)\,.
\end{equation}
In order to relate the residue of $(F',\nu)$ to $\psi_W(v_j)-P(B^{(v_j)})$ we consider the sets
$Z_j=(G_r\cap\CC_{r,(\a,b)}^\nu)\cup(E_j\setminus\CC_{r,(\a,\b)}^\nu)$, which, by isoperimetry, satisfy
\begin{eqnarray}\label{thanks iso}
P(Z_j)\!\ge\! P(B^{(|E_j\setminus\CC_{r,(\a,\b)}^\nu|)})\ge
\end{eqnarray}
Since for a.e. $r>R'$ we have
\[
P(Z_j)=P(E_j;\R^{n+1}\!\setminus\!\CC_{r,(\a,\b)}^\nu)+P(G_r;\CC_{r,(\a,b)}^\nu)+\H^n\big((E_j\Delta G_r)\cap\pa \CC_{r,(\a,b)}^\nu\big)\,,
\]
we conclude that
\begin{eqnarray*}
\psi_W(v_j)-P(B^{(v_j)})\!\!\!\!&=&\!\!\!P(E_j;\CC_{r,(\a,\b)}^\nu\!\setminus\! W)\!+P(E_j;\R^{n+1}\!\setminus\!\CC_{r,(\a,\b)}^\nu)\!-\!P(B^{(v_j)})
\\
&=&\!\!\!P(E_j;\CC_{r,(\a,\b)}^\nu\setminus W)+P(Z_j)-P(B^{(v_j)})
\\
&&\!\!\!-P(G_r;\CC_{r,(\a,b)}^\nu)-\H^n\big((E_j\Delta G_r)\cap\pa \CC_{r,(\a,b)}^\nu\big)
\end{eqnarray*}
so that $E_j\to F$ in $L^1_{\rm loc}(\R^{n+1})$ and \eqref{thanks iso} give, for a.e. $r>R'$,
\begin{eqnarray*}
&&\!\!\!\!\!\!\!\!\!\varliminf_{j\to\infty}\psi_W(v_j)-P(B^{(v_j)})\ge P(F;\CC_{r,(\a,\b)}^\nu\!\setminus\! W)-P(G_r;\CC_{r,(\a,b)}^\nu)
\\
&&-\H^n\big((F\Delta G_r)\cap\pa \CC_{r,(\a,\b)}^\nu\big)\ge P(F;\CC_r^\nu\!\setminus\! W)-P(G_r;\CC_r^\nu)-C(n,W)/r\,,
\end{eqnarray*}
thanks to \eqref{b2} and $(F\Delta G_r)\cap\pa\CC_r^\nu=(F\Delta G_r)\cap\pa\CC_{r,(\a,\b)}^\nu$. Letting $r\to\infty$, recalling \eqref{coo4}, and by $(F',\nu)\in\F$, we find $\varliminf_{j\to\infty}\psi_W(v_j)-P(B^{(v_j)})\ge-{\rm res}_W(F',\nu)\ge-\Rr(W)$. This completes the proof of \eqref{lower bound equation}, which in turn, combined with \eqref{energy upper bound}, gives \eqref{main asymptotic expansion}, and also shows that $L^1_{\rm loc}$-subsequential limits $F$ of $E_j\in{\rm Min}[\psi_W(v_j)]$ for $v_j\to\infty$ are such that, for some $\nu\in\SS^n$, $(F\cup W,\nu)\in\F$ and $F'=F\cup W\in{\rm Max}[\Rr(W)]$.

\noindent {\bf Step four:} Moving towards the proof of \eqref{f of Ev}, we prove the validity, uniformly among varifolds associated to maximizers of $\Rr(W)$, of estimates analogous to \eqref{blowdown hp check 1} and \eqref{blowdown hp check 2}.
%%%Recalling that maximizers in $\Rr(W)$ form a $L^1_{\rm loc}$-compact class in which improved convergence holds in the sense of Remark \ref{remark improved convergence}, this is obtained by applying to an arbitrary sequence of maximizers of $\Rr(W)$ the argument that was used in step three-(c) on blowdown sequences relative to a subsequential limit of relative isoperimetric sets with large volumes. With this premise in mind,
For a constant $\Gamma>2\,n\,\om_n$ to be determined later on (see \eqref{blowdown hp check 1 Ej part two}, \eqref{choose Gamma 1}, and \eqref{choose Gamma 2} below) in dependence of $n$ and $W$, and for $\s>0$, we let $M_0=M_0(n,2\,\Gamma,\s)$ and $\e_0=\e_0(n,2\,\Gamma,\s)$ be determined by Theorem \ref{theorem mesoscale criterion}. If $(F,\nu)\in{\rm Max}[\Rr(W)]$, then by Theorem \ref{thm main of residue}-(ii) we can find $R_2=R_2(W)>0$, $f\in C^\infty(\nu^\perp)$ such that
\begin{equation}
\label{basta}
(\pa F)\setminus\CC_{R_2}^\nu=\big\{x+f(x)\,\nu:x\in\nu^\perp\,,|x|>R_2\big\}\,,
\end{equation}
and such that \eqref{asymptotics of F} holds with  $\max\{|a|,|b|,|c|\}\le C(W)$ and $|\nabla f(x)|\le C_0/|x|^{n-1}$ for $|x|>R_2$. Thus $\|\nabla f\|_{C^0(\nu^\perp\setminus\DD_r^\nu)}\to 0$ as $r\to\infty$ uniformly on $(F,\nu)\in{\rm Max}[\Rr(W)]$, and there is $R_3>\max\{2\,R_2,1\}$ (depending on $W$) such that, if
$V_F=\var((\pa F)\setminus B_{R_3},1)$, then $V_F\in\V_n(0,R_3,\infty)$, and
\begin{equation}
\label{blowdown hp check 1 four}
\|{\rm bd}_{V_F}\|(\pa B_{R_3})\le \Gamma\,R_3^{n-1}\,,\qquad \|V_F\|(B_\rho\setminus B_{R_3})\le\Gamma\,\rho^n\qquad\forall\rho>R_3\,,
\end{equation}
(compare with \eqref{blowdown hp check 1}). Then, arguing as in step three-(c), or more simply by exploiting \eqref{basta} and the decay estimates \eqref{asymptotics of F}, we see that there is $L>\max\{M_0,64\}$, depending on $n$, $W$ and $\s$ only, such that, setting
\begin{equation}
\label{def of sWs}
\end{equation}
we have for some $c(n)>0$ (compare with \eqref{blowdown hp check 2})
\begin{eqnarray}
\label{blowdown hp check 2 four}
|\de_{V_F,R_3,0}(s_W(\s)/8)|\le\e_0/2\,,\qquad \inf_{r>s_W(\s)/8}\de_{V_F,R_3,0}(r)\ge-\e_0/2\,.
%\label{blowdown hp check 3 four}
%\frac1{s_W(\s)^n}\,\int_{A_{s_W(\s)/8}^{s_W(\s)/2}}\,\om_{\nu^\perp}^2\,d\|V_F\|\le\frac{\e_0}2\,,\qquad \|V_F\|\big(A_{s_W(\s)/6}^{s_W(\s)/4}\big)\ge c(n)\,.
\end{eqnarray}
\noindent {\bf Step five:} Given $E_j\in{\rm Min}[\psi_W(v_j)]$ for $v_j\to\infty$, we prove the existence of $(F,\nu)\in{\rm Max}[\Rr(W)]$ and $h_j\in C^\infty((\pa F)\setminus B_{R_2})$ such that
\begin{eqnarray}
\label{f of Ev jj}
&&(\pa E_j)\cap A^{R_1\,\l_{j}}_{4\,R_2}
=\Big\{y+h_j(y)\,\nu_F(y):y\in\pa F\Big\}\cap A^{R_1\,\l_{j}}_{4\,R_2}\,,
\\
\label{aspetta}
&&\lim_{j\to\infty}\|h_j\|_{C^1((\pa F)\cap A_{4\,R_2}^M)}=0\,,\qquad\forall M<\infty\,;
\end{eqnarray}
and that if $x_j$ satisfies $|E_j\Delta B^{(v_j)}(x_j)|=\inf_x|E_j\Delta B^{(v_j)}(x)|$, then
\begin{equation}
\label{aspetta ancora}
\lim_{j\to\infty}||x_j|^{-1}\,x_j-\nu|=0\,;
\end{equation}
finally, we prove statement (iii) (i.e., $(\pa E_j)\setminus B_{R_2}$ is diffeomorphic to an $n$-dimensional disk). By step three, there is $(F,\nu)\in{\rm Max}[\Rr(W)]$ such that, up to extracting subsequences, \eqref{convergence of Ej to F last proof} holds. By \eqref{convergence of Ej to F last proof} and \eqref{basta}, and with $s_W(\s)$ defined as in step four (see \eqref{def of sWs}) starting from $F$, we can apply Remark \ref{remark improved convergence} to find $f_j\in C^\infty(\nu^\perp)$ such that
\begin{equation}
\label{basta j}
(\pa E_j)\cap A^{s_W(\s)}_{2\,R_2}=\big\{x+f_j(x)\,\nu:x\in\nu^\perp\big\}\cap A^{s_W(\s)}_{2\,R_2}\,,
\end{equation}
for $j$ large enough (in terms of $\s$, $n$, $W$, and $F$), and such that $f_j\to f$ in $C^1(\DD_{s_W(\s)}^\nu\setminus\DD_{2\,R_2}^\nu)$.
