M|y^{\prime}|^{2}\\}$, and likewise for $F^{c}\Delta H^{c}$. Moreover, observe
that (by symmetry)

$\int_{\mathcal{B}_{\delta}}dy\,(\chi_{H}-\chi_{H^{c}})(y)\frac{\alpha_{n,s}}{|y|^{n+s}}=0\,.$

We can then write

$\displaystyle\int_{\mathcal{B}_{\delta}}dy\,(\chi_{F}-\chi_{F^{c}})(y)\frac{\alpha_{n,s}}{|y|^{n+s}}$
$\displaystyle=\int_{\mathcal{B}_{\delta}}dy\,\big{[}(\chi_{F}-\chi_{F^{c}})-(\chi_{H}-\chi_{H^{c}})\big{]}(y)\frac{\alpha_{n,s}}{|y|^{n+s}}$
$\displaystyle=O\Big{(}\int_{\mathcal{B}_{\delta}\cap(F\Delta
H)}dy\,\frac{\alpha_{n,s}}{|y|^{n+s}}\Big{)}$
$\displaystyle=O\Big{(}\int_{\mathcal{B}_{\delta}\cap\\{|y_{n}|\leq
M|y^{\prime}|^{2}\\}}dy\,\frac{\alpha_{n,s}}{|y|^{n+s}}\Big{)}\,.$

Passing to polar coordinates it is immediate to see that the last integral is
bounded by $C\int_{0}^{\delta}dr\,\frac{1}{r^{s}}=C\delta^{1-s}$. With this,
as mentioned above, we conclude our proof of (5.8). In particular, we deduce
(5.7).

We now focus on the terms in the RHS of (5.5). Thanks to (5.8), we immediately
see that

$\displaystyle\int_{\partial E}\widetilde{H}_{s}^{\varepsilon}[\partial
E](x)\Big{[}\xi^{2}(x)\text{div}_{x}(\nu_{\partial
E}(x))+\langle\nabla_{x}^{\rm tan}\xi,X\rangle_{g}-\xi\langle\nu_{\partial
E},\nabla\xi\rangle_{g}\Big{]}(x)$ $\displaystyle\hskip
28.45274pt\xrightarrow{\varepsilon\to 0}\int_{\partial
E}\widetilde{H}_{s}[\partial
E](x)\Big{[}\xi^{2}(x)\text{div}_{x}(\nu_{\partial
E}(x))+\langle\nabla_{x}^{\rm tan}\xi,X\rangle_{g}-\xi\langle\nu_{\partial
E},\nabla\xi\rangle_{g}\Big{]}(x)\,,$

Moreover, it is clear that

$\int_{\partial E}dx\,\int_{\partial
E}dy\,(\xi(x)-\xi(y))^{2}K_{\varepsilon}(p,q)\xrightarrow{\varepsilon\to
0}\int_{\partial E}dx\,\int_{\partial E}dy\,(\xi(x)-\xi(y))^{2}K(p,q)\,,$

simply by monotone convergence.
It remains to see that the second term in (5.5) converges as $\varepsilon\to
0$. Recall that $\nu_{\partial E}$ denotes a smooth extension to $M$ of the
outer normal vector to $E$. We start by applying the Leibniz (product) rule
for the divergences appearing in the integral, by which we can expand:

$\displaystyle\int_{M}dq\,(\chi_{E}-\chi_{E^{c}})(q)[\text{div}_{p}(\nu_{\partial
E}(p)K_{\varepsilon}(p,q))+\text{div}_{q}(\nu_{\partial
E}(q)K_{\varepsilon}(p,q))]=$ (5.9) $\displaystyle\hskip
21.33955pt=\text{div}_{p}(\nu_{\partial
E}(p))\int_{M}dq\,(\chi_{E}-\chi_{E^{c}})(q)K_{\varepsilon}(p,q)+\int_{M}dq\,(\chi_{E}-\chi_{E^{c}})(q)\text{div}_{q}(\nu_{\partial
E}(q))K_{\varepsilon}(p,q)$ $\displaystyle\hskip
35.56593pt+\int_{M}dq\,(\chi_{E}-\chi_{E^{c}})(q)\langle\nu_{\partial
E}(p),\nabla_{p}K_{\varepsilon}(p,q)\rangle_{g}+\int_{M}dq\,(\chi_{E}-\chi_{E^{c}})(q)\langle\nu_{\partial
E}(q),\nabla_{q}K_{\varepsilon}(p,q)\rangle_{g}\,.$

Thanks to the smoothness of $\nu_{\partial E}(p)$, we can Taylor–expand
$\text{div}_{q}(\nu_{\partial E}(q))=\text{div}_{p}(\nu_{\partial
E}(p))+h(p,q)$, with $h(p,q)=O({\rm dist}(p,q))$. We can then write the first
two terms on the right hand side as

$\displaystyle 2\text{div}_{p}(\nu_{\partial
E}(p))\int_{M}dq\,(\chi_{E}-\chi_{E^{c}})(q)K_{\varepsilon}(p,q)+\int_{M}dq\,(\chi_{E}-\chi_{E^{c}})(q)h(p,q)K_{\varepsilon}(p,q)\,.$

The first integral (which corresponds to $\widetilde{H}_{s}^{\varepsilon}(p)$)
converges as $\varepsilon\to 0$ by (5.8). The second integral can be shown to
converge simply by dominated convergence (without the need for cancellations),
since combining Lemma 3.10 (see also Proposition 4.9) and the fact that
$h(p,q)=O({\rm dist}(p,q))$ we can bound
$h(p,q)K_{\varepsilon}(p,q)\leq\frac{C}{{\rm dist}^{n+s-1}(p,q)}$, and the
latter is Lebesgue integrable in $q$.

We now focus on the last two terms on the RHS of (5.9). These are the most
challenging ones, since it is not obvious at all at first sight how they can
be made to converge as $\varepsilon\to 0$ (observe that $|\nabla_{p}K(p,q)|$
is of order $\frac{1}{{\rm dist}^{n+s+1}(p,q)}$, which is $1+s$ orders away
from being Lebesgue integrable), and necessarily one needs to exploit some
cancellations. In fact, the two terms need to be considered together, i.e. we
write them as

$\int_{M}dq\,(\chi_{E}-\chi_{E^{c}})(q)[\nu_{\partial
E}(p)\nabla_{p}K_{\varepsilon}(p,q)+\nu_{\partial
E}(q)\nabla_{q}K_{\varepsilon}(p,q)]\,.$

Let $\psi_{\nu_{\partial E}}^{t}$ denote the flow of the extended vector field
$\nu_{\partial E}$, which is defined on all of $M$, at time $t$. Note that
this is not related in any way to $\psi_{X}^{t}$, the flow of the vector field
$X$ that we fixed at the beginning, and which does not appear at all in the
integral we are considering. The first crucial observation is that we can
actually write

$\displaystyle\nu_{\partial E}(p)\nabla_{p}K(p,q)+\nu_{\partial
E}(q)\nabla_{q}K(p,q)$
$\displaystyle=\frac{d}{dt}\big{|}_{t=0}[K(\psi_{\nu_{\partial
E}}^{t}(p),q)+K(p,\psi_{\nu_{\partial E}}^{t}(q))]$
$\displaystyle=\frac{d}{dt}\big{|}_{t=0}K(\psi_{\nu_{\partial
E}}^{t}(p),\psi_{\nu_{\partial E}}^{t}(q))$
$\displaystyle=:K^{\prime}(p,q)\,.$

This new kernel $K^{\prime}(p,q)$ can be bounded by $K(p,q)$, thanks to (3.18)
of Proposition 3.11 (which is stated for $K$, but the proof of which works
exactly the same for $K_{\varepsilon}$). We then need to make sense of
$\int_{M}dq\,(\chi_{E}-\chi_{E^{c}})(q)K^{\prime}(p,q)$ as a singular
integral, and for that, the second crucial observation is that $K^{\prime}$ is
actually symmetric at first order:

Claim. Denoting $K^{\prime}(x,y):=K^{\prime}(\varphi(x),\varphi(y))$, we have
that $|K^{\prime}(x,x+z)-K^{\prime}(x,x-z)|\leq\frac{C}{|z|^{n+s-1}}$.

###### Proof of the claim.

Since $K^{\prime}(p,q)=K^{\prime}(q,p)$, we can rewrite this as
$|K^{\prime}(x,x+z)-K^{\prime}(x-z,x)|\leq\frac{C}{|z|^{n+s-1}}$.
Equivalently, defining the constant vector field $v=\frac{z}{|z|}$, this
corresponds to
$|K^{\prime}(x-z+|z|v,x+|z|v)-K^{\prime}(x-z,x)|\leq\frac{C}{|z|^{n+s-1}}$.
Consider the vector field $V=v\eta$, where $\eta$ is a cutoff with small
support around $x$, and let $\psi_{V}^{t}$ denote the flow of $V$ at time $t$;
then, $(x-z+|z|v,x+|z|v)=(\psi_{V}^{|z|}(x-z),\psi_{V}^{|z|}(x))$, so that our
desired inequality can be rewritten as
$|K^{\prime}(\psi_{V}^{|z|}(x-z),\psi_{V}^{|z|}(x))-K^{\prime}(x-z,x)|\leq\frac{C}{|z|^{n+s-1}}$.
By the fundamental theorem of calculus, the LHS is

$\displaystyle\Big{|}K^{\prime}(\psi_{V}^{|z|}(x-z),\psi_{V}^{|z|}(x))-K^{\prime}(x-z,x)\Big{|}$
$\displaystyle=\Big{|}\int_{0}^{|z|}dt\,\frac{d}{dt}K^{\prime}(\psi_{V}^{t}(x-z),\psi_{V}^{t}(x))\Big{|}$
$\displaystyle\leq\int_{0}^{|z|}dt\,\Big{|}\frac{d}{dt}K^{\prime}(\psi_{V}^{t}(x-z),\psi_{V}^{t}(x))\Big{|}$
$\displaystyle\leq C|z|\sup_{0\leq
t\leq|z|}\Big{|}\frac{d}{dt}K^{\prime}(\psi_{V}^{t}(x-z),\psi_{V}^{t}(x))\Big{|}\,.$

Finally, arguing as in the proof of Proposition 3.11 (recalling that
$K^{\prime}(p,q)=\frac{d}{dt}\big{|}_{t=0}K(\psi_{\nu_{\partial
E}}^{t}(p),\psi_{\nu_{\partial E}}^{t}(q))$), we see that we can bound

$\sup_{0\leq
t\leq|z|}\Big{|}\frac{d}{dt}K^{\prime}(\psi_{V}^{t}(x-z),\psi_{V}^{t}(x))\Big{|}\leq
CK(x-z,x)\leq\frac{C}{|z|^{n+s}}\,,$

which concludes the proof of the claim. ∎

With this claim at hand, the proof of the convergence of

$\int_{M}dq\,(\chi_{E}-\chi_{E^{c}})(q)[\nu_{\partial
E}(p)\nabla_{p}K_{\varepsilon}(p,q)+\nu_{\partial
E}(q)\nabla_{q}K_{\varepsilon}(p,q)]=\int_{M}dq\,(\chi_{E}-\chi_{E^{c}})(q)K^{\prime}_{\varepsilon}(p,q)$

to

$\int_{M}dq\,(\chi_{E}-\chi_{E^{c}})(q)K^{\prime}(p,q)$

follows like the proof of the convergence of
$\widetilde{H}_{s}^{\varepsilon}[\partial
E](p)=\int_{M}dq\,(\chi_{E}-\chi_{E^{c}})(q)K_{\varepsilon}(p,q)$ to
$\widetilde{H}_{s}[\partial E](p)$. The only difference is that, in place of
the full cancellation property

$\int_{\mathcal{B}_{\delta}}dy\,(\chi_{H}-\chi_{H^{c}})(y)\frac{\alpha_{n,s}}{|y|^{n+s}}=0\,,$

one needs to estimate

$\int_{\mathcal{B}_{\delta}}dy\,(\chi_{H}-\chi_{H^{c}})(y)K^{\prime}_{\varepsilon}(0,y)$

by some function of $\delta$ which goes to zero as $\delta\to 0$. To see that
the latter holds, using the symmetry of the Lebesgue measure and the claim, we
compute

$\displaystyle\int_{\mathcal{B}_{\delta}}dy\,\chi_{H}(y)K^{\prime}_{\varepsilon}(0,y)$
$\displaystyle=\int_{\mathcal{B}_{\delta}}dy\,\chi_{H}(-y)K^{\prime}_{\varepsilon}(0,-y)$
$\displaystyle=\int_{\mathcal{B}_{\delta}}dy\,\chi_{H^{c}}(y)K^{\prime}_{\varepsilon}(0,-y)$
$\displaystyle=\int_{\mathcal{B}_{\delta}}dy\,\chi_{H^{c}}(y)K^{\prime}_{\varepsilon}(0,y)+O\Big{(}\int_{\mathcal{B}_{\delta}}dy\,|K^{\prime}_{\varepsilon}(0,-y)-K^{\prime}_{\varepsilon}(0,y)|\Big{)}$
$\displaystyle=\int_{\mathcal{B}_{\delta}}dy\,\chi_{H^{c}}(y)K^{\prime}_{\varepsilon}(0,y)+O\Big{(}\int_{\mathcal{B}_{\delta}}dy\,\frac{1}{|y|^{n+s-1}}\Big{)}$
$\displaystyle=\int_{\mathcal{B}_{\delta}}dy\,\chi_{H^{c}}(y)K^{\prime}_{\varepsilon}(0,y)+O\Big{(}\delta^{1-s}\Big{)}\,.$

Bringing the first term on the RHS to the LHS, we see that

$\int_{\mathcal{B}_{\delta}}dy\,(\chi_{H}-\chi_{H^{c}})(y)K^{\prime}_{\varepsilon}(0,y)=O\Big{(}\delta^{1-s}\Big{)}$

as desired. ∎

The formulas above motivate the following definitions.

######  Definition 5.2.

For a set $E$ as in Proposition 5.1, as indicated there the fractional mean
curvature $H_{s}[\partial E](p)$ at a point $p\in\partial E$ is defined as

$\widetilde{H}_{s}[\partial
E](p)=\int_{M}dq\,(\chi_{E^{c}}-\chi_{E})(q)K_{s}(p,q)\,.$

In particular, by arbitrariness of $\xi$ in (5.1), $E$ is an $s$-minimal
surface (recall Definition 3.1) iff $\widetilde{H}_{s}[\partial E](p)=0$ for
every $p\in\partial E$.
Denoting $\sigma:=1-s$, we also set

$H_{s}[\partial E](p)=\sigma\widetilde{H}_{s}[\partial E](p)\,.$

Then, up to a multiplicative dimensional constant, $H_{s}[\partial E]$
converges as $s\to 1$ to the classical mean curvature $H[\partial E]$, as we
will see later.

######  Definition 5.3.

The second variation $\delta^{2}\textnormal{Per}_{s}(E)$ of an $s$-minimal
surface $E$ is defined as the quadratic form acting on test functions $\xi\in
C^{1}(\partial E)$ by

$\displaystyle\delta^{2}\textnormal{Per}_{s}(E)[\xi,\xi]=\iint_{\partial
E\times\partial E}$ $\displaystyle|\xi(x)-\xi(y)|^{2}K(p,q)$ (5.10)
$\displaystyle-\int_{\partial
E}\,dx\,\xi^{2}(x)\,p.v.\int_{M}\,dy\,(\chi_{E}-\chi_{E^{c}})(y)\Big{[}\textnormal{div}_{x}(\nu(x)K(p,q))+\textnormal{div}_{y}(\nu(y)K(p,q))\Big{]}\,.$
(5.11)

If $X$ is a $C^{1}$ vector field on $M$, set $\xi=X\cdot\nu$ on $\partial E$.
By Proposition 5.1, since $\widetilde{H}_{s}[\partial E](p)=0$ we then have
that
$\frac{d^{2}}{dt^{2}}\Big{|}_{t=0}\textnormal{Per}_{s}(\psi_{X}^{t}(E))=\delta^{2}\textnormal{Per}_{s}(E)[\xi,\xi]$.

###  5.2 A first classical perimeter estimate for $s$-minimal surfaces with
bounded index

In this section, we will prove the following classical perimeter estimate in
all dimensions:

######  Theorem 5.4 (Perimeter estimate).

Let $p\in M$, $s_{0}\in(0,1)$, $s\in(s_{0},1)$ and assume that $M$ satisfies
the flatness assumption ${\rm FA}_{2}(M,g,1,p_{0},\varphi)$. Let $E$ be a
$C^{2}$ $s$-minimal surface with index bounded by $m$ in
$\varphi(\mathcal{B}_{1})$.

Then, there exists a constant $C=C(n,s_{0},m)$ such that

$\textnormal{Per}(E;\varphi(\mathcal{B}_{1/2}))\leq\frac{C}{1-s}\,.$ (5.12)

If moreover we add the assumption that

$\textnormal{Per}_{s}(E;\varphi(\mathcal{B}_{1}))\leq\frac{\kappa}{1-s}=\frac{\kappa}{\sigma}$
(5.13)

for some $\kappa>0$, then we get the improved estimate

$\textnormal{Per}(E;\varphi(\mathcal{B}_{1/2}))\leq
C\frac{1+\sqrt{\kappa}}{\sqrt{\sigma}}\,.$ (5.14)

######  Remark 5.5.

The proof is similar to the one in [18], which deals with a BV estimate for
solutions of Allen–Cahn with bounded index.

######  Remark 5.6.

In the case where
$\textnormal{Per}_{s}(E;\varphi(\mathcal{B}_{1}))\leq\frac{\kappa}{\sigma}$,
for $n=3$ we will eventually show (see Proposition 5.29) that
$\textnormal{Per}(E;\varphi(\mathcal{B}_{1/2}))\leq C(\kappa)$, which is what
one naturally expects: Recall that, at least for a fixed smooth set $E$,
$\displaystyle\lim_{s\to 1}(1-s){\rm Per}_{s}(E)=\gamma_{n}{\rm Per}(E)$, and
a similar convergence will be obtained for sequences $E_{k}$ with bounded
index. Nevertheless, the non-optimal perimeter bound in Theorem 5.4 will prove
very helpful in the proof of the sharp result.

Before giving the proof of Theorem 5.4, we need some preliminary results that
will be repeatedly used during the article. The first one is a lemma which
deals with the finite index property in the nonlocal case. For critical points
of local functionals, it is a standard fact that having Morse index bounded by
$m$ implies stability in one out of every $m+1$ disjoint open sets. For
nonlocal functionals this is not true anymore, but in one of the sets we can
obtain a weaker, quantitative lower bound on the second derivative which we
will refer to as almost stability.

