We obtain a full resolution result for minimizers in the exterior isoperimetric problem with respect to a compact obstacle in the large volume regime $v\to\infty$. This is achieved by the study of a Plateau-type problem with free boundary (both on the compact obstacle and at infinity) which is used to identify the first obstacle-dependent term (called isoperimetric residue) in the energy expansion, as $v\to\infty$, of the exterior isoperimetric problem. A crucial tool in the analysis of isoperimetric residues is a new “mesoscale flatness criterion” for hypersurfaces with bounded mean curvature, which we obtain as a development of ideas originating in the theory of minimal surfaces with isolated singularities.

§ INTRODUCTION

§.§ Overview
Given a compact set $W\subset\R^{n+1}$ ($n\ge 1$), we consider the classical exterior isoperimetric problem associated to $W$, namely

\begin{equation}
\label{psiv}
\psi_W(v)=\inf\big\{P(E;\Om):E\subset\Om=\R^{n+1}\setminus W\,,|E|=v\big\}\,,\qquad v>0\,,
\end{equation}

in the large volume regime $v\to\infty$. Here $|E|$ denotes the volume (Lebesgue measure) of $E$, and $P(E;\Om)$ the (distributional) perimeter of $E$ relative to $\Om$, so that $P(E;\Om)=\H^n(\Om\cap\pa E)$ whenever $\pa E$ is locally Lipschitz. Relative isoperimetric problems are well-known for their analytical <cit.> and geometric <cit.> relevance.

They are also important in physical applications: beyond the obvious example of capillarity theory [18], exterior isoperimetry at large volumes provides an elegant approach to the Huisken-Yau theorem in general relativity, see [14].

When $v\to\infty$, we expect minimizers $E_v$ in (<ref>) to closely resemble balls of volume $v$. Indeed, by minimality and isoperimetry, denoting by $B^{(v)}(x)$ the ball of center $x$ and volume $v$, and with $B^{(v)}=B^{(v)}(0)$, we find that

\begin{equation}
\label{basic energy estimate}
\lim_{v\to\infty}\frac{\psi_W(v)}{P(B^{(v)})}=1\,.
\end{equation}

Additional information can be obtained by combining (<ref>) with quantitative isoperimetry [21, 22]: if $0<|E|<\infty$, then

\begin{equation}
\label{quantitative euclidean isop}
P(E)\ge P(B^{(|E|)})\Big\{1+ c(n)\,\inf_{x\in\R^{n+1}}\,\Big(\frac{|E\Delta B^{(|E|)}(x)|}{|E|}\Big)^2\Big\}\,.
\end{equation}

The combination of (<ref>) and (<ref>) shows that minimizers $E_v$ in $\psi_W(v)$ are close in $L^1$-distance to balls. Based on that, a somehow classical argument exploiting the local regularity theory of perimeter minimizers shows the existence of $v_0>0$ and of a function $R_0(v)\to 0^+$, $R_0(v)\,v^{1/(n+1)}\to\infty$ as $v\to\infty$, both depending on $W$, such that,

Quantitative isoperimetry gives no information on how $W$ affects $\psi_W(v)$ for $v$ large.

if $E_v$ is a minimizer of (<ref>) with $v>v_0$, then (see Figure <ref>)

\begin{eqnarray}
\label{basic C1 estimate}
&&\mbox{$(\pa E_v)\setminus B_{R_0\,v^{1/(n+1)}}\subset$ a $C^1$-small normal graph over $\pa B^{(v)}(x)$},
\\\nonumber
&&\mbox{for some $x\in\R^{n+1}$ with $|x|=(v/\om_{n+1})^{1/(n+1)}+{\rm o}(v^{1/(n+1)})$ as $v\to\infty$}\,;
\end{eqnarray}

here $\om_m$ stands for the volume of the unit ball in $\R^m$, $B_r(x)$ is the ball of center $x$ and radius $r$ in $\R^{n+1}$, and $B_r=B_r(0)$. The picture of the situation offered by (<ref>) and (<ref>) is thus incomplete under one important aspect: it offers no information related to the specific “obstacle” $W$ under consideration – in other words, two different obstacles are completely unrecognizable from (<ref>) and (<ref>) alone.

The first step to obtain obstacle-dependent information on $\psi_W$ is studying $L^1_{\rm loc}$-subsequential limits $F$ of exterior isoperimetric sets $E_v$ as $v\to\infty$. Since the mean curvature of $\pa E_v$ has order $v^{-1/(n+1)}$ as $v\to\infty$ in $\Om$, each $\pa F$ is easily seen to be a minimal surface in $\Om$. A finer analysis leads to establish a more useful characterization of such limits $F$ as minimizers in a “Plateau's problem with free boundary on the obstacle and at infinity”, whose negative is precisely defined in (<ref>) below and denoted by $\Rr(W)$. We call $\Rr(W)$ the isoperimetric residue of $W$ because it captures the “residual effect” of $W$ in (<ref>), as expressed by the limit identity

\begin{equation}
\label{main energy estimate}
\lim_{v\to\infty}\,\psi_W(v)-P(B^{(v)})=-\Rr(W)\,.
\end{equation}

The study of the geometric information about $W$ stored in $\Rr(W)$ is particularly interesting: roughly, $\Rr(W)$ is close to an $n$-dimensional sectional area of $W$, although its precise value is elusively determined by the behavior of certain “plane-like” minimal surfaces with free boundary on $W$. The proof of (<ref>) itself requires proving a blowdown result for such exterior minimal surfaces, and then extracting sharp decay information towards hyperplane blowdown limits. In particular, in the process of proving (<ref>), we shall prove the existence of a positive $R_2$ (depending on $n$ and $W$ only) such that for every maximizer $F$ of $\Rr(W)$, $(\pa F)\setminus B_{R_2}$ is the graph of a smooth solution to the minimal surfaces equation. An application of Allard's regularity theorem [3] leads then to complement (<ref>) with the following “local” resolution formula: for every $S>R_2$, for $v$ is large in terms of $n$, $W$ and $S$,

\begin{eqnarray}
\label{local C1 estimate}
&&\mbox{$(\pa E_v)\cap \big(B_S\setminus B_{R_2}\big)\subset$ a $C^1$-small normal graph over $\pa F$},
\\
\nonumber
&&\mbox{where $F$ is optimal for  the isoperimetric residue $\Rr(W)$ of $W$}\,.
\end{eqnarray}

Interestingly, this already fine analysis gives no information on $\pa E_v$ in the mesoscale region $B_{R_0(v)\,v^{1/(n+1)}}\setminus B_S$ between the  resolution formulas (<ref>) and (<ref>). To address this issue, we are compelled to develop what we have called a mesoscale flatness criterion for hypersurfaces with bounded mean curvature. This kind of statement is qualitatively novel with respect to the flatness criteria typically used in the study of blowups and blowdowns of minimal surfaces – although it is clearly related to those tools at the mere technical level – and holds promise for applications to other geometric variational problems. In the study of the exterior isoperimetric problem, it allows us to prove the existence of positive constants $v_0$ and $R_1$, depending on $n$ and $W$ only, such that if $v>v_0$ and $E_v$ is a minimizer of $\psi_W(v)$, then

\begin{eqnarray}  \nonumber
&&\mbox{$(\pa E_v)\cap \big(B_{R_1\,v^{1/(n+1)}}\setminus B_{R_2}\big)\subset$ a $C^1$-small normal graph over $\pa F$},
\\\label{main C1 estimate}
&&\mbox{where $F$ is optimal for the isoperimetric residue $\Rr(W)$ of $W$}\,.
\end{eqnarray}

The key difference between (<ref>) and (<ref>) is that the domain of resolution given in (<ref>) overlaps with that of (<ref>): indeed, $R_0(v)\to0^+$ as $v\to\infty$ implies that $R_0\,v^{1/(n+1)}< R_1\,v^{1/(n+1)}$ for $v>v_0$. As a by-product of this overlapping and of the graphicality of $\pa F$ outside of $B_{R_2}$, we deduce that boundaries of exterior isoperimetric sets, outside of $B_{R_2}$, are diffeomorphic to $n$-dimensional disks. Finally, when $n\le 6$, and maximizers $F$ of $\Rr(W)$ have locally smooth boundaries in $\Om$, (<ref>) can be propagated up to the obstacle itself; see Remark <ref> below.

Concerning the rest of this introduction: In section <ref> we present our analysis of isoperimetric residues, see Theorem <ref>. In section <ref> we gather all our results concerning exterior isoperimetric sets with large volumes, see Theorem <ref>. Finally, we present the mesoscale flatness criterion in section <ref> and the organization of the paper in section <ref>.

§.§ Isoperimetric residues
To define $\Rr(W)$ we introduce the class

\[
\F
\]

of those pairs $(F,\nu)$ with $\nu\in\SS^n$ ($=$ the unit sphere of $\R^{n+1}$) and $F\subset\R^{n+1}$ a set of locally finite perimeter in $\Om$ (i.e., $P(F;\Om')<\infty$ for every $\Om'\cc\Om$), contained in slab around $\nu^\perp=\{x:x\cdot\nu=0\}$, and whose boundary (see Remark <ref> below) has full projection over $\nu^\perp$ itself: i.e., for some $\a,\b\in\R$,

\begin{eqnarray}\label{def Sigma nu 1}
&&\pa F\subset \big\{x:\a< x\cdot\nu<\b\big\}\,,
\\\label{def Sigma nu 2}
&&\pp_{\nu^\perp}(\pa F)=\nu^\perp:=\big\{x:x\cdot\nu=0\big\}\,,
\end{eqnarray}

where $\pp_{\nu^\perp}(x)=x-(x\cdot\nu)\,\nu$, $x\in\R^{n+1}$. In correspondence to $W$ compact, we define the residual perimeter functional, ${\rm res}_W:\F\to\R\cup\{\pm\infty\}$, by

\[
{\rm res}_W(F,\nu)=\varlimsup_{R\to\infty}\om_n\,R^n-P(F;\CC_R^\nu\setminus W)\,,\qquad (F,\nu)\in\F\,,
\]

where $\CC_R^\nu=\{x\in\R^{n+1}:|\pp_{\nu^\perp}(x)|<R\}$ denotes the (unbounded) cylinder of radius $R$ with axis along $\nu$ – and where the limsup is actually a monotone decreasing limit thanks to (<ref>) and (<ref>) (see (<ref>) below for a proof). For a reasonably “well-behaved” $F$, e.g. if $\pa F$ is the graph of a Lipschitz function over $\nu^\perp$, $\om_n\,R^n$ is the (obstacle-independent) leading order term of the expansion of $P(F;\CC_R^\nu\setminus W)$ as $R\to\infty$, while ${\rm res}_W(F,\nu)$ is expected to capture the first obstacle-dependent “residual perimeter” contribution of $P(F;\CC_R^\nu\setminus W)$ as $R\to\infty$.  The isoperimetric residue of $W$ is then defined by maximizing ${\rm res}_W$ over $\F$, so that

\begin{equation}
\label{def RW}
\Rr(W)=\sup_{(F,\nu)\in\F}\,{\rm res}_W(F,\nu)\,;
\end{equation}

If $(F,\nu)\in\F$ then $F$ is contained in a slab around $\nu^\perp$ and is such that $\pa F$ has full projection over $\nu^\perp$. Only the behavior of $\pa F$ outside $W$ matters in computing ${\rm res}_W(F,\nu)$. The perimeter of $F$ in $\mathbf{C}_R^\nu\setminus W$ (depicted as a bold line) is compared to $\om_n\,R^n(=$perimeter of a half-space orthogonal to $\nu$ in $\mathbf{C}_R^\nu$); the corresponding “residual” perimeter as $R\to\infty$, is ${\rm res}_W(F,\nu)$.

Figure <ref>. Clearly $\Rr(\l\,W)=\l^n\,\Rr(W)$ if $\l>0$, and $\Rr(W)$ is trapped between the areas of the largest hyperplane section and directional projection of $W$, see (<ref>) below. In the simple case when $n=1$ and $W$ is connected, $\Rr(W)=\diam(W)$ by (<ref>) and (<ref>) below, although, in general, $\Rr(W)$ does not seem to admit a simple characterization, and it is finely tuned to the near-to-the-obstacle behavior of “plane-like” minimal surfaces with free boundary on $W$. Our first main result collects these (and other) properties of isoperimetric residues and of their maximizers.

If $W\subset\R^{n+1}$ is compact, then there are $R_2$ and $C_0$ positive and depending on $W$ with the following property.

(i): If $\Ss(W)=\sup\{\H^n(W\cap\Pi):\mbox{$\Pi$ is a hyperplane in $\R^{n+1}$}\}$ and $\P(W)=\sup\{\H^n(\pp_{\nu^{\perp}}(W)):\nu\in\SS^n\}$, then we have

\begin{eqnarray}
\label{R larger than S}
\Ss(W)\le \Rr(W)\le \P(W)\,.
\end{eqnarray}

(ii): The family ${\rm Max}[\Rr(W)]$ of maximizers of $\Rr(W)$ is non-empty. If $(F,\nu)\in{\rm Max}[\Rr(W)]$, then $F$ is a perimeter minimizer with free boundary in $\Om=\R^{n+1}\setminus W$, i.e.

\begin{equation}\label{local perimeter minimizer}
P(F;\Omega \cap B) \leq P(G;\Omega \cap B)\,,\qquad\mbox{$\forall F\Delta G\cc B$, $B$ a ball}\,;
\end{equation}

and if $\Rr(W)>0$, then $\pa F$ is contained in the smallest slab $\{x:\a\le x\cdot\nu\le\b\}$ containing $W$, and there are $a,b\in\R$, $c\in\nu^\perp$ with $\max\{|a|,|b|,|c|\}\le C_0$ and $f\in C^\infty(\nu^\perp)$ such that

\begin{equation}
\label{main residue graphicality of F}
\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(\partial F) \setminus \CC^\nu_{R_2}=\big\{x+f(x)\,\nu:x\in\nu^\perp\,,|x|>R_2\big\}\,,
\end{equation}

\begin{eqnarray}\nonumber
&&f(x)=a\,,\hspace{5.4cm} (n=1)
\\\label{asymptotics of F}
&&\Big|f(x)-\Big(a+\frac{b}{|x|^{n-2}}+\frac{c\cdot x}{|x|^n}\Big)\Big|\le\frac{C_0}{|x|^n}\,,\qquad (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_2\,.
\end{eqnarray}

(iii): At fixed diameter, isoperimetric residues are maximized by balls, i.e.

\begin{equation}\label{optimal RW}
\Rr(W) \leq \omega_n \big(\diam W/2\big)^n\,= \Rr\big({\rm cl}\big(B_{\diam W /2}\big)\big)\,,
\end{equation}

where $\cl(X)$ denotes topological closure of $X\subset\R^{n+1}$. Moreover, if equality holds in (<ref>) and $(F,\nu)\in{\rm Max}[\Rr(W)]$, then (<ref>) holds with $b=0$ and $c=0$, and setting $\Pi=\big\{y:y\cdot\nu=a\big\}$, we have

\begin{equation}
\label{characterization 2}
(\partial F)\setminus W=\Pi \setminus\cl\big(B_{\diam W/2}(x)\big)\,,
\end{equation}

for some $x\in\Pi$. Finally, equality holds in (<ref>) if and only if there are a hyperplane $\Pi$ and a point $x\in\Pi$ such that

\begin{eqnarray}\label{characterization 1}
\partial B_{\diam W/2}(x) \cap \Pi\subset W\,,
\end{eqnarray}

i.e., $W$ contains an $(n-1)$-dimensional sphere of diameter $\diam(W)$, and

\begin{eqnarray}
\label{characterization 3}
&&\mbox{$\Om\setminus \big(\Pi \setminus \cl\big(B_{\diam W/2}(x)\big)\big)$}
\\\nonumber
&&\mbox{has exactly two unbounded connected components}.
\end{eqnarray}

The assumption $\Rr(W)>0$ is quite weak: indeed, if $\Rr(W)=0$, then $W$ is purely $\H^n$-unrectifiable; see Proposition <ref> in the appendix. For the role of the topological condition (<ref>), see Figure
The obstacle $W$ (depicted in grey) is obtained by removing a cylinder $\CC_r^{e_{n+1}}$ from a ball $B_{d/2}$ with $d/2>r$. In this way $d=\diam(W)$ and $B_{d/2}$ is the only ball such that (<ref>) can hold. Hyperplanes $\Pi$ satisfying (<ref>) are exactly those passing through the center of $B_{d/2}$, and intersecting $W$ on a $(n-1)$-dimensional sphere of radius $d/2$. For every such $\Pi$, $\Om\setminus(\Pi\setminus B_{d/2})$ has exactly one unbounded connected component, and  (<ref>) does not hold.

In the physical dimension $n=2$, and provided $\Om$ has boundary of class $C^{1,1}$, maximizers of $\Rr(W)$ are $C^{1,1/2}$-regular up to the obstacle, and smooth away from it. More generally, condition (<ref>) implies that $M=\cl(\Om\cap\pa F)$ is a smooth hypersurface with boundary in $\Omega\setminus\Sigma$, where $\Sigma$ is a closed set such that $\Sigma\cap\Om$ is empty if $1\le n\le 6$, is locally discrete in $\Om$ if $n=7$, and is locally $\H^{n-7}$-rectifiable in $\Om$ if $n\ge 8$; see, e.g. <cit.>, [29]. Of course, by (<ref>), $\Sigma\setminus B_{R_2}=\emptyset$ in every dimension.  Moreover, justifying the initial claim concerning the case $n=2$, if we assume that $\Om$ is an open set with $C^{1,1}$-boundary, then $M$ is a $C^{1,1/2}$-hypersurface with boundary in $\R^{n+1}\setminus\Sigma$, with boundary contained in $\pa\Om$, $\Sigma\cap\pa\Om$ is $\H^{n-3+\e}$-negligible for every $\e>0$, and Young's law $\nu_F\cdot\nu_\Om=0$ holds on $(M\cap\pa\Om)\setminus\Sigma$; see, e.g. [24, 23, 12, 13].

An interesting open direction is finding additional geometric information on $\Rr(W)$, e.g. in the class of convex obstacles.

We adopt the convention that any set of locally finite perimeter $F$ in $\Om$ open is tacitly modified on and by a set of zero Lebesgue measure so to entail $\Om\cap\pa F=\Om\cap\cl(\pa^*F)$, where $\pa^*F$ is the reduced boundary of $F$ in $\Om$; see <cit.>. Under this normalization, local perimeter minimality conditions like (<ref>) (or (<ref>) below) imply that $F\cap\Om$ is open in $\R^{n+1}$; see, e.g. <cit.>.