With $R_3$ as in step four and with the goal of applying Theorem \ref{theorem mesoscale criterion} to the varifolds
$V_j=\var((\pa E_j)\setminus B_{R_3},1)$, we notice that $V_j\in\V_n(\Lambda_j,R_3,\infty)$, for some $\Lambda_j\le \Lambda_0\,\l_{j}^{-1}$ (thanks to \eqref{uniform lambda minimality}). In particular, by \eqref{def of sWs}, $s_W(\s)$ satisfies the ``mesoscale bounds'' (compare with \eqref{mesoscale bounds})
\begin{equation}
\label{mesoscale bounds end}
\e_0\,(4\,\Lambda_j)^{-1}>s_W(\s)>\max\{M_0,64\}\,R_3
\end{equation}
provided $j$ is large. Moreover, by $R_3>2\,R_2$ and $s_W(\s)/8>2\,R_2$, by \eqref{basta}, \eqref{basta j} and $f_j\to f$ in $C^1$, we exploit \eqref{blowdown hp check 1 four} and \eqref{blowdown hp check 2 four} to deduce
\begin{eqnarray}
\label{blowdown hp check 1 Ej part one}
\|{\rm bd}_{V_j}\|(\pa B_{R_3})&\le&(2\,\Gamma)\,R_3^{n-1}\,,
\\
\label{blowdown hp check 2 Ej part one}
\end{eqnarray}
We claim that, up to increasing $\Gamma$ (depending on $n$ and $W$), we can entail
\begin{equation}
\label{blowdown hp check 1 Ej part two}
\|V_j\|(B_\rho\setminus B_{R_3}) \le\Gamma\,\rho^n\,,\qquad\forall \rho>R_3\,.
\end{equation}
Indeed, by Theorem \ref{thm existence and uniform min}-(i), for some positive $\Lambda_0$ and $s_0$ depending on $W$ only,  $E_j$ is a $(\Lambda_0\,\l_{j}^{-1},s_0\,\l_{j})$-perimeter minimizer with free boundary in $\Om$. Comparing $E_j$ to $E_j\setminus B_r$ by  \eqref{uniform lambda minimality}, for every $r<s_0\,\l_{j}$,
\begin{equation}
\label{choose Gamma 1}
P(E_j;\Om\cap B_r)\le C(n)\,\big(r^n+\Lambda_0\,\l_{j}^{-1}\,r^{n+1}\big)\le C(n,W)\,r^n\,;
\end{equation}
since, at the same time, if $r>s_0\,\l_{j}$, then
\begin{equation}
\label{choose Gamma 2}
P(E_j;\Om\cap B_r)\le P(E_j;\Om)=\psi_W(v_j)\le P(B^{(v_j)})\le C(n)\,s_0^{-n}\,r^n\,,
\end{equation}
by combining \eqref{choose Gamma 1} and \eqref{choose Gamma 2} we find \eqref{blowdown hp check 1 Ej part two}. With \eqref{blowdown hp check 1 Ej part one} and \eqref{blowdown hp check 1 Ej part two} at hand, we can also show that
\begin{equation}
\label{blowdown hp check 2 Ej part two}
\end{equation}
Indeed, by $s_W(\s)=2\,L\,R_3$ and by $\Lambda_j\le \Lambda_0\,\l_{j}^{-1}$,
\begin{eqnarray*}
\\
&&\le(\Lambda_0/\l_{j})\,\int_{R_3}^{s_W(\s)/8}\!\!\!\!\rho^{-n}\,\|V_j\|(B_\rho\setminus B_{R_3})\,d\rho
\le\frac{\Lambda_0\,R_3\,\Gamma}{\l_{j}}\,\Big(\frac{L}{4}-1\Big)\le\frac{\e_0}3\,,
\end{eqnarray*}
provided $j$ is large enough. To complete checking that Theorem \ref{theorem mesoscale criterion} can be applied to every $V_j$ with $j$ large enough, we now consider the quantities
\[
\]
and prove that, for a constant $\tau_0$ depending on $n$ and $W$ only, we have
\begin{equation}
\label{Rstar true}
R_{*j}\ge \tau_0\,\l_{j}\,;
\end{equation}
in particular, provided $j$ is large enough, \eqref{Rstar true} implies immediately
\begin{equation}
\label{Rstar check}
R_{*j}\ge 4\,s_W(\s)\,,
\end{equation}
which was the last assumption in Theorem \ref{theorem mesoscale criterion} that needed to be checked. To prove \eqref{Rstar true}, we pick $\tau_0$ such that
\begin{equation}
\label{geometry of B1}
\big|\tau_0^{-n}\,\H^n\big(B_{\tau_0}(z)\cap\pa B^{(1)}\big)-\om_n\big|\le \e_0/2\,,\qquad\forall z\in\pa B^{(1)}\,.
\end{equation}
(Of course this condition only requires $\tau_0$ to depend on $n$.) By definition of $x_j$ and by \eqref{limsupmax goes to zero take 2}, and up to extracting a subsequence, we have $x_j\to z_0$ for some $z_0\in\pa B^{(1)}$. In particular, setting $\rho_j=\tau_0\,\l_{j}$, we find
\begin{eqnarray*}
\rho_j^{-n}\,\|V_j\|(B_{\rho_j}\setminus B_{R_3})\!\!\!&=&\!\!\!\tau_0^{-n}\,P\big((E_j-x_j)/\l_{j}\,;\,B_{\tau_0}(-x_j)\setminus B_{R_3/\rho_j}(-x_j)\big)
\\
&&\to\tau_0^{-n}\,\H^n\big(B_{\tau_0}(-z_0)\cap\pa B^{(1)}\big)\le \om_n+(\e_0/2)\,,
\end{eqnarray*}
thus proving that, for $j$ large enough,
\begin{eqnarray*}
&&\!\!\!\!\!\!\!\!\!\de_{V_j,R_3,\Lambda_j}(\rho_j)\!\!\ge\!\!-\frac{\e_0}2+\frac{1}{n\,\rho_j^n}\,\int\,x\cdot\nu^{\rm co}_{V_j}\,d\,{\rm bd}_{V_j}-\Lambda_j\,\int_{R_3}^{\rho_j}\frac{\|V_j\|(B_{\rho}\setminus B_{R_3})}{\rho^n}\,d\rho
\\
\ge-\frac{\e_0}2-\frac{C_*(n,W)}{\tau_0^n\,\l_{j}}-C_{**}(n,W)\,\tau_0\,,
\end{eqnarray*}
where we have used \eqref{blowdown hp check 1 Ej part one}, $\spt\,{\rm bd}_{V_j}\subset\pa B_{R_3}$, and \eqref{blowdown hp check 1 Ej part two}. Therefore, provided we pick $\tau_0$ depending on $n$ and $W$ so that $C_{**}\,\tau_0\le\e_0/4$, and then we pick $j$ large enough to entail $(C_*(n,W)/\tau_0^n)\l_{j}^{-1}\le\e_0/4$, we conclude that if $r\in(R_3,\rho_j]$, then
$\de_{V_j,R_3,\Lambda_j}(r)\ge  \de_{V_j,R_3,\Lambda_j}(\rho_j)\ge-\e_0$,
where in the first inequality we have used Theorem \ref{theorem 7.17AA exterior lambda}-(i) and the fact that $V_j\in\V_n(\Lambda_j,R_3,\infty)$. In summary, by \eqref{blowdown hp check 1 Ej part one} and \eqref{blowdown hp check 1 Ej part two} (which give \eqref{Gamma bounds}), by \eqref{mesoscale bounds end} (which gives \eqref{mesoscale bounds} with $s=s_W(\s)/8$), and by \eqref{blowdown hp check 2 Ej part two} and \eqref{Rstar check} (which imply, respectively, \eqref{mesoscale delta small s8} and \eqref{mesoscale Rstar larger s4}) we see that Theorem \ref{theorem mesoscale criterion}-(i) can be applied with $V=V_j$ and $s=s_W(\s)/8$ provided $j$ is large in terms of $\s$, $n$, $W$ and the limit $F$ of the $E_j$'s. Thus, setting
\begin{equation}\label{def of Starj}
\end{equation}
and noticing that by \eqref{Rstar true} and $\Lambda_j\le \Lambda_0\,\l_{j}^{-1}$ we have
\begin{equation}
\label{Starj big}
S_{*j}\ge 16\,R_1\,\l_{j}\,,
\end{equation}
(for $R_1$ depending on $n$ and $W$ only) we conclude that, for $j$ large, there are $K_j\in\H$ and $u_j\in\X_\s(\S_{K_j},\s_W(\s)/32,R_1\,\l_{j})$, such that
\begin{equation}
\label{par Ej}
(\pa E_j)\cap A_{s_W(\s)/32}^{R_1\,\l_{j}}=\S_{K_j}\big(u_j,s_W(\s)/32,R_1\,\l_{j}\big)\,.