######  Definition 5.7 (Almost stability).

Let $\Omega\subset M$ open and $E\subset M$ be an $s$-minimal surface in
$\Omega$. Given $\Lambda\in\mathbb{R}$, we say that $E$ is $\Lambda$-almost
stable in $\Omega$ if

$\delta^{2}\textnormal{Per}_{s}(E)[\xi,\xi]\geq-\Lambda\Big{(}\int_{\partial
E\cap\Omega}|\xi|\Big{)}^{2}\,\quad\forall\,\xi\in C_{c}^{1}(\partial
E\cap\Omega)\,.$ (5.15)

######  Lemma 5.8 (Finite Morse index and almost stability).

Let $\Omega\subset M$ open and let $E\subset M$ be an $s$-minimal surface in
$\Omega$ with Morse index at most $m$. Consider a collection
$\mathcal{U}_{1},\dotsc,\mathcal{U}_{m+1}$ of $(m+1)$ disjoint open subsets of
$\Omega$, and set

$\Lambda:=m\max_{i\neq j}\sup_{\mathcal{U}_{i}\times\mathcal{U}_{j}}K(p,q)\,.$

Then, there is (at least) one set $\mathcal{U}_{k}$ among
$\mathcal{U}_{1},\dotsc,\mathcal{U}_{m+1}$ such that $E$ is $\Lambda$-almost
stable in $\mathcal{U}_{k}$, that is

$\delta^{2}\textnormal{Per}_{s}(E)[\xi,\xi]\geq-\Lambda\Big{(}\int_{\partial
E\cap\mathcal{U}_{k}}|\xi|\Big{)}^{2}\,\quad\forall\,\xi\in C_{c}^{1}(\partial
E\cap\mathcal{U}_{k})\,.$

###### Proof.

We prove the Lemma just for $m=1$ for the sake of clarity, the proof goes on
exactly the same for general $m$. Let $\xi_{1}\in C_{c}^{\infty}(\partial
E\cap\mathcal{U}_{1})$ and $\xi_{2}\in C_{c}^{\infty}(\partial
E\cap\mathcal{U}_{2})$. Testing the second variation of the fractional
perimeter, the expression of which we gave in Definition 5.3, with linear
combinations of $\xi_{1}$ and $\xi_{2}$ gives (by expanding the square in the
Sobolev part of the expression)

$\displaystyle\delta^{2}\textnormal{Per}_{s}(E)[a\xi_{1}+b\xi_{2},a\xi_{1}+b\xi_{2}]=\,$
$\displaystyle
a^{2}\delta^{2}\textnormal{Per}_{s}(E)[\xi_{1},\xi_{1}]+b^{2}\delta^{2}\textnormal{Per}_{s}(E)[\xi_{2},\xi_{2}]$
$\displaystyle\hskip 28.45274pt-2ab\iint_{(\partial
E\cap\mathcal{U}_{1})\times(\partial
E\cap\mathcal{U}_{2})}\xi_{1}(p)\xi_{2}(q)K(p,q)\,.$

Since by assumption $K(p,q)\leq\Lambda$ for all
$(p,q)\in\mathcal{U}_{1}\times\mathcal{U}_{2}$, the interaction term can be
bounded as

$\begin{split}-2ab\iint_{(\partial E\cap\mathcal{U}_{1})\times(\partial
E\cap\mathcal{U}_{2})}\xi_{1}(p)\xi_{2}(q)K(p,q)&\leq
2ab\Lambda\|\xi_{1}\|_{L^{1}(\partial
E\cap\mathcal{U}_{1})}\|\xi_{2}\|_{L^{1}(\partial E\cap\mathcal{U}_{2})}\\\
&\leq a^{2}\Lambda\|\xi_{1}\|_{L^{1}(\partial
E\cap\mathcal{U}_{1})}^{2}+b^{2}\Lambda\|\xi_{2}\|_{L^{1}(\partial
E\cap\mathcal{U}_{2})}^{2}.\end{split}$

Hence

$\displaystyle\delta^{2}\textnormal{Per}_{s}(E)[a\xi_{1}+b\xi_{2},a\xi_{1}+b\xi_{2}]$
$\displaystyle\leq
a^{2}\Big{(}\underbrace{\delta^{2}\textnormal{Per}_{s}(E)[\xi_{1},\xi_{1}]+\Lambda\|\xi_{1}\|_{L^{1}(\partial
E\cap\mathcal{U}_{1})}^{2}}_{=:F_{1}(\xi_{1})}\Big{)}$ $\displaystyle\hskip
28.45274pt+b^{2}\Big{(}\underbrace{\delta^{2}\textnormal{Per}_{s}(E)[\xi_{2},\xi_{2}]+\Lambda\|\xi_{2}\|_{L^{1}(\partial
E\cap\mathcal{U}_{2})}^{2}}_{=:F_{2}(\xi_{2})}\Big{)}.$ (5.16)

We want to show that either $F_{1}(\xi_{1})\geq 0$ for all $\xi_{1}\in
C_{c}^{\infty}(\partial E\cap\mathcal{U}_{1})$ or $F_{2}(\xi_{2})\geq 0$ for
all $\xi_{2}\in C_{c}^{\infty}(\partial E\cap\mathcal{U}_{2})$. Suppose
neither of these two held, then there would exist $\xi_{1}$ and $\xi_{2}$ such
that $F_{1}(\xi_{1})<0$ and $F_{2}(\xi_{2})<0$. This would imply, however,
that (5.16) is negative for all $(a,b)\neq(0,0)$, thus contradicting that the
Morse index of $u$ is at most one. ∎

######  Remark 5.9.

Almost-stability improves upon zooming in. More precisely, if $E$ is a
$\Lambda$-almost stable $s$-minimal set in $\Omega\subset M$ for $(M,g)$, it
is a $(r^{n+s}\Lambda)$-almost stable $s$-minimal set in $\Omega$ for
$(M,\frac{1}{r^{2}}g)$. This follows from (a) in Remark 3.8, since it
immediately gives that the $s$-perimeter of any set just rescales by a
constant factor $(\frac{1}{r})^{n-s}$ under the scaling of the metric, which
shows that the scaling preserves the criticality/$s$-minimality condition and
it transforms (as seen by rescaling both sides of equation (5.15))
$\Lambda$-almost stability into $(r^{n+s}\Lambda)$-almost stability.

We first prove Theorem 5.4 in the almost-stable case.

######  Proposition 5.10 (Almost-stable case).

Let $p\in M$, $s_{0}\in(0,1)$, $s\in(s_{0},1)$ and assume that $M$ satisfies
the flatness assumption ${\rm FA}_{2}(M,g,1,p_{0},\varphi)$. Let $E$ be a
$C^{2}$ $\Lambda$-almost stable $s$-minimal surface in
$\varphi(\mathcal{B}_{1})$.

Then, there exists a constant $C=C(n,s_{0},\Lambda)$ such that

$\textnormal{Per}(E;\varphi(\mathcal{B}_{1/2}))\leq\frac{C}{1-s}\,.$ (5.17)

If moreover we add the assumption that

$\textnormal{Per}_{s}(E;\varphi(\mathcal{B}_{1}))\leq\frac{\kappa}{1-s}=\frac{\kappa}{\sigma}$
(5.18)

for some $\kappa>0$, then we get the improved estimate

$\textnormal{Per}(E;\varphi(\mathcal{B}_{1/2}))\leq
C\frac{1+\sqrt{\kappa}}{\sqrt{\sigma}}\,.$ (5.19)

###### Proof.

Step 1. We first prove that there exists a $\Lambda_{0}>0$, depending only on
$n$ and $s_{0}$, such that if $E$ is $\Lambda$-almost stable in
$\varphi(\mathcal{B}_{1})\subset M$ (see Definition 5.7) with
$\Lambda\leq\Lambda_{0}$, then the Proposition is true.

The proof goes as follows. Let $X$ be a $C^{1}$ vector field, compactly
supported on $\varphi(\mathcal{B}_{1})$, and define $\xi=\langle
X,\nu_{\partial E}\rangle_{g}$ on $\partial E$. Considering the test functions
$\xi$ and $|\xi|$ for the second variation formula (recall Definition 5.3),
and applying the $\Lambda$-almost stability assumption to bound
$\delta^{2}\text{Per}_{s}(E)[|\xi|,|\xi|]$ from below appropriately (see
Definition 5.7, and observe that it can be extended to Lipschitz functions
such as $|\xi|$ by standard approximation arguments), we get

$\delta^{2}\text{Per}_{s}(E)[\xi,\xi]-\delta^{2}\text{Per}_{s}(E)[|\xi|,|\xi|]\leq\delta^{2}\text{Per}_{s}(E)[\xi,\xi]+\Lambda\Big{(}\int_{\partial
E}|\xi|\Big{)}^{2}\,.$ (5.20)

On the other hand, both second variations in the left hand side have the same
“local parts”, since obviously $\xi^{2}=|\xi|^{2}$. Also for this reason,
looking at the expression in Definition 5.3, one also sees that the difference
of the terms of the form $\iint_{\partial E\times\partial E}$ has a very
simple form, since only the cross-terms appearing after expanding the squares
do not cancel out. Precisely, we obtain that

$\delta^{2}\text{Per}_{s}(E)[\xi,\xi]-\delta^{2}\text{Per}_{s}(E)[|\xi|,|\xi|]=2\iint_{\partial
E\times\partial E}\Big{[}|\xi(p)||\xi(q)|-\xi(p)\xi(q)\Big{]}K(p,q)\,.$ (5.21)

Using the identity $|a||b|-ab=2a_{+}b_{-}+2a_{-}b_{+}$, which holds for real
numbers $a,b$ and their positive and negative parts, together with the
symmetry of $K$ we find that

$\delta^{2}\text{Per}_{s}(E)[\xi,\xi]-\delta^{2}\text{Per}_{s}(E)[|\xi|,|\xi|]=8\iint_{\partial
E\times\partial E}\xi_{+}(p)\xi_{-}(q)K(p,q)\,.$

Thanks to Proposition 3.10, we can bound $K(p,q)\geq c>0$ for
$(p,q)\in\varphi(\mathcal{B}_{1})\times\varphi(\mathcal{B}_{1})$. Applying
this to (5.21), we then find that

$c\Big{(}\int_{\partial E}dp\,\xi_{+}(p)\Big{)}\Big{(}\int_{\partial
E}dq\,\xi_{-}(q)\Big{)}\leq\delta^{2}\text{Per}_{s}(E)[\xi,\xi]-\delta^{2}\text{Per}_{s}(E)[|\xi|,|\xi|]\,.$

Recall now that $\xi=X\cdot\nu$, where $X$ is a $C^{1}$ vector field supported
on $\varphi(\mathcal{B}_{1})$. By the divergence theorem, we then have that

$\Big{|}\int_{\partial E}\xi(p)\,dp\Big{|}=\Big{|}\int_{\partial
E}X\cdot\nu(p)\,dp\Big{|}=\Big{|}\int_{E}\textnormal{div}(X)\Big{|}\leq|B_{2}|C_{X}\,.$

Therefore, putting together all the above, we can bound

$\displaystyle\Big{(}\int_{\partial E}\Big{|}\xi\Big{|}\Big{)}^{2}$
$\displaystyle=\Big{(}\int_{\partial E}(X\cdot\nu)_{+}+\int_{\partial
E}(X\cdot\nu)_{-}\Big{)}^{2}$ $\displaystyle=\Big{(}\int_{\partial
E}(X\cdot\nu)_{+}-\int_{\partial
E}(X\cdot\nu)_{-}\Big{)}^{2}+4\Big{(}\int_{\partial
E}(X\cdot\nu)_{+}\Big{)}\Big{(}\int_{\partial E}(X\cdot\nu)_{-}\Big{)}$
$\displaystyle=\Big{(}\int_{\partial
E}X\cdot\nu\Big{)}^{2}+4\Big{(}\int_{\partial
E}(X\cdot\nu)_{+}\Big{)}\Big{(}\int_{\partial E}(X\cdot\nu)_{-}\Big{)}$
$\displaystyle\leq
C+C\delta^{2}\text{Per}_{s}(E)[\xi,\xi]+C\Lambda\Big{(}\int_{\partial
E}|\xi|\Big{)}^{2}\,.$

In case $\Lambda_{0}$ is small enough (say, so that $C\Lambda_{0}\leq 1/2$),
we can absorb the last term of the inequality on the LHS, which after taking
square roots in the expression gives

$\displaystyle\int_{\partial E}\Big{|}X\cdot\nu\Big{|}=\int_{\partial
E}\Big{|}\xi\Big{|}$
$\displaystyle\leq\Big{(}C+C\delta^{2}\text{Per}_{s}(E)[\xi,\xi]\Big{)}^{1/2}\,.$

We now make the precise choice of $X$. We consider
$X_{1}=\varphi_{*}(e_{1}\eta)$, $X_{2}=\varphi_{*}(e_{2}\eta)$, … , up to
$X_{n}=\varphi_{*}(e_{n}\eta)$, where $e_{i}$ is the $i$-th basis coordinate
vector in $\mathbb{R}^{n}$ and
$\chi_{\mathcal{B}_{1/2}}\leq\eta\leq\chi_{\mathcal{B}_{3/4}}$ is a standard
cutoff. We denote then $\xi_{i}=\langle X_{i},\nu_{\partial E}\rangle_{g}$.
Adding up the inequalities that we obtain for these choices of $X$, and since
the $e_{1},...,e_{n}$ form an orthonormal basis of vectors for
$\mathbb{R}^{n}$, together with the flatness assumptions we obtain the
perimeter in the LHS:

$\displaystyle\text{Per}(E,\varphi(\mathcal{B}_{1/2}))$
$\displaystyle\leq\sum_{i}\Big{(}C+C\delta^{2}\text{Per}_{s}(E)[\xi_{i},\xi_{i}]\Big{)}^{1/2}=\sum_{i}\Big{(}C+C\frac{d^{2}}{dt^{2}}\Big{|}_{t=0}\text{Per}_{s}(\psi_{X_{i}}^{t}(E))\Big{)}^{1/2}\,.$

On the other hand, applying Lemma 3.12 we can get rid of the second
derivatives on the RHS, obtaining

$\displaystyle\text{Per}(E,\varphi(\mathcal{B}_{1/2}))$
$\displaystyle\leq\Big{(}C+C\text{Per}_{s}(E,\varphi(\mathcal{B}_{1}))\Big{)}^{1/2}\,.$

Under the assumption (5.18), i.e. assuming a constant bound for the fractional
perimeter $\text{Per}_{s}(E)$, we conclude (5.19) as desired.

The former is the only case we will need in this article. If, on the other
hand, we did not assume (5.18) and wished to obtain (5.17) instead, we would
need to continue the argument as in the proof of [18, Proposition 3.14].
First, we would use the interpolation result in (3.20) and the inequality
$ab\leq\frac{a^{2}}{\delta}+\delta b^{2}$, valid for any $\delta>0$, to get

$\displaystyle\text{Per}(E,\varphi(\mathcal{B}_{1/2}))$
$\displaystyle\leq\frac{C}{(1-s)\delta}+C\delta\text{Per}(E,\varphi(\mathcal{B}_{1}))\,.$

Rescaling and translating this inequality, see Step 2 next for full details in
a similar argument, we would find that

$\displaystyle r^{1-n}\text{Per}(E,\varphi(\mathcal{B}_{r/2}(x)))$
$\displaystyle\leq\frac{C}{(1-s)\delta}+C\delta
r^{1-n}\text{Per}(E,\varphi(\mathcal{B}_{r}(x)))$

for all $x\in\mathcal{B}_{3/4}$ and $r\leq 1/8$. Using then a covering lemma
from [48], see [18, Lemma 3.18] for the exact statement, we would then
conclude that

$\displaystyle\text{Per}(E,\varphi(\mathcal{B}_{1/2}))$
$\displaystyle\leq\frac{C}{1-s}$

as desired.

Step 2. We obtain the Proposition in the $\Lambda$-almost stable case, for any
$\Lambda$.

We need to get rid of the requirement in Step 1 that $\Lambda\leq\Lambda_{0}$;
this will follow from a scaling + finite covering argument. Let $0<r<1/2$ be a
small radius to be chosen later depending only on $\Lambda$. Given
$x_{k}\in\mathcal{B}_{1/2}$, define $\varphi_{x_{k},r}(z):=\varphi(x_{k}+rz)$,
for $z\in\mathcal{B}_{1}$. By (c) in Remark 3.8 and our flatness assumptions,
the rescaled manifold $(M,\frac{1}{r^{2}}g)$ satisfies the flatness
assumptions ${\rm
FA}_{\ell}(M,\frac{1}{r^{2}}g,1,\varphi(x_{k}),\varphi_{x_{k},r})$. Moreover,
since by assumption $E$ is a $\Lambda$-almost stable $s$-minimal set in
$\varphi_{x_{k},r}(\mathcal{B}_{1})$ for $(M,g)$, by Remark 5.9 it is a
$(r^{n+s}\Lambda)$-almost stable $s$-minimal set in
$\varphi_{x_{k},r}(\mathcal{B}_{1})$ for $(M,\frac{1}{r^{2}}g)$.
Assume for concreteness that we are assuming (5.18) and we want to prove
(5.19); the proof of (5.17) is even simpler. Letting $r$ small enough so that
$r^{n+s}\Lambda\leq\Lambda_{0}$, the above discussion shows that $E$ satisfies
the almost-stability requirement in Step 1 on
$\varphi_{x_{k},r}(\mathcal{B}_{1})$ for $(M,\frac{1}{r^{2}}g)$. Moreover,
applying (3.29) with $r_{1}=r$ and rescaling shows that (up to changing the
value of $\kappa$), the condition (5.18) also holds for $E$ in the rescaled
setting. Therefore, by Step 1 we find that (5.19) holds on
$(M,\frac{1}{r^{2}}g)$. Scaling back this information, this gives that

$\displaystyle\textnormal{Per}(E;\varphi(\mathcal{B}_{r/2}(x_{k})))$
$\displaystyle\leq C(n,s_{0})\frac{1+\sqrt{\kappa}}{\sqrt{\sigma}}r^{n-1}$
(5.22)
$\displaystyle=C(n,s_{0},\Lambda)\frac{1+\sqrt{\kappa}}{\sqrt{\sigma}}\,,$

where the perimeter is now taken on $(M,g)$. We can now conclude by covering
$\mathcal{B}_{1/2}$ with a finite number $C(n,\Lambda)$ of balls
$\mathcal{B}_{r/2}(x_{k})$ as above, so that summing over $k$ the perimeter
bounds on each of them we conclude that

$\textnormal{Per}(E;\varphi(\mathcal{B}_{1/2}))\leq
C(n,s_{0},\Lambda)\frac{1+\sqrt{\kappa}}{\sqrt{\sigma}}$

as desired. ∎

The proof of Theorem 5.4 in the finite index case will need the following
covering-type lemma, which was devised in [18] to exhibit estimates in the
finite Morse index case by iteratively reducing to the almost-stable case. It
was inspired by the proof of [26, Proposition 2.6]. Essentially, it says that
if every time we take $(m+1)$ cubes which are sufficiently far from each other
we are able to prove an estimate of power-type on at least one of them, then
the estimate actually holds everywhere. The proof consists in subdividing a
cube at scale 1 into small subcubes, adding the estimate on the small subcubes
(except for a “bad” set, which will consist of at most $m$ subcubes and their
close neighbours), and then iterating the procedure dyadically by subdividing
the “bad” subcubes once again into smaller subcubes and starting the argument
again.

######  Lemma 5.11 ([18]).

Let $n\geq 1$, $m\geq 0$, $\theta\in(0,1)$, $D_{0}>0$ and $\beta>0$. Let
$\mathcal{S}:\mathfrak{B}\to[0,+\infty)$ be a subadditive333Meaning
subadditive for finite unions of (hyper)cubes. function defined on the family
$\mathfrak{B}$ of the (hyper)cubes contained in
$\mathcal{Q}_{1}(0)\subset\mathbb{R}^{n}$. Denote by
$\mathcal{Q}_{r}(x)\subset\mathbb{R}^{n}$ the closed (hyper)cube of center $x$
and side $r$, and assume that

  * (i)

$\displaystyle\sup_{\\{x\,:\,\mathcal{Q}_{r}(x)\in\mathfrak{B}\\}}\mathcal{S}(\mathcal{Q}_{r}(x))\to
0$ as $r\to 0$.

  * (ii)

Whenever
$\mathcal{Q}_{r}(x_{0}),\mathcal{Q}_{r}(x_{1}),\dotsc,\mathcal{Q}_{r}(x_{m})\subset\mathcal{Q}_{1}(0)$
are $(m+1)$ disjoint cubes of the same side at pairwise distance at least
$D_{0}r$, then

$\exists\,i\in\\{0,1,\dotsc,m\\}\hskip 7.0pt\textit{such that}\hskip
7.0pt\mathcal{S}(\mathcal{Q}_{\theta r}(x_{i}))\leq r^{\beta}M_{0}\,.$

Then

$\mathcal{S}(\mathcal{Q}_{1/2}(0))\leq CM_{0}\,,$

for some $C=C(n,\theta,m,\beta,D_{0})>0$.