§.§ Resolution of exterior isoperimetric sets
Denoting the family of minimizers of $\psi_W(v)$ by ${\rm Min}[\psi_W(v)]$, our second main result is as follows:

If $W\subset\R^{n+1}$ is compact, then ${\rm Min}[\psi_W(v)]\ne\emptyset\,\,\forall v>0$. Moreover, if $\Rr(W)>0$, then

\begin{equation}
\label{main asymptotic expansion}
\lim_{v\to\infty}\psi_W(v)-P(B^{(v)})=-\Rr(W)\,,
\end{equation}

and, depending on $n$ and $W$ only, there are $v_0$, $C_0$, $R_1$, and $R_2$ positive, and $R_0(v)$ with $R_0(v)\to 0^+$, $R_0(v)\,v^{1/(n+1)}\to\infty$ as $v\to\infty$, such that, if $E_v\in {\rm Min}[\psi_W(v)]$ and $v>v_0$, then:

(i): $E_v$ determines $x\in\R^{n+1}$ and $u\in C^\infty(\pa B^{(1)})$ such that

\begin{eqnarray}
\label{nonsharp rates}
&&\frac{|E_v\Delta B^{(v)}(x)|}v \le \frac{C_0}{v^{1/[2(n+1)]}}\,,
\\\label{x and u of Ev}
&&(\pa E_v)\setminus B_{R_0(v)\,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)\,v^{1/(n+1)}}\,,
\end{eqnarray}

where, for any $G\subset\R^{n+1}$ with locally finite perimeter, $\nu_G$ is the outer unit normal to $G$;

(ii): $E_v$ determines $(F,\nu)\in{\rm Max}[\Rr(W)]$ and $f\in C^\infty((\pa F)\setminus B_{R_2})$ with

\begin{equation}
\label{f of Ev}
(\pa E_v)\cap A_{R_2}^{R_1\,v^{1/(n+1)}}=\big\{y+f(y)\,\nu_F(y):y\in\pa F\big\}\cap A_{R_2}^{R_1\,v^{1/(n+1)}}\,;
\end{equation}

(iii): $(\pa E_v)\setminus B_{R_2}$ is diffeomorphic to an $n$-dimensional disk;

(iv): Finally, with $(x,u)$ as in (<ref>) and $(F,\nu,f)$ as in (<ref>),

\begin{eqnarray*}
{\rm Min}[\psi_W(v)]}\Big\{\Big|\frac{|x|}{v^{1/(n+1)}}-\frac{1}{\omega_{n+1}^{1/(n+1)}}\Big|\,,\Big|\nu-\frac{x}{|x|}\Big|\,,
\|u\|_{C^1(\pa B^{(1)})}\Big\}=0\,,
\\
&&\lim_{v\to\infty}\sup_{E_v\in{\rm Min}[\psi_W(v)]}\,\|f\|_{C^1(B_M\cap\pa F)}=0\,,\hspace{2cm}\forall M>R_2\,.
\end{eqnarray*}

By Remark <ref> and a covering argument, if $n\le 6$, $\de>0$, and $v>v_0(n,W,\de)$, then (<ref>) holds with $B_{R_1\,v^{1/(n+1)}}\setminus I_\de(W)$ in place of $B_{R_1\,v^{1/(n+1)}}\setminus B_{R_2}$, where $I_\de(W)$ is the open $\de$-neighborhood of $W$. Similarly, when $\pa\Om\in C^{1,1}$ and $n=2$ (and thus $\Om\cap\pa F$ is regular up to the obstacle), we can find $v_0$ (depending on $n$ and $W$ only) such that (<ref>) holds with $B_{R_1\,v^{1/(n+1)}}\cap \Om$ in place of $B_{R_1\,v^{1/(n+1)}}\setminus B_{R_2}$, that is, graphicality over $\pa F$ holds up to the obstacle itself.

If $W$ is convex and $J$ is an half-space, then $\psi_W(v)\ge\psi_J(v)$ for every $v>0$, with equality for $v>0$ if and only if $\pa W$ contains a flat facet supporting an half-ball of volume $v$; see [4, 20]. Since $\psi_J(v)=P(B^{(v)})/2^{1/(n+1)}$ and $\psi_W(v)-P(B^{(v)})\to-\Rr(W)$ as $v\to\infty$, the bound $\psi_W(v)\ge\psi_J(v)$ is far from optimal if $v$ is large. Are there stronger global bounds than $\psi_W\ge\psi_J$ on convex obstacles? Similarly, it would be interesting to quantify the convergence towards $\Rr(W)$ in (<ref>), or even that of $\pa E_v$ towards $\pa B^{(v)}$ and $\pa F$ (where (<ref>) should not to be sharp).

§.§ The mesoscale flatness criterion
We work with with hypersurfaces $M$ whose mean curvature is bounded by $\Lambda\ge0$ in an annulus $B_{1/\Lambda}\setminus\ov{B}_R$, $R\in(0,1/\Lambda)$. Even without information on  $M$ inside $B_R$ (where $M$ could have a non-trivial boundary, or topology, etc.) the classical proof of the monotonicity formula can be adapted to show the monotone increasing character on  $r\in(R,1/\Lambda)$ of

\begin{eqnarray}\nonumber
\Theta_{M,R,\Lambda}(r)&=&
\frac{\H^n\big(M\cap(B_r\setminus B_R)\big)}{r^n}+\frac{R}{n\,r^n}\,\int_{M\cap\pa B_R}\frac{|x^{TM}|}{|x|}\,d\H^{n-1}
\\\label{theta smooth}
+\Lambda\,\int_R^r\frac{\H^n\big(M\cap(B_\rho\setminus B_R)\big)}{\rho^n}\,d\rho\,,\,\,\,\,
\end{eqnarray}

(here $x^{TM}={\rm proj}_{T_xM}(x)$);

moreover, if $\Theta_{M,R,\Lambda}$ is constant over $(a,b)\subset(R,1/\Lambda)$, then $M\cap(B_b\setminus\ov{B}_a)$ is a cone. Since the constant density value corresponding to $M=H\setminus B_R$, $H$ an hyperplane through the origin, is $\om_n$ (as a result of a double cancellation which also involves the “boundary term” in $\Theta_{H\setminus B_R,R,0}$), we consider the area deficit

\begin{equation}
\label{def of delta}
\de_{M,R,\Lambda}(r)=\om_n-\Theta_{M,R,\Lambda}(r)\,,\qquad r\in(R,1/\Lambda)\,,
\end{equation}

which defines a decreasing quantity on $(R,1/\Lambda)$. Here we use the term “deficit”, rather than the more usual term “excess”, since $\de_{M,R,\Lambda}$ does not necessarily have non-negative sign (which is one of the crucial property of “excess quantities” typically used in $\e$-regularity theorems, see, e.g., <cit.>). Recalling that $A_r^s=B_s\setminus\cl(B_r)$ if $s>r>0$, we are now ready to state the following “smooth version” of our mesoscale flatness criterion (see Theorem <ref> below for the varifold version).

If $n\ge 2$, $\Gamma\ge 0$, and $\s>0$, then there are $M_0$ and $\e_0$ positive and depending on $n$, $\Gamma$ and $\s$ only, with the following property. Let $\Lambda\ge0$, $R\in(0,1/\Lambda)$, and $M$ be a smooth hypersurface with mean curvature bounded by $\Lambda$ in $A^{1/\Lambda}_R$, and with

\begin{equation}
\label{meso intro bounds Gamma}
\H^{n-1}\big(M\cap\pa B_{R}\big)\le\Gamma\,R^{n-1}\,,
\quad
\sup_{\rho\in(R,1/\Lambda)}\frac{\H^n\big(M\cap(B_\rho\setminus B_R)\big)}{\rho^n}\le\Gamma\,.
\end{equation}

If there is $s>0$ such that

\begin{eqnarray}\label{meso intro range of s}
&&\max\{M_0,64\}\,R<s<\frac{\e_0}{4\,\Lambda}\,,% \\  \label{meso intro flat hp}
%&& |\de_{M,R,\Lambda}(s/8)|\le \e_0\,,
%\qquad \H^n\big(M\cap A^{s/6}_{s/4}\big)>0\,,
\end{eqnarray}

\begin{equation}
\label{meso intro flat hp}
|\de_{M,R,\Lambda}(s/8)|\le \e_0\,,%\qquad
%\frac1{s^n}\,\int_{M\cap A_{s/2}^{s/8}}\,\Big(\frac{|y\cdot\nu_H|}{|\p_H y|}\Big)^2\,d\,\H^n_y\le\e_0\,,
\end{equation}

and if, setting,

\begin{equation}
\label{meso intro propagation}
R_*=\sup\Big\{\rho\ge\frac{s}8: \de_{M,R,\Lambda}(\rho)\ge -\e_0\Big\}\,,\qquad S_*=\min\Big\{R_*,\frac{\e_0}{\Lambda}\Big\}\,,
\end{equation}

we have $R_*>4\,s$ (and thus $S_*>4\,s$), then

\begin{eqnarray}
\label{meso intro conclusion}
&&M\cap A_{s/32}^{S_*/16}=\big\{x+f(x)\,\nu_K: x\in K\big\}\cap A_{s/32}^{S_*/16}\,,
\\
\nonumber
&&\sup\big\{|x|^{-1}\,|f(x)|+|\nabla f(x)|:x\in K\big\}\le C(n)\,\s\,,
\end{eqnarray}

for a hyperplane $K$ with $0\in K$ and unit normal $\nu_K$, and for $f\in C^1(K)$.

The first condition in (<ref>) implicitly requires $R$ to be sufficiently small in terms of $1/\Lambda$, as it introduces a mesoscale $s$ which is both small with respect to $1/\Lambda$ and large with respect to $R$. The condition in (<ref>) express the flatness of $M$ at the mesoscale $s$ in terms of its area deficit. The final key assumption, $R_*>4\,s$, express the requirement that the area deficit does not decrease too abruptly, and stays above $-\e_0$ at least up to the scale $4\,s$. Under these assumptions, graphicality with respect to a hyperplane $K$ is inferred on an annulus whose lower radius $s/32$ has the order of the mesoscale $s$, and whose upper radius $S_*/16$ can be as large as the decay of the area deficit allows (potentially up to $\e_0/16\,\Lambda$ if $R_*=\infty$), but in any case not too large with respect to $1/\Lambda$.

If $M$ is a hypersurface containing the origin, so that, formally speaking, $R=0$, and the tangent cone of $M$ there is a plane, Theorem 1 reduces to Allard's theorem [3]. Similarly, if $\Lambda=0$ and the exterior minimal hypersurface $M$ has a planar tangent cone at infinity, we recover the exterior blow-down results stated in  [34, 35]. In particular, although the motivation for Theorem 1 comes from scenarios where both $R$ and $\Lambda$ are positive, it can also be viewed as a general framework containing as special cases the blow-up and blow-down flatness criteria for hypersurfaces with planar tangent cones.

The statement is sharp in the sense that for a surface “with bounded mean curvature and non-trivial topology inside a hole”, flatness can only be established on a mesoscale which is both large with respect to the size of the hole and small with respect to the size of the inverse mean curvature. An example is provided by unduloids $M_\e$ with waist size $\e$ and mean curvature $n$ in $\R^{n+1}$; see

A half-period of an unduloid with mean curvature $n$ and waist size $\e$ in $\R^{n+1}$. By (<ref>), the flatness of $M_\e$ is no smaller than ${\rm O}(\e^{2(n-1)/n})$, and is exactly ${\rm O}(\e^{2(n-1)/n})$ on an annulus sitting in the mesoscale ${\rm O}(\e^{(n-1)/n})$. This mesoscale is both very large with respect to waist size $\e$, and very small with respect to the size of the inverse mean curvature, which is order one.

Figure <ref>. A “half-period” of $M_\e$ is the graph $\{x+f_\e(x)\,e_{n+1}:x\in\R^n\,,\e<|x|<R_\e\}$ of

\begin{equation}
\label{def of veps}
\end{equation}

where $\e$ and $R_\e$ are the only solutions of $r^{n-1}=r^n-\e^n+\e^{n-1}$. Clearly $f_\e$ solves $-\Div(\nabla f_\e/\sqrt{1+|\nabla f_\e|^2})=n$ with $f_\e=0$, $|\nabla f_\e|=+\infty$ on $\{|x|=\e\}$, and $|\nabla f_\e|=+\infty$ on $\{|x|=R_\e\}$, where $R_\e=1-{\rm O}(\e^{n-1})$; moreover, $\min|\nabla f_\e|$ is achieved at $r={\rm O}(\e^{(n-1)/n})$, and if $r\in(a\,\e^{(n-1)/n},b\,\e^{(n-1)/n})$ for some $b>a>0$, then $|\nabla f_\e|={\rm O}_{a,b}(\e^{2(n-1)/n})$. Thus, the horizontal flatness of $M_\e$ is no smaller than ${\rm O}(\e^{2(n-1)/n})$, and has that exact order on a scale which is both very large with respect to the hole ($\e^{(n-1)/n}>\!\!\!>\e$) and very small with respect to the inverse mean curvature ($\e^{(n-1)/n}<\!\!\!<1$).

Exterior isoperimetric sets $E_v$ with large volume $v$ have small constant mean curvature of order $\Lambda=\Lambda_0(n,W)/v^{1/(n+1)}$. We will work with “holes” of size $R=R_3(n,W)$, for some $R_3$ sufficiently large with respect to the radius $R_2$ appearing in Theorem <ref>-(ii), and determined through the sharp decay rates (<ref>).
The decay properties of $F$ towards $\{x:x\cdot\nu=a\}$ when $(F,\nu)$ is a maximizer of $\Rr(W)$, the $C^1$-proximity of $\pa E$ to $\pa B^{(v)}(x)$ for $|x|\approx(\om_{n+1}/v)^{1/(n+1)}$, and the $C^1$-proximity of $\pa E$ to $\pa F$ for some optimal $(F,\nu)$ on bounded annuli of the form $A^{2\,R_3}_{R_2}$ are used in checking that (<ref>) holds with $\Gamma=\Gamma(n,W)$, that $E_v$ is flat in the sense of (<ref>), and, most importantly, that the area deficit $\de_{M,R,\Lambda}$ of $M=(\pa E_v)\setminus B_{R_3}$ lies above $-\e_0$ up to scale $r={\rm O}(v^{1/(n+1)})$ (which is the key information to deduce $R_*\approx1/\Lambda$), and thus obtain overlapping domains of resolutions in terms of $\pa B^{(v)}(x)$ and $\pa F$.

While Theorem <ref> seems clearly applicable to other problems, there are situations where one may need to develop considerably finer “mesoscale flatness criteria”. For example, consider the problem of “resolving” almost CMC boundaries undergoing bubbling [8, 11, 10]. When the oscillation of the mean curvature around a constant $\Lambda$ is small, such boundaries are close to finite unions of mutually tangent spheres of radius $n/\Lambda$, and can be covered by $C^1$-small normal graphs over such spheres away from their tangency points up to distance $\e/\Lambda$, with $\e=\e(n)$, and provided the mean curvature oscillation is small in terms of $\e$. For propagating flatness up to a distance directly related to the oscillation of the mean curvature, one would need a version of Theorem <ref> for “double” spherical graphs; in the setting of blowup/blowdown theorems, this would be similar to passing to the harder case of multiplicity larger than one.

From the technical viewpoint, Theorem <ref> fits into the framework set up by Allard and Almgren in [1] for the study of blowups and blowdowns of minimal surfaces with tangent integrable cones. At the same time, as exemplified by Remark <ref>, Theorem <ref> really points in a different direction,  since it pertains to situations where neither blowup or blowdown limits make sense.

Another interesting point is that, in [1], the area deficit $\de_{M,R,\Lambda}$ is considered with a sign, non-positive for blowups, and non-negative for blowdowns, see <cit.>. A key insight here is that for hypersurfaces where the deficit changes sign, graphicality obtained through small negative (or positive) deficit nevertheless persists past the scale where $\delta_{M,R,\Lambda}$ vanishes, and possibly much farther depending on the surface in question; this is actually crucial for obtaining overlapping domains of resolutions in statements like (<ref>) and (<ref>).

Proving Theorem <ref> in higher codimension and with arbitrary integrable minimal cones should be possible with essentially the same proof presented here. We do not pursue this extension because, first, only the case of hypersurfaces and hyperplanes is needed in studying $\psi_W(v)$; and, second, in going for generality, one should work in the framework set up by Simon in [32, 34, 36], which, at variance with the simpler Allard–Almgren's framework used here, allows one to dispense with the integrability assumption. In this direction, we notice that Theorem <ref> with $\Lambda=0$ and $R_*=+\infty$ is a blowdown result for exterior minimal surfaces (see also Theorem <ref>-(ii, iii)). A blowdown result for exterior minimal surfaces is outside the scope of <cit.> which pertains to entire minimal surfaces, but it is claimed, with a sketch of proof, on <cit.> as a modification of <cit.>. It should be mentioned that, to cover the case of exterior minimal surfaces, an additional term of the form $C\int_{\Sigma}(\dot u(t))^{2}$ should be added on the right side of  assumption <cit.>. This additional term seems not to cause difficulties with the rest of the arguments leading to <cit.>. Thus Simon's approach, in addition to giving the blowdown analysis of exterior minimal surfaces, should also be viable for generalizing our mesoscale flatness criterion.

§.§ Organization of the paper
In section <ref> we prove Theorem <ref> (actually, its generalization to varifolds, i.e. Theorem <ref>). In section <ref> we prove those parts of Theorem <ref> which follow simply by quantitative isoperimetry (i.e., they do not require isoperimetric residues nor our mesoscale flatness analysis); see Theorem <ref>. Section <ref> is devoted to the study of isoperimetric residues and of their maximizers, and contains the proof Theorem <ref>. We also present there a statement, repeatedly used in our analysis, which summarizes some results from [31]; see Proposition <ref>. Finally, in section <ref>, we prove the energy expansion (<ref>) and those parts of Theorem <ref> left out in section <ref> (i.e., statements (ii, iii, iv)). This final section is, from a certain viewpoint, the most interesting part of the paper: indeed, it is only the detailed examination of those arguments that clearly illustrates the degree of fine tuning of the preliminary analysis of exterior isoperimetric sets and of maximizers of isoperimetric residues which is needed in order to allow for the application of the mesoscale flatness criterion.

Acknowledgements: Supported by NSF-DMS RTG 1840314, NSF-DMS FRG 1854344, and NSF-DMS 2000034. We thank William Allard and Leon Simon for clarifications on [1] and [34] respectively, and Luca Spolaor for his comments on some preliminary drafts of this work.

§ A MESOSCALE FLATNESS CRITERION FOR VARIFOLDS
In section <ref> we introduce the class $\V_n(\Lambda,R,S)$ of varifolds used to reformulate Theorem <ref>, see Theorem <ref>. In sections <ref>-<ref> we present two reparametrization lemmas (Lemma <ref> and Lemma <ref>) and some “energy estimates” (Theorem <ref>) for spherical graphs; in section <ref> we state the monotonicity formula in $\V_n(\Lambda,R,S)$ and some energy estimates involving the monotonicity gap;

in section <ref>, we prove Theorem <ref>.

§.§ Statement of the criterion
Given an $n$-dimensional integer rectifiable varifold $V=\var(M,\theta)$ in $\R^{n+1}$, defined by a locally $\H^n$-rectifiable set $M$, and by a multiplicity function $\theta:M\to\N$ (see [33]), we denote by $\|V\|=\theta\,\H^n\llcorner M$ the weight of $V$, and by $\de V$ the first variation of $V$, so that
$\de V(X)=\int\,\Div^T\,X(x)\,dV(x,T)=\int_M\,\Div^M\,X(x)\,\theta\,d\H^n_x$ for every $X\in C^1_c(\R^{n+1};\R^{n+1})$. Given $S>R>0$ and $\Lambda\ge0$, we consider the family

\[
\V_n(\Lambda,R,S)\,,
\]

of those $n$-dimensional integral varifolds $V$ with $\spt\,V\subset\R^{n+1}\setminus B_R$ and

\begin{eqnarray*}
\de V(X)=\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})\,,
\end{eqnarray*}

holds for a Radon measure ${\rm bd}_V$ in $\R^{n+1}$ supported in $\pa B_R$, and Borel vector fields $\vec{H}:\R^{n+1}\to\R^{n+1}$ with $|\vec{H}|\le \Lambda$ and $\nu_V^{\rm co}:\pa B_R\to\R^{n+1}$ with $|\nu_V^{\rm co}|=1$. We let $\M_n(\Lambda,R,S)=\{V\in\V_n(\Lambda,R,S):\mbox{$V=\var(M,1)$ for $M$ smooth}\}$, that is, $M\subset \R^{n+1}\setminus B_R$ is a smooth hypersurface with boundary in $A_R^S$, $\bd(M)\subset \pa B_R$, and $|H_M|\le\Lambda$. If $V\in\M_n(\Lambda,R,S)$, then $\vec{H}$ is the mean curvature vector of $M$, ${\rm bd}_V=\H^{n-1}\llcorner\bd(M)$, and $\nu_V^{\rm co}$ is the outer unit conormal to $M$ along $\pa B_R$. Given $V\in\V_n(\Lambda,R,S)$, we define $\Theta_{V,R,\Lambda}(r)$ as

\begin{eqnarray}\label{def theta}
\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\,;
\end{eqnarray}

$\Theta_{V,R,\Lambda}(r)$ is increasing for $r\in(R,S)$ (Theorem <ref>-(i) below), and equal to (<ref>) when $V\in\M_n(\Lambda,R,S)$. The area deficit of $V$ is then defined as in (<ref>), while given a hyperplane $H$ in $\R^{n+1}$ with $0\in H$ we call the quantity

\begin{equation}
\label{def of omega H}
\int_{A_r^s}\,\om_H(y)^2\,d\|V\|_y\,,\qquad \om_H(y)=\atan\Big(\frac{|y\cdot\nu_H|}{|\p_H y|}\Big)\,,
\end{equation}

the angular flatness of $V$ on the annulus $A_r^s=B_s\setminus\cl(B_r)$ with respect to $H$. (See (<ref>) for the notation concerning $H$.)