\end{equation}
%$u_j\in C^1(\Sigma_{K_j}\times(s_W(\s)/32,S_{*j}/16))$, such that
%  \begin{eqnarray}\nonumber
%  &&\!\!\!\!\!\!\!\!\!\!\!\!(\spt\,V_j)\cap A_{s_W(\s)/32}^{S_{*j}/16}=\Big\{r\,\frac{\om+u_j(r,\om)\,\nu_{K_j}}{\sqrt{1+u_j(r,\om)^2}}:(\om,r)\in \S_{K_j}\times\Big(\frac{s_W(\s)}{32},\frac{S_{*j}}{16}\Big)\Big\}
%  \\\label{mesoscale thesis graphicality end}
%  &&\sup_{\S_{K_j}\times(s_W(\s)/32,S_{*j}/16)}\Big\{|u_j|+|\nabla^{K_j\cap\SS^n}u_j|+|r\,\pa_ru_j|\Big\}\le \s\,,
Similarly, by \eqref{blowdown hp check 1 four} and \eqref{blowdown hp check 2 four}, thanks to Theorem \ref{theorem mesoscale criterion}-(ii) we have
\begin{equation}
\label{par F}
(\pa F)\cap \big(\R^{n+1}\setminus B_{s_W(\s)/32}\big)=\Sigma_{\nu^\perp}\big(u,s_W(\s)/32,\infty\big)\,,
\end{equation}
for $u\in\X_{\s'}(\S_{\nu^\perp},s_W(\s)/32,\infty)$ for every $\s'>\s$. Now, by $E_j\to F$ in $L^1_{\rm loc}(\R^{n+1})$, \eqref{par Ej} and \eqref{par F} can hold only if $|\nu_{K_j}-\nu|\le \zeta(\s)$ for a function $\zeta$, depending on $n$ and $W$ only, such that $\zeta(\s)\to 0$ as $\s\to 0^+$. In particular (denoting by $\s_0^*$, $\e_0^*$ and $C_0^*$ the dimension dependent constants originally introduced in Lemma \ref{lemma step one} as $\s_0$, $\e_0$ and $C_0$) we can find $\s_1=\s_1(n,W)\le\s_0^*$ such that if $\s<\s_1$, then $\e_0^*\ge \zeta(\s)\ge |\nu_{K_j}-\nu|$, and correspondingly, Lemma \ref{lemma step one}-(i) can be used to infer the existence of
$u_j^*\in\X_{C_0\,(\s+\zeta(\s))}(\S_{\nu^\perp},s_W(\s)/32,2\,R_1\,\l_{j})$ such that, for $j$ large,
\begin{eqnarray}
\nonumber
\S_{\nu^\perp}\big(u_j^*,s_W(\s)/32,2\,R_1\,\l_{j}\big)&=&\S_{K_j}\big(u_j,s_W(\s)/32,2\,R_1\,\l_{j}\big)
\\
&=&(\pa E_j)\cap A_{s_W(\s)/32}^{2\,R_1\,\l_{j}}\,.
\label{star 2 jj}
\end{eqnarray}
By \eqref{basta j} and Lemma \ref{lemma D1}, \eqref{star 2 jj} implies cylindrical graphicality: more precisely, provided $\s_1$ is small enough, there are $g_j\in C^1(\nu^\perp)$ such that
\begin{eqnarray}
\label{what about gj 1}
&&\sup_{x\in\nu^\perp}\{|g_j(x)|\,|x|^{-1},|\nabla g_j(x)|\}\le C\,\big(\s+\zeta(\s)\big)\,,
\\
\label{what about gj 2}
&&(\pa E_j)\cap A^{R_1\,\l_{j}}_{2\,R_2}
=\big\{x+g_j(x)\,\nu:x\in\nu^\perp\big\}\cap A^{R_1\,\l_{j}}_{2\,R_2}\,.
\end{eqnarray}
At the same time, by \eqref{basta}, \eqref{asymptotics of F}, and up to further increasing $R_2$ and decreasing $\s_1$, we can exploit Lemma \ref{lemma D2} in the appendix to find  $h_j\in C^1(G(f))$, $G(f)=\{x+f(x)\,\nu:x\in\nu^\perp\}$, such that
\[
\big\{x+g_j(x)\,\nu:x\in\nu^\perp\big\}\setminus B_{4\,R_2}
=\big\{z+h_j(z)\,\nu_F(z):z\in G(f)\big\}\setminus B_{4\,R_2}
\]
which, combined with \eqref{basta} and \eqref{what about gj 2} shows that
\begin{equation}\nonumber
(\pa E_j)\cap A^{R_1\,\l_{j}}_{4\,R_2}
=\big\{z+h_j(z)\,\nu_F(z):z\in\pa F\big\}\cap A^{R_1\,\l_{j}}_{4\,R_2}
\end{equation}
that is \eqref{f of Ev jj}. By $E_j\to F$ in $L^1_{\rm loc}(\R^{n+1})$, we find $h_j\to 0$ in $L^1((\pa F)\cap A_{4\,R_2}^M)$ for every $M<\infty$, so that, by elliptic regularity, \eqref{aspetta} follows. We now recall that, by Theorem \ref{thm existence and uniform min}-(ii), $(\pa E_j)\setminus B_{R_0(v_j)\,\l_{j}}$ coincides with
\begin{eqnarray}\nonumber
&&\!\!\!\!\!\!\!\!\big\{y+ \l_{j}w_j\big((y-x_j)/\l_{j}\big)\,\nu_{B^{(v_j)}(x_j)}(y):y\in\pa B^{(v_j)}(x_j)\big\}\setminus B_{R_0(v_j)\,\l_{j}}
\\\label{x and u of Ev take 2 jjj}
&&\!\!\!\!\!\!\!\!\mbox{for $\|w_j\|_{C^1(\pa B^{(1)})}\to 0$ and $R_0(v_j)\to 0$}\,.
\end{eqnarray}
The overlapping of \eqref{what about gj 2} and \eqref{x and u of Ev take 2 jjj} (i.e., the fact that $R_0(v_j)<R_1$ if $j$ is large enough) implies statement (iii). Finally, combining \eqref{what about gj 1} and \eqref{what about gj 2} with \eqref{x and u of Ev take 2 jjj} and $\|w_j\|_{C^1(\pa B^{(1)})}\to 0$ we deduce the validity of \eqref{aspetta ancora}. More precisely, rescaling by $\l_j$ in \eqref{what about gj 1} and \eqref{what about gj 2} and setting $E_j^*=E_j/\l_j$, we find $g_j^*\in C^1(\nu^\perp)$ such that, for every $j\ge j_0(\s)$ and  $\s<\s_1$,
\begin{eqnarray}
\label{what about gj 1 star}
&&\sup_{x\in\nu^\perp}\{|g_j^*(x)||x|^{-1},|\nabla g_j^*(x)|\}\le C\,\big(\s+\zeta(\s)\big)\,,
\\
\label{what about gj 2 star}
&&(\pa E_j^*)\cap A^{R_1}_{2\,R_2/\l_j}=\big\{x+g_j^*(x)\,\nu:x\in\nu^\perp\big\}\cap A^{R_1}_{2\,R_2/\l_j}\,,
\end{eqnarray}
while rescaling by $\l_j$ in \eqref{x and u of Ev take 2 jjj} and setting $z_j=x_j/\l_j$ we find
\begin{eqnarray}
\label{x and u of Ev take 2 jjjjj}
&&\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!(\pa E_j^*)\setminus B_{R_0(v_j)}
\!\!=\!\big\{z_j+z+w_j(z)\,\nu_{B^{(1)}}(z)::y\in\pa B^{(1)}(z_j)\big\}\!\setminus \!B_{R_0(v_j)}
\end{eqnarray}
where $||z_j|-\om_{n+1}^{1/(n+1)}|\to 0$ thanks to \eqref{limsupmax goes to zero take 2}. Up to subsequences, $z_j\to z_0$, where $|z_0|=\om_{n+1}^{1/(n+1)}$. Should $z_0\ne|z_0|\,\nu$, then picking $\s$ small enough in terms of $|\nu-(z_0/|z_0|)|>0$ and picking $j$ large enough, we would then be able to exploit \eqref{what about gj 1 star} to get a contradiction with $\|w_j\|_{C^1(\pa B^{(1)})}\to 0$.