With the lemma at hand, we can finally give the

###### Proof of Theorem 5.4.

As in previous arguments, we assume for concreteness that we are assuming
(5.18) and we want to prove (5.19); the proof of (5.17) is again even simpler.
Let
$\mathcal{Q}_{r}(x_{0}),\mathcal{Q}_{r}(x_{1}),\dotsc,\mathcal{Q}_{r}(x_{m})\subset\mathcal{Q}_{1}(0)$
be $(m+1)$ disjoint cubes of the same side $r$ and at pairwise distance at
least $D_{0}r$. By Lemmas 5.8 and 3.10, we immediately see that there exists
some $\ell\in\\{0,\dotsc m\\}$ such that the inequality

$\delta^{2}{\rm
Per}_{s}(E)[\xi,\xi]\geq-\frac{Cm}{(D_{0}r/2)^{n+s}}\|\xi\|^{2}_{L^{1}(\partial
E\cap\varphi(\mathcal{B}_{r/2}(x_{\ell})))}=:-\frac{\Lambda}{(r/2)^{n+s}}\|\xi\|^{2}_{L^{1}(\partial
E\cap\varphi(\mathcal{B}_{r/2}(x_{\ell})))}$ (5.23)

holds for all $\xi\in C_{c}^{\infty}(\partial
E\cap\varphi(\mathcal{B}_{r/2}(x_{\ell})))$ and some $C=C(n,s_{0})$; we have
used that $\mathcal{B}_{r/2}(x_{l})\subset\mathcal{Q}_{r}(x_{l})$. Considering
the rescaled manifold $\widehat{M}:=(M,(2/r)^{2}g)$, by Remarks 3.8 and 5.9
the flatness assumption ${\rm
FA}_{2}(M,(2/r)^{2}g,1,q_{\ell},\varphi_{x_{\ell},r/2})$ holds and $E$ is a
$\Lambda$-almost stable $s$-minimal surface in
$\varphi_{x_{\ell},r/2}(\mathcal{B}_{1}(x_{\ell}))\subset\widehat{M}$.
Therefore, we can apply Proposition 5.10 (selecting $D_{0}$ be large enough
that $\Lambda=\frac{Cm}{D_{0}^{n+s}}\leq\Lambda_{0}$) and get that

$\textnormal{Per}(E;\varphi_{x_{\ell},1/2}(\mathcal{B}_{1/2}))\leq
C\frac{1+\sqrt{\kappa}}{\sqrt{\sigma}}\,,$

which scaling back to $M$ shows that

$\mathcal{S}(\mathcal{Q}_{r/C}(x_{\ell}))\leq\textnormal{Per}(E;\varphi(\mathcal{B}_{r/4}(x_{l})))\leq
C\frac{1+\sqrt{\kappa}}{\sqrt{\sigma}}r^{n-1}$

for some $C=C(n,s_{0},m)$. Defining the subadditive function

$\mathcal{S}(\mathcal{Q}):={\rm
Per}(E;\varphi(\frac{1}{2\sqrt{n}}\mathcal{Q}))\,,$

where $\lambda\mathcal{Q}:=\\{\lambda y\,:\,y\in\mathcal{Q}\\}\,,$ this shows
that (ii) in Lemma 5.11 is satisfied, with $\beta=n-1$,
$M_{0}=C\frac{1+\sqrt{\kappa}}{\sqrt{\sigma}}$, and $\theta$ and $D_{0}$
depending only on $n$, $s_{0}$ and $m$. Moreover, (i) in Lemma 5.11 trivially
holds, since $E$ is just a fixed set with finite perimeter and thus has small
perimeter on small enough subsets. Hence, applying the Lemma we find that

$\mathcal{S}(\mathcal{Q}_{1/2}(0))={\rm
Per}(E;\varphi(\frac{1}{4\sqrt{n}}\mathcal{Q}))\leq
C\frac{1+\sqrt{\kappa}}{\sqrt{\sigma}}\,,$

which after a simple scaling+covering argument shows that

${\rm Per}(E;\varphi(\mathcal{B}_{1/2}))\leq
C\frac{1+\sqrt{\kappa}}{\sqrt{\sigma}}$

as well. ∎

###  5.3 Estimates for almost-stable $s$-minimal surfaces

####  5.3.1 $L^{-2}$ separation estimate for almost-stable graphs

In all this section we will always assume that $n\geq 2$ and

$l:=n+2\,.$ (5.24)

We will prove our estimates under the assumption that

${\rm FA}_{\ell}(M,g,4,p_{0},\varphi)\hskip 7.0pt\mbox{is satisfied}$ (5.25)

and that there is $\kappa<\infty$ such that $E\subset M$ is an $s$-minimal set
in $\varphi(\mathcal{B}_{2})$ with a fractional perimeter bound

$\textnormal{Per}_{s}(E;\varphi(\mathcal{B}_{2}))\leq\frac{\kappa}{1-s}=\frac{\kappa}{\sigma}\,.$
(5.26)

Recall that in (5.25), $\varphi:\mathcal{B}_{4}\to M$ denotes a chart
parametrization. We will suppose, in addition, that defining444The notation
$\mathcal{B}_{r}^{\prime}$ denotes the $(n-1)$-th dimensional Euclidean ball
of radius $r$ and centered at $0\in\mathbb{R}^{n-1}$.
$\Omega_{0}:=\mathcal{B}_{1}^{\prime}\times(-1,1)\subset\mathcal{B}_{2}\subset\mathbb{R}^{n}$,

$\displaystyle\varphi^{-1}(\partial E)\mbox{ coincides with }\
\Gamma=\bigcup_{i=1}^{N}\,\Gamma_{i}=\bigcup_{i=1}^{N}\,\\{x_{n}=g_{i}(x^{\prime})\\}\
\mbox{ inside }\Omega_{0},\quad{\mbox{for some
}}g_{i}:\mathcal{B}_{1}^{\prime}\to\mathbb{R}$ (5.27)
$\displaystyle{\mbox{satisfying}}\quad
g_{1}<g_{2}<\cdots<g_{N}\qquad\mbox{and}\qquad\|\nabla
g_{i}\|_{L^{\infty}(\mathcal{B}_{1}^{\prime})}+\|D^{2}g_{i}\|_{L^{\infty}(\mathcal{B}_{1}^{\prime})}<\delta\,.$
(5.28)

Here the number $N$ of graphs is some arbitrarily large positive integer. We
will shortly be able to deduce that, in a smaller cylinder contained in
$\Omega_{0}$, the number of graphical layers is actually bounded by a constant
depending only on the choice of constants above (but independent of $\sigma$,
so that it is uniform as $s\to 1$).

We now record a simple comparison lemma for graphs verifying flatness
assumptions as in our hypotheses.

######  Lemma 5.12 ([20, Lemma 3.3]).

Let $1-s=\sigma\in(0,1/2)$. Assume that $E$ is a set such that (5.25) and
(5.27)–(5.28) hold. Moreover, assume that for some $i<j$ we have
$\min_{\overline{B_{1}^{\prime}}}(g_{j}-g_{i})\leq\widetilde{\delta}$ for some
$\widetilde{\delta}>0$, and let555Note that, in terms of the standard Beta and
Gamma functions,
$c_{n,s}=\frac{\mathcal{H}^{n-2}(\mathbb{S}^{n-2})}{2}{\mathrm{B}}\Bigl{(}\frac{n-1}{2},\frac{1+s}{2}\Bigr{)}=\frac{\mathcal{H}^{n-2}(\mathbb{S}^{n-2})\Gamma\bigl{(}\frac{n-1}{2}\bigr{)}\Gamma\bigl{(}\frac{1+s}{2}\bigr{)}}{2\Gamma\bigl{(}\frac{n+s}{2}\bigr{)}}=\frac{\mathcal{H}^{n-2}(\mathbb{S}^{n-2})}{n-1}+O(\sigma)$
is universally comparable to $1$.

$c_{n,s}:=\mathcal{H}^{n-2}(\mathbb{S}^{n-2})\int_{0}^{\infty}(1+t^{2})^{-\frac{n+s}{2}}\,t^{n-2}\,dt.$

Then, it follows that

$\big{(}1-c_{\delta,\widetilde{\delta}}\big{)}\frac{c_{n,s}\;\sigma}{|g_{j}(x^{\prime})-g_{i}(x^{\prime})|^{1+s}}\leq\int_{\Gamma_{j}}\frac{\sigma}{|x-y|^{n+s}}\,dy\leq\big{(}1+c_{\delta,\widetilde{\delta}}\big{)}\frac{c_{n,s}\;\sigma}{|g_{j}(x^{\prime})-g_{i}(x^{\prime})|^{1+s}}$

for all $x\in\Gamma_{i}\cap B_{3/4}^{\prime}\times\mathbb{R}$, where
$c_{\delta,\widetilde{\delta}}\leq
C(\sqrt{\delta}+\widetilde{\delta})\downarrow 0$ as
$\delta,\widetilde{\delta}\downarrow 0$.

######  Proposition 5.13.

Let $E$ be $\Lambda$-almost stable $s$-minimal in $\varphi(\mathcal{B}_{2})$,
and assume that (5.25)–(5.28) hold true, with $(1-s)=\sigma\in(0,1/2)$. There
exists $C<\infty$, depending only on $\Lambda$, $n$, and the constants in
(5.25)–(5.28), such that for every $i$ such that

$\min_{\overline{B}_{1/2}^{\prime}}|g_{i}|\leq\frac{1}{2},$

we have that

$\sigma\int_{B^{\prime}_{1/2}}\frac{dx^{\prime}}{|g_{i+1}-g_{i}|^{1+s}}\leq
C.$ (5.29)

###### Proof.

Observe that $B^{\prime}_{1}\times[-1,1]\subset\mathcal{B}_{2}$. Let $X$ be a
$C^{1}$ vector field of the form $\varphi_{*}(e_{n}\eta)$, where $e_{n}$ is
the $n$–th basis coordinate vector of $\mathbb{R}^{n}$ (i.e. pointing
”upwards”, in the direction in which the $\Gamma_{i}$ are graphical) and
$\eta$ is a standard cutoff verifying
$\chi_{\mathcal{B}_{1/2}^{\prime}\times[-3/4,3/4]}\leq\eta\leq\chi_{\mathcal{B}_{2/3}^{\prime}\times[-4/5,4/5]}$.
Define the function $\xi=\langle X,\nu_{\partial E}\rangle_{g}$ on $\partial
E$. As in (5.20)–(5.21), considering the test functions $\xi$ and $|\xi|$, the
$\Lambda$-almost stability assumption gives

$2\iint_{\partial E\times\partial
E}\Big{[}|\xi(p)||\xi(q)|-\xi(p)\xi(q)\Big{]}K(p,q)\leq\delta^{2}\text{Per}_{s}(E)[\xi,\xi]+\Lambda\Big{(}\int_{\partial
E}|\xi|\Big{)}^{2}\,.$ (5.30)

Now, we have defined

$\xi(p)=\langle X,\nu_{\partial
E}\rangle_{g}(p)=(\eta\circ\varphi^{-1})(p)\langle\varphi_{*}(e_{n}),\nu_{\partial
E}\rangle_{g}(p)\,.$ (5.31)

Recall that, by assumptions (5.27) and (5.28), $\varphi^{-1}(\partial E)$ is a
union of very flat graphs $\Gamma_{j}$ of the form
$\Gamma_{j}=\\{x_{n}=g_{j}(x^{\prime})\\}$. Together with the flatness
assumptions on the metric, this implies that

$|\langle\varphi_{*}(e_{n}),\nu_{\partial E}\rangle_{g}|\geq
c_{1}|e_{n}\cdot\varphi_{*}^{-1}(\nu_{\partial E})|\geq c_{2}\,,$

where $c_{2}>0$ depends only on $\delta$ from (5.28). Furthermore,
$\langle\varphi_{*}(e_{n}),\nu_{\partial E}\rangle_{g}$ changes sign on
$\varphi(\Gamma_{i})$ and $\varphi(\Gamma_{i+1})$, since the consecutive
layers $\Gamma_{i}$ and $\Gamma_{i+1}$ have normal vectors pointing in
(almost) opposite directions.
These simple facts and (5.31) imply that, for
$p\in\varphi(\Gamma_{i}\cap\mathcal{B}_{1/2}^{\prime}\times[-3/4,3/4])$ and
$q\in\varphi(\Gamma_{i+1}\cap\mathcal{B}_{1/2}^{\prime}\times[-3/4,3/4])$,
then

$\Big{[}|\xi(p)||\xi(q)|-\xi(p)\xi(q)\Big{]}=2|\xi(p)||\xi(q)|\geq
2c_{2}^{2}[(\eta\circ\varphi^{-1})(p)][(\eta\circ\varphi^{-1})(q)]\geq
2c_{2}^{2}\,.$

Substituting into the original inequality (5.30), we deduce that

$8c_{2}^{2}\iint_{\Sigma_{i}\times\Sigma_{i+1}}K(p,q)\leq\delta^{2}\text{Per}_{s}(E)[\xi,\xi]+\Lambda\Big{(}\int_{\partial
E}|\xi|\Big{)}^{2}\,,$

where
$\Sigma_{j}:=\varphi(\Gamma_{j}\cap\mathcal{B}_{1/2}^{\prime}\times[-3/4,3/4])$.
So, passing to coordinates with $\varphi$, using Lemma 3.10 and Lemma 5.12 we
obtain

$c\sigma\int_{B^{\prime}_{1/2}}\frac{1}{|g_{i+1}(x^{\prime})-g_{i}(x^{\prime})|^{1+s}}dx^{\prime}\leq\sigma\delta^{2}\text{Per}_{s}(E)[\xi,\xi]+\Lambda\sigma\Big{(}\int_{\partial
E}|\xi|\Big{)}^{2}\,.$ (5.32)

On the other hand, the right hand side in this inequality is bounded: To
estimate the first term, recalling Definition 5.3, by Lemma 3.12 and the
assumption (5.26) we can bound

$\sigma\delta^{2}\text{Per}_{s}(E)[\xi,\xi]=\sigma\frac{d^{2}}{dt^{2}}\Big{|}_{t=0}\textnormal{Per}_{s}(\phi_{X}^{t}(E))\leq\sigma
C\textnormal{Per}_{s}(E)\leq C\kappa\,.$ (5.33)

To estimate the second term (corresponding to the error introduced by almost-
stability), we apply the classical perimeter bound (5.18) given by Proposition
5.10, since then we can bound

$\sigma\Lambda\Big{(}\int_{\partial E}|\xi|\Big{)}^{2}\leq
C\sigma\Lambda\textnormal{Per}^{2}(E,\mathcal{B}^{\prime}_{2/3}\times[-4/5,\,4/5])\leq
C(\Lambda,n,s_{0})\,.$ (5.34)

To be precise, Proposition 5.10 is stated on a ball of radius one. We can
cover $\mathcal{B}^{\prime}_{2/3}\times[-4/5,\,4/5]$ with a finite number
$C_{n}$ of balls $\mathcal{B}_{1/10}(x_{k})$ so that
$\mathcal{B}_{1/5}(x_{k})\subset\mathcal{B}^{\prime}_{1}\times[-1,1]$ for all
$1\leq k\leq C_{n}$. As in Step 2 of the proof of Proposition 5.10, applying
Proposition 5.10 on the manifold $(M,5^{2}g)$ and scaling back we see that

$\textnormal{Per}(E;\mathcal{B}_{1/5}(x_{k}))\leq
C\frac{1+\sqrt{\kappa}}{\sqrt{\sigma}}\,,$

which summing over $k$ shows that

$\textnormal{Per}(E,\mathcal{B}^{\prime}_{2/3}\times[-4/5,\,4/5])\leq
C(\Lambda,n,s_{0})\frac{1+\sqrt{\kappa}}{\sqrt{\sigma}}$

thus giving (5.34).

Applying (5.33) and (5.34) to the RHS of (5.32), we conclude that

$\sigma\int_{B^{\prime}_{1/2}}\frac{1}{|g_{i+1}(x^{\prime})-g_{i}(x^{\prime})|^{1+s}}dx^{\prime}\leq
C\,,$

where $C$ has the right dependencies. ∎

The next lemma gives a rough lower bound for the separation between layers.
Any bound which is polynomial in $\sigma$ would serve our purposes here;
later, we will upgrade it to the optimal rate $\sigma^{1/2}$ in the case
$n=3$.

######  Lemma 5.14.

Let $E\subset M$ be $\Lambda$-almost stable $s$-minimal in
$\varphi(\mathcal{B}_{2})$, and assume that (5.25)–(5.28) hold true, with
$(1-s)=\sigma\in(0,1/2)$. There exists $C<\infty$, depending only on
$\Lambda$, $n$, and the constants in (5.25)–(5.28), such that for every $i$
such that

$\min_{\overline{B}_{1/2}^{\prime}}|g_{i}|\leq\frac{1}{2},$

we have that

$\inf_{B_{1/2}^{\prime}}(g_{i+1}-g_{i})\geq\frac{\sigma^{2}}{C}.$

###### Proof.

Let $i$ be as in the statement, and assume that there exists a point
$z^{\prime}\in\mathcal{B}^{\prime}_{1/2}$ such that
$g_{i+1}(z^{\prime})-g_{i}(z^{\prime})=\tau^{2}\sigma^{2}$ for some
$\tau\in(0,1)$. Our goal will be to bound $\tau$ away from zero (hence we may
assume without loss of generality that $\tau$ is very small whenever needed).

By the assumption on $i$, we have that $|g_{i}(z^{\prime})|<3/4$. Let
$z:=(z^{\prime},g_{i}(z^{\prime}))$ and $\varrho:=\tau\sigma$. Define
$\varphi_{z,\varrho}:=\varphi(z+\varrho\,\cdot\,)$. In the coordinates given
by $\varphi_{z,\varrho}$, $\Gamma$ is simply transforming into the rescaled
set $\widetilde{\Gamma}:=\frac{\Gamma-z}{\varrho}\cap\mathcal{B}_{1}$, in the
sense that
$(\varphi_{z,\varrho}^{-1}\circ\varphi)(\Gamma\cap\mathcal{B}_{\varrho}(z))=\widetilde{\Gamma}$.

By (c) in Remark 3.8 and (5.25), if $\rho$ is sufficiently small the rescaled
manifold $(M,\frac{1}{\varrho^{2}}g)$ satisfies the flatness assumptions ${\rm
FA}_{\ell}(M,\frac{1}{\varrho^{2}}g,1,\varphi(z),\varphi_{z,\varrho})$.
Moreover, by the same argument we gave in Step 2 of the proof of Proposition
5.10, $E$ is a $\Lambda$-almost stable $s$-minimal set for $(M,\varrho^{2}g)$
in $\varphi_{z,\varrho}(\mathcal{B}_{1})$; in fact, it is even
$\varrho^{n+s}\Lambda$-almost stable. Likewise, thanks to (3.29) the
fractional perimeter bound (5.26) is satisfied in this rescaled setting, up to
changing the value of $\kappa$. Finally, we also observe that the “layers”
$\\{x_{n}=g_{j}(x^{\prime})\\}$ become, in the coordinates given by
$\varphi_{z,\varrho}$,

$\\{x_{n}=\widetilde{g}_{j}(x^{\prime})\\}\qquad\mbox{where
}\|D^{2}\widetilde{g}_{j}\|_{L^{\infty}}\leq\varrho\delta,\quad\mbox{for all
}j=1,\dots,N.$

Therefore, we can apply Proposition 5.13 in our rescaled situation and deduce
that

$\sigma\int_{B_{1/2}^{\prime}}\frac{1}{|\widetilde{g}_{i+1}(x^{\prime})-\widetilde{g}_{i}(x^{\prime})|^{1+s}}\,dx^{\prime}\leq
C.$

On the other hand, since the point $z$ is mapped to $0$, we have that

$\widetilde{g}_{i}(0)=0\quad\mbox{and}\quad\widetilde{g}_{i+1}(0)\leq\varrho.$

Combining this with the bound for the $D^{2}\widetilde{g}_{j}$, a Taylor
expansion immediately gives that, for $\delta,\varrho$ sufficiently small,

$0\leq\widetilde{g}_{i+1}-\widetilde{g}_{i}\leq 2\varrho\quad\mbox{in
}B_{1}^{\prime}.$

Therefore, we also get a lower bound

$\frac{1}{\varrho^{1+s}}\leq
C\int_{B_{1/2}^{\prime}}\frac{1}{|\widetilde{g}_{i+1}(x^{\prime})-\widetilde{g}_{i}(x^{\prime})|^{1+s}}\,dx^{\prime}\,.$

Combining the above, we deduce that

$\frac{\sigma}{\varrho^{1+s}}\leq C\,,$

which since $\varrho=\tau\sigma$ gives that $\sigma^{-s}\leq C\tau^{1+s}$. In
particular, this shows that $\tau$ is bounded away from zero with a uniform
bound for $\sigma\in(0,1/2)$, as desired. ∎

From this rough separation estimate, we can already deduce that the number of
layers inside a smaller cylinder is bounded.

######  Lemma 5.15.

Let the same assumptions as in Lemma 5.14 hold. Then, defining $N_{2}$ as the
number of layers of $\varphi^{-1}(\partial E)$ intersecting
$\mathcal{B}_{1/2}^{\prime}\times[-1/2,1/2]$, there exists $C<\infty$,
depending only on $\Lambda$, $n$ and the constants in (5.25)–(5.28), such that

$N_{2}\leq C\,.$

Therefore, we deduce that

$\textnormal{Per}(E;\varphi(\mathcal{B}_{1/2}^{\prime}\times[-1/2,1/2]))\leq
C\,.$

###### Proof.

$N_{2}$ corresponds to the number of indices $i$ such that

$\min_{\overline{B}_{1/2}^{\prime}}|g_{i}|\leq\frac{1}{2}.$

Since for any such $i$ the associated $\Gamma_{i}$ is a graph with uniform
estimates (by (5.27)–(5.28)), bounding the number $N_{2}$ of such $i$
immediately gives a bound for the total classical perimeter. We thus focus on
bounding $N_{2}$, which will be possible thanks to every layer $\Gamma_{i}$
“contributing a positive amount to the fractional perimeter” due to the rough
separation estimate between layers.

Claim. Let $i$ be such that

$\min_{\overline{B}_{1/2}^{\prime}}|g_{i}|\leq\frac{1}{2},$

and let $j\neq i$. Then there is $c>0$ small enough such that, given points
$x\in\Gamma_{i}$ and $y\in\Gamma_{j}$, the balls $\mathcal{B}_{r}(x)$ and
$\mathcal{B}_{r}(y)$ are disjoint for $r=c\sigma^{4}$.