If $n\ge 2$, $\Gamma\ge 0$, and $\s>0$ then there are positive constants $M_0$ and $\e_0$, depending on $n$, $\Gamma$ and $\s$ only, with the following property. If $\Lambda\ge0$, $R\in(0,1/\Lambda)$, $V\in\V_n(\Lambda,R,1/\Lambda)$,

\begin{equation}
\label{Gamma bounds}
\|{\rm bd}_V\|(\pa B_R)\le\Gamma\,R^{n-1}\,,\qquad \sup_{\rho\in(R,1/\Lambda)}\frac{\|V\|(B_\rho\setminus B_R)}{\rho^n}\le\Gamma\,.
\end{equation}

and for some $s>0$ we have

\begin{eqnarray}
\label{mesoscale bounds}
\\
\label{mesoscale delta small s8}
&&|\de_{V,R,\Lambda}(s/8)|\le \e_0\,,
\\
\label{mesoscale Rstar larger s4}
R_*:=\sup\Big\{\rho\ge\frac{s}8: \de_{V,R,\Lambda}(\rho)\ge -\e_0\Big\}\ge 4\,s\,,
\end{eqnarray}

then (i): if $S_*=\min\{R_*,\e_0/\Lambda\}<\infty$,
then there is an hyperplane $K\subset\R^{n+1}$ with $0\in K$ and $u\in C^1((K\cap\SS^n)\times(s/32,S_*/16))$ with

\begin{eqnarray}\nonumber
&&(\spt\,V)\cap A_{s/32}^{S_*/16}=\Big\{r\,\frac{\om+u(r,\om)\,\nu_K}{\sqrt{1+u(r,\om)^2}}:\om\in K\cap\SS^n\,,r\in\big(s/32,S_*/16\big)\Big\}
\\\label{mesoscale thesis graphicality}
&&\sup_{(K\cap\SS^n)\times\big(s/32,S_*/16\big)}\Big\{|u|+|\nabla^{K\cap\SS^n}u|+|r\,\pa_ru|\Big\}\le C(n)\,\s\,;
\end{eqnarray}

(ii): if $\Lambda=0$ and $\de_{V,R,0}\ge -\varepsilon_0$ on $(s/8,\infty)$, then $\de_{V,R,0}\ge 0$ on $(s/8,\infty)$, (<ref>) holds with $S_*=\infty$, and one has decay estimates, continuous in the radius, of the form

\begin{eqnarray}
\label{decay deficit for exterior minimal surfaces}
\!\!\!\!\!\!\de_{V,R,0}(r)\!\!&\le& \!\!C(n)\,\Big(\frac{s}{r}\Big)^\a\,\de_{V,R,0}\Big(\frac{s}8\Big)\,,\qquad\forall r>\frac{s}4\,,
\\
\label{decay flatness for exterior minimal surfaces}
\!\!\!\!\!\!\frac1{r^n}\,\int_{A_r^{2\,r}}\,\om_K^2\,d\|V\|\!\!&\le&\!\! C(n)\,(1+\Gamma)\,\Big(\frac{s}{r}\Big)^\a\,\de_{V,R,0}\Big(\frac{s}8\Big)\,,\qquad\forall r>\frac{s}4\,,
\end{eqnarray}

for some $\a(n)\in(0,1)$.

In Theorem <ref>, graphicality is formulated in terms of the notion of spherical graph (see section <ref>) which is more natural than the usual notion of “cylindrical graph” in setting up the iteration procedure behind Theorem <ref>. Spherical graphicality in terms a $C^1$-small $u$ as in (<ref>) translates into cylindrical graphicality in terms of $f$ as in (<ref>) with
$f(x)/|x|\approx u(|x|,\hat{x})$ and $\nabla_{\hat x} f(x)-(f(x)/|x|)\approx |x|\,\pa_r\,u(|x|,\hat{x})$  for $x\ne 0$ and $\hat{x}=x/|x|$; see, in particular, Lemma <ref> in appendix <ref>.

§.§ Spherical graphs
We start setting up some notation. We denote by

\[
\H
\]
the family of the oriented hyperplanes $H\subset\R^{n+1}$ with $0\in\H$, so that choice of $H\in\H$ implies that a unit normal vector $\nu_H$ to $H$. Given $H\in\H$, we set
\begin{equation}
\label{seeee}
\Sigma_H=H\cap\SS^n\,,\qquad \p_H:\R^{n+1}\to H\,,\qquad \q_H:\R^{n+1}\to H^\perp\,,
\end{equation}
for the equatorial sphere defined by $H$ on $\SS^n$ and for the orthogonal projections of $\R^{n+1}$ onto $H$ and onto $H^\perp=\{t\,\nu_H:t\in\R\}$. We set
\[
\X_\s(\Sigma_H)=\big\{u\in C^1(\S_H):\|u\|_{C^1(\S_H)}<\s\big\}\,,\qquad\s>0\,.
\]
Clearly there is $\s_0=\s_0(n)>0$ such that if $H\in\H$ and $u\in\X_{\s_0}(\S_H)$, then
\[
f_u(\om)=\frac{\om+u(\om)\,\nu_H}{\sqrt{1+u(\om)^2}}\,,\qquad \om\in\S_H\,,
\]
defines a diffeomorphism of $\S_H$ into an hypersurface $\S_H(u)\subset\SS^n$, namely
\begin{equation}
\label{def of spherical graph}
\S_H(u)=f_u(\Sigma_H)=\Big\{\frac{\om+u(\om)\,\nu_H}{\sqrt{1+u(\om)^2}}:\om\in\S_H\Big\}\,.
\end{equation}
We call $\S_H(u)$ a {\bf spherical graph} over $\S_H$. Exploiting the fact that $\S_H$ is a minimal hypersurface in $\SS^n$ and that if $\{\tau_i\}_i$ is a local orthonormal frame on $\S_H$ then $\nu_H\cdot\nabla_{\tau_i}\tau_j=0$, a second variation computation (see, e.g., \cite[Lemma 2.1]{EngelSpolaVelichGandT}) gives, for $u\in\X_\s(\S_H)$,
\[\Big|\H^{n-1}(\S_H(u))-n\,\om_n-\frac12\,\int_{\S_H}\!\!\!|\nabla^{\S_H} u|^2-(n-1)\,u^2\Big|\le\! C(n)\,\s\!\int_{\S_H}\!\!\!u^2+|\nabla^{\Sigma_H} u|^2\,,
\](where $n\,\om_n=\H^{n-1}(\S_H)=\H^{n-1}(\S_H(0))$). We recall that $u\in L^2(\S_H)$ is a unit norm Jacobi field of $\S_H$ (i.e., a zero eigenvector of $\Delta^{\S_H}+(n-1)\,\Id$ with unit $L^2(\S_H)$-norm) if and only if there is $\tau\in\SS^n$ with $\tau\cdot\nu_H=0$ and   $u(\om)=c_0(n)\,(\om\cdot\tau)$ ($\om\in\S_H$) for $c_0(n)=(n/\H^{n-1}(\S_H))^{1/2}$. We denote by $E^0_{\S_H}$ the orthogonal projection operator of $L^2(\Om)$ onto the span of the Jacobi fields of $\S_H$. The following lemma provides a way to reparameterize spherical graphs over equatorial spheres so that the projection over Jacobi fields is annihilated.

\begin{lemma}\label{lemma step one sigma}
There exist constants $C_0$, $\e_0$ and $\s_0$, depending on the dimension $n$ only, with the following properties:

\noindent {\bf (i):} if $H,K\in\H$, $|\nu_H-\nu_K|\le\e<\e_0$, and $u\in\X_\s(\S_H)$ for $\s<\s_0$, then the map $T_u^K:\S_H\to\S_K$ defined by
\[
\]
is a diffeomorphism between $\S_H$ and $\S_K$, and $v_u^K:\S_K\to\R$ defined by
\begin{equation}
\label{vuK def}
\end{equation}
is such that
\begin{eqnarray}
\label{for later}
&&v_u^K\in\X_{C(n)\,(\s+\e)}(\S_K)\,,\quad \S_H(u)=\S_K(v_u^K)\,,
\\
\label{auto 4 pre}
&&\Big|\int_{\S_K}(v_u^K)^2-\int_{\S_H}u^2\Big|\le C(n)\,\Big\{|\nu_H-\nu_K|^2+\int_{\S_H}u^2\Big\}\,.
\end{eqnarray}

\noindent {\bf (ii):} if $H\in\H$ and $u\in\X_{\s_0}(\Sigma_H)$, then there exist $K\in\H$ with $|\nu_H-\nu_K|<\e_0$ and $v\in\X_{C_0\,\s_0}(\Sigma_K)$ such that
\begin{eqnarray}\label{auto 1}
\\\label{auto 2}
\\\label{auto 3}
&&|\nu_K-\nu_H|^2\le C_0(n)\,\int_{\Sigma_H}\,\big(E_{\Sigma_H}^0[u]\big)^2\,,
\\\label{auto 4}
&&\Big|\int_{\Sigma_K}v^2-\int_{\Sigma_H}u^2\Big|\le C_0(n)\,\int_{\S_H}u^2\,.
\end{eqnarray}
%%%%%%%%%%%  \noindent {\rm (iii):} if $H,K\in\H$, $\nu_H\cdot\nu_K>0$, $|\nu_H-\nu_K|\ge\e_0$, $u\in\X_{\s_0}(\Sigma_H)$, and $v\in\X_{\s_0}(\Sigma_K)$, then
%%%%%%%%%%%  \[
%%%%%%%%%%%  \H^n\big(Z_H(u)\Delta Z_K(v)\big)\ge\frac1{C_0(n)}\,,
%%%%%%%%%%%  \]
%%%%%%%%%%%  where
%%%%%%%%%%%  \[
%%%%%%%%%%%  Z_H(u)=\Big\{\frac{\om+t\,\nu_H}{\sqrt{1+t^2}}:\om\in H\,,t<u(\om)\Big\}\,,
%%%%%%%%%%%  \]
%%%%%%%%%%%  denotes the spherical epigraph of $u$.
\end{lemma}

\begin{remark}
{\rm It may seem unnecessary to present a detailed proof of Lemma \ref{lemma step one sigma}, as we are about to do,  given that, when $\Sigma_H$ is replaced by a generic integrable minimal surface $\Sigma$ in $\SS^n$, similar statements are found in the first four sections of \cite[Chapter 5]{allardalmgrenRADIAL}. However, two of those statements, namely \cite[5.3(4), 5.3(5)]{allardalmgrenRADIAL}, seem not to be correct; and the issue requires clarification, since those statements are used in the iteration arguments for the blowup/blowdown theorems \cite[Theorem 5.9/Theorem 9.6]{allardalmgrenRADIAL}; see, e.g., the second displayed chain of inequalities on \cite[Page 254]{allardalmgrenRADIAL}. To explain this issue {\bf we momentarily adopt the notation of [1]}. In \cite[Chapter 5]{allardalmgrenRADIAL} they consider a family of minimal surfaces $\{M_t\}_{t\in U}$ in $\SS^n$ obtained as diffeomorphic images of a minimal surface $M=M_0$. The parameter $t$ ranges in an open ball $U\subset\R^j$, where $j$ is the dimension of the space of Jacobi fields of $M$. Given a vector field $Z$ in $\SS^n$, defined on and normal to $M_t$, they denote by $F_t(Z)$ the diffeomorphism of $M_t$ into $\SS^n$ obtained by combining $Z$ with the exponential map of $\SS^n$ (up to lower than second order corrections in $Z$, this is equivalent to taking the graph of $Z$ over $M_t$, and then projecting it back on $\SS^n$, which is what we do, following [32], in \eqref{def of spherical graph}). Then, in \cite[5.2(2)]{allardalmgrenRADIAL}, they define $\Lambda_t$ as the family of those $Z$ such that ${\rm Image}(F_t(Z))={\rm Image}(F_0(W))$ for some vector field $W$ normal to $M$, and, given $t,u\in U$ and $Z\in\Lambda_t$, they define $F_t^u:\Lambda_t\to\Lambda_u$ as the map between such classes of normal vector fields with the property that ${\rm Image}(F_t(Z))={\rm Image}(F_u(F_t^u(Z)))$: in particular, $F_t^u(Z)$ is the vector field that takes $M_u$ to the same surface to which $Z$ takes $M_t$. With this premise, in \cite[5.3(5)]{allardalmgrenRADIAL} they say that if $t,u\in U$, and $Z\in\Lambda_t$, then
\begin{equation}
\label{from AA}
\Big|\int_{M_u}|F_t^u(Z)|^2-\int_{M_t}|Z|^2\Big|\le C\,|t-u|\,\int_{M_t}|Z|^2\,,
\end{equation}
for a constant $C$ depending on $M$ only. Testing this with $Z=0$ (notice that $0\in\Lambda_t$ by \cite[5.3(1)]{allardalmgrenRADIAL}) one finds $F_t^u(0)=0$, and thus $M_t={\rm Image}(F_t(0))={\rm Image}(F_u(F_t^u(0)))={\rm Image}(F_u(0))=M_u$. In particular, $M_u=M_t$ for every $t,u\in U$, that is, $\{M_t\}_{t\in U}$ consists of a single surface, $M$ itself. But this is never the case since $\{M_t\}_{t\in U}$ always contains, to the least, every sufficiently small rotation of $M$ in $\SS^n$. An analogous problem is contained in \cite[5.3(4)]{allardalmgrenRADIAL}. Coming back to our notation, the analogous estimate to \eqref{from AA} in our setting would mean that, for every $H,K\in\H$ with $|\nu_K-\nu_H|<\e_0$ and $u\in\X_{\s_0}(\S_H)$, $v_u^K$ defined in \eqref{vuK def} satisfies
\begin{equation}
\label{equivalent to from AA}
\Big|\int_{\S_K}(v_u^K)^2-\int_{\Sigma_H}u^2\Big|\le C(n)\,|\nu_H-\nu_K|\,\int_{\S_H}u^2\,,
\end{equation}
which again gives a contradiction if $u=0$. A correct estimate, analogous in spirit to \eqref{equivalent to from AA} and still sufficiently precise to be used in iterations, is \eqref{auto 4 pre} in Lemma \ref{lemma step one sigma}. There should be no difficulty in adapting our proof to the more general context of integrable cones, and then in using the resulting generalization of \eqref{auto 4 pre} to implement the iterations needed in \cite[Theorem 5.9, Theorem 9.6]{allardalmgrenRADIAL}.}
\end{remark}

\begin{proof}
[Proof of Lemma \ref{lemma step one sigma}] The constants $\e_0$ and $\s_0$ in the statement will be such that $\s_0=\e_0/C_*$ for a sufficiently large dimension dependent constant $C_*$.

\noindent {\bf Step one:} To prove statement (i), let $H,K\in\H$, $|\nu_H-\nu_K|\le\e<\e_0$ and $u\in\X_{\s}(\S_H)$ with $\s<\s_0$. Setting (for $\om\in\S_H$ and $x\in\R^{n+1}\setminus\{0\}$)
\[
g_u^K(\om)=\p_K\om+u(\om)\,\p_K\nu_H\,,\qquad \Phi(x)=x/|x|\,,
\]
we have $T_u^K=\Phi\circ g_u^K$, and, if $u$ is identically $0$,
\[
g_0^K(\om)=\p_K\om\,,\qquad T_0^K(\om)=\frac{\p_K\om}{|\p_K\om|}\,,\qquad\forall\om\in\S_H\,.
\]
By $|\p_K\nu_H|^2=1-(\nu_H\cdot\nu_K)^2\le 2\,(1-(\nu_H\cdot\nu_K))=|\nu_H-\nu_K|^2$,
\begin{eqnarray*}
\\
\end{eqnarray*}
In particular, $|g_u^K|\ge 1-\s_0\,\e_0\ge 1/2$, and since $\Phi$ and $\nabla\Phi$ are Lipschitz continuous on $\{|x|\ge 1/2\}$, we find
\begin{equation}
\label{guK and TuK near g0K and T0K}
\max\big\{\|g_u^K-g_0^K\|_{C^1(\S_H)},\|T_u^K-T_0^K\|_{C^1(\S_H)}\big\}\le C(n)\, \|u\|_{C^1(\S_H)}\,|\nu_H-\nu_K|\,.
\end{equation}
Similarly, since $\om\cdot\nu_K=\om\cdot(\nu_K-\nu_H)$ for $\om\in\S_H$, we find that
\begin{equation}
\label{g0K and T0K close to id}
\|g_0^K-{\rm id}\|_{C^1(\S_H)}\le C(n)\,|\nu_H-\nu_K|\,,\qquad \|T_0^K-{\rm id}\|_{C^1(\S_H)}\le C(n)\,|\nu_H-\nu_K|\,,
\end{equation}
and we thus conclude that $T_u^K$ is a diffeomorphism between $\S_H$ and $\S_K$. As a consequence, the definition \eqref{vuK def} of  $v_u^K$ is well-posed, and \eqref{for later} immediately follows (in particular, $\S_H(u)=\S_K(v_u^K)$ is deduced easily from \eqref{vuK def} and \eqref{def of spherical graph}). Finally, if we set $F_u^K(\om)=v_u^K(T_u^K(\om))^2\,J^{\S_H}\,T_u^K(\om)$ ($\om\in\S_H$), then
\begin{eqnarray*}
\int_{\S_K}(v_u^K)^2-\int_{\S_H}u^2=\int_{\S_H}\Big(\frac{\nu_K\cdot(\om+u\,\nu_H)}{|g_u^K(\om)|}\Big)^2\,J^{\S_H}\,T_u^K(\om)-u^2\,,
\end{eqnarray*}
where, using again $|\om\cdot\nu_K|\le|\nu_H-\nu_K|$ for every $\om\in\S_H$, we find
\begin{eqnarray*}
&&|J^{\S_H}T_u^K(\om)-1|\le C(n)\,\|T_u^K-{\rm id}\|_{C^1(\S_H)}\le C(n)\,|\nu_H-\nu_K|\,,
\\
\\
&&\hspace{2.9cm}\le C\,\big(|\nu_H-\nu_K|^2+u^2\big)\,,
\\
\\
\\
\le |\nu_K-\nu_K|^2+2\,u^2\,|\nu_H-\nu_K|+2\,|u|\,|\nu_H-\nu_K|    \le C\,\big(|\nu_H-\nu_K|^2+u^2\big)\;
\end{eqnarray*}
and thus, \eqref{auto 4 pre}, thanks to
\begin{eqnarray*}\hspace{-1cm}
\int_{\S_H}\!\!|J^{\S_H}\,T_u^K-1|\,u^2
\\
&&\hspace{2cm}+2\,\int_{\S_H}\,\Big|1-\frac1{|g_u^K|^2}\Big|\,u^2\le C(n)\,\Big(|\nu_H-\nu_K|^2+\int_{\S_H}u^2\Big)\,.
\end{eqnarray*}