\noindent {\bf Conclusion}: Theorem \ref{thm existence and uniform min} implies  Theorem \ref{thm main psi}-(i), and \eqref{main asymptotic expansion} was proved in step three. Should Theorem \ref{thm main psi}-(ii), (iii), or (iv) fail, then we could find a sequence $\{(E_j,v_j)\}_j$ contradicting the conclusions of either step five or Theorem \ref{thm existence and uniform min}. We have thus completed the proof of Theorem \ref{thm main psi}.
\end{proof}

\appendix

\section{Proof of Theorem \ref{theorem 7.15AA main estimate lambda}}\label{appendix proof of expansion mean curvature} We assume $H\in\H$, $\Lambda\ge0$, $\eta_0>\eta>0$,  $(r_1,r_2)$ and $(r_3,r_4)$ are $(\eta,\eta_0)$-related as in \eqref{r1r2r3r4}, and $u\in\X_{\s}(\Sigma_H,r_1,r_2)$ is such that $\S_H(u,r_1,r_2)$ has mean curvature bounded by $\Lambda$ in $A_{r_1}^{r_2}$. We want to find $\s_0$ and $C_0$, depending on $n$, $\eta_0$, and $\eta$ only, such that, if $\max\{1,\Lambda\,r_2\}\,\s\le\s_0$, then
\begin{equation}
\label{lambdaAA 7.14 proof}
\Big|\H^n(\Sigma_H(u,r_3,r_4))-\H^n(\Sigma_H(0,r_3,r_4))\Big|\le C_0\,\int_{\Sigma_H\times(r_1,r_2)}\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!r^{n-1}\,\big(u^2+\Lambda\,r\,|u|\big);
\end{equation}
and such that, if there is $r\in(r_1,r_2)$ s.t. $E_{\Sigma_H}^0[u_r]=0$ on $\Sigma_H$, then
\begin{equation}
\label{lambdaAA 7.15(5) proof}
\int_{\Sigma_H\times(r_3,r_4)} \!\!\!\!\!\!\!\!\!r^{n-1}\,u^2 \le C(n)\,\Lambda\,r_2\,(r_2^n-r_1^n)+ C_0\,\int_{\Sigma_H\times(r_1,r_2)}\!\!\!\!\!\!\!\!\!r^{n-1}\,(r\,\pa_ru)^2 \,.
\end{equation}
%\label{lambdaAA 7.15(5) proof}
% \int_{\Sigma_H\times(r_3,r_4)} r^{n-1}\,\big(u^2+|\nabla^{\S_H}u|^2\big)\le C(n)\,\Lambda\,r_2\,(r_2^n-r_1^n)+ C_0\,\int_{\Sigma_H\times(r_1,r_2)}\,r^{n-1}\,(r\,\pa_ru)^2 \,.
We make three preliminary considerations: {\bf (i):} By \cite[4.5(8)]{allardalmgrenRADIAL}
\begin{eqnarray}\label{second order area radial}
\\
\nonumber
&&\hspace{3cm}-\frac12\int_{\S_H\times(r_1,r_2)}\!\!\!\!\!\!\!\!\!r^{n-1}\,\big(|\nabla^{\Sigma_H} u|^2+(r\,\pa_ru)^2-(n-1)\,u^2\big)\Big|
\\
\nonumber
&&\hspace{3cm}\le C(n)\,\s\,\int_{\S_H\times(r_1,r_2)}\!\!\!\!\!\!\!\!\!r^{n-1}\big(u^2+|\nabla^{\Sigma_H} u|^2+(r\,\pa_ru)^2\big)\,.
\end{eqnarray}
Similarly, by the last displayed formula on \cite[Page 236]{allardalmgrenRADIAL} and by \cite[Lemma 4.9(1)]{allardalmgrenRADIAL}, if $\vphi=\psi^2\,w$, $w\in C^1(\S_H\times(r_1,r_2))$ and $\psi\in C^1(r_1,r_2)$, then
\begin{eqnarray}\label{second order first variation radial}
\\\nonumber
&&\hspace{0.5cm}-\int_{\Sigma_H\times(r_1,r_2)}\!\!\!\!\!\!\!\!\!\!r^{n-1}\,\Big\{\nabla^{\Sigma_H}u\cdot\nabla^{\S_H}\vphi+(r\,\pa_ru)\,(r\,\pa_r \vphi)-(n-1)\,u\,\vphi\Big\}\Big|
\\\nonumber
&\le& C(n)\,\s\,\int_{\Sigma_H\times(r_1,r_2)}\!\!\!\!\!\!\!\!\!\!r^{n-1}\,\psi^2\,\big\{|\nabla^{\Sigma_H}u|^2+|\nabla^{\Sigma_H}w|^2
\\\nonumber
\end{eqnarray}
which is the second order expansion of the first variation of the area at $\S_H(u,r_1,r_2)$ along outer variations in spherical coordinates of the form $\vphi=\psi^2\,w$, $\psi=\psi(r)$. {\bf (ii):} For the sake of brevity, given $\zeta:(r_1,r_2)\to\R$ a radial function, $u,v:\Sigma_H\times(r_1,r_2)\to\R$, $X,Y:\Sigma_H\times(r_1,r_2)\to\R^m$, we set
\begin{eqnarray*}
Q_\zeta(u,v)=\int_{\Sigma_H\times(r_1,r_2)}\!\!\!\!\!\!\!\!\!\!\!\!r^{n-1}\,\zeta(r)^2\,u\,v\,,\quad Q_\zeta(X,Y)=\int_{\Sigma_H\times(r_1,r_2)}\!\!\!\!\!\!\!\!\!\!\!\!r^{n-1}\,\zeta(r)^2\,X\cdot Y\,,
\end{eqnarray*}
and $Q_\zeta(u)=Q_\zeta(u,u)$, $Q_\zeta(X)=Q_\zeta(X,X)$. {\bf (iii):} The following two estimates (whose elementary proof is contained in \cite[Lemma 7.13]{allardalmgrenRADIAL}) hold: whenever $v\in C^1(\Sigma_H\times (r_1,r_2))$, we have
\begin{eqnarray}\label{elementary 1}
\int_{\S_H\times(r_1,r_2)}\!\!\!\!\!\!\!\!\!\!r^{n-1}\,v^2\le C(n,\eta,\eta_0)\,\Big\{  \int_{\S_H\times(r_1,r_2)}\!\!\!\!\!\!\!\!\!\!r^{n-1}\,(r\,\pa_rv)^2
+  \int_{\S_H\times(r_3,r_4)}\!\!\!\!\!\!\!\!\!\!\!\!r^{n-1}\,v^2\Big\},
\end{eqnarray}
and, provided there is $r\in[r_1,r_2]$ such that $v_r=0$ on $\S_H$, we have
\begin{equation}\label{elementary 2}
\int_{\S_H\times(r_1,r_2)}r^{n-1}\,v^2\le C(n,\eta_0)\,  \int_{\S_H\times(r_1,r_2)}r^{n-1}\,(r\,\pa_rv)^2\,.
\end{equation}
We are now ready for the proof. Compared to \cite[Chapter 4]{allardalmgrenRADIAL}, the main difference is that we replace \cite[Lemma 4.10]{allardalmgrenRADIAL} with \eqref{going back to psi w plug in}.

\noindent {\bf Step one:} We prove that there is $h:\Sigma_H\times(r_1,r_2)\to[-\Lambda,\Lambda]$ such that for every $w\in C^1(\S_H\times(r_1,r_2))$ and $\psi\in C^1(r_1,r_2)$ we have
\begin{eqnarray}\nonumber
\le C(n)\s_0\big(Q_\psi(u)+Q_\psi(w)+Q_\psi(\nabla^{\Sigma_H} u)
\\\label{going back to psi w plug in}
&&\hspace{2cm}+Q_\psi(\nabla^{\Sigma_H} w)+Q_{r\,\psi}(\pa_ru)+Q_{r\,\psi}(\pa_rw)+Q_{r\,\psi'}(w)\big)\,.
\end{eqnarray}
where $T_\psi(u,w)=Q_\psi(\nabla^{\Sigma_H} u,\nabla^{\Sigma_H} w)+Q_{r}\big(\pa_ru,\pa_r[\psi^2w]\big)-(n-1)\,Q_\psi(u,w)$.