###### Proof of the claim.

Indeed, if $\mathcal{B}_{r}(x)\cap\mathcal{B}_{r}(y)\neq\emptyset$, then in
particular $|y^{\prime}-x^{\prime}|\leq 2r$ and
$|g_{j}(y^{\prime})-g_{i}(x^{\prime})|\leq 2r$. But then, by Lemma 5.14 and
(5.28),

$\displaystyle 2c\sigma^{4}$ $\displaystyle=2r$
$\displaystyle\geq|g_{j}(y^{\prime})-g_{i}(x^{\prime})|$
$\displaystyle\geq|g_{j}(x^{\prime})-g_{i}(x^{\prime})|-|g_{j}(y^{\prime})-g_{j}(x^{\prime})|$
$\displaystyle\geq\frac{\sigma^{4}}{C}-\delta|y^{\prime}-x^{\prime}|$
$\displaystyle\geq\frac{\sigma^{4}}{C}-2\delta r$
$\displaystyle=\frac{\sigma^{4}}{C}-2\delta c\sigma^{4}\,,$

or in other words $\frac{1}{C}\leq 2(1+\delta)c$, which is a contradiction for
$c=\frac{1}{2C(2+\delta)}$ or smaller. ∎

Now that the claim has been proved, denote $F=\varphi^{-1}(E)$, and let $r$ be
as in the claim. Now, thanks to the uniform $C^{2}$ estimates on our graphs,
for $s$ sufficiently close to $1$ (i.e. $\sigma$ close to $0$) we have the
following: if $x\in\Gamma_{i}$, then both $|F\cap\mathcal{B}_{r}(x)|\geq
1/4|\mathcal{B}_{r}(x)|$ and $|F^{c}\cap\mathcal{B}_{r}(x)|\geq
1/4|\mathcal{B}_{r}(x)|$ hold. This is true since satisfying uniform $C^{2}$
estimates means that $\Gamma_{i}$ separates from its tangent hyperplane at $x$
at most $Cr^{2}$ inside $\mathcal{B}_{r}(x)$, so that for $\sigma$ (and thus
$r$) sufficiently small $\Gamma_{i}$ actually divides the ball
$\mathcal{B}_{r}(x)$ in two pieces of almost equal volume. Denote

$\textnormal{Per}_{s}|_{\mathcal{B}_{r}(x)}(F):=\iint_{(F\cap\mathcal{B}_{r}(x))\times(F^{c}\cap\mathcal{B}_{r}(x))}\frac{1}{|x-y|^{n+s}}\,dx\,dy\,.$

Then, by the relative fractional isoperimetric inequality (see [28]) together
with $\min\\{|F\cap\mathcal{B}_{r}(x)|,|F^{c}\cap\mathcal{B}_{r}(x)|\\}\geq
1/4|\mathcal{B}_{r}(x)|$, we have that

$\textnormal{Per}_{s}|_{\mathcal{B}_{r}(x)}(F)\geq\frac{c}{1-s}r^{n-s}=\frac{c}{\sigma}r^{n-s}$

with $c(n)>0$. Moreover, since $\Gamma_{i}$ is a graph over the
$(n-1)$-dimensional ball of radius $1/2$ in $\mathbb{R}^{n-1}$, we can find at
least $k\geq c/r^{n-1}$ disjoint balls of radius $r$ centered at points
$x_{1}^{i}\in\Gamma_{i},...,x_{k}^{i}\in\Gamma_{i}$ (indeed, it is clear that
one can arrange at least $k\geq c/r^{n-1}$ points in a square lattice inside
the Euclidean ball of radius $1/2$ in $\mathbb{R}^{n-1}$ with distance $r$
between adjacent points, and then it suffices to take their images under the
graph parametrisation $g_{i}$). Repeating this for each $i=1,...,N_{2}$ and
observing that any pair of balls obtained in this way will be disjoint (by
construction if they belong to the same $\Gamma_{i}$, and by the choice of $r$
at the beginning of the proof if they belong to $\Gamma_{i}$ and $\Gamma_{j}$
with $i\neq j$), we can bound

$\displaystyle\textnormal{Per}_{s}(F,\mathcal{B}^{\prime}_{1/2}\times(-3/4,\,3/4))$
$\displaystyle\geq\sum_{1\leq i\leq N_{2}}\sum_{1\leq l\leq
k}\textnormal{Per}_{s}|_{\mathcal{B}_{r}(x_{l}^{i})}(F)$ $\displaystyle\geq
N_{2}k\frac{c}{\sigma}r^{n-s}$ $\displaystyle\geq
N_{2}\frac{c}{\sigma}r^{1-s}$
$\displaystyle=N_{2}\frac{c}{\sigma}\sigma^{-4\sigma}$ $\displaystyle\geq
N_{2}\frac{c}{2\sigma}\,.$

In the last line we are taking $s$ sufficiently close to $1$ (equiv.
$\sigma=1-s$ sufficiently close to $0$) so that

$\sigma^{-4\sigma}=1+O(\sigma|\log(\sigma)|)\geq\frac{1}{2}\,.$

In other words, we have found that the number $N_{2}$ of layers is bounded by

$N_{2}\leq
C(1-s)\textnormal{Per}_{s}(F,\mathcal{B}^{\prime}_{1/2}\times(-3/4,\,3/4))\,.$

Moreover, by Lemma 3.10 and the flatness of the metric it is immediate that

$\textnormal{Per}_{s}(F,\mathcal{B}^{\prime}_{1/2}\times(-3/4,\,3/4))\leq
C\textnormal{Per}_{s}(E,\varphi(\mathcal{B}^{\prime}_{1/2}\times(-3/4,\,3/4)))\leq
C\textnormal{Per}_{s}(E,\varphi(\Omega_{0}))\,.$

Combining these two things, by assumption (5.26) we then deduce that
$N_{2}\leq C\kappa$, which concludes the proof. ∎

We now introduce the following observation, which will be very useful since it
will permit us to relate volume and boundary integrals through integration by
parts while allowing us to reuse our known estimates for the family of kernels
of the type $K_{s}$ (defined through (3.1)):

######  Lemma 5.16.

The identity

$K_{s}(p,q)=-\textnormal{div}_{q}(\nabla_{q}K_{s-2})(p,q)$

holds.

###### Proof.

By definition,
$K_{s}(p,q)=\frac{s/2}{\Gamma(1-s/2)}\int_{t}\frac{H_{t}}{t^{1+s/2}}\,dt$. We
start by computing, using the identity $\Gamma(1+z)=z\Gamma(z)$:

$\displaystyle K_{s-2}(p,q)$
$\displaystyle=\frac{(s-2)/2}{\Gamma(1-(s-2)/2)}\int_{t}H_{t}(p,q)/t^{1+(s-2)/2}$
$\displaystyle=-\frac{1-s/2}{\Gamma(1+[1-s/2])}\int_{t}H_{t}(p,q)/t^{s/2}$
$\displaystyle=-\frac{1}{\Gamma(1-s/2)}\int_{t}H_{t}(p,q)/t^{s/2}\,.$

Therefore, by an application of the heat equation and integration by parts (in
time), we can compute

$\displaystyle\textnormal{div}_{q}(\nabla_{q}K_{s-2})(p,q)$
$\displaystyle=\Delta_{q}K_{s-2}(p,q)$
$\displaystyle=-\frac{1}{\Gamma(1-s/2)}\int_{t}\Delta_{p}H_{t}(p,q)/t^{s/2}$
$\displaystyle=-\frac{1}{\Gamma(1-s/2)}\int_{t}(\partial_{t}H_{t}(p,q))/t^{s/2}$
$\displaystyle=-\frac{s/2}{\Gamma(1-s/2)}\int_{t}H_{t}(p,q)/t^{1+s/2}$
$\displaystyle=-K_{s}(p,q)\,,$

giving the claimed result. ∎

This allows us to make the following ”generalised” definition, justified by
the remark that follows it:

######  Definition 5.17.

The fractional mean curvature (or nonlocal mean curvature, or NMC) of a two-
sided regular hypersurface $\Sigma\subset M$ with choice of normal vector
$\nu_{\Sigma}$, where $\Sigma\subset M$ is not assumed to be part of the
boundary of a set, at $p\in\Sigma$ is defined as

$H_{s}[\Sigma](p):=2\sigma\int_{\Sigma}\langle\nu_{\Sigma}(q),\nabla_{q}K_{s-2}(p,q)\rangle_{g}\,dq\,.$

######  Remark 5.18.

Definition 5.17 is compatible with Definition 5.2, in the sense that given a
regular open domain $E$, setting $\Sigma=\partial E$ and $\nu_{\Sigma}$ the
outer normal vector gives the same value for $H_{s}[\Sigma]$ with both
definitions. This is immediately seen by an application of the divergence
theorem together with Lemma 5.16.

We can now deduce the desired $L^{-2}$ bound for the separation between
layers, as well as the decay in $L^{2}$ of the nonlocal mean curvature of each
of the individual layers as $s\to 1$. We remark that we are making no
assumptions on the dimension $n$ in this entire section.

######  Proposition 5.19.

Let $E\subset M$ be $\Lambda$-almost stable $s$-minimal in
$\varphi(\mathcal{B}_{2})$, and assume that (5.25)–(5.28) hold true, with
$(1-s)=\sigma\in(0,1/2)$. There exists $C<\infty$, depending only on
$\Lambda$, $n$ and the constants in (5.25)–(5.28), such that for every $i$
such that

$\min_{\overline{B}_{1/2}^{\prime}}|g_{i}|\leq\frac{1}{2},$

we have that

$\sigma\int_{B^{\prime}_{1/2}}\frac{dx^{\prime}}{|g_{i+1}-g_{i}|^{2}}\leq C.$
(5.35)

Moreover, defining $f_{i}:=H_{s}[\varphi(\Gamma_{i})]$, we have that

$\|f_{i}\|_{L^{2}(\varphi(\Gamma_{i}\cap[B^{\prime}_{1/2}\times(-1,1)]))}\leq
C\sqrt{\sigma}.$ (5.36)

###### Proof.

Step 1. In Proposition 5.13 we have shown that

$\sigma\int_{B^{\prime}_{1/2}}\frac{dx^{\prime}}{|g_{i+1}-g_{i}|^{1+s}}\leq
C.$

To reach (5.35), we need to upgrade the exponent in the denominator from
$(1+s)$ to $2$. For this, it suffices to notice that since by Lemma 5.14 we
have that $(g_{i+1}-g_{i})\in(\sigma^{4}/C,1)$, we can then estimate

$|g_{j}-g_{i}|^{s}=\bigl{(}1+O(\sigma|\log\sigma|)\bigr{)}|g_{j}-g_{i}|.$
(5.37)

Step 2. We deduce (5.36) from (5.35).

As before, we use the notation
$\Omega_{0}=\mathcal{B}_{1}^{\prime}\times(-1,1)\subset\mathbb{R}^{n}$. We
want to split the contributions of each of the individual layers of $E$ to the
total nonlocal mean curvature. For that, define the set
$A_{i}=\varphi(\\{x_{n}<g_{i}(x^{\prime})\\}\cap\Omega_{0})$, so that
$\partial
A_{i}=\varphi(\Gamma_{i})\cup\varphi(\partial\Omega_{0}\cap\\{x_{n}<g_{i}(x^{\prime})\\})$.
In other words, in coordinates $A_{i}$ comprises all the space below the graph
$\Gamma_{i}$, and the set $\partial A_{i}$ corresponds to a portion of the
layer $\Gamma_{i}$ (plus some extra piece contained in $\partial\Omega_{0}$).
Define also the sets $E_{i}=A_{i}\setminus A_{i-1}$, so that (up to possibly
interchanging $E$ with its complement) in particular it follows that
$E\cap\varphi(\Omega_{0})=\bigcup_{i\in 2\mathbb{Z}}E_{i}$.
Let $i$ be such that
$\min_{\overline{B}_{1/2}^{\prime}}|g_{i}|\leq\frac{1}{2}$, with $i$ even (the
odd case is essentially identical), and let
$p\in\varphi(\Gamma_{i}\cap\\{x^{\prime}\in\mathcal{B}^{\prime}_{1/2}\\})$.
With the definitions above, it is immediate to see that we can split

$\displaystyle\int_{\varphi(\Omega_{0})}dq\,(\chi_{E}-\chi_{E^{c}})(q)K_{s}(p,q)=\int_{\varphi(\Omega_{0})}dq\,(\chi_{A_{i}}-\chi_{A_{i}^{c}})(q)K_{s}(p,q)$
$\displaystyle-2\sum_{j<i,\,j-i\,odd}\int_{\varphi(\Omega_{0})}dq\,\chi_{E_{j}}(q)K_{s}(p,q)$
$\displaystyle+2\sum_{j>i,\,j-i\,even}\int_{\varphi(\Omega_{0})}dq\,\chi_{E_{j}}(q)K_{s}(p,q)\,.$

On the other hand, since $E$ is an $s$-minimal surface in particular it has
zero NMC at $p$, so that recalling Definition 5.2 we have that (we omit $dq$
in the notation from now on)

$\int_{\varphi(\Omega_{0})}(\chi_{E}-\chi_{E^{c}})K_{s}(p,q)+\int_{\varphi(\Omega_{0})^{c}}(\chi_{E}-\chi_{E^{c}})K_{s}(p,q)=\int_{M}(\chi_{E}-\chi_{E^{c}})K_{s}(p,q)=0\,.$

Combining both expressions, we can then write

$\displaystyle-\int_{\varphi(\Omega_{0})}(\chi_{A_{i}}-\chi_{A_{i}^{c}})K_{s}(p,q)=-\int_{\varphi(\Omega_{0})^{c}}(\chi_{E}-\chi_{E^{c}})K_{s}(p,q)$
$\displaystyle-2\sum_{j<i,\,j-i\,odd}\int_{\varphi(\Omega_{0})}\chi_{E_{j}}K_{s}(p,q)$
$\displaystyle+2\sum_{j>i,\,j-i\,even}\int_{\varphi(\Omega_{0})}\chi_{E_{j}}K_{s}(p,q)\,.$

Applying the divergence theorem on the left hand side using Lemma 5.16, we
then find that

$\displaystyle
H_{s}[\varphi(\Gamma_{i})](p)=-2\sigma\sum_{j<i,j-i\,even}\int_{\varphi(\Omega_{0})}\chi_{E_{j}}K_{s}(p,q)+2\sigma\sum_{j>i,j-i\,even}\int_{\varphi(\Omega_{0})}\chi_{E_{j}}K_{s}(p,q)+\sigma{\rm
Ext}(p)\,,$ (5.38)

where

${\rm
Ext}(p):=-\int_{\varphi(\Omega_{0})^{c}}(\chi_{E}-\chi_{E^{c}})K_{s}(p,q)-\int_{\varphi(\partial\Omega_{0})}\langle\nu_{\partial\Omega_{0}}(q),\nabla_{q}K_{s-2}(p,q)\rangle_{g}\,d\mathcal{H}^{n-1}(q)\,.$

In other words, (5.38) says that, since the NMC of $E$ is zero, the
“individual” NMC of $\varphi(\Gamma_{i})$ in the sense of Definition 5.17 is
equal to the contribution from the other layers plus an exterior error.

Now, it is clear that the $E_{j}$ with $j<i$ are contained in
$A_{i-1}:=\varphi(\\{x_{n}<g_{i-1}(x^{\prime})\\}\cap\Omega_{0})$, and
likewise the $E_{j}$ with $j>i$ are contained in
$B_{i+1}:=\varphi(\\{x_{n}>g_{i+1}(x^{\prime})\\}\cap\Omega_{0})$, so that

$\Big{|}H_{s}[\varphi(\Gamma_{i})]\Big{|}(p)\leq
2\sigma\Big{|}\int_{A_{i-1}}K_{s}(p,q)\Big{|}+2\sigma\Big{|}\int_{B_{i+1}}K_{s}(p,q)\Big{|}+\sigma\Big{|}{\rm
Ext}\Big{|}(p)\,.$

Using this observation, passing to coordinates with $\varphi^{-1}$ (letting
$x:=\varphi^{-1}(p)$, and using Lemma 3.10 and the flatness assumptions) and
then integrating by parts (once again thanks to Lemma 5.16/the divergence
theorem, applied also with the Euclidean kernel $\frac{1}{|x-y|^{n+s}}$), we
can bound

$\displaystyle\Big{|}H_{s}[\varphi(\Gamma_{i})]\Big{|}(p)$ $\displaystyle\leq
2\sigma\Big{|}\int_{A_{i-1}}K_{s}(p,q)\Big{|}+2\sigma\Big{|}\int_{B_{i+1}}K_{s}(p,q)\Big{|}+\sigma\Big{|}{\rm
Ext}\Big{|}(p)$ $\displaystyle\leq
C\sigma\int_{\varphi^{-1}(A_{i-1})}dy\,\frac{1}{|x-y|^{n+s}}+C\sigma\int_{\varphi^{-1}(B_{i+1})}dy\,\frac{1}{|x-y|^{n+s}}+\sigma\Big{|}{\rm
Ext}\Big{|}(p)$ $\displaystyle\leq
C\sigma\int_{\Gamma_{i-1}}d\mathcal{H}^{n-1}(y)\frac{1}{|x-y|^{n+s-1}}+C\sigma\int_{\Gamma_{i+1}}d\mathcal{H}^{n-1}(y)\frac{1}{|x-y|^{n+s-1}}+\sigma{\rm
Ext}_{2}(p)\,.$

Here, the new exterior error term is

$\displaystyle{\rm Ext}_{2}(p)$
$\displaystyle=\Big{|}\int_{\varphi(\Omega_{0})^{c}}dq\,(\chi_{E}-\chi_{E^{c}})K_{s}(p,q)\Big{|}+\Big{|}\int_{\varphi(\partial\Omega_{0})}\langle\nu_{\Sigma}(q),\nabla_{q}K_{s-2}(p,q)\rangle_{g}\,d\mathcal{H}^{n-1}(q)\Big{|}$
$\displaystyle\ \
+\Big{|}\int_{\partial\Omega_{0}\cap\\{x_{n}<g_{i-1}(x^{\prime})\\}}\frac{1}{|x-y|^{n+s-1}}\,d\mathcal{H}^{n-1}(q)\Big{|}+\Big{|}\int_{\partial\Omega_{0}\cap\\{x_{n}>g_{i+1}(x^{\prime})\\}}\frac{1}{|x-y|^{n+s-1}}\,d\mathcal{H}^{n-1}(q)\Big{|}\,.$

We can bound the first term in ${\rm Ext}_{2}(p)$ by a uniform constant using
(3.17) (with $\alpha=0$). Moreover, by an application of (3.16), we can bound
the (differentiated) kernel in the second term by a uniform constant (since
$\textnormal{dist}(p,q)\geq\frac{1}{C}>0$ for all $q$ in
$\varphi(\Omega_{0})^{c}$), and likewise for the two last terms involving
$\frac{1}{|x-y|^{n+s-1}}$ (since $|x-y|\geq\frac{1}{C}$ for all $y$ in
$\partial\Omega_{0}$). Therefore, ${\rm Ext}_{2}(p)\leq C$.
Substituting this above and using Lemma 5.12 and (5.37), we can estimate

$\displaystyle\Big{|}H_{s}[\varphi(\Gamma_{i})]\Big{|}(p)$ $\displaystyle\leq
C\sigma\int_{\Gamma_{i-1}}\frac{1}{|x-y|^{n+s-1}}+C\sigma\int_{\Gamma_{i+1}}\frac{1}{|x-y|^{n+s-1}}+\sigma{\rm
Ext_{2}}(p)$ $\displaystyle\leq
C\frac{\sigma}{|g_{i}(x^{\prime})-g_{i-1}(x^{\prime})|^{s}}+C\frac{\sigma}{|g_{i+1}(x^{\prime})-g_{i}(x^{\prime})|^{s}}+C\sigma$
$\displaystyle\leq
C\frac{\sigma}{|g_{i}(x^{\prime})-g_{i-1}(x^{\prime})|}+C\frac{\sigma}{|g_{i+1}(x^{\prime})-g_{i}(x^{\prime})|}+C\sigma\,.$
(5.39)

Applying the $L^{-2}$ estimate in (5.35), we conclude the desired bound for
the $L^{2}$ norm of $H_{s}[\varphi(\Gamma_{i})]$. ∎

####  5.3.2 Uniform $C^{2,\alpha}$ estimates for almost-stable graphs in
dimension $n=3$

In this section we will prove that (for $n=3$) almost-stability implies a
uniform $C^{2,\alpha}$ bound for the layers of our surface. We remark that
this is the only section of the article in which the restriction to dimension
$3$ is used.
Our strategy is a bit different from the one in [20], and it allows us to
avoid using the nonlocal mean curvature equation for graphs, which could be
complicated to implement in a Riemannian setting. We show instead that, even
before upgrading the regularity of our layers to a uniform $C^{2,\alpha}$
estimate, the local and nonlocal mean curvatures of our layers are much closer
to each other than their natural size. This allows us to argue using the
classical minimal graph equation instead of the nonlocal one in Proposition
5.24, which is the crucial step in dimension $n=3$ which allows to decouple
the separation estimate from the regularity estimates. Moreover, the
$C^{2,\alpha}$ regularity result in [20] is only obtained for $\alpha$
sufficiently small, whereas our strategy allows us to obtain it for any
$\alpha\in(0,1)$.
With an eye towards future directions, using that one can show the proximity
of the nonlocal mean curvatures to the local ones before upgrading the
regularity of our layers to the uniform $C^{2,\alpha}$ estimate will probably
be essential to extend our results to dimensions $4\leq n\leq 7$. Indeed, it
hints at the possibility of using the Toda system (obtained in [20] under the
assumption of uniform $C^{2,\alpha}$ convergence as $s\to 1$ to a hyperplane)
prior to knowing this desired uniform regularity.