\noindent {\bf Step two:} We prove (ii). If $E_{\Sigma_H}^0[u]=0$, then we conclude with $K=H$, $v=u$. We thus assume $\g^2=\int_{\Sigma_H}\,(E_{\Sigma_H}^0[u])^2>0$, and pick an orthonormal basis $\{\phi_H^i\}_{i=1}^n$ of $L^2(\S_H)\cap\{E_{\Sigma_H}^0=0\}$ with $E_{\Sigma_H}^0[u]=\g\,\phi_H^1$ and $\g=\int_{\Sigma_H}u\,\phi_H^1\ne 0$. This corresponds to choosing an orthonormal basis $\{\tau_H^i\}_{i=1}^n$ of $H$ such that
\begin{equation}
\label{what are the jacobi fields}
\phi_H^i(\om)=c_0(n)\,\om\cdot\tau_H^i\,,\qquad\om\in\S_H\,,
\end{equation}
for $c_0(n)=(n/\H^{n-1}(\S_H))^{1/2}$. For each $K\in\H$ with $\dist_{\SS^n}(\nu_H,\nu_K)<\e_0$ we define an orthonormal basis $\{\tau_K^i\}_{i=1}^n$ of $K$ by parallel transport of $\{\tau_H^i\}_{i=1}^n\subset H\equiv T_{\nu_H}\SS^n$ to $K\equiv T_{\nu_K}\SS^n$. The maps $\nu\mapsto\tau^i(\nu):=\tau_{K(\nu)}^i$ define an orthonormal frame $\{\tau^i\}_{i=1}^n$ of $\SS^n$ on the open set $A=B_{\e_0}^{\SS^n}(\nu_H)=\{\nu\in\SS^n:\dist_{\SS^n}(\nu,\nu_H)<\e_0\}$. We denote by $\rho_H^K$ the rotation of $\R^{n+1}$ which takes $H$ into $K$ by setting $\rho_H^K(\tau_H^i)=\tau_K^i$ and $\rho_H^K(\nu_H)=\nu_K$. By the properties of parallel transport we have that
\begin{equation}
\label{small rotation}
\|\rho_H^K-{\rm Id}\|_{C^0(\Sigma_K)}\le C(n)\,\dist_{\SS^n}(\nu_H,\nu_K)\le C(n)\,\e_0\,.
\end{equation}
Finally, we define an $L^2(\Sigma_K)$-orthonormal basis $\{\phi_K^i\}_{i=1}^n$ of $L^2(\S_K)\cap\{E_{\Sigma_K}^0=0\}$ by setting
$\phi_K^i(\om)=c_0(n)\,\om\cdot\tau_K^i$ ($\om\in\S_K$), and correspondingly consider the map $\Psi_u:A\to\R^n$ defined by setting
\[
\Psi_u(\nu)=\Big(\int_{\S_{K(\nu)}}v_u^{K(\nu)}\,\phi_{K(\nu)}^1,\dots,\int_{\S_{K(\nu)}}v_u^{K(\nu)}\,\phi_{K(\nu)}^n\Big)\,,\qquad\nu\in A\,,
\]
where $v_u^{K(\nu)}$ is well-defined for every $\nu\in A$ thanks to step one. {\bf We now claim} the existence of $\nu_*\in A$ such that
$\Psi_u(\nu_*)=0$.  By the area formula, \eqref{vuK def}, and $\q_{K(\nu)}[e]=\nu\cdot e$, we find
\begin{eqnarray*}
\int_{\S_H}\!\!\!v_u^{K(\nu)}(T_u^{K(\nu)})\,\phi_{K(\nu)}^j(T_u^{K(\nu)})\,J^{\S_H}T_u^{K(\nu)}
\\
\!\!\!\!c_0(n)\,\int_{\S_H}\!\!\nu\cdot(\om+u\,\nu_H)\Big(\rho_H^{K(\nu)}[\tau_H^j]\cdot\frac{\p_K(\om+u\,\nu_H)}{|\p_K(\om+u\,\nu_H)|^2}\Big)
\end{eqnarray*}
so that \eqref{guK and TuK near g0K and T0K} gives
\begin{eqnarray}
\label{psiu vicina psi0}
&&\!\!\!\!\!\!\!\!\|\Psi_u-\Psi_0\|_{C^1(A)} \le C(n)\,\s_0\,,\qquad\mbox{where}
\\\nonumber
\end{eqnarray}
By definition of $A$ and by \eqref{g0K and T0K close to id} and \eqref{small rotation},
\begin{eqnarray}\nonumber
&&\sup_{\nu\in A}\sup_{\om\in\S_H}\,\Big|\tau_H^j\cdot\om- \Big(\rho_H^{K(\nu)}[\tau_H^j]\cdot\frac{\p_K\om}{|\p_K\om|^2}\Big)\,\,
J^{\S_H}\Big[\frac{\p_K\om}{|\p_K\om|}\Big]\Big|\le C(n)\,\e_0\,,
\\\label{psi0 vicina psistar}
&&\mbox{and thus}\,\,\|\Psi_0-\Psi_*\|_{C^1(A)}\le C(n)\,(\s_0+\e_0)\,,
\end{eqnarray}
where $\Psi_*:A\to\R^n$ is defined by $e_j\cdot\Psi_*(\nu)=c_0(n)\,\int_{\S_H}
(\nu\cdot\om)\,(\tau_H^j\cdot\om)\,d\H^{n-1}_\om$ ($\nu\in A$).
Recalling that $\{\tau^i\}_{i=1}^n$ is an orthonormal frame of $\SS^n$ on $A$, with $\nabla_{\tau^i}\nu=\tau^i(\nu)=\tau_{K(\nu)}^i=\rho_H^{K(\nu)}[\tau^i_H]$, we find
\begin{eqnarray*}
\\
\end{eqnarray*}
By \eqref{small rotation}, \eqref{psiu vicina psi0} and \eqref{psi0 vicina psistar} we conclude that
\begin{eqnarray}\label{pino 1}
&&\|\Psi_u-\Psi_*\|_{C^1(A)}\le C(n)\,(\s_0+\e_0)\,,
\\\label{pino 2}
&&\big\|\nabla^{\SS^n}\Psi_u-c_0(n)^{-1}\,\sum_{j=1}^n\,e_j\otimes\tau^j\big\|_{C^0(A)}\le C(n)\,(\s_0+\e_0)\,.
\end{eqnarray}
Let us finally consider the map $h:A\times[0,1]\to\R^n$,
\[
h(\nu,t)=h_t(\nu)=t\,\Psi_*(\nu)+(1-t)\,\Psi_u(\nu)\,,\qquad(\nu,t)\in A\times[0,1]\,,
\]
which defines an homotopy between $\Psi_*$ and $\Psi_u$. By \eqref{pino 1} and \eqref{pino 2} we see that if $\nu\in\pa A$, that is, if $\dist_{\SS^n}(\nu,\nu_H)=\e_0$, then, denoting by $[\nu_H,\nu]_s$ the unit-speed length minimizing geodesic from $\nu_H$ to $\nu$, considering that $[\nu_H,\nu]_s\in A$ for every $s\in(0,\e_0$), and that $\SS^n$ is close to be flat in $A$, we find
\begin{eqnarray*}
\\
\end{eqnarray*}
provided $\s_0=\e_0/C_*$ is small enough with respect to $\e_0$ (i.e., provided $C_*$ is large), $\e_0$ is small in terms of $c_0$, and where we have used $\Psi_*(\nu_H)=0$ and
\begin{equation}
\label{pino 3}
|\Psi_u(\nu_H)|=|\g|=\Big|\int_{\S_H}u\,\phi^1_H\Big|\le C(n)\,\s_0\,,
\end{equation}
to deduce $|h_t(\nu_H)|\le C(n)\,\s_0$. This proves that
$0\not\in \pa\,h_t(\pa A)$ for every $t\in[0,1]$,
so that $\deg(h_t,A,0)$ is independent of $t\in[0,1]$. In particular, $h_0=\Psi_u$ and $h_1=\Psi_*$ give
where we have used $\Psi_*(\nu_H)=0$ and the fact that, up to decreasing the value of $\e_0$, $\Psi_*$ is injective on $A$. By $\deg(\Psi_u,A,0)=1$, there is $\nu_*\in A$ such that $\Psi_u(\nu_*)=0$, {\bf as claimed}. With $K=K(\nu_*)$ and $v=v_u^K$ we deduce \eqref{auto 1} from \eqref{for later} and \eqref{auto 2} from $\Psi_u(\nu_*)=0$. By \eqref{pino 2} and \eqref{pino 3}, if $\eta=\dist_{\SS^n}(\nu_*,\nu_H)$, then
\begin{eqnarray*}
\\
\ge\Big(\frac1{c_0(n)}-C(n)\,(\e_0+\s_0)\Big)\,\eta
\ge \frac{|\nu_*-\nu_H|}{2\,c_0(n)}\,\,,
\end{eqnarray*}
that is \eqref{auto 3}. Finally, \eqref{auto 4} follows from \eqref{auto 3} and \eqref{auto 4 pre}.
\end{proof}

\subsection{Energy estimates for spherical graphs over annuli}\label{subsection energy estimates on annuli} Given $H\in\H$ and $0<r_1<r_2$ we let $\X_\s(\Sigma_H,r_1,r_2)$ be the class of those $u\in C^1(\Sigma_H\times(r_1,r_2))$ such that, setting $u_r=u(\cdot,r)$, one has
$u_r\in\X_\s(\Sigma_H)$ for every $r\in(r_1,r_2)$ and $|r\,\pa_r u|\le\s$ on $\Sigma_H\times(r_1,r_2)$. If $u\in\X_\s(\Sigma_H,r_1,r_2)$, then the spherical graph of $u$ over $\Sigma_H\times(r_1,r_2)$, given by
\[
\Sigma_H(u,r_1,r_2)=\Big\{r\,\frac{\om+u_r(\om)\,\nu_H}{\sqrt{1+u_r(\om)^2}}:\om\in\Sigma_H\,,r\in(r_1,r_2)\Big\}\,,
\]
is an hypersurface in $A_{r_1}^{r_2}$. It is useful to keep in mind that
$\Sigma_H(0,r_1,r_2)=\{r\,\om:\om\in\Sigma\,,r\in(r_1,r_2)\}=H\cap A_{r_1}^{r_2}$ is a flat annular region of area $\om_n\,(r_2^n-r_1^n)$, and that if $\s<\s_1=\s_1(n)$, then
\begin{equation}
\label{equuivalence between u square and omega square}
\frac1{C(n)}\,\int_{\S_H(u,r_1,r_2)}\!\!\!\!\om_H^2\,d\H^n\le \int_{\S_H\times(r_1,r_2)}\!\!r^{n-1}\,u^2\le C(n)\,\int_{\S_H(u,r_1,r_2)}\!\!\!\!\om_H^2\,d\H^n\,.
\end{equation}

\begin{lemma}\label{lemma step one}
There are $\e_0$, $\s_0$, $C_0$ positive, depending on $n$ only, such that:

\noindent {\bf (i):} if $H,K\in\H$, $\nu_H\cdot\nu_K>0$, $|\nu_H-\nu_K|=\e<\e_0$, $u\in\X_{\s}(\S_H,r_1,r_2)$, and $\s<\s_0$, then there is $v\in \X_{C_0(\s+\e)}(\S_H,r_1,r_2)$ such that $\S_K(v,r_1,r_2)=\S_H(u,r_1,r_2)$.

\noindent {\bf (ii):} if $H\in\H$, $u\in\X_{\s_0}(\Sigma_H,r_1,r_2)$, and $(a,b)\cc(r_1,r_2)$, then there exist $K\in\H$, $v\in\X_{C_0\,\s_0}(\Sigma_K,r_1,r_2)$, and $r_*\in[a,b]$ such that
\begin{eqnarray}\label{biro 1}
\\\label{biro 2}
\\\label{biro 3}
&&|\nu_H-\nu_K|^2\le C_0(n)\,\min_{\rho\in[a,b]}\int_{\Sigma_H}\,\big(E_{\Sigma_H}^0[u_\rho]\big)^2\,.
\end{eqnarray}
Moreover,  for every $r\in(r_1,r_2)$,
\begin{equation}
\label{biro 4}
\Big|\int_{\Sigma_K}(v_r)^2-\int_{\Sigma_H}(u_r)^2\Big|\le C_0(n)\,\Big\{
\min_{\rho\in[a,b]}\int_{\Sigma_H}\,(u_\rho)^2+\int_{\Sigma_H}(u_r)^2\Big\}\,.
\end{equation}
\end{lemma}

\begin{proof}
We prove statement (i). If $|\nu_H-\nu_K|=\e<\e_0$, since $u_r\in\X_{\s}(\S_H)$ for every $r\in(r_1,r_2)$, by Lemma \ref{lemma step one sigma}-(i) we see that $T_r:\S_H\to\S_K$,
\begin{equation}
\label{tr}
\end{equation}
is a diffeomorphism between $\S_H$ and $\S_K$, and $v_r:\S_K\to\R$,
\begin{equation}
\label{vrtr}
\end{equation}
satisfies $v_r\in\X_{C_0\,(\s+\e)}(\S_K)$, $\S_H(u_r)=\S_K(v_r)$ for every $r\in(r_1,r_2)$, and
\begin{equation}
\label{biro 5}
\Big|\int_{\S_K}\!\!\!(v_r)^2-\int_{\S_H}\!\!\!(u_r)^2\Big|\le C(n)\,\Big\{|\nu_H-\nu_K|^2+\int_{\S_H}\!\!\!(u_r)^2\Big\}\,.
\end{equation}
Since $u\in\X_{\s}(\Sigma_H,r_1,r_2)$, and $T_r$ and $v_r$ depend smoothly on $u_r$,  setting $v(\om,r):=v_r(\om)$ we have $\S_H(u,r_1,r_2)=\S_K(v,r_1,r_2)$ (by $\S_H(u_r)=\S_K(v_r)$ for every $r\in(r_1,r_2)$), and $v\in\X_{C_0\,(\s+\e)}(\Sigma_H,r_1,r_2)$ ($|r\,\pa_rv_r|\le C_0(\s+\e)$ is deduced by differentiation in \eqref{tr} and \eqref{vrtr}, and by $|u_r|,|r\,\pa_ru_r|<\s$).
%This proves \eqref{star 2}.
%, and \eqref{star 3} is easily deduced from the area formula.

\noindent {\bf Step two:} We prove (ii). Let
and let $r_*\in[a,b]$ be such that the minimum $\g$ is achieved at $r=r_*$. If $\g=0$, then we set $K=H$ and $v=u$. If $\g>0$, then we apply Lemma \ref{lemma step one sigma}-(ii) to $u_{r_*}\in\X_{\s_0}(\S_H)$, and find $K\in\H$ with $|\nu_K-\nu_H|<\e_0$ and $v_{r_*}\in\X_{C_0\,s_0}(\S_K)$ such that $\Sigma_H(u_{r_*})=\Sigma_K(v_{r_*})$ and
\begin{eqnarray}
\label{biro 6}
\\\label{biro 7}
&&|\nu_K-\nu_H|^2\le C_0(n)\,\int_{\Sigma_H}\,\big(E_{\Sigma_H}^0[u_{r_*}]\big)^2=C_0(n)\,\g\,,
\\\label{biro 8}
&&\Big|\int_{\Sigma_K}(v_{r_*})^2-\int_{\Sigma_H}(u_{r_*})^2\Big|\le C_0(n)\,\int_{\S_H}(u_{r_*})^2\,.
\end{eqnarray}
Since $v_{r_*}=v(\cdot,r_*)$ for $v$ constructed in step one starting from $u$, $H$ and $K$, we deduce \eqref{biro 4} by \eqref{biro 5} and \eqref{biro 7}, while \eqref{biro 6} is \eqref{biro 2}.
\end{proof}

We will use two basic ``energy estimates'' for spherical graphs over annuli. To streamline the application of these estimates to diadic families of annuli we consider intervals $(r_1,r_2)$ and $(r_3,r_4)$ are {\bf $(\eta,\eta_0)$-related}, meaning that
\begin{equation}
\label{r1r2r3r4}
r_2=r_0(1+\eta_0)\,,\quad r_1=r_0(1-\eta_0)\,,\quad   r_4=r_0(1+\eta)\,,\quad r_3=r_0(1-\eta)\,,
\end{equation}
for some $\eta_0>\eta>0$, and with $r_0=(r_1+r_2)/2=(r_3+r_4)/2$; in particular, $(r_3,r_4)$ is contained in, and concentric to, $(r_1,r_2)$. The case $\Lambda=0$ of the following statement is the codimension one, equatorial spheres case of \cite[Lemma 7.14, Theorem 7.15]{allardalmgrenRADIAL}.

\begin{theorem}[Energy estimates for spherical graphs]\label{theorem 7.15AA main estimate lambda}
If $n\ge 2$ and $\eta_0>\eta>0$, then there are $\s_0=\s_0(n,\eta_0,\eta)$ and $C_0=C_0(n,\eta_0,\eta)$ positive, with the following property. If $H\in\H$, $\Lambda\ge0$, and $u\in\X_{\s}(\Sigma_H,r_1,r_2)$ is such that $\max\{1,\Lambda\,r_2\}\,\s\le\s_0$ and $\Sigma_H(u,r_1,r_2)$ has mean curvature bounded by $\Lambda$ in $A_{r_1}^{r_2}$, then, whenever $(r_1,r_2)$ and $(r_3,r_4)$ are $(\eta,\eta_0)$-related as in \eqref{r1r2r3r4},
\[
\Big|\H^n(\Sigma_H(u,r_3,r_4))-\H^n(\Sigma_H(0,r_3,r_4))\Big|\le C_0\,\int_{\Sigma_H\times(r_1,r_2)}\!\!\!\!\!\!\!\!\!\!r^{n-1}\,\big(u^2+\Lambda\,r\,|u|\big)\,.
\]
Moreover, if there is $r\in(r_1,r_2)$ s.t. $E_{\Sigma_H}^0 u_r=0$ on $\Sigma_H$, then we also have
\[
\int_{\Sigma_H\times(r_3,r_4)} r^{n-1}\,u^2
\le C(n)\,\Lambda\,r_2\,(r_2^n-r_1^n)+ C_0\,\int_{\Sigma_H\times(r_1,r_2)}\,r^{n-1}\,(r\,\pa_ru)^2\,.
\]
% \label{lambdaAA 7.15(5)}
% &&\int_{\Sigma_H\times(r_3,r_4)} r^{n-1}\,\big(u^2+|\nabla^{\S_H}u|^2\big)
% \\\nonumber
% &&\hspace{2cm}\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{theorem}

\begin{proof}
See appendix \ref{appendix proof of expansion mean curvature}.
\end{proof}

\subsection{Monotonicity for exterior varifolds with bounded mean curvature}\label{subsection monotonicity of exterior varifolds}
The following theorem states the monotonicity of $\Theta_{V,R,\Lambda}$ for $V\in\V_n(\Lambda,R,S)$, and provides, when $V$ corresponds to a spherical graph, a quantitative lower bound for the gap in the associated monotonicity formula; the case $\Lambda=0$, $R=0$ is contained in \cite[Lemma 7.16, Theorem 7.17]{allardalmgrenRADIAL}.

\begin{theorem}\label{theorem 7.17AA exterior lambda} {\bf (i):} If $V\in\V_n(\Lambda,R,S)$, then
\begin{equation}
\label{lambdamonotonicity of theta ext}
\mbox{$\Theta_{V,R,\Lambda}$ is increasing on $(R,S)$}\,.
\end{equation}