%  \\\nonumber
%  &=& Q_\psi(\nabla^{\Sigma_H} u,\nabla^{\Sigma_H} w)+Q_{r\,\psi}(\pa_ru,\pa_rw)-(n-1)\,Q_\psi(u,w)+2\,Q_r(\psi\,\pa_ru,\psi'w)\,.\end{eqnarray}
We start rewriting \eqref{second order first variation radial} as
\begin{eqnarray}\label{going back to psi w}
\\\nonumber
&&\le C(n)\,\s\,\big(Q_\psi(u)+Q_\psi(w)+Q_\psi(\nabla^{\Sigma_H} u)+Q_\psi(\nabla^{\Sigma_H} w)
\\\nonumber
\end{eqnarray}
If $F_{u+t\,\vphi}:{\Sigma_H}\times(r_1,r_2)\to{\Sigma_H}(u+t\,\vphi,r_1,r_2)$, $\vphi=\psi^2\,w$, is given by
\[
\]
then $\{\Phi_t=F_{u+t\,\vphi}\circ (F_u)^{-1}\}_{t\in[0,1]}$ are diffeomorphisms on ${\Sigma_H}(u,r_1,r_2)$, with $\Phi_t(\S_H(u,r_1,r_2))=\Sigma_H(u+t\,\vphi,r_1,r_2)$ and $\dot{\Phi}_0=(d/dt)_{t=0}\Phi_t$. Since $\S_H(u,r_1,r_2)$ has mean curvature bounded by $\Lambda$ in $A_{r_1}^{r_2}$, for some bounded function $h:\Sigma_H\times(r_1,r_2)\to[-\Lambda,\Lambda]$ we have
\begin{eqnarray*}
\frac{d}{dt}\Big|_{t=0}\,\H^n(\S(u+t\vphi,r_1,r_2))
\\
&=&\Lambda\,\int_{\Sigma_H\times(r_1,r_2)}\!\!\!\!\!\!\!\!\!\!\!h\,\dot{\Phi}_0\big(F_u\big)\cdot\star\big(\pa_r F_u\wedge\,\wedge_{i=1}^{n-1}\pa_i F_u\big)\,,
\end{eqnarray*}
where $\pa_i=\nabla_{\tau_i}$ for a local orthonormal frame $\{\tau_i\}_{i=1}^{n-1}$ in $\Sigma_H$, and where $\star$ is the Hodge star-operator (so that $\star\,(v_1\wedge v_2...\wedge v_n)$ is a normal vector to the hyperplane spanned by the $v_i$s, with length equal to the $n$-dimensional volume of the parallelogram defined by the $v_i$s, and whose orientation depends on the ordering of the $v_i$s themselves). We can compute the initial velocity $\dot{\Phi}_0$ of $\{\Phi_t\}_{t\in[0,1]}$ by noticing that
$\Phi_t\big(F_u(\om,r)\big)=r\,(1+(u+t\,\vphi)^2)^{-1/2}\,(\om+(u+t\,\vphi)\,\nu_H)$, so that,
\begin{eqnarray*}
\dot\Phi_0(F_u)&=&\frac{d}{dt}\Big|_{t=0}\,r\,\frac{\om+(u+t\,\vphi)\,\nu_H}{\sqrt{1+(u+t\,\vphi)^2}}
\\
&=&r\,\big(-u\,\vphi\,\om+\vphi\,\nu_H)+r\,\s\,{\rm O}\big(\psi^2\,(u^2+w^2)\big)\,.
\end{eqnarray*}
At the same time
\begin{eqnarray*}
\pa_rF_u\!\!&=&\!\!\!\frac{\om+u\,\nu_H}{\sqrt{1+u^2}}\!+r\,\pa_r\Big(\frac{\om+u\,\nu_H}{\sqrt{1+u^2}}\Big)
\!\!=\!\frac{\om+u\,\nu_H}{\sqrt{1+u^2}}\!-\frac{r\,u\,\pa_ru\,\om}{(1+u^2)^{3/2}}
+\!\frac{r\,\pa_r u\,\nu_H}{(1+u^2)^{3/2}}
\\
&=&\!\!\!\big(1-(u^2/2)-u\,r\,\pa_ru\big)\,\om+\big(u+r\,\pa_ru\big)\,\nu_H+\s\,{\rm O}(u^2+(r\pa_r u)^2)
\\
&=&\!\!\!A\,\om+B\,\nu_H+\s\,{\rm O}(u^2+(r\pa_r u)^2)\,,
\\
\frac{\pa_iF_u}r\!\!&=&\!\!\!\pa_i\Big(\frac{\om+u\,\nu_H}{\sqrt{1+u^2}}\Big)=\frac{\tau_i}{\sqrt{1+u^2}}-\frac{u\,\pa_iu}{(1+u^2)^{3/2}}\,\om
\\
&=&\big(1-(u^2/2)\big)\,\tau_i-u\,\pa_iu\,\om+\pa_iu\,\nu_H+\s\,{\rm O}(u^2+(\pa_iu)^2)
\\
&=& C\,\tau_i+ E_i\,\om+F_i\,\nu_H+\s\,{\rm O}(u^2+(\pa_iu)^2)
\end{eqnarray*}
so that, with $\Xi=\wedge_{i=1}^{n-1}\tau_i$, $\hat\tau_i=\wedge_{j\ne i}\tau_j$, and $P(u)^2=u^2+|\nabla^{\Sigma_H} u|^2+(r\pa_r u)^2$,
\begin{eqnarray*}
&&\!\!\!\!\!\!\!\!\frac{\pa_r F_u\wedge\wedge_{i=1}^{n-1}\pa_iF_u}{r^{n-1}}
\!=\!\big(A\,\om+B\,\nu_H\big)\!\wedge\!\wedge_{i=1}^{n-1}\big(C\,\tau_i+E_i\,\om+F_i\,\nu_H\big)
\!+\!\s{\rm O}\big(P(u)^2\big)
\\
+G_i\,\big(\om\wedge\nu_H\wedge\hat\tau_i\big)+\s{\rm O}\big(P(u)^2\big),
\end{eqnarray*}
for a coefficient $G_i$ which we do not need to compute. Indeed, $\star(\om\wedge\nu_H\wedge\hat\tau_i)$, being parallel to $\tau_i$, is orthogonal to $\om$ and $\nu_H$, so that
\begin{eqnarray*}
&&r^{-n}\,\dot{\Phi}_0\big(F_u(r,\om)\big)\cdot\star\big(\pa_r F_u\wedge\,\wedge_{i=1}^{n-1}\pa_i F_u\big)
\\
&=&\!\!\!\!\big[\big(-u\,\vphi\,\om+\vphi\,\nu_H)+\s\,{\rm O}\big(\psi^2(u^2+v^2)\big)\big]\cdot
\\
+\s\,{\rm O}\big(P(u)^2\big)\big]
\\
\!\!+\s\,{\rm O}\big(\psi^2\,(w^2+P(u)^2)\big)
\\
&=&\!\!\!\!\vphi+\s\,{\rm O}\big(\psi^2\,(w^2+P(u)^2)\big)
\end{eqnarray*}
In particular, since $|h|\le\Lambda$,
\begin{eqnarray*}
=\int_{\S\times(r_1,r_2)}\!\!\!\!\!\!\!\!h\,\,\dot{\Phi}_0\big(F_u\big)\cdot\star\big(\pa_r F_u\wedge\,\wedge_{i=1}^{n-1}\pa_i F_u\big)
\\
\s\Lambda r_2{\rm O}\big(Q_\psi(u)+Q_\psi(w)+Q_\psi(\nabla^{\Sigma_H} u)+Q_{r\,\psi}(\pa_ru)\big)\,.
\end{eqnarray*}
Plugging this estimate into \eqref{going back to psi w}, and  by $\max\{1,\Lambda\,r_2\}\,\s\le\s_0$, we find \eqref{going back to psi w plug in}.

\noindent {\bf Step two:} We prove that
\begin{eqnarray}\label{lemma 411(1)}
Q_\psi(\nabla^{\Sigma_H} u)+Q_{r\,\psi}(\pa_ru)\le Q_\psi(|u|,\Lambda\,r)+C(n)\,\big(Q_{\psi}(u)+Q_{r\,\psi'}(u)\big)\,.
\end{eqnarray}
By $Q_r(\psi\,\pa_ru,\psi'u)\le Q_{r\,\psi}(\pa_ru)/4+C\,Q_{r\,\psi'}(u)$ and by \eqref{going back to psi w plug in}$_{w=u}$ we find
\begin{eqnarray*}\label{proof of 411(1)}
&&Q_\psi(\nabla^{\Sigma_H} u)+Q_{r\,\psi}(\pa_ru)\le
\\\nonumber
&&+C(n)\,\s_0\,\big(Q_\psi(u)+Q_{r\,\psi'}(u)+Q_\psi(\nabla^{\Sigma_H} u)+Q_{r\,\psi}(\pa_ru)\big)\,.
\end{eqnarray*}
which implies \eqref{lemma 411(1)} provided $\s_0$ is small enough.