We begin the section by showing a rough $C^{2,\alpha}$ bound which explodes as
$\sigma\to 0$, but which will allow us to compare the nonlocal and local mean
curvatures.

######  Proposition 5.20.

Let $E\subset M$ be $\Lambda$-almost stable $s$-minimal in
$\varphi(\mathcal{B}_{2})$, and assume that (5.25)–(5.28) hold true, with
$(1-s)=\sigma\in(0,1/8)$. Let $\alpha\in(0,1/4)$. Then, for every $i$ such
that

$\min_{\overline{B}_{1/2}^{\prime}}|g_{i}|\leq\frac{1}{2},$

we have that, for every $\alpha\in(0,1/4)$, there exists a constant $C<\infty$
depending only on $\Lambda$, $n$, $s_{0}$, $\alpha$ and the constants in
(5.25)–(5.28), and in particular independent of $s$, such that

$\|g_{i}\|_{C^{2,\alpha}(\mathcal{B}_{1/4}^{\prime})}\leq
C\sigma^{-12\alpha}.$ (5.40)

###### Proof.

Step 1. We first show the Hölder estimate
$\|H_{s}[\varphi(\Gamma_{i})](\cdot)\|_{C^{0,1/2}(\varphi(\Gamma_{i}\cap[B^{\prime}_{1/2}\times(-1,1)]))}\leq
C\sigma^{-3}$.
We start by using Lemma 5.14 in (5.39), which gives the rough pointwise
estimate

$\Big{|}H_{s}[\varphi(\Gamma_{i})]\Big{|}(p)\leq C\sigma^{-1}\,.$ (5.41)

To find a higher-order bound, we differentiate both sides of equation (5.38)
over $\mathbf{\Gamma_{i}}$. Doing this, and then reproducing the computations
that led to (5.39) from (5.38), we can compute

$\displaystyle|\nabla_{p}^{\varphi(\Gamma_{i})}H_{s}[\varphi(\Gamma_{i})]|(p)$
$\displaystyle\leq
2\sigma\sum_{j<i,j-i\,odd}\int_{\varphi(\Omega_{0})}dq\,\chi_{E_{j}}(q)|\nabla_{p}^{\varphi(\Gamma_{i})}K_{s}(p,q)|$
$\displaystyle\ \ \ \
+2\sigma\sum_{j>i,j-i\,even}\int_{\varphi(\Omega_{0})}dq\,\chi_{E_{j}}(q)|\nabla_{p}^{\varphi(\Gamma_{i})}K_{s}(p,q)|+\sigma|\nabla^{\varphi(\Gamma_{i})}{\rm
Ext}_{1}|(p)$ $\displaystyle\leq
2\sigma\int_{A_{i-1}}|\nabla_{p}^{\varphi(\Gamma_{i})}K_{s}(p,q)|+2\sigma\int_{B_{i+1}}|\nabla_{p}^{\varphi(\Gamma_{i})}K_{s}(p,q)|+C\sigma$
$\displaystyle\leq
C\sigma\int_{\varphi^{-1}(A_{i-1})}\frac{1}{|x-y|^{n+s+1}}+C\sigma\int_{\varphi^{-1}(B_{i+1})}\frac{1}{|x-y|^{n+s+1}}+C\sigma$
$\displaystyle\leq
C\sigma\int_{\Gamma_{i-1}}\frac{1}{|x-y|^{n+s}}+C\sigma\int_{\Gamma_{i+1}}\frac{1}{|x-y|^{n+s}}+C\sigma$
$\displaystyle\leq
C\frac{\sigma}{|g_{i}(0)-g_{i-1}(0)|^{2}}+C\frac{\sigma}{|g_{i+1}(0)-g_{i}(0)|^{2}}+C\sigma\,.$
(5.42)

Using Lemma 5.14 once again, we get the pointwise bound

$|\nabla_{p}^{\varphi(\Gamma_{i})}H_{s}[\varphi(\Gamma_{i})]|(p)\leq
C\sigma^{-3}\,.$ (5.43)

Combining equations (5.41) and (5.43), we find that
$\|H_{s}[\varphi(\Gamma_{i})](\cdot)\|_{C^{0,1/2}(\Gamma_{i}\cap
B^{\prime}_{1/2}\times(-1,1))}\leq C\sigma^{-3}$. We could have argued by
interpolation between Hölder spaces and obtained a bound of order
$\sigma^{-2}$, but we do not care about the precise negative exponent at this
point.

Step 2. Schauder estimates and interpolation.
Define $A_{i}=\varphi(\\{x_{n}<g_{i}(x^{\prime})\\}\cap\Omega_{0})$, so that
$\varphi^{-1}(\partial
A_{i})\cap[B^{\prime}_{1/2}\times(-1/2,1/2)]=\Gamma_{i}\cap[B^{\prime}_{1/2}\times(-1/2,1/2)]$
is a $C^{2}$ graph over $B^{\prime}_{1/2}$ parametrised by $g_{i}$. By Step 1,
it moreover verifies that

$\|H_{s}[\varphi(\Gamma_{i})](\cdot)\|_{C^{0,1/2}(\varphi(\Gamma_{i}\cap[B^{\prime}_{1/2}\times(-3/4,\,3/4)]))}\leq
C\sigma^{-3}.$

Then, by the Schauder estimate for the nonlocal mean curvature equation in
[27] (in its version for graphs), we deduce that
$\|g_{i}\|_{C^{2,1/4}(\mathcal{B}_{1/4}^{\prime})}\leq C\sigma^{-3}$. On the
other hand, by assumption (5.28) we have that
$\|g_{i}\|_{C^{2}(\mathcal{B}_{1}^{\prime})}\leq C$. By interpolation, we
deduce that

$\|g_{i}\|_{C^{2,\alpha}(\mathcal{B}_{1/4}^{\prime})}\leq
C(\alpha)\sigma^{-12\alpha}$

for every $\alpha\in(0,1/4)$, as desired. ∎

The next proposition gives a precise representation for the gradient of the
singular kernel $K_{s}$ up to a lower order error. The fact that we manage to
get an error of two orders less will be crucial in the proof of Lemma 5.23.

######  Proposition 5.21.

Assume that the flatness assumptions ${\rm FA}_{\ell}(M,g,4,p,\varphi)$ are
satisfied, where $\varphi$ corresponds to the restriction of the exponential
map ${\rm exp}_{p}$ to $\mathcal{B}_{4}\subset\mathbb{R}^{n}$. Then, the
difference between the gradient of $K_{s}$ and the gradient of its Euclidean
representative is of two orders less. More precisely, using the notation in
Proposition 3.9, there is $r=r(n)>0$ such that if $y\in\mathcal{B}_{r}(0)$
then

$\displaystyle\nabla_{y}K_{s}(p,\varphi_{p}(y))$
$\displaystyle=\nabla_{y}\frac{\alpha_{n,s}}{|y|^{n+s}}+O\Big{(}\frac{1}{|y|^{n+s-1}}\Big{)}$
$\displaystyle=-(n+s)\alpha_{n,s}\,\frac{y}{|y|^{n+s+2}}+O\Big{(}\frac{1}{|y|^{n+s-1}}\Big{)}\,.$

###### Proof.

The proof is given in Appendix B. ∎

The next proposition simply records the standard fact that having flatness
assumptions centered at some point implies that, in a smaller neighbourhood,
normal coordinates centered at any other point also induce flatness
assumptions. See also Remark 1.6 in [45].

######  Proposition 5.22.

Assume that the flatness assumptions ${\rm FA}_{\ell}(M,g,4,p_{0},\varphi)$
are satisfied. Then, there exists a small constant $r_{\circ}=r_{\circ}(n)>0$
such that for every $x\in\mathcal{B}_{2}$ and $r\in(0,r_{0})$, letting
$\varphi_{q}:\mathcal{B}_{r}\to M$ denote the restriction of the exponential
map $exp_{q}$ centered at $q=\varphi(x)$, then the flatness assumptions
$FA_{l}(M,g,r,q,\varphi_{q})$ are also satisfied. Moreover, the change of
coordinates $F_{q}=\varphi^{-1}\circ\varphi_{q}$ is a well-defined
diffeomorphism onto its image and both $F_{q}$ and $F_{q}^{-1}$ have bounded
$C^{3}$ norm.

The next lemma shows that the local and nonlocal mean curvatures, of a
uniformly $C^{2}$ graph whose $C^{2,\alpha}$ seminorm is allowed to degenerate
with a small rate as $\sigma\to 0$, are nevertheless close to each other.

######  Lemma 5.23.

Let $1-s=\sigma\in(0,1/2)$, and assume that ${\rm
FA}_{\ell}(M,g,4,p_{0},\varphi)\hskip 7.0pt\mbox{is satisfied}$. Let
$\alpha>0$, and let $g:\mathcal{B}^{\prime}_{1/2}\to(-1/2,1/2)$ be a
$C^{2,\alpha}$ function with

$\|\nabla
g\|_{L^{\infty}(\mathcal{B}_{1/2}^{\prime})}+\|D^{2}g\|_{L^{\infty}(\mathcal{B}_{1/2}^{\prime})}<\delta\mbox{
and }[g]_{C^{2,\alpha}}\leq C\sigma^{-{1/4}}\,.$

Define $\Gamma$ as the graph of $g$ in
$\mathcal{B}^{\prime}_{1/2}\times(-1/2,1/2)$, so that $\varphi(\Gamma)$ is a
$C^{2,\alpha}$ hypersurface in $M$. Then, the nonlocal mean curvature (NMC) of
$\varphi(\Gamma)$ is close to its classical mean curvature (or local mean
curvature, LMC) with a small error. Precisely, if $p=\varphi(x)$ with
$x\in\Gamma\cap[\mathcal{B}^{\prime}_{1/4}\times(-1/2,1/2)]$, then

$H_{s}[\varphi(\Gamma)](p)=c_{\circ}H[\varphi(\Gamma)](p)+O(\sigma^{3/4})\,,$

where
$c_{\circ}:=\frac{\mathcal{H}^{n-2}(\partial\mathcal{B}_{1}^{\prime})}{2(n-1)}$.

###### Proof.

Step 1. We begin by showing that the Riemannian NMC of the set and the
Euclidean NMC of its representative in normal coordinates are close.
Let $x\in\Gamma\cap[\mathcal{B}^{\prime}_{1/4}\times(-1/2,1/2)]$ be as in the
statement of the lemma. By Proposition 5.22, we can consider normal
coordinates centered at $p=\varphi(x)$ and then our setting still applies.
More precisely, $FA_{l}(M,g,r,p,\varphi_{p})$ is satisfied for some $r$
sufficiently small. Moreover,

$\hat{\Gamma}:=\Big{[}\varphi_{p}^{-1}\circ\varphi(\Gamma)\Big{]}\cap\Big{[}\mathcal{B}^{\prime}_{r/2}\times(-r/2,r/2)\Big{]}$

is still a $C^{2,\alpha}$ surface, which (possibly after a rotation and making
$r$ sufficiently smaller) is given in
$\mathcal{B}^{\prime}_{r/2}\times(-r/2,r/2)$ by the graph of some function
$\hat{g}:\mathcal{B}^{\prime}_{r/2}\to(-r/2,r/2)$ with

$\hat{g}(0)=|\nabla\hat{g}(0)|=0\,,$ (5.44)

and satisfying the bounds

$\|\nabla\hat{g}\|_{L^{\infty}(\mathcal{B}_{r/2}^{\prime})}+\|D^{2}\hat{g}\|_{L^{\infty}(\mathcal{B}_{r/2}^{\prime})}<C\delta\mbox{
and }[\hat{g}]_{C^{2,\alpha}}\leq C\sigma^{-{1/4}}\,.$

Now, thanks to taking normal coordinates, we have that $g_{ij}(0)=\delta_{ij}$
and $\nabla g_{ij}(0)=0$. This will simplify the computations that will
follow. We start by writing

$\displaystyle\frac{1}{2}H_{s}[\varphi(\Gamma)](p)$
$\displaystyle=\frac{1}{2}H_{s}[\varphi_{p}(\hat{\Gamma})](p)+\sigma\int_{\varphi(\Gamma)\setminus\varphi_{p}(\hat{\Gamma})}\langle\nu_{\Sigma}(q),\nabla_{q}^{M}K_{s-2}(p,q)\rangle_{g}\,d\mathcal{H}^{n-1}(q)$
$\displaystyle=\frac{1}{2}H_{s}[\varphi_{p}(\hat{\Gamma})](p)+O(\sigma)\,,$
(5.45)

where we have used (3.16) with $s-2$ instead of $s$ to bound the second term.
This shows that we can work with $\hat{\Gamma}$ and $\varphi_{p}$ instead of
$\Gamma$ and $\varphi$ without any loss of generality. Now, passing to
coordinates we can write

$\displaystyle\frac{1}{2}H_{s}[\varphi_{p}(\hat{\Gamma})](p)$
$\displaystyle=\sigma\int_{\varphi_{p}(\hat{\Gamma})}\langle\nu_{\varphi_{p}(\hat{\Gamma})}(q^{\prime}),\nabla_{q^{\prime}}^{M}K_{s-2}(p,q^{\prime})\rangle_{g}\,d\mathcal{H}^{n-1}(q^{\prime})$
$\displaystyle=\sigma\int_{\hat{\Gamma}}\langle\nu_{\varphi_{p}(\hat{\Gamma})}(\varphi_{p}(y)),\nabla_{q^{\prime}}^{M}K_{s-2}(\varphi_{p}(0),\varphi_{p}(y))\rangle_{g(\varphi_{p}(y))}\sqrt{|g_{ij}^{\hat{\Gamma}}|(y)}\,d\mathcal{H}^{n-1}(y)\,.$
(5.46)

A simple Riemannian computation, which we give in Appendix A, allows us to
compute the terms in the inner product, obtaining that

$\displaystyle\frac{1}{2}H_{s}[\varphi(\hat{\Gamma})](p)$
$\displaystyle=\sigma\int_{\hat{\Gamma}}\frac{1}{\sqrt{\nu_{\hat{\Gamma}}^{t}g^{-1}\nu_{\hat{\Gamma}}}}[\nu_{\hat{\Gamma}}(y)]^{t}g^{-1}(y)\nabla_{y}^{\mathbb{R}^{n}}K_{s-2}(\varphi_{p}(0),\varphi_{p}(y))\,\sqrt{|g_{ij}^{\hat{\Gamma}}|(y)}\,d\mathcal{H}^{n-1}(y)\,.$
(5.47)

Here all products involving $g$ denote products of matrices. Now, since
$g(y)=Id+O(|y|^{2})$ thanks to having taken normal coordinates, we can
exchange all instances of the Riemannian metric $g$ by the identity metric up
to a controlled quadratic error. More precisely, we can estimate

$\displaystyle\frac{1}{2}H_{s}[\varphi(\hat{\Gamma})](p)$
$\displaystyle=\sigma\int_{\hat{\Gamma}}\big{[}\nu_{\hat{\Gamma}}(y)\cdot\nabla_{y}^{\mathbb{R}^{n}}K_{s-2}(\varphi_{p}(0),\varphi_{p}(y))\big{]}\,d\mathcal{H}^{n-1}(y)$
(5.48) $\displaystyle\hskip
17.07182pt+O\Big{(}\sigma\int_{\hat{\Gamma}}|y|^{2}|\nabla_{y}^{\mathbb{R}^{n}}K_{s-2}(\varphi_{p}(0),\varphi_{p}(y))|\,d\mathcal{H}^{n-1}(y)\Big{)}\,,$

where $\cdot$ denotes the usual Euclidean scalar product.
Proposition 5.21 (applied with $s-2$ instead of $s$) allows us now to compare
the gradient of the kernel $K_{s-2}$ with the one of the corresponding
Euclidean kernel. We reach

$\displaystyle\frac{1}{2}H_{s}[\varphi(\hat{\Gamma})](p)$
$\displaystyle=-(n+s-2)\alpha_{n,(s-2)}\sigma\int_{\hat{\Gamma}}\frac{\nu_{\hat{\Gamma}}(y)\cdot
y}{|y|^{n+s}}\,d\mathcal{H}^{n-1}(y)$ $\displaystyle\hskip
17.07182pt+O\Big{(}\sigma\int_{\hat{\Gamma}}|y|^{2}\frac{1}{|y|^{n+s-1}}\,d\mathcal{H}^{n-1}(y)\Big{)}\,.$

Setting $c(n,s)=-\frac{2(n+s-2)}{s}\alpha_{n,(s-2)}$ and bounding the second
integral by a uniform constant, we can then write

$\displaystyle H_{s}[\varphi(\hat{\Gamma})](p)$
$\displaystyle=c(n,s)\sigma\int_{\hat{\Gamma}}\frac{\nu_{\hat{\Gamma}}(y)\cdot
y}{|y|^{n+s}}\,d\mathcal{H}^{n-1}(y)$ $\displaystyle\hskip
17.07182pt+O(\sigma)\,.$

Now, the last expression is exactly the expression for the Euclidean nonlocal
mean curvature of $\hat{\Gamma}$, plus an $O(\sigma)$ error term. In other
words, together with (5.45) we have found that

$H_{s}[\varphi(\Gamma)](p)=H_{s}[\hat{\Gamma}](0)+O(\sigma)\,,$

where the NMC on the left and right hand sides are respectively on $M$ and on
$\mathbb{R}^{n}$.

Step 2. We now show that
$H_{s}[\hat{\Gamma}](0)=c_{\circ}H[\hat{\Gamma}](0)+O(\sigma^{3/4})$, with
$c_{\circ}:=\frac{\mathcal{H}^{n-2}(\partial\mathcal{B}_{1}^{\prime})}{2(n-1)}$.
This is an exclusively Euclidean computation and which we reproduce from [20],
with the caveat that we are not assuming uniform $C^{2,\alpha}$ bounds for our
graphs but rather bounds which degenerate as $\sigma\to 0$.

Observe first that, from the assumption that

$\hat{g}(0)=|\nabla\hat{g}(0)|=0\,,$ (5.49)

we have that

$H[\hat{\Gamma}](0)=-\Delta\hat{g}(0).$

Thus our goal is to relate $H_{s}[\hat{\Gamma}](0)$ with $\Delta\hat{g}(0)$.

Recall that

$H_{s}[\hat{\Gamma}](0)=c(n,s)\sigma\int_{\hat{\Gamma}}\frac{\nu_{\hat{\Gamma}}(y)\cdot
y}{|y|^{n+s}}\,d\mathcal{H}^{n-1}(y)=c(n,s)\sigma\int_{\mathcal{B}_{r/2}^{\prime}}\frac{(-\nabla\hat{g}(y^{\prime}),1)\cdot(y^{\prime},\hat{g}(y^{\prime}))}{(|y^{\prime}|^{2}+\hat{g}(y^{\prime})^{2})^{\frac{n+s}{2}}}\,dy^{\prime}.$

Now, on the one hand, we have

$I_{1}:=\sigma\int_{B_{\varrho}^{\prime}}\frac{(-\nabla\hat{g}(y^{\prime}),1)\cdot(y^{\prime},\hat{g}(y^{\prime}))}{(|y^{\prime}|^{2}+\hat{g}(y^{\prime})^{2})^{\frac{n+s}{2}}}\,dy^{\prime}=(1+O(\varrho^{2}))\sigma\int_{B_{\varrho}^{\prime}}\frac{\hat{g}(y^{\prime})-\nabla\hat{g}(y^{\prime})\cdot
y^{\prime}}{|y^{\prime}|^{n+s}}\,dy^{\prime},$

where we have used that
$|y^{\prime}|^{2}+\hat{g}(y^{\prime})^{2}=|y^{\prime}|^{2}(1+\hat{g}(y^{\prime})^{2}/|y^{\prime}|^{2})=|y|^{2}(1+O(\varrho^{2}))$
for $y^{\prime}\in B_{\varrho}^{\prime}$, thanks to (5.49).