\noindent {\bf (ii):} There is $\s_0(n)$ such that, if $V\in\V_n(\Lambda,R,S)$ and, for some $H\in\H$, $u\in\X_\s(\Sigma,r_1,r_2)$ with $\s\le\s_0(n)$, and $(r_1,r_2)\subset(R,S)$, we have
\begin{eqnarray}
\label{V corresponds to u monoto lambda}
\mbox{$V$ corresponds to $\Sigma_H(u,r_1,r_2)$ in $A_{r_1}^{r_2}$}\,,
\end{eqnarray}
\begin{eqnarray}
\label{lemma 7.16AA monoto lambda}
\int_{\S_H\times(r_1,r_2)}\!\!\!\!\!\!\!\!r^{n-1} (r\,\pa u_r)^2\le C(n)\,r_2^n\,
\Big\{\Theta_{V,R,\Lambda}(r_2)-\Theta_{V,R,\Lambda}(r_1)\Big\}\,.
\end{eqnarray}
{\bf (iii):} Finally, given $\eta_0>\eta>0$, there exist $\s_0$ and $C_0$ depending on $n$, $\eta_0$, and $\eta$ only, such that if the assumptions of part (i) and part (ii) hold and, in addition to that, we also have $\max\{1,\Lambda\,r_2\}\,\s\le\s_0$ and
\begin{equation}
\label{lambdaAA 7.15(4) for 7.17 hole}
\mbox{$\exists\,r\in(r_1,r_2)$ s.t. $E_{\Sigma_H}^0 u_r=0$ on $\Sigma_H$}\,,
\end{equation}
then, whenever $(r_1,r_2)$ and $(r_3,r_4)$ are $(\eta,\eta_0)$-related as in \eqref{r1r2r3r4}, we have
\begin{eqnarray}\label{lambdaAA tesi 7.17 hole}
\\\nonumber
&&\hspace{2cm}\le C_0\,r_2^n\,\Big\{\Theta_{V,R,\Lambda}(r_2)-\Theta_{V,R,\Lambda}(r_1)+(\Lambda\,r_2)^2\Big\}\,.
\end{eqnarray}
%as well as
%\begin{eqnarray}\label{W12 estimate}
%  &&\int_{\S_H\times(r_3,r_4)}r^{n-1}\,\big(u^2+|\nabla^{\S_H}u|^2+(r\,\pa_ru)^2\big)
%  \\\nonumber
%  &&\hspace{2cm}\le C_0\,r_2^n\,\Big\{\Theta_{V,R,\Lambda}(r_2)-\Theta_{V,R,\Lambda}(r_1)+(\Lambda\,r_2)^2\Big\}\,,
\end{theorem}

\begin{proof}
We give details of the proof of (i) when $V\in\M_n(\Lambda,R,S)$ (whereas the general case is addressed as in \cite[Section 17]{SimonLN}). By the coarea formula, the divergence theorem and $|\vec{H}|\le\Lambda$, for a.e. $\rho>R$,
\begin{eqnarray}\nonumber
\frac{d}{d\rho}\,\frac{\|V\|(B_\rho\setminus B_R)}{\rho^n}&=&
\frac1{\rho^n}\,\int_{M\cap\pa B_\rho}\frac{|x|\,d\H^{n-1}}{|x^{TM}|}
-\frac{n\,\H^n(M\cap (B_\rho\setminus B_R))}{\rho^{n+1}}
\\\nonumber
\frac1{\rho^n}\,\int_{M\cap\pa B_\rho}\frac{|x|\,d\H^{n-1}}{|x^{TM}|}-\frac1{\rho^n}\int_{M\cap(B_\rho\setminus B_R)}\,\frac{x}{\rho}\cdot\vec{H}\,d\H^n
\\\nonumber
\!\!\!\!-\frac{1}{\rho^{n+1}}\Big\{\int_{M\cap\pa B_\rho}\!\!\!\!\!\nu^{{\rm co}}_M\cdot x\,d\H^{n-1}+\int_{M\cap\pa B_{R}}\!\!\!\!\!\nu^{{\rm co}}_M\cdot x\,d\H^{n-1}\Big\}
\\\nonumber
&\ge& \frac1{\rho^n}\,\int_{M\cap\pa B_\rho}\Big(\frac{|x|}{|x^{TM}|}-\frac{|x^{TM}|}{|x|}\Big)\,d\H^{n-1}
\\\nonumber
&& \!\!\!\!-\frac{1}{\rho^{n+1}}\,\int_{M\cap\pa B_R}\!\!\!\!\!\nu^{{\rm co}}_M\cdot x\,d\H^{n-1}
-\Lambda\,\frac{\H^n(M\cap(B_\rho\setminus B_R))}{\rho^n}
\\\label{conelimit}
={\rm Mon}(V,\rho)+\frac{d}{d\rho}\,\frac1{n\,\rho^n}\,\int x\cdot\nu^{{\rm co}}_V\,d\,{\rm bd}_V-\Lambda\,\frac{\|V\|(B_\rho\setminus B_R)}{\rho^n}\,\,\,\,\,\,\,\,\,\,\,
\end{eqnarray}
where ${\rm Mon}(V,\rho)=(d/d\rho)\int_{B_\rho\setminus B_R}\,|x^\perp|^2\,|x|^{-n-2}\,d\|V\|$. Since ${\rm Mon}(V,\rho)\ge0$, this proves \eqref{lambdamonotonicity of theta ext}. Assuming now \eqref{V corresponds to u monoto lambda}, by using \cite[Lemma 3.5(6)]{allardalmgrenRADIAL} as done in the proof of \cite[Lemma 7.16]{allardalmgrenRADIAL}, we see that, under \eqref{V corresponds to u monoto lambda},
\[
C(n)\,r_2^n\,\int_{r_1}^{r_2}{\rm Mon}(V,\rho)\,d\rho\ge \int_{\Sigma_H\times(r_1,r_2)}r^{n-1}\,(r\,\pa_ru)^2\,,
\]
thus proving (ii). To prove (iii), we set $a=r_0\,(1-(\eta+\eta_0)/2)$ and $b=r_0\,(1+(\eta+\eta_0)/2)$, so that $(a,b)$ and $(r_3,r_4)$ are $(\eta,(\eta+\eta_0)/2)$-related, and $(r_1,r_2)$ and $(a,b)$ are $((\eta+\eta_0)/2,\eta_0)$-related (in particular, $(r_3,r_4)\subset (a,b)\subset (r_1,r_2)$). By suitably choosing $\s_0$ in terms of $n$, $\eta$ and $\eta_0$, we can apply Theorem \ref{theorem 7.15AA main estimate lambda} with $(r_3,r_4)$ and $(a,b)$, so to find (with $C=C(n,\eta_0,\eta)$)
\begin{eqnarray*}
&&\Big|\H^n(\S(u,r_3,r_4))-\H^n(\S(0,r_3,r_4))\Big|\le C\,\int_{\S_H\times(a,b)} r^{n-1}\big(u^2+\Lambda\,r\,|u|\big)
\\
&&\hspace{3cm}\le C\,\Big\{(\Lambda\,b)^2\,(b^n-a^n)+\int_{\S_H\times(a,b)} r^{n-1}\,u^2\Big\}\,.
\end{eqnarray*}
Thanks to \eqref{lambdaAA 7.15(4) for 7.17 hole} we can apply Theorem \ref{theorem 7.15AA main estimate lambda} with
$(a,b)$ and $(r_1,r_2)$ to find
\[
\int_{\S_H\times(a,b)} r^{n-1}\,u^2\le
C\,\Big\{(\Lambda\,r_2)^2\,(r_2^n-r_1^n)+\int_{\S_H\times(r_1,r_2)} r^{n-1}\,(r\,\pa_ru)^2\Big\}\,.
\]
%%\int_{\S_H\times(a,b)} r^{n-1}\,\big(u^2+|\nabla^{\S_H}u|^2\big)\le
%%C(n,\eta_0,\eta)\,\Big\{(\Lambda\,r_2)^2\,(r_2^n-r_1^n)+\int_{\S_H\times(r_1,r_2)} r^{n-1}\,(r\,\pa_ru)^2\Big\}\,.
We find \eqref{lambdaAA tesi 7.17 hole} by \eqref{lemma 7.16AA monoto lambda} and $(\Lambda\,b)^2\,(b^n-a^n)\le (\Lambda\,r_2)^2\,r_2^n$.
\end{proof}

\subsection{Proof of the mesoscale flatness criterion}\label{subsection mesoscale flatness criterion} As a final preliminary result to the proof of Theorem \ref{theorem mesoscale criterion}, we prove the following lemma, where Allard's regularity theorem is combined with a compactness argument to provide the basic graphicality criterion used throughout the iteration. The statement should be compared to \cite[Lemma 5.7]{allardalmgrenRADIAL}.

\begin{lemma}[Graphicality lemma]\label{lemma graphicality lambda}
Let $n\ge 2$. For every $\s>0$, $\Gamma\ge0$, $(\l_3,\l_4)\cc (\l_1,\l_2)\cc (0,1)$, and $(\eta_1,\eta_2)\cc(0,1)$, there are positive constants $\e_1$ and $M_1$, depending only on $n$, $\sigma$, $\Gamma$, $(\l_1,\l_2)$, $(\l_3,\l_4)$, and $(\eta_1,\eta_2)$, and $\e_2$ and $M_2$, depending only on $n$, $\sigma$, $\Gamma$, $\l_1$, and $(\eta_1,\eta_2)$, with the following properties.\par%If $\Lambda\ge0$, $R\in(0,1/\Lambda)$, $V\in\V_n(\Lambda,R,1/\Lambda)$,
\noindent {\bf (i):} If $\Lambda\ge0$, $R\in(0,1/\Lambda)$, $V\in\V_n(\Lambda,R,1/\Lambda)$,
\begin{equation}\label{gamma bound lemma statement}
\|{\rm bd}_V\|(\pa B_{R})\le\Gamma\,R^{n-1}\,,\qquad \sup_{\rho\in(R,1/\Lambda)}\frac{\|V\|(B_\rho\setminus B_R)}{\rho^n}\le\Gamma\,,
\end{equation}
there exists $r>0$ such that
\begin{eqnarray}
\label{conditions on r lambda}
&&\hspace{-0.2cm}\max\{M_1,64\}\,R\le r\le \frac{\e_1}\Lambda\,,
\\
\label{ext hp step two 1 lambda}
\\\label{ext hp step two 3 lambda}
\end{eqnarray}
and if, for some $K\in\H$, we have
\begin{equation}
\label{ext hp step two 2 lambda}
\frac1{r^n}\,\int_{A_{\l_1\,r}^{\l_2\,r}}\,\om_K^2\,d\|V\|\le\e_1\,,
\end{equation}
then there exists $u\in\X_\s(\Sigma_K,\eta_1 \, r,\eta_2 \, r)$ such that
\[
\mbox{$V$ corresponds to $\Sigma_K(u,\eta_1 \, r,\eta_2 \, r)$ on $A_{\eta_1 \, r}^{\eta_2 \, r}$}\,.
\]
\noindent {\bf (ii):} If $\Lambda$, $R$, and $V$ are as in (i), \eqref{gamma bound lemma statement} holds, and there exists $r$ such that \begin{eqnarray}
\label{alternative conditions on r lambda}
&&\hspace{0.5cm}\max\{M_2,64\}\,R\le r\le \frac{\e_2}\Lambda\,,
\\
\label{alternative hypo to angular flatness}
&&\max\{|\de_{V,R,\Lambda}(\lambda_1\, r)|,|\de_{V,R,\Lambda}(r)|\}\le\e_2\,,
%\\\label{ext hp step two 3 lambda}
\end{eqnarray}
then there exists $K\in \mathcal{H}$ and $u\in\X_\s(\Sigma_K, \eta_1\, r, \eta_2\, r)$ such that
\[
\mbox{$V$ corresponds to $\Sigma_K(u, \eta_1 \, r , \eta_2 \, r)$ on $A_{\eta_1 \, r}^{\eta_2 \, r}$}\,.
\]
\end{lemma}

\begin{proof} \textit{Step one}: As a preliminary, we first show that if $V$ is a stationary, $n$-dimensional, integer rectifiable varifold in $B_1$ such that
\begin{equation}\label{plane mass}
\|V\|(B_1)\leq \omega_n\,,\quad \mathrm{spt}\, V \cap A_{\beta_1}^{\beta_2}\subset K\,,\quad\textup{and}\quad \mathrm{spt}\, V \cap A_{\beta_1}^{\beta_2} \neq \emptyset\,,
\end{equation}
for some $K\in\mathcal{H}$ and $0<\beta_1<\beta_2\leq 1$,
%\begin{equation}\label{step one annular containment}
%\mathrm{spt}\, V \cap A_{\beta_1}^{\beta_2}\subset K\,,\quad \mathrm{spt}\, V \cap A_{\beta_1}^{\beta_2} \neq \emptyset\,,
then $V = \var (K\cap B_1,\left.1\right|_{K\cap B_1})$.