\noindent {\bf Step three:} We prove that, if $w:\S_H\times(r_1,r_2)\to\R$ is slice-wise orthogonal to $u-w$, in the sense that $\int_{\Sigma_H}w_r\,(u_r-w_r)=0$, $\int_{\Sigma_H}\pa_rw_r\,(\pa_ru_r-\pa_rw_r)=0$, and $\int_{\Sigma_H}\nabla^{\Sigma_H} w_r\cdot(\nabla^{\Sigma_H} u_r-\nabla^{\Sigma_H} w_r)=0$ for every $r\in(r_1,r_2)$, then
\begin{equation}
\label{lemma 411(2)}
|T_\psi(u,w)|\le Q_\psi(|w|,\Lambda\,r)+C(n)\,\s_0\,\big(Q_\psi(u)+Q_{r\,\psi'}(u)+Q_\psi(|u|,\Lambda\,r)\big)\,.
\end{equation}
Indeed, by slice-wise orthogonality, we find that $Q_\zeta(w)\le Q_\zeta(u)$, $Q_\zeta(\pa_rw)\le Q_\zeta(\pa_ru)$ and $Q_\zeta(\nabla^{\Sigma_H} w)\le Q_\zeta(\nabla^{\Sigma_H} u)$ whenever $\zeta:(r_1,r_2)\to\R$ is radial. Therefore \eqref{going back to psi w plug in} gives
$|T_\psi(u,w)|\le Q_\psi(|w|,\Lambda\,r)+C(n)\,\s_0\,R_\psi(u)$, with $R_\psi(u)=Q_\psi(u)+Q_\psi(\nabla^{\Sigma_H} u)+Q_{r\,\psi}(\pa_ru)+Q_{r\,\psi'}(u)$. Combining this with \eqref{lemma 411(1)} we get \eqref{lemma 411(2)}.

\noindent {\bf Step four:} We prove \eqref{lambdaAA 7.14 proof}. Let now $\psi$ be a cut-off function between $(r_3,r_4)$ and $(r_1,r_2)$, so that with $Z_\psi(u)=Q_\psi(\nabla^{\S_H}u)+Q_\psi(u)+Q_{r\,\psi}(\pa_ru)$,
\begin{eqnarray*}
\Big|\int_{\S_H\times(r_3,r_4)}r^{n-1}\big\{|\nabla^{\Sigma_H}u|^2-(n-1)\,u^2+(r\,\pa_ru)^2\big\}\Big|
\le Z_\psi(u)\,.
\end{eqnarray*}
If $A(u)=\H^n(\S_H(u,r_3,r_4))-\H^n(\S_H(0,r_3,r_4))$, then by \eqref{second order area radial} with $(r_3,r_4)$ in place of $(r_1,r_2)$, we find
\begin{eqnarray*}
&&|A(u)|\le Z_\psi(u)+C(n)\,\s\,Z_\psi(u)\le Q_\psi(|u|,\Lambda\,r)+C(n)\,\big(Q_{\psi}(u)+Q_{r\,\psi'}(u)\big)
\\
\end{eqnarray*}
where in the last inequality we have used  \eqref{lemma 411(1)}. We deduce
\begin{equation}\label{zephyr}
|A(u)|\le C(n)\,\big(Q_\psi(|u|,\Lambda\,r)+Q_{\psi}(u)+Q_{r\,\psi'}(u)\big)\,,
\end{equation}
and \eqref{lambdaAA 7.14 proof} follows (with $C_0=C_0(n,\eta_0,\eta)$ by the properties of $\psi$).

\noindent {\bf Step five:} We finally prove that, if $E^0_{\Sigma_H}[u_{r_*}]=0$ for some $r_*\in(r_1,r_2)$, then \eqref{lambdaAA 7.15(5) proof} holds, that is
%%%\label{main estimate 0}
%%%  &&\int_{\Sigma_H\times(r_3,r_4)}r^{n-1}\,\big(u^2+|\nabla^{\S_H}u|^2\big)
%%%  \\\nonumber
%%%  &&\le C(n)\,\Lambda\,r_2\,(r_2^n-r_1^n)+ C(n,\eta_0,\eta)\,\int_{\S\times(r_1,r_2)}\,r^{n-1}\,(r\,\pa_ru)^2\,,
%%%%whenever $(r_1,r_2)$ and $(r_3,r_4)$ are $(\eta_0,\eta)$-related. To this end let us set $b=r_0\,(1+(\eta_0+\eta)/2)$ and $a=r_0\,(1-(\eta_0+\eta)/2)$. If we let $\psi$ be a cut-off function between $(r_3,r_4)$ and $(a,b)$ in \eqref{lemma 411(1)}, then we obtain
%%  \label{mmm}
%%  \int_{\S\times(r_3,r_4)}r^{n-1}\,|\nabla^{\S_H}u|^2\le C(n)\,\Lambda\,b\,(b^n-a^n)+C(n,\eta_0,\eta)\,\int_{\S\times(a,b)}r^{n-1}\,u^2\,,
%%and thus \eqref{main estimate 0} follows if we prove
%%  \label{main estimate}
%%  \int_{\Sigma_H\times(a,b)}r^{n-1}\,u^2\le C(n)\,\Lambda\,r_2\,(r_2^n-r_1^n)+
%% C(n,\eta_0,\eta)\,\int_{\S\times(r_1,r_2)}\,r^{n-1}\,(r\,\pa_ru)^2\,.
\begin{equation}
\label{main estimate}
\int_{\Sigma_H\times(r_3,r_4)}\!\!\!\!\!\!\!\!\!\!\!\!\!\!r^{n-1}\,u^2\le C(n)\,\Lambda\,r_2\,(r_2^n-r_1^n)+
\end{equation}
Define $u^+,u^-,u^0:\Sigma_H\times(r_1,r_2)\to\R$ by setting, for $r\in(r_1,r_2)$,
$(u^+)_r=E^+_{\Sigma_H}[u_r]$, $(u^-)_r=E^-_{\Sigma_H}[u_r]$ and $(u^0)_r=E^0_{\Sigma_H}[u_r]$,
where $E^\pm_{\Sigma_H}$ denote the $L^2(\Sigma_H)$-orthogonal projections on the spaces of positive/negative eigenvectors of the Jacobi operator of $\Sigma_H$, and where $E^0_{\Sigma_H}$ is the $L^2(\Sigma_H)$-orthogonal projection onto the space of the Jacobi fields of $\Sigma_H$. Since $(u^0)_{r_*}=0$, we can directly apply \eqref{elementary 2} with $v=u^0$ and deduce that
\begin{equation}
\label{deh 0}
\int_{\Sigma_H\times(r_1,r_2)}r^{n-1}\,(u^0)^2\le C(n,\eta_0)\,\int_{\Sigma_H\times(r_1,r_2)}r^{n-1}\,(r\,\pa_ru^0)^2\,.
\end{equation}
By the orthogonality relations between $u^0_r$, $u^+_r$ and $u^-_r$ we have that
\begin{eqnarray}\label{deh 1}
\\\label{deh 2}
\int_{\Sigma_H\times(r_1,r_2)}\!\!\!\!\!\!\!\!\!\!r^{n+1}\,\big((\pa_ru^0)^2+(\pa_ru^+)^2+(\pa_ru^-)^2\big).
\end{eqnarray}
By the spectral theorem, for every $r\in(r_1,r_2)$ we have
C_1(n)^{-1}\,\int_{\Sigma_H}(u^-)_r^2\le\,\int_{\Sigma_H}\,(n-1)\,(u^-)_r^2-|\nabla^{\Sigma_H} (u^-)_r|^2\,,
which, multiplied by $r^{n-1}\,\psi^2$, gives
\begin{eqnarray*}
C_1(n)^{-1}\,Q_\psi(u^-)&\le& (n-1)\,Q_\psi(u^-)-Q_\psi(\nabla^{\Sigma_H}u^-)
\\
\\
\end{eqnarray*}
where in the second to last identity we have used that $w=u^-$ is slice-wise orthogonal to $w-u$; in particular, by \eqref{lemma 411(2)} with $w=u^-$, we find
\begin{eqnarray}\nonumber
&& C_1(n)^{-1}\,Q_\psi(u^-)\le
\,.