Note that our assumptions on $[\hat{g}]_{C^{2,\alpha}}$ give the Taylor
expansions

$\big{|}\hat{g}(y^{\prime})-\tfrac{1}{2}y^{\prime}\cdot
D^{2}\hat{g}(0)y^{\prime}\big{|}\leq
C\sigma^{-1/4}|y^{\prime}|^{2+\alpha}\quad\mbox{and}\quad\big{|}\nabla\hat{g}(y^{\prime})-D^{2}\hat{g}(0)y^{\prime}\big{|}\leq
C\sigma^{-1/4}|y^{\prime}|^{1+\alpha}.$

Hence, choosing $\varrho=\sigma$, and using that
$\sigma^{\sigma}=\exp(\sigma\log\sigma)=1+O(\sigma|\log\sigma|)$ as
$\sigma\downarrow 0$, we obtain

$\begin{split}I_{1}&=(1+O(\varrho^{2}))\sigma\left(\int_{B_{\varrho}^{\prime}}\frac{-\tfrac{1}{2}y^{\prime}\cdot
D^{2}\hat{g}(0)y^{\prime}}{|y^{\prime}|^{n+s}}\,dy^{\prime}+O\Big{(}C\sigma^{-1/4}\int_{B_{\varrho}^{\prime}}\frac{2|y^{\prime}|^{2+\alpha}}{|y^{\prime}|^{n+s}}\,dy^{\prime}\Big{)}\right)\\\
&=(1+O(\varrho^{2}))\left(\frac{{\rm
trace}(D^{2}\hat{g}(0))}{2(n-1)}\sigma\int_{B_{\varrho}^{\prime}}\frac{|y^{\prime}|^{2}}{|y^{\prime}|^{n+s}}\,dy^{\prime}+\sigma^{3/4}O(\varrho^{\alpha+1-s})\right)\\\
&=(1+O(\varrho^{2}))\left(-\Delta\hat{g}(0)\frac{\mathcal{H}^{n-2}(\partial
B_{1}^{\prime})}{2(n-1)}\sigma\frac{\varrho^{1-s}}{1-s}+\sigma^{3/4}O(\varrho^{\alpha+1-s})\right)\\\
&=-c_{\circ}\Delta\hat{g}(0)+O(\sigma|\log\sigma|+\sigma^{3/4+\alpha}).\end{split}$

On the other hand, using that
$|\hat{g}(y^{\prime})|+|y^{\prime}\cdot\nabla\hat{g}(y^{\prime})|\leq\frac{1}{2}|y^{\prime}|^{2}+|y^{\prime}|^{2}$
(thanks to our assumptions on $\hat{g}$), we obtain that

$I_{2}:=\sigma\int_{B_{r/2}^{\prime}\setminus
B_{\varrho}^{\prime}}\frac{(-\nabla\hat{g}(y^{\prime}),1)\cdot(y^{\prime},\hat{g}(y^{\prime}))}{(|y^{\prime}|^{2}+\hat{g}(y^{\prime})^{2})^{\frac{n+s}{2}}}\,dy^{\prime}$

can be bounded by

$\big{|}I_{2}\big{|}\leq\sigma\int_{B_{r/2}^{\prime}\setminus
B_{\varrho}^{\prime}}\frac{\frac{3}{2}|y^{\prime}|^{2}}{|y^{\prime}|^{n+s}}\,dy^{\prime}=C\sigma\int_{\varrho}^{1}r^{-s}\,dr=C(1-\sigma^{\sigma})=O(\sigma|\log\sigma|).$

Since $H_{s}[\hat{\Gamma}](0)=I_{1}+I_{2}$, these considerations complete the
proof of Step 2.

Step 3. Conclusion of the proof.

Combining Steps 1 and 2, we have seen that

$H_{s}[\varphi(\Gamma)](p)=c_{0}H[\hat{\Gamma}](0)+O(\sigma^{3/4})\,.$

On the other hand, the identity

$H[\hat{\Gamma}](0)=H[\varphi_{p}(\hat{\Gamma})](p)$ (5.50)

holds, where the left and right mean curvatures are computed on
$\mathbb{R}^{n}$ and on $M$ respectively, thanks to having taken normal
coordinates centered at $p$; a proof is given in Appendix A. Therefore, we
conclude that

$H_{s}[\varphi(\Gamma)](p)=c_{0}H[\varphi_{p}(\hat{\Gamma})](p)+O(\sigma^{3/4})=c_{0}H[\varphi(\Gamma)](p)+O(\sigma^{3/4})$

as desired. ∎

With the results up to now, we can show the optimal separation estimate
between layers in dimension $3$.

######  Proposition 5.24.

Let $n=3$. Assume that (5.25)–(5.28) are satisfied, and let $E\subset M$ be
$\Lambda$-almost stable $s$-minimal in $\varphi(\mathcal{B}_{2})$, with
$(1-s)=\sigma\in(0,1/8)$. Then, there exists $c>0$, depending only on
$\Lambda$, $n$ and the constants in (5.25)–(5.28), such that for every $i$
such that

$\min_{\overline{B}_{1/2}^{\prime}}|g_{i}|\leq\frac{1}{2},$

we have the separation estimate

$\inf_{B^{\prime}_{1/2}}(g_{i+1}-g_{i})\geq c\sigma^{1/2}>0\,.$ (5.51)

Moreover, we have the pointwise mean curvature estimates

$\|H_{s}[\varphi(\Gamma_{i})]\|_{L^{\infty}(\varphi(\Gamma_{i}\cap[B^{\prime}_{1/2}\times(-1,1)]))}\leq
C\sqrt{\sigma}\quad\mbox{and}\quad\|H[\varphi(\Gamma_{i})]\|_{L^{\infty}(\varphi(\Gamma_{i}\cap[B^{\prime}_{1/2}\times(-1,1)]))}\leq
C\sqrt{\sigma}.$ (5.52)

###### Proof.

Step 1. Let us show the pointwise separation estimate in (5.51).
It is well known that the classical mean curvature can be written, for a graph
in coordinates, as an operator depending on the graph and the coefficients of
the metric. Moreover, the difference of the parametrising functions of two
parallel graphs satisfies an elliptic equation. The precise statement is that,
for $x^{\prime}\in B^{\prime}_{3/4}$ and $v:=g_{i+1}-g_{i}$, we have that (see
for example [22, page 237])

$\displaystyle h(x´):$
$\displaystyle=H[\varphi(\Gamma_{i+1})](\varphi(x^{\prime},g_{i+1}(x^{\prime})))-H[\varphi(\Gamma_{i})](\varphi(x^{\prime},g_{i}(x^{\prime})))$
(5.53)
$\displaystyle=a_{ij}(x^{\prime})v_{x_{i}x_{j}}(x")+b_{i}(x^{\prime})v_{x_{i}}+cv,$

where the $a_{ij}$ form a uniformly elliptic matrix and $b_{i}$ and $c$ are
uniformly bounded.
Moreover, we know from Proposition 5.19 that the NMCs
$H_{s}[\varphi(\Gamma_{i})]$ and $H_{s}[\varphi(\Gamma_{i+1})]$ are bounded in
$L^{2}$ by $C\sqrt{\sigma}$, which together with the combination of (5.40)
(with $\alpha=\frac{1}{48}$, so that $12\alpha=1/4$) and Lemma 5.23 gives that
the LMCs $H[\varphi(\Gamma_{i})]$ and $H[\varphi(\Gamma_{i+1})]$ are bounded
in $L^{2}$ by $C\sqrt{\sigma}$ as well. Thus (5.53) gives that

$\|h\|_{L^{2}(B_{3/4}^{\prime})}\leq C\sqrt{\sigma}\,.$

On the other hand, by the Harnack inequality for uniformly elliptic operators,
see [34, Theorems 8.17 and 8.18], we have that

$\sup_{B_{1/2}^{\prime}}v\leq
C\inf_{B_{1/2}^{\prime}}v+C\|h\|_{L^{q}(B_{3/4}^{\prime})}$

for every $q>n^{\prime}/2$, with $n^{\prime}:=n-1$. For $n=3$ we may take
$q=2$, thus obtaining that

$\sup_{B_{1/2}^{\prime}}(g_{i+1}-g_{i})\leq
C\inf_{B_{1/2}^{\prime}}(g_{i+1}-g_{i})+C\sqrt{\sigma}.$ (5.54)

We now argue as in [20, Proposition 3.5]. Still in dimension $n=3$ (and thus
$n^{\prime}=2$), assume that
$(g_{i+1}-g_{i})(x_{\circ}^{\prime})=\delta\sqrt{\sigma}$ for some
$x^{\prime}_{\circ}\in B^{\prime}_{1/2}$ and let us prove a lower bound for
$\delta$.

For $r\in(0,1/4)$ we now dilate around
$x_{\circ}=(x_{\circ}^{\prime},g_{i}(x_{\circ}^{\prime}))\in\Gamma_{i}$ and we
obtain new surfaces $\Gamma_{i,r}:=\frac{1}{r}(\Gamma_{i}-x_{\circ})$ which
have graphical expressions $x_{n}=g_{i,r}(x^{\prime})$ in
$\mathcal{B}_{1}^{\prime}\times(-1,1)$, where
$g_{i,r}(x^{\prime}):=\frac{1}{r}g_{i}(x_{\circ}^{\prime}+rx^{\prime})$. As we
have used several times, our assumptions (5.25)–(5.28) and the
$\Lambda$-almost stability condition are all preserved in this rescaled
setting, considering the manifold $(M,\frac{1}{r^{2}}g)$ and the new
parametrisation $\varphi(rx+x_{0})$; see for example Step 2 of the proof of
Proposition 5.10.
Then, reapplying the argument above we deduce that (5.54) also holds with
$g_{i}$ replaced by $g_{i,r}$. Since
$(g_{i+1,r}-g_{i,r})(0)=\frac{1}{r}(g_{i+1}-g_{i})(x_{\circ}^{\prime})=\frac{\delta\sqrt{\sigma}}{r}$,
this gives that

$\sup_{B_{r/2}^{\prime}(x_{\circ}^{\prime})}(g_{i+1}-g_{i})=r\sup_{B_{1/2}^{\prime}}(g_{i+1,r}-g_{i,r})\leq
Cr\left(\frac{\delta\sqrt{\sigma}}{r}+\sqrt{\sigma}\right).$

In other words, for all $r\geq\delta$ we have that

$\sup_{B_{r/2}^{\prime}(x_{\circ}^{\prime})}(g_{i+1}-g_{i})\leq
Cr\sqrt{\sigma}.$

But then using again Proposition 5.19 and the fact that $n^{\prime}=2$, we
obtain that

$C\geq\int_{B_{1/4}^{\prime}(x_{\circ}^{\prime})}\frac{\sigma}{(g_{i+1}-g_{i})^{2}}\geq\frac{\sigma}{C}\int_{\delta}^{1/4}\frac{r\,dr}{(Cr\sqrt{\sigma})^{2}}=\frac{1}{C}\int_{\delta}^{1/4}\frac{dr}{r}\geq\frac{|\log\delta|}{C},$
(5.55)

which proves that $\delta\geq c>0$.

Step 2. Now that we know the pointwise separation estimate in (5.51), using it
in equation (5.39) gives the pointwise bound for the NMC,

$\|H_{s}[\varphi(\Gamma_{i})]\|_{L^{\infty}(\varphi(\Gamma_{i}\cap
B^{\prime}_{1/2}\times(-1,1)))}\leq C\sqrt{\sigma}.$

Combining this once again with Lemma 5.23, which we can apply thanks to (5.40)
in Proposition 5.20, we complete the proof of (5.52). ∎

We finally obtain the uniform $C^{2,\alpha}$ estimate under the assumptions in
the section.

######  Corollary 5.25.

Let $n=3$. Assume that (5.25)–(5.28) are satisfied, and let $E\subset M$ be
$\Lambda$-almost stable $s$-minimal in $\varphi(\mathcal{B}_{2})$, with
$(1-s)=\sigma\in(0,1/8)$. Given $\alpha\in(0,1)$, there exists $C>0$,
depending only on $\alpha$, $\Lambda$, $n$ and the constants in (5.25)–(5.28),
such that for every $i$ such that

$\min_{\overline{\mathcal{B}}_{1/2}^{\prime}}|g_{i}|\leq\frac{1}{2},$

we have that $\|g_{i}\|_{C^{2,\alpha}(\mathcal{B}_{1/2}^{\prime})}\leq C$.

###### Proof.

In (5.42) from Proposition 5.20 we found the pointwise bound

$|\nabla_{p}^{\varphi(\Gamma_{i})}H_{s}[\varphi(\Gamma_{i})](p)|\leq
C\frac{\sigma}{|g_{i}(0)-g_{i-1}(0)|^{2}}+C\frac{\sigma}{|g_{i+1}(0)-g_{i}(0)|^{2}}+C\sigma\,,$

which combined with the separation estimate (5.51) shows that

$|\nabla_{p}^{\varphi(\Gamma_{i})}H_{s}[\varphi(\Gamma_{i})](p)|\leq C\,.$

On the other hand, by (5.52) in Proposition 5.24 we know that

$\|H_{s}[\varphi(\Gamma_{i})]\|_{L^{\infty}(\varphi(\Gamma_{i}\cap[B^{\prime}_{1/2}\times(-1,1)]))}\leq
C\sqrt{\sigma}.$

Then, by interpolation between Hölder spaces, we deduce that

$\|H_{s}[\varphi(\Gamma_{i})]\|_{C^{\beta}(\varphi(\Gamma_{i}\cap[B^{\prime}_{1/2}\times(-1,1)]))}\leq
C\sigma^{(1-\beta)/2}$

for all $\beta\in(0,1)$.
By the Schauder estimate for the nonlocal mean curvature equation in [27] (in
its version for graphs), since the NMC $H_{s}$ is an operator of order $1+s$,
we deduce that $\|g_{i}\|_{C^{2,1+s+\beta-2}(\mathcal{B}_{1/2}^{\prime})}\leq
C(\beta)$. Given $\alpha$ as in the statement, choose some $\beta>\alpha$ in
$(0,1)$, say $\beta=\alpha+(1-\alpha)/2$. Choosing then $s_{0}=s_{0}(\alpha)$
in $(0,1)$ large enough, so that $1+s_{0}+\beta-2\geq\alpha$, we conclude that

$\|g_{i}\|_{C^{2,\alpha}(\mathcal{B}_{1/2}^{\prime})}\leq C(\alpha)$

for all $s\in[s_{0},1)$.
The proof for the range $s\in(7/8,s_{0})$ instead, in which
$\sigma>(1-s_{0})>0$, is essentially trivial at this point. First, by the
lower separation bounds that we have found (here any rough bound, such as the
one in Lemma 5.14, suffices), there is a uniform $r_{\circ}>0$ such that our
surface is a single graph in balls of radius $r_{\circ}$ for this range of
$s$. Then, differentiating (5.38) two times and arguing exactly as in the
proof of Proposition 5.20, we get a uniform pointwise bound

$\|\nabla^{2,\varphi(\Gamma_{i})}H_{s}[\varphi(\Gamma_{i})]\|_{L^{\infty}(\varphi(\Gamma_{i}\cap[B^{\prime}_{1/2}\times(-1,1)]))}\leq
C\,,$

or for any chosen derivative of $H_{s}$ for that matter. By interpolation, in
particular we get a uniform bound

$\|H_{s}[\varphi(\Gamma_{i})]\|_{C^{1,1/4}(\varphi(\Gamma_{i}\cap[B^{\prime}_{1/2}\times(-1,1)]))}\leq
C\,,$

which again by the Schauder estimate in [27] gives a bound for the
$C^{3,1+s+(1+1/4)-3}=C^{3,s-3/4}$ norm of $g_{i}$. Since $s\geq 7/8>3/4$, in
particular we find a bound for the $C^{3}$ norm of $g_{i}$, thus also a bound
for its $C^{2,\alpha}$ norm as desired. This concludes the proof. ∎

######  Lemma 5.26.

Let $\gamma_{n}$ be as in (3.7), and let
$\Omega_{0}=\mathcal{B}_{1}^{\prime}\times(-1,1)\subset\mathbb{R}^{n}$. Assume
that (5.25)–(5.28) are satisfied, and let $E\subset M$ be $\Lambda$-almost
stable $s$-minimal in $\varphi(\mathcal{B}_{2})$, with
$(1-s)=\sigma\in(0,1/8)$. Moreover, assume that the metric $g$ (in
coordinates) and the Euclidean metric are $\delta$-close in $\mathcal{B}_{4}$,
in the sense that

$\sup_{x\in\mathcal{B}_{4}}\sup_{i,j}|g_{ij}(x)-\delta_{ij}(x)|\leq\delta\,.$

Then, there exists a function $\varepsilon(t)$, depending only on $\Lambda$
and $n$, such that

$|(1-s)\textnormal{Per}_{s}(E:\varphi(\Omega_{0}))-\gamma_{n}\textnormal{Per}(E:\varphi(\Omega_{0}))|\leq\varepsilon(\sigma)+\varepsilon(\delta)[\textnormal{Per}(E:\varphi(\Omega_{0}))+(1-s)\textnormal{Per}_{s}(E:\varphi(\Omega_{0}))]$

where $\varepsilon(t)$ indicates some function of $t$ satisfying $\lim_{t\to
0}\varepsilon(t)=0$.

###### Proof.

Let $A_{i}=\varphi(\\{x_{n}<g_{i}(x^{\prime})\\}\cap\Omega_{0})$,
$B_{i}=\varphi(\\{x_{n}>g_{i}(x^{\prime})\\}\cap\Omega_{0})$, and
$E_{i}=A_{i}\setminus A_{i-1}$. In particular (up to possibly interchanging
$E$ with its complement) it follows that
$E\cap\varphi(\Omega_{0})=\bigcup_{i\in 2\mathbb{Z}}E_{i}$. We can write the
fractional perimeter as the contribution of each subgraph $A_{i}$ plus an
error:

$\displaystyle{\rm Per}_{s}(E;\Omega_{0})$
$\displaystyle=2\int_{E}\int_{E^{c}}K_{s}(p,q)dV_{p}dV_{q}$
$\displaystyle=\sum_{i\in
2\mathbb{Z}}2\int_{E_{i}}\int_{E^{c}}K_{s}(p,q)dV_{p}dV_{q}$
$\displaystyle=\sum_{i\in
2\mathbb{Z}}2\int_{E_{i}}\int_{E_{i-1}}K_{s}(p,q)dV_{p}dV_{q}+\sum_{i\in
2\mathbb{Z}}2\int_{E_{i}}\int_{E_{i+1}}K_{s}(p,q)dV_{p}dV_{q}+O\Big{(}\sum_{i\in
2\mathbb{Z}}2\int_{E_{i}}\int_{A_{i-3}\cup
B_{i+2}}K_{s}(p,q)dV_{p}dV_{q}\Big{)}$
$\displaystyle=\sum_{j\in\mathbb{Z}}2\int_{E_{j}}\int_{E_{j-1}}K_{s}(p,q)dV_{p}dV_{q}+O\Big{(}\sum_{i\in
2\mathbb{Z}}\int_{E_{i}}\int_{A_{i-3}\cup
B_{i+2}}K_{s}(p,q)dV_{p}dV_{q}\Big{)}$
$\displaystyle=\sum_{j\in\mathbb{Z}}2\int_{A_{j-1}}\int_{A_{j-1}^{c}}K_{s}(p,q)dV_{p}dV_{q}+O\Big{(}\sum_{i\in\mathbb{Z}}\int_{E_{i}}\int_{A_{i-2}\cup
B_{i+1}}K_{s}(p,q)dV_{p}dV_{q}\Big{)}$
$\displaystyle=\sum_{j\in\mathbb{Z}}{\rm
Per}_{s}(A_{j};\Omega_{0})+O\Big{(}\sum_{i\in\mathbb{Z}}\int_{E_{i}}\int_{A_{i-2}\cup
B_{i+1}}K_{s}(p,q)dV_{p}dV_{q}\Big{)}$
$\displaystyle=\sum_{j\in\mathbb{Z}}{\rm
Per}_{s}(A_{j};\Omega_{0})+O\Big{(}\sum_{i\in\mathbb{Z}}\int_{A_{i}}\int_{B_{i+1}}K_{s}(p,q)dV_{p}dV_{q}\Big{)}$

Thanks to the separation estimates, ${\rm dist}(A_{i},B_{i+1})\geq
c\sqrt{\sigma}$, so that passing to coordinates in $q$ and using Lemma 3.10 we
can bound

$\displaystyle\int_{A_{i}}\int_{B_{i+1}}K_{s}(p,q)dV_{p}dV_{q}$
$\displaystyle\leq\int_{A_{i}}\int_{\varphi(\mathcal{B}_{2})\setminus
B_{c\sqrt{\sigma}}^{c}(p)}K_{s}(p,q)dV_{p}dV_{q}$
$\displaystyle\leq\int_{A_{i}}\int_{\\{|\varphi^{-1}(p)-y|\geq
c\sqrt{\sigma}\\}}\frac{C}{|\varphi^{-1}(p)-y|^{n+s}}dV_{p}dy$
$\displaystyle\leq\frac{C}{|c\sqrt{\sigma}|^{s}}$
$\displaystyle\leq\frac{C}{\sqrt{\sigma}}$
$\displaystyle\leq\frac{C}{\sigma}\varepsilon(\sigma)\,.$

A more precise computation would lead to the much better bound
$C\log{\sigma}$, but this suffices for our purposes since it shows that

$\displaystyle(1-s){\rm Per}_{s}(E;\varphi(\Omega_{0}))$
$\displaystyle=\sum_{j\in\mathbb{Z}}(1-s){\rm
Per}_{s}(A_{j};\varphi(\Omega_{0}))+\varepsilon(\sigma)\,.$ (5.56)

Moreover, we obviously have

$\displaystyle{\rm Per}(E;\varphi(\Omega_{0}))$
$\displaystyle=\sum_{j\in\mathbb{Z}}{\rm Per}(A_{j};\varphi(\Omega_{0}))\,.$
(5.57)

Now, thanks to the assumption on the metric,

$|{\rm Per}(A_{j};\varphi(\Omega_{0}))-{\rm
Per}(\varphi^{-1}(A_{j});\Omega_{0})|\leq\varepsilon(\delta){\rm
Per}(A_{j};\varphi(\Omega_{0}))\leq\varepsilon(\delta)\,,$ (5.58)

and likewise

$|(1-s){\rm Per}_{s}(A_{j};\varphi(\Omega_{0}))-(1-s){\rm
Per}_{s}(\varphi^{-1}(A_{j});\Omega_{0})|\leq\varepsilon(\delta)\,.$ (5.59)

On the other hand, the $\varphi^{-1}(A_{j})$ are sets with uniformly $C^{2}$
(graphical) boundaries and which are transversal to $\partial\Omega_{0}$ in a
quantified way thanks to their graphical structure. We can then apply [16,
Lemma 11] separately to each of the $\varphi^{-1}(A_{j})$, obtaining that

$|(1-s){\rm Per}_{s}(A_{j};\varphi(\Omega_{0}))-{\rm
Per}(A_{j};\varphi(\Omega_{0}))|\leq\varepsilon(\sigma)\,.$ (5.60)

Combining (5.56)–(5.60), we conclude the desired result. ∎

####  5.3.3 Uniform $C^{2,\alpha}$ regularity for almost-stable surfaces in
dimension $n=3$

In the previous section we have concluded that assuming bounds on the almost-
stability constant and the fractional perimeter, $s$-minimal surfaces
satisfying uniform $C^{2}$ estimates actually satisfy a uniform $C^{2,\alpha}$
estimate. Thanks to this “a priori” type result, and a
compactness/contradiction argument together with a global classification
result, we will now prove that uniform $C^{2}$ (and thus also $C^{2,\alpha}$)
estimates actually hold unconditionally.