Let $\beta'\in (\beta_1,\beta_2)$ and $\varphi_1$, $\varphi_2\in C^\infty(\mathbb{R}^{n+1};[0,1])$ be such that $\spt\, \varphi_1 \subset B_{\beta_2}$, $\left.\varphi_1\right|_{B_{\beta'}}\equiv 1$, and $\varphi_1+\varphi_2\equiv 1$. As a consequence of \eqref{plane mass} and the stationarity of $V$ in $B_{\beta_2}$, for $X\in  C_c^1(\mathbb{R}^{n+1}\setminus (K \cap (\overline{B}_{\beta_2} \setminus B_{\beta'}))$, we have
\begin{eqnarray}\notag
\delta (V\mres B_{\beta'})(X)&=& \int_{B_{\beta'}}\Div^{M}(\varphi_1 X)+\Div^{M}(\varphi_2 X)\,d\|V\| \\ \notag%\\ \notag %+ \delta (V\mres B_{\lambda'})(\varphi_2 X)
&=&\int_{B_{\beta_2}}\Div^{M}(\varphi_1 X)\,d\|V\|
\end{eqnarray}
%$V\mres B_{\lambda}$ is stationary in $\mathbb{R}^{n+1}\setminus (K \cap (\overline{B}_1 \setminus B_{\lambda}))$.
Then by the convex hull property \cite[Theorem 19.2]{SimonLN}, $\mathrm{spt}\,(V \mres B_{\beta'}) \subset K$.
%\begin{equation}\label{wigging out junior}
%\mathrm{spt}\,(V \mres B_{\beta'})\subset \textup{convex hull of }(K \cap (\overline{B_{\beta_2}}\setminus B_{\beta'})) \subset K\,.
By the constancy theorem \cite[Theorem 41.1]{SimonLN}, $V \mres B_{\beta_2}=\var ( K \cap B_{\beta_2},\theta)$ for some constant $\theta$. Furthermore, since $V$ assigns non-trivial mass to $B_{\beta_2}$ by \eqref{plane mass} and is integer rectifiable, $\theta\geq 1$. Therefore $0\in \spt \|V\|$, and the monotonicity formula gives $\omega_n \leq \lim_{r\to 0^+}\|V\|(B_r)r^{-n} \leq \|V\|(B_1)\leq \omega_n$. Thus $V$ is a stationary, $n$-dimensional, integer rectifiable varifold in $B_1$ with constant area ratios $\omega_n$ and $\spt V \cap A_{\beta_1}^{\beta_2}\subset K$, so $V = \var (K\cap B_1,\left.1\right|_{K\cap B_1})$.
\par
\noindent\textit{Step two}: We prove item (i) by contradiction. If it were false, we could find $\s>0$, $\Gamma\ge0$, $(\l_3,\l_4)\cc(\l_1,\l_2)\cc(0,1)$, $(\eta_1,\eta_2)\subset (0,1)$, with $K_j\in H$, positive numbers $R_j$, $\Lambda_j<1/R_j$, $r_j$, and $W_j\in\V_n(\Lambda_j,R_j,1/\Lambda_j)$  such that
$\|W_j\|\big(A_{\l_3\,r_j}^{\l_4\,r_j}\big)>0$, $\|{\rm bd}_{W_j}\|(\pa B_{R_j})\le\Gamma\,R_j^{n-1}$, $\|W_j\|(B_\rho\setminus B_{R_j})\le\Gamma\,\rho^n$ for every $\rho\in(R_j,1/\Lambda_j)$, and $\rho_j=R_j/r_j\to 0$, $r_j\,\Lambda_j\to 0$, $\de_{W_j,R_j,\Lambda_j}(r_j)\to 0$, and $r_j^{-n}\int_{B_{\l_2\,r_j}\setminus B_{\l_1\,r_j}} \om_{K_j}^2\,d\|W_j\|\to 0$, but there is no $u\in\X_{\s}(\Sigma_{K_j},\eta_1\,r_j,\eta_2\,r_j)$ with the property that $W_j$ corresponds to $\Sigma_{K_j}(u,\eta_1\,r_j,\eta_2\,r_j)$ on $A_{\eta_1\,r_j}^{\eta_2\,r_j}$.
Hence, setting $V_j=W_j/r_j$, no $u\in\X_{\s}(\Sigma_{K_j},\eta_1,\eta_2)$ can exist such that $V_j$ corresponds to $\Sigma_{K_j}(u,\eta_1,\eta_2)$ on $A_{\eta_1}^{\eta_2}$, despite the fact that each $V_j$ belongs to $\V_n(r_j\,\Lambda_j,\rho_j,1/(r_j\,\Lambda_j))$ and satisfies
\begin{eqnarray}\nonumber
\frac{\|{\rm bd}_{V_j}\|(\pa B_{\rho_j})}{\rho_j^{n-1}}\le\Gamma\,,\,\,
\sup_{\rho\in(\rho_j,1/(\Lambda_j\,r_j))}\!\!\frac{\|V_j\|(B_\rho\setminus B_{\rho_j})}{\rho^n}\le\Gamma\,,
\\ \label{wigged out}
&&\hspace{2cm}\lim_{j\to\infty}\max\big\{\delta_{V_j,\rho_j,r_j\,\Lambda_j}(1)\,,\,\,\, \int_{A_{\l_1}^{\l_2}}\,\om_{K_j}^2\,d\|V_j\|\big\}=0\,.
\end{eqnarray}
Clearly we can find $K\in\H$ such that, up to extracting subsequences, $K_j\cap B_1\to K\cap B_1$ in $L^1(\R^{n+1})$. Similarly, by \eqref{wigged out}, we can find an $n$-dimensional integer rectifiable varifold $V$ such that $V_j\weak V$ as varifolds in $B_1\setminus\{0\}$. Since the bound on the distributional mean curvature of $V_j$ on $B_{1/(\Lambda_j\,r_j)}\setminus \ov{B}_{\rho_j}$ is $r_j\,\Lambda_j$, and since $\rho_j\to 0^+$ and $r_j\,\Lambda_j\to 0^+$, it also follows that $V$ is stationary in $B_1\setminus \{0\}$, and thus, by a standard argument and since $n\ge 2$, on $B_1$. By $\|V_j\|(A_{\l_3}^{\l_4})>0$, for every $j$ there is $x_j\in A_{\l_3}^{\l_4}\cap\spt\,V_j$,  so that, up to extracting subsequences, $x_j\to x_0$ for some $x_0\in \ov{A}_{\l_3}^{\l_4}\cap\spt\,V$.
By $(\l_3,\l_4)\cc  (\l_1,\l_2)$, there is $\rho>0$ such that $B_\rho(x_0)\subset A_{\l_1}^{\l_2}$, hence
\begin{equation}\label{nontrivial V step 2}
\|V\|(A_{\l_1}^{\l_2})\ge\|V\|(B_\rho(x_0))\ge \om_n\,\rho^n>0\,,
\end{equation}
%$\|V\|(A_{\l_1}^{\l_2})\ge\|V\|(B_\rho(x_0))\ge \om_n\,\rho^n>0$,
thus proving $V\,\llcorner\, A_{\l_1}^{\l_2}\ne\emptyset$. By this last fact, by $\om_K=0$ on $(\spt\,V)\cap A_{\l_1}^{\l_2}$, and by the constancy theorem \cite[Theorem 41.1]{SimonLN}, we have
\begin{equation}\label{support in plane step two}
A_{\l_1}^{\l_2}\cap\spt\,V=A_{\l_1}^{\l_2}\cap K\,.
\end{equation}
%Since $V$ is stationary in $B_1$, we conclude that $B_1\cap K\subset B_1\cap\spt\,V$, so that,
At the same time, since $\|{\rm bd}_{V_j}\|(\pa B_{\rho_j})\le\Gamma\,\rho_j^{n-1}$ and $\|V_j\|(B_\rho\setminus B_{\rho_j})\le\Gamma\,\rho^n$ for every $\rho\in(\rho_j,1/(\Lambda_j\,r_j))\supset (\rho_j\,1)$, by \eqref{wigged out},
\begin{eqnarray}\notag
\om_n&=&\!\!\!\!\lim_{j\to\infty}\|V_j\|(B_1\setminus B_{\rho_j})-
\frac{\rho_j}n\,\|\de V_j\|(\pa B_{\rho_j})+\Lambda_j\,r_j\,\int_{\rho_j}^1\,\frac{\|V_j\|(B_\rho\setminus B_{\rho_j})}{\rho^n}\,d\rho
\\ \label{less mass than plane step two}
\end{eqnarray}
Since $V$ is stationary in $B_1$ and integer rectifiable, and since \eqref{nontrivial V step 2} and \eqref{less mass than plane step two} imply \eqref{plane mass} with $\lambda_1=\beta_1$ and $\lambda_2=\beta_2$, the first step yields $V = \mathbf{var}\,(K\cap B_1,\left.1\right|_{K\cap B_1})$. By Allard's regularity theorem and by $V_j\weak V$ as $j\to\infty$ we deduce the existence of a sequence $\{u_j\}_j$, with $u_j\in\X_{\s_j}(\Sigma_K,\eta_1,\eta_2)$ for some $\s_j\to 0$ as $j\to\infty$, such that $V_j$ corresponds to $\Sigma_K(u_j,\eta_1,\eta_2)$ in $A_{\eta_1}^{\eta_2}$ for $j$ large enough. As soon as $j$ is large enough to give $\s_j<\s$, we have reached a contradiction.\par
%so that $\|V\|(B_1)=\om_n$ and thus $B_1\cap K=B_1\cap(\spt V)$. By Allard's regularity theorem and $V_j\weak V$, there are $\s_j\to 0$ and $u_j\in\X_{\s_j}(\Sigma_K,1/32,1/2)$ such that $V_j$ corresponds to $\Sigma_K(u_j,1/32,1/2)$ in $A_{1/32}^{1/2}$ for $j$ large, which is a contradiction as soon as $\s_j<\s$.
\noindent\textit{Step three}: For item (ii), we again argue by contradiction. Should the lemma be false, then we could find $\s>0$, $\Gamma\ge0$, $\l_1\in (0,1)$, $(\eta_1,\eta_2)\subset (0,1)$, %with $(\l_3,\l_4)\cc(\l_1,\l_2)\cc(0,1)$ with $\l_1\ge 1/32$, $K_j\in H$,
positive numbers $R_j$, $\Lambda_j<1/R_j$, $r_j$, and, by the same rescaling as in step two, $V_j\in\V_n(r_j\,\Lambda_j,\rho_j,1/(r_j\,\Lambda_j))$ with
\begin{align}\label{gamma bounds}
&\frac{\|{\rm bd}_{V_j}\|(\pa B_{\rho_j})}{\rho_j^{n-1}}\le\Gamma\,,\qquad
\sup_{\rho\in(\rho_j,1/(\Lambda_j\,r_j))}\frac{\|V_j\|(B_\rho\setminus B_{\rho_j})}{\rho^n}\le\Gamma\,,
\\  \label{lambdaext contra 1}
&\lim_{j\to\infty}\max\Big\{\rho_j=\frac{R_j}{r_j}\,,\,\, r_j\,\Lambda_j\,,\,\,
|\delta_{V_j,\rho_j,r_j\,\Lambda_j}(1)|\,,\,\, |\delta_{V_j,\rho_j,r_j\,\Lambda_j}(\lambda_1)|\Big\}=0\,,
\end{align}
such that there exists no $u\in\X_{\s}(\Sigma_{K_j},\eta_1 ,\eta_2)$ with the property that $V_j$ corresponds to $\Sigma_{K_j}(u,\eta_1,\eta_2 )$ on $A_{\eta_1}^{\eta_2 }$.
%Clearly we can find $K\in\H$ such that, up to extracting subsequences and as $j\to\infty$, $K_j\cap B_1\to K\cap B_1$ in $L^1(\R^{n+1})$. Similarly,
As in step two, we can find an $n$-dimensional integer rectifiable varifold $V{\color{black}=\mathbf{var}\,(M,\theta)}$ such that $V_j\weak V$ as varifolds in $B_1\setminus\{0\}$ and $V$ is stationary on $B_1$. If for some $K\in \mathcal{H}$, $V  =\mathbf{var}\,(K\cap B_1,\left.1\right|_{K\cap B_1})$, then using Allard's theorem as in the proof of (i), we have a contradiction. So we prove $V  =\mathbf{var}\,(K\cap B_1,\left.1\right|_{K\cap B_1})$.
\par
{\color{black}For every $r\in [\lambda_1,1]$, using $\rho_j\to 0^+$ and $r_j \Lambda_j \to 0^+$ in conjunction with \eqref{gamma bounds}, and then the monotonicity of $\delta_{V_j, \rho_j, r_j\,\Lambda_j}$ and \eqref{lambdaext contra 1}, we have
\begin{eqnarray}\notag
\varlimsup_{j\to \infty}\bigg|\omega_n - \frac{\|V_j\|(B_{r}\setminus B_{\rho_j})}{r^n} \bigg| &=& \varlimsup_{j\to \infty}\big|\delta_{V_j, \rho_j, r_j\,\Lambda_j}(r)\big|\\ \notag
&\leq& \lim_{j\to\infty}\max_{r\in\{\lambda_1,1\}}\Big\{|\delta_{V_j,\rho_j,r_j\,\Lambda_j}(r)|\Big\}=0\,.
\end{eqnarray}
Thus the convergence $V_j \rightharpoonup V$ and the monotonicity of $\|V \|(B_r)/r^n$ yield
\begin{equation}\label{density for V}
\|V\|(B_{r})=\omega_n r^n\quad\forall r\in(\lambda_1,1)
\quad
\textup{and}\quad
%\begin{equation}\label{plane mass step three}
\| V \|(B_1) %= \lim_{r\to 1^-}\|V\|(B_r)
= \omega_n\,.
\end{equation}
%Reasoning from the monotonicity formula,
%In turn, by the monotonicity formula,
%    \int_{B_1 \setminus \overline{B}_\lambda} \frac{|x^\perp|^{2}}{|x|^{n+2}}\,d\|V\|=0\,,
%from which
%it can be shown that
By \eqref{density for V}, $V\mres( B_1\setminus \overline{B}_{\lambda_1})=\var(C,\theta_C) \mres ( B_1\setminus \overline{B}_{\lambda_1})$ for some locally $\mathcal{H}^n$-rectifiable cone $C\subset\mathbb{R}^{n+1}$ and zero homogeneous $\theta_C:C\to \mathbb{N}$. %; see e.g. the proof in \cite[Theorem 19.3]{SimonLN}.
%By the conical approximation result \cite[Theorem 5.6.1]{SimonLN2} and the fact that $V$ is integer rectifiable, for each $\xi \in \mathbb{S}^n$, the density $\theta_{V}( \tau \xi)$ is a constant, say $\alpha(\xi)\in \mathbb{N}\cup\{ 0\}$, when $\tau\in (2\lambda, 1)$. Letting $C$ be the varifold cone corresponding to the integer rectifiable varifold $\mathbf{var}\,(\{x:\alpha(x/|x|)>0\},\alpha(x/|x|) )$, we know that since $V$ is stationary on $B_1\setminus B_{2\lambda}$, $C$ is stationary on $\mathbb{R}^{n+1}\setminus \{0\}$ and thus on $\mathbb{R}^{n+1}$. % To see this, let $\{\varphi_m \}$ be an partition of unity on $(0,\infty)$ subordinate to the open cover $\{(\lambda \,2^{m+1},\,2^m) \}_{m\in \mathbb{Z}}$. For any $X\in C_c^1(\mathbb{R}^{n+1}\setminus \{0\})$, decomposing $X=\sum_m \varphi_m X$, rescaling, and using the fact that $V$ is stationary on $B_1\setminus B_{2\lambda}$, we find
%    \delta C(X) = \sum_{m=1}^\infty \delta C(\varphi_m X) = 0\,.
%As a consequence, $C$ is stationary in $\mathbb{R}^{n+1}\setminus \{0\}$ and, since $n\geq 2$, in $\mathbb{R}^{n+1}$.
Now since the integer rectifiable varifold cone $\var(C,\theta_C)$ is stationary in $B_1\setminus \overline{B}_{\lambda_1}$, it is stationary in $\mathbb{R}^{n+1}$ by $n\geq 2$, and due to \eqref{density for V}, satisfies $\int_{C\cap B_1} \theta_C\, d\mathcal{H}^n = \omega_n$. Therefore $C =K$ for some $K\in  \mathcal{H}$, and $\theta_C\equiv 1$. From the definition of $C$, it follows that
\begin{equation}\label{annuluar containment}
\mathrm{spt}\,V \cap (B_1\setminus \overline{B}_{\lambda_1}) \subset K\,.
\end{equation}
Finally, \eqref{density for V} and \eqref{annuluar containment} give \eqref{plane mass} with $\beta_1=\lambda_1$, $\beta_2=1$. The result of step one then completes the proof that $V  =\mathbf{var}\,(K\cap B_1,\left.1\right|_{K\cap B_1})$.
\end{proof}

\begin{proof}[Proof of Theorem \ref{theorem mesoscale criterion}]
{\color{black}The proof proceeds in four steps, which we outline here. Precise statements can be found at the beginning of each step. First, we assume that $\delta_{V,R,\Lambda}(s/8)\geq 0$, and prove that $C^1$-graphicality can be propagated from $s/32$ to an upper radius $S_+/16\leq S_*/16$ as long as $\delta_{V,R,\Lambda}(S_+)$ remains non-negative and $S_+\leq \varepsilon_0/\Lambda$. This is then enough to prove the exterior blow-down result in part (ii) of Theorem \ref{theorem mesoscale criterion} in step two. In the third step, we argue that if $\delta_{V,R,\Lambda}(s/8)\leq 0$, then $C^1$-graphicality can be propagated inwards from $S_*/2$ down to $s/32$. The details in this step are quite similar to the first, so we summarize them. Finally, the first and third steps are combined in step four to conclude the proof Theorem \ref{theorem mesoscale criterion}-(i), in which there are no sign restrictions on the deficit.}\par
\noindent\textit{Step one}: In this step, given $n\geq 2$, $\Gamma\geq 0$, and $\s>0$, we prove the existence of $\e_0$ and $M_0$ (specified below in \eqref{choice of e0 2} and \eqref{choice of M0 2}, and depending on $n$, $\Gamma$, and $\s$) such that if \eqref{Gamma bounds}, \eqref{mesoscale bounds}, \eqref{mesoscale delta small s8} and \eqref{mesoscale Rstar larger s4} hold with $\e_0$ and $M_0$, and in addition,
\begin{equation}\label{step one positive deficit assumption}
0\leq \delta_{V,R,\Lambda}(s/8)\leq \varepsilon_0\,,
\end{equation}
then there exists $K_+\in \mathcal{H}$ and $u_+\in \X_\sigma(\Sigma_{K_+},s/32,S_{+}/16
)$ such that
\begin{equation}\label{step one conclusion}
\textup{$V$ corresponds to $\Sigma_{K_+}(u_+,s/32,S_{+}/16)$ on $A_{s/32}^{S_{+}/16}$,}
\end{equation}
\begin{equation}\label{S+ definition}
R_+= \max \Big\{\sup\Big\{\rho\ge\frac{s}8: \de_{V,R,\Lambda}(\rho)\ge 0\Big\}, 4\,s\Big\} \,,\quad
S_+= \min \Big\{R_+,\frac{\varepsilon_0}{\Lambda} \Big\}\geq 4\,s \,.
\end{equation}\par
We start by imposing some constraints on the constants $\e_0$ and $M_0$. For the finite set
\begin{equation}
\label{def of J lambda}
\end{equation}
we let $\s_0=\s_0(n)$ be such that Lemma \ref{lemma step one}-(ii), Theorem \ref{theorem 7.15AA main estimate lambda}, and Theorem \ref{theorem 7.17AA exterior lambda}-(ii,iii) hold for every $(\eta_0,\eta)\in J$, Lemma \ref{lemma step one}-(i) holds for $\s<\s_0$, and
\begin{equation}
\label{choice of sigma0}
\s_0\le\frac{\s_1}{C_0}\qquad\mbox{for $\s_1(n)$ as in \eqref{equuivalence between u square and omega square}, and
$C_0(n)$ as in Lemma \ref{lemma step one}-(ii)}\,;
\end{equation}
we shall henceforth assume, without loss of generality, that
\[
\s<\s_0\,.
\]
Moreover, for $\e_1$ and $M_1$ as in Lemma \ref{lemma graphicality lambda}-(i) and $C_0$ as in Lemma \ref{lemma step one}, we let
\begin{eqnarray}
\notag
M_0'&\ge&\max\Big\{M_1\Big(n,\frac{\s}{2\,C_0},\Gamma,\Big(\frac1{8},\frac12\Big),\Big(\frac16,\frac14\Big),\Big(\frac{1}{32},\frac{1}{2}\Big)\Big),\\  \label{choice of M0}
&&\qquad M_1\Big(n,\frac{\s}{2\,C_0},\Gamma,\Big(\frac1{16},\frac18\Big),\Big(\frac{3}{32},\frac{7}{64}\Big),\Big(\frac{1}{32},\frac{1}{2}\Big)\Big)\Big\}\,,
\end{eqnarray}
\begin{eqnarray}
\notag
\e_0'&\le&
\min\Big\{\e_1\Big(n,\frac{\s}{2\,C_0},\Gamma,\Big(\frac1{8},\frac12\Big),\Big(\frac16,\frac14\Big),\Big(\frac{1}{32},\frac{1}{2}\Big)\Big),\\ \label{choice of eps0}
&& \qquad \e_1\Big(n,\frac{\s}{2\,C_0},\Gamma,\Big(\frac1{16},\frac18\Big),\Big(\frac{3}{32},\frac{7}{64}\Big),\Big(\frac{1}{32},\frac{1}{2}\Big)\Big)\Big\}\,.
\end{eqnarray}
We also assume that%, up a dimensional constant to be specified in the upcoming \eqref{root bound 2},
\begin{equation}\label{ability to rotate}
C(n,\Gamma)(\e_0')^{1/2} \leq \min \Big\{ \e_0 , \frac{\s}{2\,C_0}\Big\}\,,
\end{equation}
%where $\tau\in(0,1)$ is a dimensional constant specified in \eqref{tau definition}, $C(n,\Gamma)$ is larger than the $C(n,\Gamma)$'s in \eqref{sum bound} and \eqref{root bound 2},
where $C(n,\Gamma)$ is specified in \eqref{root bound 2}, $C_0$ is as in Lemma \ref{lemma step one}, and $\e_0$ is smaller than both of the $n$-dependent $\e_0$'s appearing in Lemma \ref{lemma step one sigma} and Lemma \ref{lemma step one}. Lastly, we choose $\overline{\s}>0$ such that
\begin{equation}\label{sprime le s and e0}
\overline{\s}\leq \min \Big\{ \frac{\s}{2\,C_0}, \sqrt{\e_0'/\omega_n} \Big\}\,,
\end{equation}
and then, for $\e_2$, $M_2$ as in Lemma \ref{lemma graphicality lambda}-(ii), we choose $\e_0$ and $M_0$ so that
\begin{equation}\label{choice of e0 2}
\e_0 \leq \min \Big\{%\frac{1-\tau^{1/2}}{C(n,\Gamma)}\,
\e_0'\,, \e_2\Big(n,\overline{\s},\Gamma,\frac{1}{8} ,\Big(\frac{1}{32},\frac{1}{2}\Big)\Big) \Big\}
\end{equation}
%where $\tau\in(0,1)$ is a dimensional constant specified in \eqref{tau definition}, $C(n,\Gamma)$ is larger than the $C(n,\Gamma)$'s in \eqref{sum bound} and \eqref{root bound 2}, and
\begin{equation}\label{choice of M0 2}
M_0 \geq \max \Big\{M_0'\,, M_2\Big(n,\overline{\s},\Gamma,\frac{1}{8},\Big(\frac{1}{32},\frac{1}{2}\Big) \Big) \Big\}\,.
\end{equation}
Let us now recall that, by assumption, $V\in\V_n(\Lambda,R,1/\Lambda)$ is such that
\begin{equation}
\label{bounds needed for proof meso}
\|{\rm bd}_V\|(\pa B_{R})\le\Gamma\,R^{n-1}\,,\qquad \sup_{\rho\in(R,1/\Lambda)}\frac{\|V\|(B_\rho\setminus B_R)}{\rho^n}\le\Gamma\,;
\end{equation}
in particular, by Theorem \ref{theorem 7.17AA exterior lambda}-(i),
\begin{equation}
\label{is decreasing}
\mbox{$\de_{V,R,\Lambda}$ is decreasing on $(R,1/\Lambda)$}\,.
\end{equation}
Moreover, we are assuming the existence of $s$ with $\max\{64,M_0\}\,R<s<\e_0/4\,\Lambda$ such that
\begin{eqnarray}\label{ext bd easy 1 proof lambda}
&&|\de_{V,R,\Lambda}(s/8)|\le \e_0\,,
\\\label{ext bd easy 3 proof lambda}
&&R_*=\sup\Big\{\rho\ge\frac{s}8: \de_{V,R,\Lambda}(\rho)\ge -\e_0\Big\}\ge 4\,s\,,
%\\\label{lambdaext bd easy 2 proof}
%\label{lambdaext bd not so easy proof}
\end{eqnarray}
so that the latter inequality, together with \eqref{S+ definition}, implies
\begin{equation}\label{R+ inequality}
%\frac{\varepsilon_0}{4\,\Lambda} >
R_* \geq R_+\,.
\end{equation}
By \eqref{is decreasing}, \eqref{ext bd easy 1 proof lambda} and \eqref{R+ inequality} we have
\begin{equation}
\label{delta below eps0 in absolute value}
|\de_{V,R,\Lambda}(r)|\le\e_0\,,\qquad\forall r\in[s/8,R_+]\,.
\end{equation}
\par
By \eqref{bounds needed for proof meso}, the specification of $s$ satisfying \eqref{mesoscale bounds}, and \eqref{delta below eps0 in absolute value}, the assumptions \eqref{gamma bound lemma statement}, \eqref{alternative conditions on r lambda}, and \eqref{alternative hypo to angular flatness}, respectively, of Lemma \ref{lemma graphicality lambda}-(ii) with $r=s$, $\lambda_1=1/8$, and $(\eta_1,\eta_2)=(1/32,1/2)$ are satisfied due to our choices \eqref{choice of e0 2} and \eqref{choice of M0 2}. Setting $H_0=H$, {\color{black}where $H\in\mathcal{H}$ is from the application of Lemma \ref{lemma graphicality lambda}-(ii),} we thus find $u_0\in\X_{\overline{\s}}(\S_{H_0},s/32,s/2)$ such that %$u_0\in\X_{\overline{\s}}(\S_{H},s/32,s/2)$ such that
\begin{equation}\label{ext bd easy corresponds 0 lambda}
\mbox{\textup{$V$ corresponds to $\Sigma_{H_0}(u_0,s/32, s/2)$ on $A_{s/32}^{s/2}$.}}
\end{equation}
If it is the case that $S_+=4\,s$, we are in fact done with the proof of \eqref{step one conclusion}, since then $s/2\geq S_+ / 16$. We may for the rest of this step assume then that $S_+ > 4\,s$, so that
\begin{equation}\label{remaining definition of R+}
R_+=\sup\Big\{\rho\ge\frac{s}8: \de_{V,R,\Lambda}(\rho)\ge 0\Big\}\geq S_+ > 4\,s\,.
\end{equation}

\par

First, we observe that
%we have,
thanks to \eqref{ext bd easy corresponds 0 lambda} and then \eqref{sprime le s and e0},%, and \eqref{lambdaext bd easy 2 proof},
\begin{eqnarray}\label{ext bd decay Tj 0 lambda}
\le \omega_n \,\overline{\s}^2 \le  \e_0'\,.
%\le\frac{C(n)}{s^n}\,\int_{A_{s/8}^{s/4}}\,\om_H^2\,d\|V\|\le C(n)\,\e_0\,.
\end{eqnarray}
We let $s_j=2^{j-3}\,s$ for  $j\in\mathbb{Z}_{\geq -1}$. By \eqref{remaining definition of R+} and by $s<\e_0/4\,\Lambda\leq \e_0'/4\,\Lambda$ there exists $N\in\{j\in\N:j\ge 2\}\cup\{+\infty\}$ such that
\begin{equation}
\label{def of N}
\{0,1,..,N\}=\Big\{j\in\N: 8\,s_j\le S_+=\min\big\{R_+,\frac{\e_0'}\Lambda\big\}\Big\}\,.
\end{equation}
Notice that if $\Lambda>0$ then it must be $N<\infty$. We are now in the position to make the following:

\noindent {\bf Claim:} There exist $\tau=\tau(n)\in(0,1)$ and $\{(H_j,u_j)\}_{j=0}^{N-2}$ with $H_j\in\H$ such that, setting
\[
\]
for every $j=0,...,N-2$,
\begin{eqnarray}
\label{claim is spherical graph 1}
&&\mbox{$u_j\in\X_\s(\Sigma_{H_j},s/32,4\,s_{j-1})\cap \X_{\s/2\,C_0}(\Sigma_{H_j},s_j/4,4\,s_j)$}\,,
\\ \label{claim is spherical graph 2}
&&\mbox{$V$ corresponds to $\Sigma_{H_j}(u_j,s/32,4\,s_j)$ on $A_{s/32}^{4\,s_j}$}\,,
\end{eqnarray}
where $C_0$ is from Lemma \ref{lemma step one}, and
\begin{eqnarray}
\label{claim delta smaller eps0}
\\
\label{claim Tj smaller eps0}
T_j&\le& C(n)\,\e_0'\,;
\end{eqnarray}
additionally, for every $j=1,...,N-2$,
\begin{eqnarray}
\label{claim decay normal vectors}
\\
\label{claim deltaj smaller deltaj-1}
\de_{V,R,\Lambda}(s_j)&\le& \tau\,\big\{\de_{V,R,\Lambda}(s_{j-1})+(1+\Gamma)\,\Lambda\,s_{j-1}\big\}\,,
\\
\label{claim Tj smaller deltaj-1}
T_j&\le& C(n)\,\Big\{\de_{V,R,\Lambda}(s_{j-1})-\de_{V,R,\Lambda}(s_{j+2})+\Lambda\,s_{j-1}\Big\}\,.
\end{eqnarray}
\noindent {\bf Proof of the claim:} We argue by induction. Clearly \eqref{claim is spherical graph 1}$_{j=0}$, \eqref{claim is spherical graph 2}$_{j=0}$, \eqref{claim delta smaller eps0}$_{j=0}$ and \eqref{claim Tj smaller eps0}$_{j=0}$ are, respectively, \eqref{ext bd easy corresponds 0 lambda}, \eqref{ext bd easy 1 proof lambda} and \eqref{ext bd decay Tj 0 lambda}. This concludes the proof of the claim if $N=2$, therefore we shall assume $N\ge 3$ for the rest of the argument. To set up the inductive argument, we consider $\ell\in\N$ such that: either $\ell=0$; or $1\le\ell\le N-3$ and \eqref{claim is spherical graph 1}, \eqref{claim is spherical graph 2}, \eqref{claim delta smaller eps0}, and
\eqref{claim Tj smaller eps0} hold for $j=0,...,\ell$, and \eqref{claim decay normal vectors}, \eqref{claim deltaj smaller deltaj-1} and
\eqref{claim Tj smaller deltaj-1} hold for $j=1,...,\ell$; and prove that all the conclusions of the claim hold with $j=\ell+1$.

The validity of \eqref{claim delta smaller eps0}$_{j=\ell+1}$ is of course immediate from \eqref{delta below eps0 in absolute value} and \eqref{def of N}. Also, after proving \eqref{claim Tj smaller deltaj-1}$_{j=\ell+1}$, we will be able to combine with  \eqref{claim delta smaller eps0}$_{j=\ell+1}$ and \eqref{def of N} to deduce \eqref{claim Tj smaller eps0}$_{j=\ell+1}$. We now prove, in order,  \eqref{claim decay normal vectors}, \eqref{claim is spherical graph 1}, \eqref{claim is spherical graph 2}, \eqref{claim deltaj smaller deltaj-1}, and \eqref{claim Tj smaller deltaj-1} with $j=\ell+1$.

\noindent{\it To prove \eqref{claim decay normal vectors}$_{j=\ell+1}$}: Let $[a,b]\cc (s_\ell,s_{\ell+1})$ with $(b-a)=(s_{\ell+1}-s_\ell)/2$, so that
\begin{equation}
\label{ext sell star lambda}
\frac1{C(n)}\,\min_{r\in[a,b]}\int_{\Sigma_{H_\ell}}[u_\ell]_r^2\le
\frac1{s_{\ell+1}^n}\,\int_{s_{\ell}}^{s_{\ell+1}}\,r^{n-1}\,dr\,\int_{\Sigma_{H_\ell}}[u_\ell]_r^2= T_\ell\,.
\end{equation}
Keeping in mind \eqref{claim is spherical graph 1}$_{j=\ell}$, \eqref{claim is spherical graph 2}$_{j=\ell}$, we can apply Lemma \ref{lemma step one}-(ii) with $(r_1,r_2)=(s/32,4\,s_\ell)$ and $[a,b]\subset(s_\ell,s_{\ell+1})$ to find $H_{\ell+1}\in\H$,
\begin{equation}\label{first u ell+1 bound}
\end{equation}
(with $C_0$ as in Lemma \ref{lemma step one}-(ii)) and
\begin{equation}
\label{where is sell star}
\end{equation}
such that, thanks also to \eqref{ext sell star lambda},
\begin{eqnarray}\label{ext u ell plus one 1 lambda}
\\\label{ext u ell plus one 2 lambda}
\\\label{ext u ell plus one 3 lambda}
&&|\nu_{H_{\ell}}-\nu_{H_{\ell+1}}|^2\le C(n)\,T_\ell\,,
\\\label{ext u ell plus one 4 lambda}
&&\int_{\Sigma_{H_{\ell+1}}}[u_{\ell+1}]_r^2\le C(n)\,\Big(T_\ell+\int_{\Sigma_{H_\ell}}[u_\ell]_r^2\Big)\,\,,\qquad\forall r\in(s/32,4\,s_{\ell})\,.\hspace{1cm}
\end{eqnarray}
In particular, \eqref{ext u ell plus one 3 lambda} is \eqref{claim decay normal vectors}$_{j=\ell+1}$.

\noindent {\it To prove \eqref{claim is spherical graph 1}$_{j=\ell+1}$ and \eqref{claim is spherical graph 2}$_{j=\ell+1}$}: Notice that \eqref{first u ell+1 bound}, \eqref{ext u ell plus one 1 lambda} do not imply \eqref{claim is spherical graph 1}$_{j=\ell+1}$ and \eqref{claim is spherical graph 2}$_{j=\ell+1}$, since, in \eqref{claim is spherical graph 2}$_{j=\ell+1}$, 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 in \eqref{claim is spherical graph 1}$_{j=\ell+1}$ we are claiming that $u_{\ell+1}$ has $C^1$-norm bounded by $\s$ or $\s/2\,C_0$ (depending on the radius), 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}-(i) with $K=H_{\ell+1}$ and
\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)\,,\,\,  (\eta_1,\eta_2)=\Big(\frac{1}{32},\frac{1}{2}\Big)\,.
\end{equation}
We check the validity of \eqref{conditions on r lambda}, \eqref{ext hp step two 1 lambda}, \eqref{ext hp step two 3 lambda}, and \eqref{ext hp step two 2 lambda} with $\e_1=\e_0'$ and $M_1=M_0'$ for these choices of $r$, $\l_1$, $\l_2$, $\l_3$, $\l_4$, $\eta_1$, $\eta_2$, and $K$. Since $r=8\,s_{\ell+1}\ge s\ge\max\{M_0,64\,R\}\geq \max\{M_0',64\,R\}$, and since \eqref{def of N} and $\ell+1\le N$ give $r=8\,s_{\ell+1}\le \e_0/\Lambda\leq \e_0'/\Lambda$, 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 2}$_{j=\ell}$, 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 can be applied to $u_{\ell+1}$ thanks to \eqref{choice of sigma0}), \eqref{ext u ell plus one 1 lambda} and \eqref{claim is spherical graph 1}$_{j=\ell}$, and, then by \eqref{ext u ell plus one 4 lambda}, we have
\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}-(i), and find $v\in\X_{\s/2\,C_0}(\Sigma_{H_{\ell+1}},s_{\ell+1}/4,4\,s_{\ell+1})$ 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 2}$_{j=\ell}$, 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 2}$_{j=\ell}$ and \eqref{ext jelly lambda}, the resulting extension is such that
\begin{eqnarray}
\label{initial ul+1 bound}
&&\mbox{$u_{\ell+1}\in\X_{\s/2\,C_0}(\Sigma_{H_{\ell+1}},s_{\ell+1}/4,4\,s_{\ell+1})$}\quad\textup{and}\\ \label{V correspondence}
&&\textup{$V$ corresponds to $\Sigma_{H_{\ell+1}}(u_{\ell+1},s/32,4\,s_{\ell+1})$ on $A_{s/32}^{4\,s_{\ell+1}}$}\,.
\end{eqnarray}
The bound \eqref{initial ul+1 bound} is part of \eqref{claim is spherical graph 1}$_{j=\ell+1}$, and \eqref{V correspondence} is \eqref{claim is spherical graph 2}$_{j=\ell+1}$, so in order to complete the proof of \eqref{claim is spherical graph 1}$_{j=\ell+1}$ and \eqref{claim is spherical graph 2}$_{j=\ell+1}$, it remains to show that the $C^1$-norm of $u$ is bounded by $\sigma$ in between $s/32$ and $4\,s_{\ell}$.

\par

Towards this end, we record the following consequence of taking square roots in \eqref{claim deltaj smaller deltaj-1}$_{j=m}$ (using $\delta_{V,R,\Lambda}\geq 0$ from \eqref{def of N}) and summing over $m=1,...,i$ for any $1\leq i \leq\ell$: for $\alpha=\sum_{k=0}^\infty 2^{-k/2}$ and $\tilde{C}(n,\Gamma)=\tau^{1/2}(1+\Gamma)$,
\begin{eqnarray}\notag
S_{i}:=\sum_{m=0}^{i}\delta_{V,R,\Lambda}(s_m)^{1/2} &\leq& \tau^{1/2} \sum_{m=0}^{i-1}\delta_{V,R,\Lambda}(s_{m})^{1/2} + (1+\Gamma)(\Lambda \, s_m)^{1/2} \\ \notag &&\qquad+\delta_{V,R,\Lambda}(s_0)^{1/2}\\ \notag
&\leq& \tau^{1/2} S_{i-1} +\alpha\,\tilde{C}(n,\Gamma)(\Lambda \, s_{i-1})^{1/2} +\delta_{V,R,\Lambda}(s_0)^{1/2}\\ \label{iterated delta bound}
&\leq&\tau^{1/2} S_{i-1}+ (1+\alpha\,\tilde{C}(n,\Gamma))(\e_0')^{1/2}\,,
\end{eqnarray}
where in the last line we have used \eqref{def of N} and \eqref{delta below eps0 in absolute value}. By induction, utilizing \eqref{step one positive deficit assumption}, \eqref{choice of e0 2} for the base case and \eqref{iterated delta bound} for the induction step we have
\begin{equation}\label{sum bound}
S_{i}\leq  \frac{(1+\alpha\,\tilde{C}(n,\Gamma))(\e_0')^{1/2}}{1-\tau^{1/2}}\quad\forall \,0\leq i \leq \ell\,.
\end{equation}
%Inserting \eqref{iterated delta bound}$_{j=i}$ into \eqref{claim Tj smaller deltaj-1} and
Now by the positivity of $\delta_{V,R,\Lambda}$ and \eqref{claim Tj smaller deltaj-1}$_{j=\ell}$, for all $m=1,...,\ell$,
\begin{align}\label{iterated T bound}
T_m^{1/2} \leq C(n)\delta_{V,R,\Lambda}(s_{m-1})^{1/2}  + C(n)(\Lambda\, s_{m-1})^{1/2}\,,%\quad \forall m=1,...,\ell\,.
\end{align}
In turn, by \eqref{claim decay normal vectors}$_{j=\ell+1}$, \eqref{ext bd decay Tj 0 lambda} and \eqref{iterated T bound}, then \eqref{def of N} and \eqref{sum bound}$_{i=\ell-1}$,
\begin{eqnarray}\notag
\frac{1}{C(n)} \sum_{m=1}^{\ell+1} |\nu_{H_m}-\nu_{H_{m-1}}|
&\leq& \sum_{m=0}^{\ell} T_{m}^{1/2} \\ \notag
&\leq&   (\e_0')^{1/2}+C(n)S_{\ell-1} + \alpha\,C(n)(\Lambda\,{s_{\ell-1}})^{1/2}  \\ \label{root epsilon bound}
&\leq& C(n,\Gamma)(\e_0')^{1/2}
\end{eqnarray}
for suitable $C(n,\Gamma)$. Combining constants, we use \eqref{root epsilon bound} to see
\begin{eqnarray}\label{root bound 2}
|\nu_{H_i}-\nu_{H_{\ell+1}}| \leq C(n,\Gamma)(\e_0')^{1/2}\quad\forall i=0,...,\ell \,.
\end{eqnarray}
Now $u_i \in \X_{\s/2\,C_0}(\Sigma_{H_j},s_i/4, 4\,s_i)$ by \eqref{claim is spherical graph 1}$_{j=i}$, and $\s/2\,C_0$ and $|\nu_{H_i}-\nu_{H_{\ell+1}}|$ are small enough to apply Lemma \ref{lemma step one}-(i) by our choice of $\s$ above \eqref{choice of sigma0} and \eqref{root bound 2} with \eqref{ability to rotate}, respectively. Then we obtain $w_i$ % \in \X_{\s/2 + C_0|\nu_{H_i}-\nu_{H_{\ell+1}}|}(\Sigma_{H_{\ell+1}}, s_i/4,4\,s_i)$
corresponding to $V$ on $A_{s_i/4}^{4\,s_i}$ and in $ \X_{\s/2 + C_0|\nu_{H_i}-\nu_{H_{\ell+1}}|}(\Sigma_{H_{\ell+1}}, s_i/4,4\,s_i)$, and by \eqref{root bound 2}, \eqref{ability to rotate},
\frac{\s}{2}+C_0\,|\nu_{H_i}-\nu_{H_{\ell+1}}| \leq \frac{\s}{2}+C_0\,\frac{\s}{2\,C_0}=\s\,,
so $w_i\in \mathcal{X}_\sigma(\Sigma_{H_{\ell+1}},s_i/4,4\,s_i)$. Finally, since they represent the same surface over $\S_{H_{\ell+1}}$, $w_i=u_{\ell+1}$ on $A^{4\,s_i}_{s_i/4}$. Gathering these estimates for $i=0,...,\ell$, we have $u_{\ell+1}\in \X_\s(\S_{H_{\ell+1}},s/32, 4\,s_\ell)$, which finishes the proof of \eqref{claim is spherical graph 1}$_{j=\ell+1}$.

\par

\noindent {\it 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 2}$_{j=\ell+1}$ 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
\begin{eqnarray*}
\\
&&=\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}
\end{eqnarray*}
we have
\begin{eqnarray*}
\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\,.
\end{eqnarray*}
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
\begin{eqnarray*}
\Big|\frac{(s_{\ell}^-)^n}{s_\ell^n}\,\de(s_{\ell}^-)
\le\! C_*(n)\,\big\{\de(s_{\ell})-\de(s_{\ell+1})\big\}+C_*(n)\,(\Gamma+1)\,\Lambda\,s_\ell\,,
\end{eqnarray*}
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
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}
\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}$_{j=\ell+1}$.

\noindent {\it To prove  \eqref{claim Tj smaller deltaj-1}$_{j=\ell+1}$}: 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
\end{equation}
By \eqref{claim is spherical graph 2}$_{j=\ell+1}$ 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
\[
\int_{s_{\ell+1}}^{2\,s_{\ell+1}}\!\!\!r^{n-1}\!\int_{\Sigma_{H_{\ell+1}}}\,[u_{\ell+1}]_r^2\le C(n)\,\int_{s_\ell}^{5\,s_\ell}\!\!\!\!r^{n+1}\!\int_{\Sigma_{H_{\ell+1}}}\!\!\!\!\!\!(\pa_ru_{\ell+1})_r^2
\]
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*}
\le
\\
&&\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}, which finishes the {\bf claim}.

\noindent {\it Proof of \eqref{step one conclusion}}: We assume $S_+<\infty$ (that is either $\Lambda>0$ or $R_+<\infty$), and recall that we have already proved \eqref{step one conclusion} if $S_+=4\,s$. Otherwise, $N$ (as defined in \eqref{def of N}) is finite, with $2^N\le \frac{S_+}{s} <2^{N+1}$. By \eqref{claim is spherical graph 1}$_{j=N-2}$ and \eqref{claim is spherical graph 2}$_{j=N-2}$, we have that $u_{N-2}\in \mathcal{X}_{\s}(\S_{H_{N-2}},s/32,4\,s_{N-2})$ and $V$ corresponds to $\Sigma_{H_{N-2}}(u_{N-2},s/32,4\,s_{N-2})$ on $A_{s/32}^{4\,s_{N-2}}$. Since $4\,s_{N-2}%=4\times 2^{N-2-3}\,s
=2^{N+1}\,s/16> S_+/16$, we deduce \eqref{step one conclusion} with $K_+=H_{N-2}$ and $u_+=u_{N-2}$.

%   \noindent {\it Step two - 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 $K\in\H$ such that, along a subsequence $\{H_{j(k)}\}_k$, we have $\e_k=|\nu_K-\nu_{H_{j(k)}}|\to 0$ 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 thus 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).

%   \medskip

\noindent \noindent {\it Step two}: In this step we prove statement (ii) in Theorem \ref{theorem mesoscale criterion}. We assume that $\Lambda=0$ and that
\begin{equation}
\label{positive}
\de(r)\ge - \e_0\qquad\forall r\ge \frac{s}8\,,
\end{equation}
where we have set for brevity $\de=\de_{V,R,0}$. We must first show that \begin{equation}\label{actually positive}
\delta(r) \geq 0 \quad \forall r\ge \frac{s}{8}\,.
\end{equation}
Since $\delta$ is decreasing in $r$, it has a limit $\lim_{r\to \infty}\delta(r)=:\delta_\infty\geq -\e_0$. Next, we know that for any sequence $R_i\to \infty$, $V/R_i$ converges locally in the varifold sense to a limiting integer rectifiable varifold cone $W$. By the local varifold convergence and $n\geq 2$, $W$ is stationary in $\mathbb{R}^{n+1}$, and it is the case that $$
\delta_{W,0,0}(r)=\delta_\infty\geq -\e_0 \quad \forall r>0\,.
Up to decreasing $\e_0$ if necessary (and recalling that $\delta_{W,0,0}$ is the usual area excess multiplied by $-1$), Allard's theorem and the fact that $W/r = W$ imply that $W$ corresponds to a multiplicity one plane. In particular, it must be that $\delta_\infty=0$, which together with the monotonicity of $\delta$ yields \eqref{actually positive}.
\par
%As in the case of statement (ii)
By \eqref{actually positive}, $S_+=S_*=\infty$, and so by \eqref{claim is spherical graph 1} and \eqref{claim is spherical graph 2}, there is a sequence  $\{(H_j,u_j)\}_{j=0}^N$ but with $N=\infty$ now, satisfying
\begin{eqnarray}
\label{claim is spherical graph fine}
&&\mbox{$V$ corresponds to $\Sigma_{H_j}(u_j,s/32,4\,s_j)$ on $A_{s/32}^{4\,s_j}$}\quad\forall j\geq 0\,,
\end{eqnarray}
\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}
\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{actually 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{actually positive}. By \eqref{fine nuj}, for every $j\ge 0$, $k\ge 1$, we have $|\nu_{H_{j+k}}-\nu_{H_j}|%\le C(n)\,\sqrt{\de(s/8)}\,\sum_{h=1}^{k+1}|\nu_{H_{j-1+h}}-\nu_{H_{j-2+h}}|
\le C(n)\,\sqrt{\de(s/8)}\,\sum_{h=1}^{k+1}\big(\sqrt\tau\big)^{j-1+h}$, so that there exists $K\in\H$ such that
\begin{equation}
\label{nuK convergence}
\e_j^2:=|\nu_K-\nu_{H_j}|^2\le C(n)\,\tau^j\,\de(s/8)\,,\qquad\forall j\ge 1\,,
\end{equation}
%By compactness of $\SS^n$, we can find $K\in\H$ such that, along a subsequence $\{H_{j(k)}\}_k$, we have $\e_k=|\nu_K-\nu_{H_{j(k)}}|\to 0$ as $k\to\infty$.
In particular, for $j$ large enough, we have $\e_j<\e_0$, and thus, by Lemma \ref{lemma step one}-(i) and by \eqref{claim is spherical graph fine} we can find $v_j\in\X_{C(n)\,(\s+\e_j)}(\S_{K};s/32,4\,s_{j})$ such that
\begin{equation}
\label{claim is spherical graph jjj}
\mbox{$V$ corresponds to $\Sigma_{K}(v_{j},s/32,4\,s_{j})$ on $A_{s/32}^{4\,s_{j}}$}\,.
\end{equation}
By \eqref{claim is spherical graph jjj}, $v_{j+1}=v_{j}$ on $\Sigma_{K}\times(s/32,4\,s_{j})$. Since $s_{j}\to\infty$ we have thus found $u\in\X_{C(n)\,\s}(\S_{K};s/32,\infty)$ such that
\begin{equation}\label{lambdaext bd easy corresponds final}
\mbox{$V$ corresponds to $\Sigma_{K}(u,s/32,\infty)$ on $A_{s/32}^\infty$}\,,
\end{equation}
which is \eqref{mesoscale thesis graphicality} with $S_*=\infty$.