\end{eqnarray}
Again by slice-wise orthogonality of $w=u^-$ to $w-u$, we have
\begin{eqnarray*}
\\
&&+2\,Q_r(\psi'\,\pa_ru^-,\psi\,u^-)\le Q_{r\,\psi}(\pa_ru^-)+\frac{Q_\psi(u^-)}{2\,C_1(n)}+C(n)\,Q_{r\,\psi'}(\pa_r u^-)\,,
\end{eqnarray*}
which combined into \eqref{cookies} gives
\begin{eqnarray}\nonumber
Q_\psi(|u^-|,\Lambda\,r)+Q_{r\,\psi}(\pa_ru^-)+C(n)\,Q_{r\,\psi'}(\pa_r u^-)
\\\nonumber
\end{eqnarray}
Using H\"older inequality again we have
\begin{eqnarray*}
\\
&&Q_\psi(|u|,\Lambda\,r)\le 2\,Q_\psi(u)+C(n)\,\Lambda\,r_2\,(r_2^n-r_1^n)\,,
\\\nonumber
\mbox{so that}\!\!\!&&\frac{Q_\psi(u^-)}{4\,C_1(n)}\le Q_{r\,\psi}(\pa_ru^-)+C(n)\big(Q_{r\,\psi'}(\pa_r u^-)+\Lambda\,r_2\,(r_2^n-r_1^n)\big)
\\\nonumber
\end{eqnarray*}
Taking $\psi$ to be a cut-off function between $(r_3,r_4)$ and $(r_1,r_2)$, we find
\begin{eqnarray}\label{deh}
&&\int_{\S\times(r_3,r_4)}\!\!\!\!\!\!r^{n-1}\,(u^-)^2\le C(n)\,\Lambda\,r_2\,(r_2^n-r_1^n)
\\\nonumber
\int_{\S\times(r_1,r_2)}\!\!\!\!\!\!r^{n-1}\,(r\,\pa_ru^-)^2
\end{eqnarray}
By combining \eqref{deh 0}, \eqref{deh}, and the analogous estimate to \eqref{deh} for $u^+$ with \eqref{deh 1} and \eqref{deh 2} we find that \eqref{deh} holds with $u$ in place of $u^-$; this latter estimate,
%%% C(n,\eta_0,\eta)\,\int_{\S\times(r_1,r_2)}\!\!\!\!\!\!r^{n-1}\,(r\,\pa_ru)^2
thanks to \eqref{elementary 1},
%(with $(a,b)$ in place of $(r_3,r_4)$ -- notice that $(r_1,r_2)$ and $(a,b)$ are $(\eta_0,(\eta_0+\eta)/2)$-related, so the resulting constant is still $C=C(n,\eta_0,\eta)$)
finally gives \eqref{main estimate}.
%%%%\section{Proof of the monotonicity formula}\label{appendix proof of monotonicity} We want to prove (see the proof of Theorem \ref{theorem 7.17AA exterior lambda}-(i), that, if $V\in\V_n(\Lambda,R,S)$, that is, if  $V$ is an $n$-dimensional integral varifolds in $\R^{n+1}$ such that
%%%%&&\spt\,V\subset\R^{n+1}\setminus B_R\,,
%%%%\\\label{mono proof 1}
%%%%&&\int\,\Div^TX\,dV=\int\,X\cdot\vec{H}\,d\|V\|+\int X\cdot\nu_V^{\rm co}\,d \,\,{\rm bd}_V\,,\qquad \forall X\in C^1_c(B_S;\R^{n+1})\,,
%%%%hold with $\vec{H}:\R^{n+1}\to\R^{n+1}$ a Borel vector field such that $|\vec{H}|\le \Lambda$, ${\rm bd}_V$ a Radon measure in $\R^{n+1}$, and  $\nu_V^{\rm co}:\pa B_R\to\R^{n+1}$ a Borel vector field with
%%%%\spt\,{\rm bd}_V\subset\pa B_R\,,\qquad\mbox{$|\nu_V^{\rm co}|=1$ ${\rm bd}_V$-a.e.}\,;
%%%%\Theta_{V,R,\Lambda}(r)=\frac{\|V\|(B_r\setminus B_R)}{r^n}-\frac1{n\,r^n}\,\int x\cdot\nu^{\rm co}_V\,d\,{\rm bd}_V
%%%%+\Lambda\,\int_R^r\,\frac{\|V\|(B_\rho\setminus B_R)}{\rho^n}\,d\rho\,,
%%%%is increasing on $(R,S)$, with
%%%%\frac{d\Theta_{V,R,\Lambda}}{d\rho}\ge \frac{d}{d\rho}\int_{B_\rho\setminus B_R}\,\frac{|x^\perp|^2}{|x|^{n+2}}\,d\|V\|\,,
%%%%for a.e. $\rho\in(R,S)$ (compare to \eqref{def of Mon}). Denoting by $M$ a locally $\H^n$-rectifiable set such that  $V=\var(M,\theta)$ for some multiplicity function $\theta$, we take $\zeta\in C^1_c([0,1))$ and $s\in(R,S)$, set $r(x)=|x|$ and $X(x)=\zeta(r(x)/s)\,x$, so that $X\in C^1_c(B_S;\R^{n+1})$, and test \eqref{mono proof 1} with $X$. If $\nu_M:M\to\SS^n$ is such that $T_xM=\nu_M(x)^\perp$ at $\H^n$-a.e. $x\in M$, then
%%%%and \eqref{mono proof 1} gives
%%%%  \int_{B_S\setminus B_R}\Big\{\frac{r}{s}\,\zeta'\Big(\frac{r}s\Big)+n\,\zeta\Big(\frac{r}{s}\Big)\Big\}\,d\|V\|
%%%%  -
%%%%  \int_{B_S\setminus B_R}\frac{r}{s}\,\zeta'\Big(\frac{r}s\Big)\,(\hat{x}\cdot\nu_M)^2 \,d\|V\|
%%%%  \\&=&\int_{B_S\setminus B_R}(x\cdot\vec{H})\,\zeta\Big(\frac{r}{s}\Big)\,d\|V\|
%%%%  +\int_{\pa B_R} (x\cdot\nu^{\rm co}_V)\,\zeta\Big(\frac{R}s\Big)\,d\,{\rm bd}_V\,.
%%%%We multiply by $s^{-n-1}$ and integrate in $s$ over $(\s,\rho)\cc(R,S)$ to obtain
%%%%  \int_{B_S\setminus B_R}\,d\|V\|\,\int_\s^\rho\,\Big\{\frac{r}{s}\,\zeta'\Big(\frac{r}s\Big)+n\,\zeta\Big(\frac{r}{s}\Big)\Big\}\,\frac{ds}{s^{n+1}}
%%%%  - \int_{B_S\setminus B_R}(\hat{x}\cdot\nu_M)^2 \,d\|V\|
%%%%  \int_\s^\rho\,\frac{r}{s}\,\zeta'\Big(\frac{r}s\Big)\,\frac{ds}{s^{n+1}}\,
%%%%  \\&=&\int_{B_S\setminus B_R}(x\cdot\vec{H})\,d\|V\|  \int_\s^\rho\,\zeta\Big(\frac{r}{s}\Big)\frac{ds}{s^{n+1}}\,
%%%%  +\int_{\pa B_R} (x\cdot\nu^{\rm co}_V)\,\,d\,{\rm bd}_V\int_\s^\rho\,\zeta\Big(\frac{R}{s}\Big)\frac{ds}{s^{n+1}}\,.
%%%%We notice that
%%%%\int_{B_S\setminus B_R}\,d\|V\|\,
%%%%  \int_\s^\rho\,\Big\{\frac{r}{s}\,\zeta'\Big(\frac{r}s\Big)+n\,\zeta\Big(\frac{r}{s}\Big)\Big\}\,\frac{ds}{s^{n+1}}
%%%%=\int_{B_S\setminus B_R}\,\Big\{\frac1{\s^n}\,\zeta\Big(\frac{r}\s\Big)-\frac1{\rho^n}\,\zeta\Big(\frac{r}\rho\Big)\Big\}\,d\|V\|
%%%%&&-\int_{B_S\setminus B_R}(\hat{x}\cdot\nu_M)^2 \,d\|V\|  \int_\s^\rho\,\frac{r}{s}\,\zeta'\Big(\frac{r}s\Big)\,\frac{ds}{s^{n+1}}\,
%%%%&&=\int_{B_S\setminus B_R}\Big\{
%%%%\frac1{\rho^n}\,\zeta\Big(\frac{r}\rho\Big)-\frac1{\s^n}\,\zeta\Big(\frac{r}\s\Big)+n\,\int_\s^\rho\,\zeta\Big(\frac{r}s\Big)\,\frac{ds}{s^{n+1}}\Big\}(\hat{x}\cdot\nu_M)^2 \,d\|V\|
%%%%Therefore, letting $\zeta\to 1_{[0,1)}$ in
%%%%\int_{B_S\setminus B_R}\,\Big\{\frac1{\s^n}\,\zeta\Big(\frac{r}\s\Big)-\frac1{\rho^n}\,\zeta\Big(\frac{r}\rho\Big)\Big\}\,
%%%%&&+\int_{B_S\setminus B_R}\Big\{
%%%%\frac1{\rho^n}\,\zeta\Big(\frac{r}\rho\Big)-\frac1{\s^n}\,\zeta\Big(\frac{r}\s\Big)+n\,\int_\s^\rho\,\zeta\Big(\frac{r}s\Big)\,\frac{ds}{s^{n+1}}\Big\}(\hat{x}\cdot\nu_M)^2 \,d\|V\|
%%%%\\&=&\int_{B_S\setminus B_R}(x\cdot\vec{H})\,d\|V\|  \int_\s^\rho\,\zeta\Big(\frac{r}{s}\Big)\frac{ds}{s^{n+1}}\,
%%%%  +\int_{\pa B_R} (x\cdot\nu^{\rm co}_V)\,\,d\,{\rm bd}_V\int_\s^\rho\,\zeta\Big(\frac{R}{s}\Big)\frac{ds}{s^{n+1}}\,,
%%%%one obtains
%%%%\int_{B_S\setminus B_R}\,\Big\{\frac{1_{B_\s}}{\s^n}\,-\frac{1_{B_\rho}}{\rho^n}\Big\}\,d\|V\|
%%%%&&+\int_{B_S\setminus B_R}\Big\{
%%%%\frac{1_{B_\rho}}{\rho^n}\,-\frac{1_{B_\s}}{\s^n}\,+n\,\int_\s^\rho\,1_{[0,1)}\Big(\frac{r}{s}\Big)\,\frac{ds}{s^{n+1}}\Big\}(\hat{x}\cdot\nu_M)^2 \,d\|V\|
%%%%\\&=&\int_{B_S\setminus B_R}(x\cdot\vec{H})\,d\|V\|  \int_\s^\rho\,1_{[0,1)}\Big(\frac{r}{s}\Big)\frac{ds}{s^{n+1}}\,
%%%%  +\int_{\pa B_R} (x\cdot\nu^{\rm co}_V)\,\,d\,{\rm bd}_V\int_\s^\rho\,1_{[0,1)}\Big(\frac{R}{s}\Big)\frac{ds}{s^{n+1}}\,.