We start with a simple general lemma about writing sets with bounded
curvatures locally as graphs.

######  Lemma 5.27 (Curvature bounds imply graphicality).

Assume that the flatness assumptions ${\rm FA}_{\ell}(M,g,4,p_{0},\varphi)$
are satisfied. Let $E\subset M$ be an $s$-minimal surface such that $|{\rm
I\negthinspace I}_{\partial E}|\leq C_{1}$ in $\varphi(\mathcal{B}_{2})$ and

$\textnormal{Per}_{s}(E;\varphi(\mathcal{B}_{2}))\leq\frac{\kappa}{1-s}=\frac{\kappa}{\sigma}\,.$
(5.61)

Then, up to scaling the metric by a constant factor, performing a rotation in
coordinates and considering a different value of $\kappa$, the hypotheses
(5.25)–(5.28) are satisfied. More precisely, given $\delta>0$, there exist
$C\geq 1$ (depending only on $n,C_{1}$ and $\delta$) and some rotation
$\mathcal{R}:\mathbb{R}^{n}\to\mathbb{R}^{n}$ such that

${\rm FA}_{\ell}(M,Cg,4,p_{0},\psi)\hskip 7.0pt\mbox{is satisfied}$ (5.62)

with $\psi(x)=\varphi(\frac{1}{C}\mathcal{R}x)$, $E$ has fractional perimeter
(as a subset of $(M,Cg)$)

$\textnormal{Per}_{s}(E;\psi(\mathcal{B}_{2}))\leq\frac{C\kappa}{1-s}=\frac{C\kappa}{\sigma}\,,$
(5.63)

and moreover, denoting
$\Omega_{0}=\mathcal{B}_{1}^{\prime}\times(-1,1)\subset\mathbb{R}^{n}$ as per
usual,

$\displaystyle\psi^{-1}(\partial E)\mbox{ coincides with }\
\Gamma=\bigcup_{i=1}^{N}\,\Gamma_{i}=\bigcup_{i=1}^{N}\,\\{x_{n}=g_{i}(x^{\prime})\\}\
\mbox{ inside }\Omega_{0},\quad{\mbox{for some
}}g_{i}:\mathcal{B}_{1}^{\prime}\to\mathbb{R}$ (5.64)
$\displaystyle{\mbox{satisfying}}\quad
g_{1}<g_{2}<\cdots<g_{N}\qquad\mbox{and}\qquad\|\nabla
g_{i}\|_{L^{\infty}(\mathcal{B}_{1}^{\prime})}+\|D^{2}g_{i}\|_{L^{\infty}(\mathcal{B}_{1}^{\prime})}<\delta\,.$
(5.65)

###### Proof.

Let $x$ be the closest point in $\varphi^{-1}(\partial
E)\cap\mathcal{B}_{\varepsilon}$ to $0$, with $\varepsilon$ to be selected
later depending only on $n,C_{1}$ and $\delta$. If there is no such point, the
Proposition is trivially true with $C=\frac{4}{\varepsilon}$. Otherwise,
perform a rotation $\mathcal{R}$ in coordinates so that
$\mathcal{R}x=(0,...,0,|x|)$. Then, defining
$F=\mathcal{R}(\varphi^{-1}(E)-x)$, we have that $0\in F$ and moreover the
tangent space $T_{0}E^{\prime}$ is oriented orthogonally to the vertical
direction $e_{n}=(0,...,0,1)$, i.e. $T_{0}F=\\{y\,:\,y_{n}=0\\}$. Then, (the
boundary of) $F$ satisfies the hypotheses (5.64)–(5.65) in place of $\Gamma$,
up to possibly replacing $\mathcal{B}_{1}^{\prime}\times(-1,1)$ and $\delta$
by
$\mathcal{B}_{\frac{1}{C_{2}}}^{\prime}\times(-\frac{1}{C_{2}},\frac{1}{C_{2}})$
and $C_{2}$ in their statements for some $C_{2}=C_{2}(C_{1},n)$ — this is a
standard fact, which follows from arguing by Taylor expansion as in the proof
of Lemma 5.12 using the second fundamental form bound (which passes to $F$)
and the fact that $T_{0}F=\\{y_{n}=0\\}$. Then, we can choose
$C_{3}=C_{3}(C_{2},n,\delta)$ so that, considering the rescaled set $C_{3}F$,
the hypotheses (5.64)–(5.65) will be satisfied with
$\mathcal{B}_{1}^{\prime}\times(-2,2)$ and $\delta$ instead. Since
$\mathcal{R}x=(0,...,0,|x|)$, we deduce that the set
$C_{3}\mathcal{R}(\varphi^{-1}(E))=C_{3}F+C_{3}\mathcal{R}x$ satisfies the
same hypotheses with
$\mathcal{B}_{1}^{\prime}\times(-2+C_{3}|x|,2+C_{3}|x|)\subset\mathcal{B}_{1}^{\prime}\times(-2+C_{3}\varepsilon,2)$
instead as the domain, so that letting $\varepsilon=\frac{1}{C_{3}}$ we have
shown that $C_{3}\mathcal{R}(\varphi^{-1}(E))$ satisfies (5.64)–(5.65) with
$\mathcal{B}_{1}^{\prime}\times(-1,1)$ and $\delta$.
Defining $C=C_{3}$, so that
$\psi(x)=\varphi(\frac{1}{C_{3}}\mathcal{R}^{-1}x)$, since $\psi^{-1}(\partial
E)=C_{3}\mathcal{R}(\varphi^{-1}(E))$ we conclude that (5.64)–(5.65) hold just
as written. Moreover, as we have repeatedly used during the article, the
condition (5.62) then holds thanks to (b) in Remark 3.8, and (5.63) holds
thanks to (3.29). This finishes the proof of the lemma. ∎

We are now ready to prove the main theorem of this section.

######  Theorem 5.28.

Let $M$ be a closed manifold of dimension $3$. Let $s_{0}\in(s_{0},1)$ and
$s\in(0,1)$. Assume that the flatness assumptions ${\rm
FA}_{\ell}(M,g,4,p_{0},\varphi)$ are satisfied, and let $E$ be a
$\Lambda$-almost stable $s$-minimal surface in $\varphi(\mathcal{B}_{2})$.
Assume, moreover, that
$\textnormal{Per}_{s}(E;\varphi(\Omega_{0}))\leq\frac{\kappa}{1-s}=\frac{\kappa}{\sigma}$.
Then $E$ satisfies uniform $C^{2,\alpha}$ estimates and uniform separation
estimates of order $\sqrt{\sigma}$ in $\varphi(\mathcal{B}_{1})$, in the sense
that there are some positive constants $r$, $C$ depending on $n,\Lambda,s_{0}$
and $\kappa$ such that for all $x\in\mathcal{B}_{1}$, after a rotation
$\varphi^{-1}(\partial E)\cap\mathcal{B}_{r}(x)$ is a union of $C^{2,\alpha}$
graphs $g_{i}$ with uniform estimates $\|g_{i}\|_{C^{2,\alpha}}\leq C$ and at
vertical separation $\inf(g_{i+1}-g_{i})\geq\frac{1}{C}\sqrt{\sigma}$.

###### Proof.

The combination of Lemma 5.27 with Corollary 5.25 and Proposition 5.24 readily
shows that, in order to prove the Theorem, it suffices to see that $|{\rm
I\negthinspace I}_{\partial E}|\leq C$ in $\varphi(\mathcal{B}_{3/2})$, where
$C$ depends on the constants in the statement of the Theorem. Letting
$F:=\varphi^{-1}(\partial E)$, for convenience we will actually prove that

$|{\rm I\negthinspace I}_{\partial F}(x)|{\rm
dist}(x,\partial\mathcal{B}_{7/4})\leq C\quad\mbox{for all }x\in\partial
F\cap\mathcal{B}_{7/4},$ (5.66)

which considering $x\in\mathcal{B}_{3/2}$ implies that $|{\rm I\negthinspace
I}_{\partial E}|\leq C$ in $\varphi(\mathcal{B}_{3/2})$ for a different choice
of $C$ (determined by the flatness assumptions).
To prove (5.66) we employ a contradiction argument à la B. White (see
[White]): assume by contradiction that there exist sequences $s_{k}$,
$E_{k}\subset M_{k}$ (satisfying the assumptions of Theorem 5.28), with
associated $F_{k}=\varphi_{k}^{-1}(\partial E_{k})$, such that

$\max_{x\in\partial F_{k}\cap\varphi(B_{7/4})}|{\rm I\negthinspace
I}_{\partial F_{k}}(x)|{\rm
dist}(x,\partial\mathcal{B}_{7/4})=:\mathcal{M}_{k}\uparrow\infty,$

and let $x_{k}$ denote a sequence of points where the previous maxima are
attained.

Let us consider

$R_{k}:=\frac{\mathcal{M}_{k}}{{\rm dist}(x_{k},\partial\mathcal{B}_{7/4})}.$

By construction, $F_{k}$ satisfies $|{\rm I\negthinspace I}_{\partial
F_{k}}(x_{k})|=R_{k}$. Moreover, letting $r={\rm
dist}(x_{k},\partial\mathcal{B}_{7/4})$, we have that $|{\rm I\negthinspace
I}_{\partial F_{k}}|\leq 2R_{k}$ in $\mathcal{B}_{r/2}(x_{k})$.
Now, by Proposition 5.22, up to performing a change of coordinates we can (and
do) assume w.l.o.g. that moreover $x_{k}=0$ and $g_{ij}(0)=\delta_{ij}$, which
will simplify the proof. We should also modify the quantities above
accordingly by uniformly controlled multiplicative constants, but we omit this
for clarity since it introduces no change in the argument.
Considering the rescaled set $\widetilde{F}_{k}=R_{k}F_{k}$, we then have that
$|{\rm I\negthinspace I}_{\partial\widetilde{F}_{k}}(0)|=1$ and $|{\rm
I\negthinspace I}_{\partial\widetilde{F}_{k}}|\leq 2$ in
$\mathcal{B}_{\mathcal{M}_{k}/2}$. Moreover, defining
$\widetilde{\varphi}_{k}(\cdot)=\varphi_{k}(\frac{1}{R_{k}}\cdot)$, we can
write $\widetilde{F}_{k}=\widetilde{\varphi}^{-1}(E_{k})$, and then by (b) in
Remark 3.8 the flatness assumptions ${\rm
FA}_{\ell}(M_{k},R_{k}^{2}g_{k},4R_{k},p_{k},\widetilde{\varphi}_{k})$ are
satisfied. Furthermore, by (3.30), up to changing the value of $\kappa$ the
bound (5.26) is satisfied. We recall that $\mathcal{M}_{k}\uparrow\infty$ by
assumption.

We divide the remainder of the proof into two steps.

Step 1. Let us show first that necessarily $s_{k}\to 1$. Indeed, suppose (up
to extracting a subsequence) that $s_{k}\to s\in[s_{*},1)$. Then, thanks to
(uniform, since $s_{k}$ is bounded away from 1) layer separation estimates
—here we can even use some rough layer separation estimate as in Lemma 5.14—
and the standard $C^{3}$ regularity of fractional minimal surfaces (e.g.
arguing as in the last part of the proof of Corollary 5.25), by Ascoli–Arzelà
we would find that the $\partial\widetilde{F}_{k}$ converge locally in $C^{2}$
(as submanifolds of $\mathbb{R}^{3}$) to an embedded hypersurface
$\partial\widetilde{F}\subset\mathbb{R}^{3}$ of class $C^{2}$. Moreover, by
Remark 5.9 the $\partial E_{k}$ are $\Lambda/R_{k}^{n+s_{k}-2}$-almost stable
in $(M_{k},R_{k}^{2}g_{k})$, which since $R_{k}\to\infty$ gives immediately
that $\partial\widetilde{F}$ must be a stable $s$-minimal surface. See e.g.
Step 2 of the proof of [13, Proposition 6.1] for a similar argument on how
stability passes to the limit in the present case. Hence by the Bernstein–type
result for $C^{2}$ stable $s$-minimal surfaces in [13, Corollary 2.12],
$\widetilde{F}$ must be a half-space. On the other hand, passing to the limit
the equation $|{\rm I\negthinspace I}_{\partial\widetilde{F}_{k}}(0)|=1$ we
obtain, thanks to the $C^{2}$ convergence, that $|{\rm I\negthinspace
I}_{\partial\widetilde{F}}(0)|=1$, which is a contradiction.

Step 2. In light of Step 1, from now on we assume that $s_{k}\to 1$ (or
equivalently that $\sigma_{k}:=(1-s_{k})\to 0$). Fix $R\geq 1$, and let
$\Gamma_{k,\ell}$, with $1\leq\ell\leq N_{k}$, be the connected components of
the $C^{2}$ hypersurfaces $\partial\widetilde{F}_{k}\cap\mathcal{B}_{2R}$.

On the one hand, since $\widetilde{F}_{k}$ satisfies $|{\rm I\negthinspace
I}_{\partial\widetilde{F}_{k}}|\leq 2$, the combination of Lemma 5.27 and
Corollary 5.25 (plus a scaling+finite covering argument as usual) shows that
the $\partial\widetilde{F}_{k}$ satisfy uniform $C^{2,\alpha}$ estimates in
$\mathcal{B}_{R}$. Moreover, by Lemma 5.15 we deduce that the number of layers
$N_{k}$ of $\partial\widetilde{F}_{k}$ in $B_{R}$ is bounded. Therefore, up to
passing to a subsequence we can assume that $N=N_{k}$ is constant.
Then, by Ascoli-Arzelà, up to passing to a further subsequence (which again we
do not relabel) as $k\to\infty$ each of the $\\{\Gamma_{k,l}\\}_{l=1}^{N}$
converge simultaneously as $C^{2}$ submanifolds to limit surfaces
$\\{\widetilde{\Gamma}_{l}\\}_{l=1}^{N}$. Moreover, since for a fixed $k$ the
$\\{\Gamma_{k,l}\\}_{l=1}^{N}$ are disjoint, the $\widetilde{\Gamma}_{l}$ can
only possibly touch tangentially, but they cannot traverse each other. Now, by
(5.52) in Proposition 5.24, as $k\to\infty$ the local mean curvatures of the
$\widetilde{\varphi}(\Gamma_{k,l})$ are converging to zero, which together
with the fact that the metric $g_{ij}$ in coordinates is converging to the
Euclidean metric (recall that we could assume that $g_{ij}(0)=\delta_{ij}$,
and combine this with (3.13) with $R=R_{k}$ large) shows that the local (i.e.
classical) mean curvatures of the $\Gamma_{k,l}$ are converging to zero as
well. Thus, the $\widetilde{\Gamma}_{l}$ are classical minimal surfaces. This
implies, by the maximum principle for classical minimal surfaces, that for
$i\neq j$ the surfaces $\widetilde{\Gamma}_{i}$ and $\widetilde{\Gamma}_{j}$
are actually either identical or disjoint. Let the multiplicity $n_{l}$ denote
the number of layers that have collapsed onto the same limit $\Gamma_{l}$; it
is clear then that the $\partial\widetilde{F}_{k}$ converge in the varifold
sense to the varifold $V=\sum_{l}n_{l}\Gamma_{l}$ in $B_{R}$ (perhaps after
decreasing $R$ slightly to ensure that the $\Gamma_{l}$ are transversal to
$\partial\mathcal{B}_{R}$). Moreover, from this and Lemma 5.26, the mass
$\|V\|$ of this varifold satisfies

$\displaystyle\gamma_{n}\|V\|_{\mathcal{B}_{R}}$
$\displaystyle=\gamma_{n}\sum_{l}n_{l}\textnormal{Per}(\Gamma_{l},\mathcal{B}_{R})$
(5.67)
$\displaystyle=\gamma_{n}\lim_{k}\textnormal{Per}(\widetilde{F}_{k},\mathcal{B}_{R})$
$\displaystyle=\lim_{k}(1-s_{k})\textnormal{Per}_{s}(\widetilde{F}_{k},\mathcal{B}_{R})$
$\displaystyle=\lim_{k}(1-s_{k})\textnormal{Per}_{s}(E_{k},\widetilde{\varphi}_{k}(\mathcal{B}_{R}))\,,$

where the last fractional perimeter is computed on $(M_{k},R_{k}^{2}g_{k})$.
The last equality follows, once again, from the fact that the metric
coefficients are converging to the Euclidean ones in the coordinates given by
$\widetilde{\varphi}_{k}$, as shown by a change of variables and our estimates
for the kernel $K_{s}$ (see Proposition 4.3 and Lemma 4.4 in [18] for full
details).
Note that, in particular, from the expression above we deduce that

$\textnormal{Per}(\cup_{l}\Gamma_{l};\mathcal{B}_{R})\leq
CNR^{n-1}=CR^{n-1}\,.$ (5.68)

Claim. $V$ (or equivalently, each of the $\Gamma_{l}$) is a stable classical
minimal surface.

###### Proof of the claim.

More generally, we have the following. Given a vector field $X\in
C_{c}^{\infty}(B_{\rho};\mathbb{R}^{n})$, define
$X_{k}:=(\widetilde{\varphi}_{j})_{*}X$ and extend it by zero to a vector
field on $\widetilde{M}_{k}$. Fix $t\in\mathbb{R}$, and let $\psi_{X}^{t}$ and
$\psi_{X_{k}}^{t}$ denote their corresponding flows at time $t$. Then,

$\frac{d^{\ell}}{dt^{\ell}}(1-s_{k})\textnormal{Per}_{s_{k}}(\psi_{X_{k}}^{t}(E_{k});\widetilde{\varphi}_{k}(\mathcal{B}_{R}))\to\gamma_{n}\frac{d^{\ell}}{dt^{\ell}}\|(\psi_{X}^{t})_{\\#}V\|_{\mathcal{B}_{R}}\,\quad\mbox{as
}k\to\infty\,.$ (5.69)

The proof is as the one for (5.6), defining the single variable functions
$f_{k}(t)=(1-s_{k})\textnormal{Per}_{s_{k}}(\psi_{X_{k}}^{t}(E_{k});\widetilde{\varphi}_{k}(\mathcal{B}_{R}))$
and $f(t)=\gamma_{n}\|(\psi_{X}^{t})_{\\#}V\|_{\mathcal{B}_{R}}$, showing that
$f_{k}(t)\to f(t)$ as $k\to\infty$ thanks to Lemma 5.26 as in (5.67), and
applying Lemma 3.12 to see that the functions
$\frac{d^{\ell}}{dt^{\ell}}f_{k}(t)$ are locally uniformly bounded. See [18,
Lemma 4.4] for full details. ∎

With the claim at hand, we can apply the curvature estimates for $C^{2}$
stable minimal hypersurfaces with area bounds of type (5.68), see [45], and
deduce that $|{\rm I\negthinspace I}_{\cup_{l}\Gamma_{l}}|\leq\frac{C}{R}$ in
$B_{R/2}$. On the other hand, let $\Gamma_{k,0}$ denote the component of
$\partial\widetilde{F}_{k}$ that passes through the origin (recall that
$0\in\partial\widetilde{F}_{k}$), which verifies that $|{\rm I\negthinspace
I}_{\Gamma_{k,0}}(0)|=1$. Thanks to the $C^{2}$ convergence, we deduce that
$|{\rm I\negthinspace I}_{\Gamma_{0}}(0)|=1$ as well, where $\Gamma_{0}$
denotes the component of $\cup_{l}\Gamma_{l}$ obtained as a limit of the
$\Gamma_{k,0}$. Choosing $R$ large enough so that $\frac{C}{R}<1$, this
clearly gives a contradiction. ∎

###  5.4 Convergence of s-minimal surfaces with bounded index to classical
minimal surfaces

####  5.4.1 Estimates for $s$-minimal surfaces with bounded index

Up to now, we have obtained estimates in the almost-stable case. We will now
focus on the finite Morse index case. The pointwise $C^{2}$ estimates and
pointwise classical mean curvature estimates do not necessarily hold anymore
in the bounded index case, since we expect that neck-pinching may occur by
analogy with classical minimal surfaces, as in the case of the rescalings of a
catenoid converging to a hyperplane. On the other hand, as recorded in the
next Proposition, the powerful covering argument in Lemma 5.11 will remarkably
allow us to obtain a bound for the classical perimeter and a decay for the
$L^{1}$-norm of the mean curvature also in this case, since these are integral
quantities with a positive power-type scaling. In particular, we will use this
in the next subsection to obtain the convergence, in the sense of varifolds,
to a stationary limit in a very simple manner.