%   By arguing as in the proof of statement (ii) we see 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 2},
%   \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 $S_*=+\infty$.
%  \medskip
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)=
\\\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 $\om_K(y)=\arctan(|\nu_K\cdot\hat{y}|/|\p_K\,\hat{y}|)$, provided $\arctan$ 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 decreasing the value of $\e_0$, and recalling $\de(s/8)\le\e_0$, we conclude
\begin{equation}
\label{nuK uniform projections}
|\p_{H_j}\,\hat{y}|\ge \frac13\,,\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 Theorem \ref{theorem mesoscale criterion}-(ii).

\par

\noindent\textit{Step three}: In this step, given $n\geq 2$, $\Gamma\geq 0$, and $\s>0$, we claim the existence of $\e_0$ and $M_0$, depending only on $n$, $\Gamma$, and $\s$, such that if \eqref{Gamma bounds}, \eqref{mesoscale bounds}, \eqref{mesoscale delta small s8} and \eqref{mesoscale Rstar larger s4} hold with $\e_0$ and $M_0$, and in addition,
\begin{equation}\label{step three sign assumption}
-\varepsilon_0\leq \delta_{V,R,\Lambda}(s/8)\leq 0\,,
\end{equation}
then there exists $K_-\in\mathcal{H}$ and $u_-\in \X_\sigma(\Sigma_{K_-},s/32,S_{*}/2
)$ such that
\begin{equation}\label{step three conclusion}
\textup{$V$ corresponds to $\Sigma_{K_-}(u_+,s/32,S_{*}/2)$ on $A_{s/32}^{S_{*}/2}$,}
\end{equation}
where $S_*$ and $R_*$ are as in Theorem \ref{theorem mesoscale criterion}.
%\begin{equation}\label{recalled definition of Sstar Rstar}
%    S_*=\min\Big\{R_*,\frac{\e_0}{\Lambda}\Big\}<\infty\,,\quad R_*:=\sup\Big\{\rho\ge\frac{s}8: \de_{V,R,\Lambda}(\rho)\ge -\e_0\Big\}\ge 4\,s\,.
%  \end{equation}
The argument is quite similar to that of the first step, with minor differences due to the opposite sign of the deficit. The first is that the iteration instead begins at the outer radius $S_*$ and proceeds inwards via intermediate radii $s_j=2^{-j}S_*$, and the second is that, in the analogue of the graphicality propagation claims \eqref{claim is spherical graph 1}$_{j=\ell+1}$ and \eqref{claim is spherical graph 2}$_{j=\ell+1}$, the negative sign on $\delta_{V,R,\Lambda}$ is used to sum the ``tilting" between successive planes $H_j$ and $H_{j+1}$.

\par

\noindent\textit{Step four}: Finally, we combine steps one and three to prove statement (i) in Theorem \ref{theorem mesoscale criterion}. Before choosing the  parameters $\e_0$ and $M_0$, we need a preliminary result. We claim that for any $\e'>0$, there exists $\s'(\e')>0$ such that if $r_1<r_2$, $K_1$, $K_2\in \mathcal{H}$ with $\nu_{K_1}\cdot \nu_{K_2}\geq 0$ and accompanying $u_i\in \X_{\s'}(\S_{K_i}, r_1 , r_2)$, and $M$ is a smooth hypersurface such that $M\cap A_{r_1}^{r_2}$ corresponds to $\S_{K_i}(u_i,r_1,r_2)$ for $k=1,2$, then
\begin{align}\label{prove planes are close}
|\nu_{{K_1}}-\nu_{{K_2}}|< \e'\,.
\end{align}
This is immediate from $\nu_{K_1}\cdot \nu_{K_2}\geq 0$ the fact that the $L^\infty$-bounds on $u_i$ imply that $M$ is contained in the intersection of two cones containing $K_1$ and $K_2$, whose openings become arbitrarily narrow as $\s'\to 0$\,.

\par

Fix $n\geq 2$, $\Gamma\geq 0$, and $\s>0$; we assume without loss of generality that $\s<\s_0$, where $\s_0$ is the dimension-dependent constant from Lemma \ref{lemma step one}. We choose $\e'$ with corresponding $\s'$ according to \eqref{prove planes are close} such that, up to decreasing $\s'$ if necessary,
\begin{align}\label{were gonna tilt}
\e'<\e_0\,,\quad  C_0(\s' + \e') \leq \s\,,
\end{align}
where $\e_0$, $C_0$ are as in Lemma \ref{lemma step one}. Next, we choose $\e_0=\e_0(n,\Gamma,\s)$ and $M_0=M_0(n,\Gamma,\s)$ to satisfy several restrictions: first, $\e_0$ is smaller than the $\e_0$ from Lemma \ref{lemma step one} and each $\e_0(n,\Gamma,\s')$ from steps one and three, and $M_0$ is larger than $M_0(n,\Gamma,\s')$ from those steps; second, with $\e_2$ and $M_2$ as in Lemma \ref{lemma graphicality lambda}-(ii), we also assume that
\begin{equation}\label{step four eps0 M0 lemma}
\e_0 \leq \min\Big\{\e',\e_2\Big(n,\sigma',\Gamma,\frac{1}{16},\Big(\frac{1}{128},\frac{1}{2}\Big) \Big)\Big\},\, M_0 \geq M_2\Big(n,\sigma',\Gamma,\frac{1}{16},\Big(\frac{1}{128},\frac{1}{2}\Big) \Big)
\end{equation}
%\begin{equation}\label{step four M0 lemma}
%    M_0 \geq M_2\Big(n,\sigma,\Gamma,\frac{1}{16},\Big(\frac{1}{128},\frac{1}{2}\Big) \Big)
In the remainder of this step, we suppose that
\begin{equation}\label{step four V assumptions}
\mbox{$V$ satisfies \eqref{Gamma bounds}, \eqref{mesoscale bounds}, \eqref{mesoscale delta small s8} and \eqref{mesoscale Rstar larger s4} at mesoscale $s$}\,.
\end{equation}
In proving Theorem \ref{theorem mesoscale criterion}-(i), there are three cases depending on whether $\delta_{V,R,\Lambda}$ changes sign on $[s/8, S_*]$.

\par

\noindent\textit{Case one}: $\delta_{V,R,\Lambda}(r)\geq 0$ for all $ r\in[s/8, S_*]$. If the deficit is non-negative, then in particular
\begin{equation}\label{case one step four}
0\leq \delta_{V,R,\Lambda}(s/8)\leq \e_0
\end{equation}
and $S_*=S_+$, where $S_+$ was defined in \eqref{S+ definition}. By our choice of $\e_0$ and $M_0$ at the beginning of this step and the equivalence of \eqref{case one step four} and \eqref{step one positive deficit assumption}, step one applies and the conclusion \eqref{step one conclusion} is \eqref{mesoscale thesis graphicality}. Thus Theorem \ref{theorem mesoscale criterion}-(i) is proved.

\par

\noindent\textit{Case two}: $\delta_{V,R,\Lambda}(r)\leq 0$ for all $r\in [s/8,S_*]$. Should the deficit be non-positive in this interval, then in particular, \eqref{step three sign assumption} holds in addition to \eqref{Gamma bounds}, \eqref{mesoscale bounds}, \eqref{mesoscale delta small s8} and \eqref{mesoscale Rstar larger s4}. Therefore, by our choice of $\e_0$ and $M_0$, step three applies. The conclusion \eqref{step three conclusion} is \eqref{mesoscale thesis graphicality} (in fact with larger upper radii $S_*/2$), and Theorem \ref{theorem mesoscale criterion}-(i) is proved.

\par

\noindent\textit{Case three}: $\delta_{V,R,\Lambda}$ changes sign in $[s/8,S_*]$. By the monotonicity of $\delta_{V,R,\Lambda}$,
\begin{equation}\label{deficit changes sign}
\delta_{V,R,\Lambda}(s/8) > 0 > \delta_{V,R,\Lambda}(S_*)\,.
\end{equation}
First, by \eqref{deficit changes sign}, \eqref{step one positive deficit assumption} is satisfied, so \eqref{step one conclusion} gives $K_+\in \mathcal{H}$ and $u_+\in \X_{\sigma'}(\Sigma_{K_+},s/32,S_{+}/16
)$ such that
\begin{equation}\label{step one conclusion redux}
\textup{$V$ corresponds to $\Sigma_{K_+}(u_+,s/32,S_{+}/16)$ on $A_{s/32}^{S_{+}/16}$,}
\end{equation}
where%, utilizing the assumption that $\delta_{V,R,\Lambda}(S_*)<0$ and the fact that $S_*\leq \e_0/\Lambda$,
\begin{equation}\label{S+ definition redux}
R_+= \max \Big\{\sup\Big\{\rho\ge\frac{s}8: \de_{V,R,\Lambda}(\rho)\ge 0\Big\}, 4\,s\Big\}\,,\quad S_+= \min \Big\{R_+,\frac{\varepsilon_0}{\Lambda} \Big\}\,.
\end{equation}
If $S_+ = S_*$, then \eqref{step one conclusion redux} is \eqref{mesoscale thesis graphicality} and we are done. So we assume for the rest of this case that $S_+<S_*$, which implies $S_+\neq \e_0/\Lambda$ and thus \begin{equation}\label{R+ le Sstar}
4\,s \leq R_+=S_+<S_*\,.
\end{equation}
Next, we make the following

\par

\noindent\textbf{Claim:} There exists $K_-\in \mathcal{H}$ and $u_-\in \X_{\s'}(\Sigma_{K_-},R_+/2,S_{*}/2
)$ such that
\begin{equation}\label{step three conclusion redux}
\textup{$V$ corresponds to $\Sigma_{K_-}(u_-,R_+/2,S_{*}/2)$ on $A_{R_+/2}^{S_{*}/2}$}\,.
\end{equation}
\par

\noindent\textbf{Proof of the claim:} There are two subcases.

\par

\noindent\textit{Subcase one}: $16\,R_+ < \e_0/4\,\Lambda $ and $ 64\,R_+ < R_* $. We claim the conditions of step three are verified at $s'=16\,R_+$. First, \eqref{Gamma bounds} holds from \eqref{step four V assumptions}, and
\begin{equation}\notag
\max\{64,M_0 \}\,R< 16\,R_+ < \frac{\e_0}{4\,\Lambda}
\end{equation}
(which is \eqref{mesoscale bounds}) holds due to the assumption of the subcase and $16R_+\geq s > \max\{64,M_0 \}R$. Next, $2\, R_+< R_*/4$ by the assumption of the subcase, which combined with the monotonicity of $\delta_{V,R,\Lambda}$ and \eqref{S+ definition redux} gives $-\varepsilon_0\leq \delta_{V,R,\Lambda}(2\,R_+) \leq 0$. This implies \eqref{step three sign assumption} and \eqref{mesoscale delta small s8} with $s'=16\,R_+$. Lastly, \eqref{mesoscale Rstar larger s4} holds at $s'=16\,R_+$ since $64\,R_+ < R_*$. Thus we apply \eqref{step three conclusion} at $s'=16\,R_+$, finding \eqref{step three conclusion redux}.

\par

\noindent\textit{Subcase two}: One or both of $16\,R_+ \geq \e_0/4\,\Lambda$, $64\,R_+ \geq R_*$ hold. In this case,
\begin{equation}\label{inequality of Sstar and R_+}
64\,R_+ \geq \min \{\e_0 / \Lambda, R_* \}=S_*\,.
\end{equation}
We wish to apply Lemma \ref{lemma graphicality lambda}-(ii) with $r=S_*$, $\lambda_1=\frac{1}{16}$, $(\eta_1,\eta_2) = (\frac{1}{128},\frac{1}{2})$. By \eqref{step four V assumptions}, \eqref{gamma bound lemma statement} holds for $V$, and by \eqref{step four eps0 M0 lemma},
\eqref{step four V assumptions}, and $S_*\geq 4\,s$,
\begin{equation}\notag
\max \{M_2,64 \}\,R \leq s \leq \frac{S_*}{4} \leq S_*\leq  \frac{\e_2}{\Lambda}\,,
\end{equation}
which is \eqref{alternative conditions on r lambda}. Finally, we have $R_*\geq S_*/16\geq s/8$, so that by the definition of $R_*$, \eqref{mesoscale delta small s8}, the monotonicity of $\delta_{V,R,\Lambda}$, and \eqref{step four eps0 M0 lemma},
\begin{equation}\notag
\max\Big\{\Big|\delta_{V,R,\Lambda}\Big(\frac{S_*}{16}\Big)\Big|,|\delta_{V,R,\Lambda}(S_*)| \Big\}\leq \e_0\leq \varepsilon_2\,,
\end{equation}
which is \eqref{alternative hypo to angular flatness}. By the choices \eqref{step four eps0 M0 lemma}, Lemma \ref{lemma graphicality lambda}-(ii) applies and yields the existence of $K_-\in\mathcal{H}$ and $u_-\in \X_{\sigma'}(\Sigma_{K_-},S_*/128,S_{*}/2
)$ such that
\begin{equation}\label{step three conclusion subcase two}
\textup{$V$ corresponds to $\Sigma_{K_-}(u_+,S_*/128,S_{*}/2)$ on $A_{S_*/128}^{S_{*}/2}$,}
\end{equation}
By \eqref{inequality of Sstar and R_+}, $S_*/128 \leq R_+/2$, so \eqref{step three conclusion subcase two} implies \eqref{step three conclusion redux}. The proof of the {\bf claim} is complete.

\par

Returning to the proof of Theorem \ref{theorem mesoscale criterion}-(i) under the assumption \eqref{deficit changes sign}, we recall \eqref{R+ le Sstar} and choose $R'\in (R_+, \min\{2\,R_+,S_* \})$. Again, we want to apply Lemma \ref{lemma graphicality lambda}-(ii), this time with $ r=R'$, $\lambda_1=\frac{1}{16}$, and $(\eta_1,\eta_2) = (1/128,1/2)$. To begin with, $V$ satisfies \eqref{gamma bound lemma statement} as usual from \eqref{step four V assumptions}. Second, \eqref{alternative conditions on r lambda} holds at $R'$ by $s\leq R'\leq S_*$, \eqref{step four V assumptions}, and the choices \eqref{step four eps0 M0 lemma}. By the monotonicity of $\delta_{V,R,\Lambda}$ and $[R'/16,R']\subset [R_+/16, S_*] \subset [s/4, S_*]$, \eqref{alternative hypo to angular flatness} is valid by our choice \eqref{step four eps0 M0 lemma} of $\e_0$. The graphicality result from Lemma \ref{lemma graphicality lambda}-(ii) therefore yields $K\in \mathcal{H}$ and $u\in \X_{\s'}(\Sigma_{K},R'/128, R'/2 )$ such that
\begin{equation}\label{double graph correspondence}
\mbox{$V$ corresponds to $\S_{K}(u,R'/128,R'/2)$ on $A_{R'/128}^{R'/2}$}\,.
\end{equation}
Now $s/32\leq R'/128 < R_+/64<S_+/16$ by $R' < 2\,R_+$ and \eqref{R+ le Sstar}, and $R_+/2<R'/2 <S_*/2$, so by \eqref{step one conclusion redux} and \eqref{step three conclusion redux}, respectively, we have
\begin{eqnarray}\label{K+ corr}
&&\mbox{$V$ corresponds to $\S_{K_+}(u_+, R'/128, S_+/16)$ on $A_{R'/128}^{S_+/16}$}
\\ \label{K- corr}
&&\mbox{$V$ corresponds to $\S_{K_-}(u_-, R_+/2, R'/2)$ on $A_{R_+/2}^{R'/2}$}\,,
\end{eqnarray}
where $u_+\in \X_{\s'}(\S_{K_+},R'/128, S_+/16)$ and $u_-\in \X_{\s'}(\S_{K_-},R_+/2, R'/2)$. Furthermore, up multiplying $\nu_{K_+}$ or $\nu_{K_-}$ by minus one, we may assume $\nu_{K}\cdot \nu_{K_\pm}\geq 0$. Thus $V$ is represented by multiple spherical graphs on nontrivial annuli. %, which is where the choice of $\s'=\s'(\e')$ above \eqref{were gonna tilt} becomes useful.
By combining \eqref{double graph correspondence}, \eqref{K+ corr} and \eqref{K- corr}, $\nu_{K}\cdot \nu_{K_\pm}\geq 0$ and $\s'=\s'(\e')$, \eqref{prove planes are close} applies and gives
\begin{equation}\label{closeness of nus}
|\nu_{K}-\nu_{K_+}| < \e'\,,\quad |\nu_{K}-\nu_{K_-}| < \e'\,.
\end{equation}
But $\e'$ was chosen according to \eqref{were gonna tilt} so that Lemma \ref{lemma step one}-(i) is applicable; that is, since $\e'<\e_0$ and $\s'<\s_0$ from that lemma, we may reparametrize \eqref{step one conclusion redux} and \eqref{step three conclusion redux}, respectively, as
\begin{eqnarray}\label{K+ corr 2}
&&\mbox{$V$ corresponds to $\S_{K}(w_+, s/32, S_+/16)$ on $A_{s/32}^{S_+/16}$}
\\ \label{K- corr 2}
&&\mbox{$V$ corresponds to $\S_{K}(w_-, R_+/2, S_*/2)$ on $A_{R_+/2}^{S_*/2}$}\,,
\end{eqnarray}
\begin{equation}\notag
w_+ \in \X_{C_0(\s' + \e')}(\S_K,s/32, S_+/16)\,,\quad w_- \in \X_{C_0(\s' + \e')}(\S_K,R_+/2, S_*/2)\,.
\end{equation}
By \eqref{were gonna tilt}, $C_0(\s'+\e')\leq \s$, and by $R'/128<S_+/16<R_+/2<R'/2$, \eqref{K+ corr 2} and \eqref{K- corr 2}, we may extend the $u$ defined in \eqref{double graph correspondence} onto $\S_K \times (s/32,S_*/2) $ using $w_+$ and $w_-$ with $C^1$-norm bounded by $\s$. The resulting extension is such that \eqref{mesoscale thesis graphicality} holds, so the proof of Theorem \ref{theorem mesoscale criterion} is finished.