%%%%We compute that
%%%%so that
%%%%&&\int_{B_S\setminus B_R}\Big\{
%%%%\frac{1_{B_\rho}}{\rho^n}\,-\frac{1_{B_\s}}{\s^n}\,+n\,\int_\s^\rho\,1_{[0,1)}\Big(\frac{r}{s}\Big)\,\frac{ds}{s^{n+1}}\Big\}(\hat{x}\cdot\nu_M)^2 \,d\|V\|
%%%%&&=\int_{B_S\setminus B_R}\Big\{
%%%%\frac{1_{B_\rho}}{\max\{r,\s\}^n}-\frac{1_{B_\s}}{\s^n}\Big\}\,\,(\hat{x}\cdot\nu_M)^2 \,d\|V\|
%%%%=\int_{B_\rho\setminus B_\s}\frac{(\hat{x}\cdot\nu_M)^2}{r^n} \,d\|V\|\,,
%%%%&&\int_{B_\rho\setminus B_\s}\frac{(\hat{x}\cdot\nu_M)^2}{r^n} \,d\|V\|=
%%%%\frac{\|V\|(B_\rho\setminus B_R)}{\rho^n}-\frac{\|V\|(B_\s\setminus B_R)}{\s^n}
%%%%&&+\int_{B_S\setminus B_R}(x\cdot\vec{H})\,d\|V\|  \int_\s^\rho\,1_{[0,1)}\Big(\frac{r}{s}\Big)\frac{ds}{s^{n+1}}\,
%%%%  +\frac1n\,\Big(\frac1{\s^n}-\frac1{\rho^n}\Big)\,\int_{\pa B_R} (x\cdot\nu^{\rm co}_V)\,\,d\,{\rm bd}_V\,,
%%%%where we have used $1_{[0,1)}(R/s)=1$ for $s\in(\s,\rho)$. Finally, by
%%%%\Big|\int_{B_S\setminus B_R}(x\cdot\vec{H})\,d\|V\|  \int_\s^\rho\,1_{[0,1)}\Big(\frac{r}{s}\Big)\frac{ds}{s^{n+1}}\Big|
%%%%&\le&\Lambda\,\int_\s^\rho\,\frac{1}{s^{n+1}}\,\int_{B_s\setminus B_R}r\,d\|V\|
%%%%&\le&\Lambda\,\int_\s^\rho\,\frac{\|V\|(B_s\setminus B_R)}{s^n}\,ds\,,
%%%%we conclude as claimed that
%%%%\Theta_{V,\Lambda,R}(\rho)-\Theta_{V,\Lambda,R}(\s)\ge\int_{B_\rho\setminus B_\s}\frac{(\hat{x}\cdot\nu_M)^2}{r^n} \,d\|V\|
%%%%=\int_{B_\rho\setminus B_\s}\,\frac{|x^\perp|^2}{|x|^{n+2}}\,d\|V\|\,,
%%%%whenever $(\s,\rho)\subset(R,S)$.

\section{Spherical and cylindrical graphs}\label{appendix spherical cylindrical} We state here for the reader's convenience two technical lemmas concerning spherical and cylindrical graphs. They are both used in the last step of the proof of Theorem \ref{thm main psi}. The elementary proofs are omitted.

\begin{lemma}[Spherical graphs as cylindrical graphs]\label{lemma D1}
There are dimension independent positive constants $C$ and $\eta_0$ with the following property. If $n\ge 1$, $H\in\H$ and $u\in\X_\eta(\S_H,r_1,r_2)$ with $\eta<\eta_0$, then we have
\[
\DD_{(1-C\,\eta^2)\,r_2}^{\nu_H}\setminus \DD_{r_1}^{\nu_H}\subset\pp_H\big(\S_H(u,r_1,r_2)\big)\subset
\DD_{r_2}^{\nu_H}\setminus \DD_{(1-C\,\eta^2)\,r_1}^{\nu_H}\,,
\]
and there is $g\in C^1(H)$ such that $\sup\big\{|x|^{-1}\,|g(x)|+|\nabla g(x)|:x\in H\big\}\le C\,\eta$ and
Moreover, if $(\rho_1,\rho_2)\subset ((1+C\,\eta)\,r_1,(1-C\,\eta^2)\,r_2)$, then
$\S_H(u,\rho_1,\rho_2)=\big\{x+g(x)\,\nu_H:x\in H\big\}\cap A_{\rho_1}^{\rho_2}$.
\end{lemma}

%%%%  For brevity, let us set $S=\S_H(u,r_1,r_2)$. The maps $\ell_u,f_u:\S_H\times(r_1,r_2)\to\R$ and $P_u:\S_H\times(r_1,r_2)\to H$ defined by setting
%%%%  \[
%%%%  \ell_u(\om,r)=\frac{r}{\sqrt{1+u(\om,r)^2}}\,,\qquad P_u=\frac{r\,\om}{\sqrt{1+u(\om,r)^2}}\,,\qquad f_u=\frac{r\,u(\om,r)}{\sqrt{1+u(\om,r)^2}}\,,
%%%%  \]
%%%%  are such that $S=\{P_u(\om,r)+f_u(\om,r)\,\nu_H:(\om,r)\in\S_H\times(r_1,r_2)\}$. By elementary computations,
%%%%  \begin{equation}
%%%%    \label{ellu estimates}
%%%%    |\ell_u-r|\le C\,\eta^2\,r\,,\qquad |\pa_r\ell_u-1|\le C\,\eta^2\,,\qquad\forall(\om,r)\in\S_H\times(r_1,r_2)\,,
%%%%  \end{equation}
%%%%  from which \eqref{spherical to cylidrical inclusions} follows, as well as the fact that $P_u=\ell_u\,\om$ is a $C^1$-diffeomorphism between $\S_H\times(r_1,r_2)$ and $\pp_H(S)$. In particular, $\pp_H(S)$ is an open subset of $H$, and $g_u=((\pp_H)|_S)^{-1}:\pp_H(S)\to\R$ defines $C^1$-function with $S=\{x+g_u(x)\,\nu_H:x\in\pp_H(S)\}$ and
%%%%  \begin{equation}
%%%%    \label{differentiate me}
%%%%      g_u\big(P_u(\om,r)\big)=f_u(\om,r)\,,\qquad\forall (\om,r)\in\S_H\times(r_1,r_2)\,.
%%%%  \end{equation}
%%%%  From \eqref{differentiate me} we find
%%%%  \[
%%%%  \frac{g_u(x)}{|x|}=\frac{f_u(\om,r)}{|P_u(\om,r)|}=u(\om,r)\,,\qquad\forall x\in\pp_H(S)=P_u\big(\Sigma_H\times(r_1,r_2)\big)\,,
%%%%  \]
%%%%  which gives $|g_u(x)|\le\eta|x|$ for $x\in\pp_H(S)$. Similarly, by
%%%%  \begin{eqnarray}\label{radial Pu fu}
%%%%    &&\frac{\pa_r P_u}{|\pa_r P_u|}=\om\,,\qquad \big||\pa_r P_u|-1\big|\le C\,\eta^2\,,\qquad |\pa_r f_u|\le C\,\eta\,,
%%%%    \\\label{tangential Pu fu}