######  Proposition 5.29.

Let $M$ be a closed manifold of dimension $3$, and assume that the flatness
assumptions ${\rm FA}_{\ell}(M,g,4,p,\varphi)$ are satisfied. Let $E$ be an
$s$-minimal surface in $\varphi(\mathcal{B}_{2})$, with Morse index bounded by
$m$ and with

$\textnormal{Per}_{s}(E;\varphi(\mathcal{B}_{2}))\leq\frac{\kappa}{1-s}=\frac{\kappa}{\sigma}\,.$
(5.70)

Then, there exists $C>0$, depending only on $n$, $m$ and $\kappa$, such that

$\textnormal{Per}(E;\varphi(\mathcal{B}_{1}))\leq C$ (5.71)

and

$\int_{\partial E\cap\varphi(\mathcal{B}_{1})}|H_{\partial E}|\leq
C\sqrt{1-s}\,.$ (5.72)

###### Proof.

Step 1. We first prove (5.71) in the case where $E$ is actually
$\Lambda$-almost stable in $\varphi(\mathcal{B}_{2})$, for some fixed
$\Lambda$ (say $\Lambda=1$), with $C$ depending on $\Lambda$.

First, observe that by Theorem 5.28, we know a uniform bound on the second
fundamental form of $\partial E$ in $\varphi(\mathcal{B}_{1})$. Therefore, in
the rescaled setting given by Lemma 5.27 the hypotheses (5.62)–(5.65) are
satisfied, which shows that Lemma 5.15 is applicable in this rescaled setting
and thus (5.71) holds. Scaling back this information, what we have shown is
precisely that if $E$ is $\Lambda$-almost stable in
$\varphi(\mathcal{B}_{2})$, then it satisfies that

$\textnormal{Per}(E;\varphi(B_{\frac{1}{2C}}))\leq C\,,$

where $C$ has the right dependencies. As usual, a scaling + finite covering
argument then gives that, up to changing the value of $C$, actually

$\textnormal{Per}(E;\varphi(B_{1}))\leq C\,,$

which concludes Step 1.

Step 2. We prove (5.71) under the hypotheses of the present Proposition 5.29.

By the same arguments above, the result in Step 1 rescales appropriately upon
looking at smaller scales and centering at different points. Then, the
argument using Lemma 5.11 that we employed in the proof of Theorem 5.4
(perimeter estimate in the finite index case) to deduce the latter from
Proposition 5.10 (perimeter estimate in the $\Lambda$-almost stable case)
applies almost word-by-word to the present situation, and thus shows (5.71) in
our finite index setting.

Step 3. We also prove (5.72), which corresponds to a bound on the $L^{1}$ norm
of the mean curvature, under the hypotheses of the present Proposition 5.29.

To do so, we follow a similar strategy to Steps 1–2. First, assume that $E$ is
$\Lambda$-almost stable in $\varphi(\mathcal{B}_{2})$ for some fixed
$\Lambda$, just as in Step 1. Then (5.72) is indeed true: By the combination
of Theorem 5.28 and Lemma 5.27, after rescaling the metric appropriately the
hypotheses of Proposition 5.24 are satisfied. Applying (5.52), scaling back
the resulting information and performing a finite covering argument, we deduce
that the mean curvature $H_{\partial E}$ satisfies a uniform pointwise bound
of type $C\sqrt{\sigma}$ in $\varphi(\mathcal{B}_{1})$. Moreover, by Step 1
itself we know that $\partial E$ has bounded classical perimeter in
$\varphi(\mathcal{B}_{1})$. Therefore, integrating $H_{\partial E}$ over
$\partial E\cap\varphi(\mathcal{B}_{1})$ we see that if $E$ is
$\Lambda$-almost stable in $\varphi(\mathcal{B}_{2})$ for some fixed
$\Lambda$, then

$\int_{\partial E\cap\varphi(\mathcal{B}_{1})}|H_{\partial E}|\leq
C\sqrt{1-s}\,.$

The finite index case follows from the $\Lambda$-almost stable case, which we
have just proved, by arguing as in Step 2. Indeed, observe that the statement
we have just shown can be appropriately rescaled, upon looking at smaller
scales and centering at different points. More precisely, the $L^{1}$ norm of
the mean curvature of a hypersurface scales, upon zooming in by a factor $r$,
with power $r^{n-2}$, which is a positive power of $r$ for $n=3$ or higher.
Then, the covering argument in Step 2 of the proof of Theorem 5.4 can also be
applied in our case: Indeed, Lemma 5.11 applies to subadditive quantities
which satisfy a positive power-type scaling $r^{\beta}$, which includes the
case of the $L^{1}$ norm of the mean curvature of a hypersurface and
$\beta=n-2$. This gives the desired result. ∎

With the results up to now, we can obtain our first global result.

######  Proposition 5.30.

Let $E\subset M$ be an $s$-minimal surface with index at most $m$ and a
fractional perimeter bound $\textnormal{Per}_{s}(E;M)\leq\frac{\kappa}{1-s}$.
Then, there exists a constant $C=C(M,m,\kappa)$ such that

$\textnormal{Per}(E;M)\leq C$

and

$\int_{\partial E\cap M}|H_{\partial E}|\leq C\sqrt{1-s}\,.$

###### Proof.

Since $M$ is closed, there exists a number $\delta>0$ with the property that,
given any $p\in M$, the flatness assumptions ${\rm
FA}_{\ell}(M,g,4\delta,p,\varphi_{p})$ are satisfied for some $\varphi_{p}$;
see Remark 3.6. Having fixed such a $\delta>0$, by compactness we can find a
finite collection of points $p_{1},...,p_{N}$ such that the sets
$\varphi_{p_{1}}(\mathcal{B}_{\delta/32}(p_{1})),...,\varphi_{p_{N}}(\mathcal{B}_{\delta/32}(p_{N}))$
cover $M$. Moreover, by (3.30) in Lemma 3.17 and the assumption on
$\textnormal{Per}_{s}(E;M)$, up to having chosen $\delta$ small enough we
deduce that
$\textnormal{Per}_{s}(E;\varphi_{p_{i}}(\mathcal{B}_{\delta/16}(p_{i})))\leq\frac{C}{1-s}\delta^{n-s}$
for each $i$. Applying then Proposition 5.29, appropriately rescaled (i.e.
considering the manifold $(M,(\delta/32)^{-2}g)$, by our usual argument), and
scaling back, we deduce that

$\textnormal{Per}(E;\varphi_{p_{i}}(\mathcal{B}_{\delta/32}(p_{i})))\leq
C\delta^{n-1}$ (5.73)

and

$\int_{\partial
E\cap\varphi_{p_{i}}(\mathcal{B}_{\delta/32}(p_{i}))}|H_{\partial E}|\leq
C\sqrt{1-s}\,\delta^{n-2}\,,$

where $C$ has the dependencies indicated in the statement of the proposition.
Adding up these inequalities for $i=1,...,N$, since the
$\varphi_{p_{i}}(\mathcal{B}_{\delta/32}(p_{i}))$ cover $M$ we reach the
conclusion of the theorem. ∎

####  5.4.2 Convergence to a smooth classical minimal surface – Proof of
Theorem 1.4

This subsection proves, in several steps, the subsequential convergence (as
$s\to 1$) of sequences of $s$-minimal surfaces with bounded Morse index and
fractional perimeters to limiting smooth, classical minimal surfaces,
culminating in the proof of Theorem 1.4.
In fact, using the results in the previous section, we can readily prove the
convergence to a limit stationary varifold. The remainder of the section will
then focus on proving its regularity.

######  Theorem 5.31.

Let $M$ be a closed manifold of dimension $3$. Let $s_{i}\to 1^{-}$ as
$i\to\infty$, and let $E_{i}\subset M$ be an associated sequence of
$s_{i}$-minimal surfaces with index $\leq m$ and
$\limsup_{i}(1-s_{i})\textnormal{Per}_{s_{i}}(E_{i};M)\leq C_{1}$. Then, a
subsequence of the $\partial E_{i}$ converges, in the varifold sense, to a
limit integral stationary varifold $V$. Moreover, the $\partial E_{i}$ in the
subsequence converge in the Hausdorff sense to $\Sigma:={\rm supp}(V)$.

###### Proof.

Step 1. Varifold convergence.
The convergence in the varifold sense of a subsequence (not relabeled) of the
$\partial E_{i}$ to a limit integral varifold $\Sigma$ follows directly from
Proposition 5.30, by Allard’s compactness theorem ([1]). To check the
stationarity, let $X$ be a smooth vector field on $M$, and denote its flow at
time $t$ by $\psi_{X}^{t}$. By Proposition 5.30, there is a uniform constant
$C$ such that we can bound

$\Bigg{|}\frac{d}{dt}\Big{|}_{t=0}\textnormal{Per}(\psi_{X}^{t}(E_{i});M)\Bigg{|}=\Bigg{|}\int_{\partial
E_{i}}X\cdot\vec{H}_{\partial E_{i}}\Bigg{|}\leq\|X\|_{C^{0}}\int_{\partial
E_{i}}|H_{\partial E_{i}}|\leq C\sqrt{1-s_{i}}\|X\|_{C^{0}}\,.$ (5.74)

Letting $i\to\infty$, since $\sqrt{1-s_{i}}\to 0$ the convergence in the
varifold sense shows that the limit varifold $V$ has vanishing first
variation, or in other words it is stationary.

Step 2. Convergence in Hausdorff distance.

Let $\delta>0$ be such that the flatness assumptions ${\rm
FA}_{\ell}(M,g,4\delta,p,\varphi_{p})$ are satisfied for every $p\in M$, and
such that $\varphi_{p}$ corresponds to the exponential map in normal
coordinates; see Remark 3.6. Up to considering the rescaled manifold
$(M,\frac{1}{\delta^{2}}g)$ instead, we can also just assume that actually
${\rm FA}_{\ell}(M,g,4,p,\varphi_{p})$ holds.
Now, by the varifold convergence proved in Step 1, to obtain the Hausdorff
convergence it suffices to prove that the $E_{i}$ satisfy lower perimeter
estimates, meaning that there exists a constant $c=c(M)>0$ such that (for $i$
large enough) if $p\in\partial{E_{i}}$ then

$cr_{2}^{n-1}\leq\textnormal{Per}(E;\varphi_{p}(\mathcal{B}_{r_{2}}))$

for all $0<r_{2}<c$. To prove that this holds, fix $i$, and let
$p\in\partial{E_{i}}$. Since the fixed set $E_{i}$ has a smooth boundary,
performing a blow-up of $E_{i}$ around $p$ we see that the corresponding
sequence of rescalings converges locally (in normal coordinates, say) in
$H^{s_{i}/2}$ norm to a half-space in $\mathbb{R}^{n}$. Scaling back this
information, this shows that

$\liminf_{r_{1}\to
0}\frac{\textnormal{Per}_{s_{i}}(E_{i};\varphi_{p}(\mathcal{B}_{r_{1}}))}{r_{1}^{n-s}}=c(n,s_{i})\,,$

where the universal constant

$c(n,s_{i}):=\textnormal{Per}_{s_{i}}(\\{x_{n}\leq
0\\};\mathcal{B}_{1})\geq\frac{c(n)}{1-s_{i}}>0$

comes from the contribution in $\mathcal{B}_{1}\subset\mathbb{R}^{n}$ to the
fractional perimeter of a half-space. Multiplying by $(1-s_{i})$ and applying
Lemma 3.17, we obtain that

$c(n)\leq(1-s_{i})\liminf_{r_{1}\to
0}\frac{\textnormal{Per}_{s_{i}}(E_{i};\varphi_{p}(\mathcal{B}_{r_{1}}))}{r_{1}^{n-s}}\leq
C(1-s_{i})+C\frac{(1-s_{i})\textnormal{Per}_{s_{i}}(E_{i};\varphi_{p}(\mathcal{B}_{r_{2}}))}{r_{2}^{n-s}}$

for all $r_{2}\leq 1$. Selecting $i$ large enough so that $C(1-s_{i})\leq
c(n)/4$, we deduce that

$cr_{2}^{n-s}\leq(1-s_{i})\textnormal{Per}_{s_{i}}(E_{i};\varphi_{p}(\mathcal{B}_{r_{2}}))$

for some $c=c(n)>0$. Combining this with the interpolation result of (3.20),
we conclude that

$cr_{2}^{n-1}\leq\textnormal{Per}(E_{i};\varphi_{p}(\mathcal{B}_{r_{2}}))\,,$

which is precisely what we wanted to prove. ∎

The rest of the section focuses on proving the smoothness of $\Sigma$. We
begin with some preliminary results. The first one is the following lemma,
which shows that the Morse index can concentrate at most at $m$ many points
along a subsequence.

######  Lemma 5.32 (Almost stability outside of $m$ points).

Let $E_{i}$ be as in Theorem 5.31. Then, up to passing to a subsequence
$E_{i_{j}}$, there exist points $p_{1},...,p_{l}$ in $M$ with $l\leq m$ such
that the following holds: Given any $p\neq p_{1},...,p_{l}$, there exists a
radius $r_{p}>0$ such that the $E_{i_{j}}$ are eventually $\Lambda_{p}$-almost
stable on $B_{r_{p}}(p)$ for some $\Lambda_{p}<\infty$.

###### Proof.

Define the concentration scale by

$R(p,i,C)=\sup\\{r>0:\mbox{ a.s. ineq. }\eqref{lamasineq}\mbox{ holds for
}E_{i}\mbox{ on }B_{r}(p)\mbox{ with constant }C\\}\,.$

Define also the ”nonlocal index concentration set” of the $E_{i}$ (or
equivalently of the associated sequence of indices $\\{i\\}$) as the set

$IC(\\{i\\})=\\{p\in M\ :\ \limsup_{C}\,\liminf_{i}\,R(p,i,C)\,=\,0\\}.$

Step 1. We show that, up to passing to a subsequence, the nonlocal index
concentration set has at most $m$ points.
Indeed, assume by contradiction that every subsequence has at least $m+1$
points in its concentration set. In particular, for our entire sequence
$E_{i}$ there exist $p_{1},...,p_{m+1}$ in the nonlocal index concentration
set. We focus on $p_{1}$. Given $j\in\mathbb{N}$, define $C_{j}=j$ and
$\varepsilon_{j}=1/j$; by definition, there exists a subsequence of indices
$\\{i_{j}^{1}\\}_{j}$ such that $R(p_{1},i_{j}^{1},C_{j})<\varepsilon_{j}$, or
in other words $R(p_{1},i_{j}^{1},j)<1/j$. This implies that $q_{1}:=p_{1}$
belongs to the nonlocal index concentration set of the subsequence of surfaces
$E_{i_{j}}$, i.e. $q_{1}\in IC(\\{i_{j}^{1}\\}_{j})$.
Now, if $m>0$ (otherwise the Lemma is trivial), by our contradiction
assumption $E_{i_{j}^{1}}$ needs to have another point $q_{2}$ in its index
concentration set, since it must have at least $(m+1)\geq 2$ such points in
total. Therefore, there exists some subsequence of the $\\{i_{j}^{1}\\}_{j}$,
denoted by $\\{i_{j}^{2}\\}_{j}$, such that $R(q_{2},i_{j}^{2},j)<1/j$.
Moreover, since $i_{j}^{2}\geq i_{j}^{1}$ it is immediate to see that
$R(q_{1},i_{j}^{2},j)<1/j$ still holds.
Now, both $q_{1}$ and $q_{2}$ are in the index concentration set
$IC(\\{i_{j}^{2}\\}_{j})$. If $(m+1)>2$, by our contradiction assumption there
must exist another point $q_{3}$ which is also in the set, and we can proceed
as before; iterating this argument $(m+1)$ times, we find $q_{1},...,q_{m+1}$
and a subsequence $\\{i_{j}^{m+1}\\}_{j}$ as above such that
$R(q_{1},i_{j}^{m+1},j)<1/j,...,R(q_{m+1},i_{j}^{m+1},j)<1/j$.
Select now $\delta$ small enough so that the balls
$B_{\delta}(q_{1}),...,B_{\delta}(q_{m+1})$ are at a positive distance $d>0$
from each other. The almost-stability inequality in Lemma 5.7, applied with
$E=E_{i_{j}^{m+1}}$ and $\mathcal{U}_{k}=B_{\delta}(q_{k})$ for $k=1,...,m+1$,
says that $E_{i_{j}^{m+1}}$ is $\Lambda$-almost stable on one of the balls,
say $B_{\delta}(q_{k_{j}})$, with a constant $\Lambda<\infty$ independent of
$j$. Then, up to passing to a subsequence, we can assume that the
$E_{i_{j}^{m+1}}$ are actually all $\Lambda$-almost stable on the same ball,
say $B_{\delta}(q_{1})$. In other words, $\delta\leq
R(q_{1},i_{j}^{m+1},\Lambda)$ for all $j$. Taking $j$ large enough so that
$j\geq\Lambda$, we deduce that

$\delta\leq R(q_{1},i_{j}^{m+1},\Lambda)\leq R(q_{1},i_{j}^{m+1},j)<1/j\,.$

But then, by taking $j$ large enough so that also $1/j<\delta$, we reach a
contradiction. This concludes the proof of the claim in Step 1.

Step 2. Conclusion of the proof.
Let $\\{i_{j}\\}_{j}$ be the subsequence given by the statement of Step $1$,
and let $p_{1},...,p_{l}$ be all the points in the nonlocal index
concentration set of the $E_{i_{j}}$, so that $l\leq m$. Consider a point $p$
different from the $p_{1},...,p_{l}$, so that it is not in the nonlocal index
concentration set. Then, by definition, there exist a radius $r_{p}>0$ and
numbers $\Lambda(p)$ and $j_{0}(p)$ such that, for all $j\geq j_{0}$, we have
that $R(p,i_{j},\Lambda)\geq r_{p}$, or in other words the $E_{i_{j}}$ are
eventually $\Lambda(p)$-almost stable on $B_{r_{p}}(p)$. This concludes the
proof of the lemma. ∎

The next proposition shows the regularity of $\Sigma$, the support of $V$,
outside of the finitely many points where the index can concentrate.

######  Proposition 5.33.

Let $V$ be the limit integral stationary varifold obtained in Theorem 5.31 as
a subsequential limit of the $\partial E_{i}$. Then there exist points
$p_{1},...,p_{l}$ in $M$, with $l\leq m$, such that $\Sigma={\rm supp}(V)$
corresponds to a smooth classical minimal surface outside of
$p_{1},...,p_{l}$. Moreover, up to passing to a subsequence, the $\partial
E_{i}$ converge as (multi-sheeted) $C^{2}$ local normal graphs over any
compact subset of $\Sigma\setminus\\{p_{1},...,p_{l}\\}$.

###### Proof.

Let $p_{1},...,p_{l}$ be as in Lemma 5.32, and pass to a subsequence (still
labeled $E_{i}$) verifying the conclusions of both Theorem 5.31 and Lemma
5.32. Then, given $p\neq p_{1},...,p_{l}$, for $i$ large enough the $E_{i}$
are $\Lambda(p)$-almost stable in $B_{r_{p}}(p)$ for some $\Lambda(p)<\infty$
and $r_{p}>0$. By Theorem 5.28, which (after scaling) we can apply with some
$\varphi_{p}$ up to choosing a smaller value for $r_{p}$, the $E_{i}$ satisfy
then uniform $C^{2,\alpha}$ estimates in $B_{r_{p}}(p)$. Moreover, by
Proposition 5.29 (rescaled appropriately), the $E_{i}$ have perimeter bounded
by $Cr_{p}^{n-1}$ inside the ball.
Therefore, by Lemma 5.27, up to making $r_{p}$ even smaller the set
$\varphi^{-1}(\partial E_{i})$ is made of only a finite, bounded number
$d_{i}$ of uniformly $C^{2,\alpha}$ graphs $\\{\Gamma_{i,l}\\}_{l=1}^{d_{i}}$.
Moreover, by (5.52) the mean curvatures of the $\varphi(\Gamma_{i,l})$ are
going to $0$ as $s\to 1$. Then, by standard Ascoli-Arzelà arguments, the
$E_{i}$ converge as $C^{2}$ submanifolds to $\Sigma={\rm supp}(V)$ in
$B_{r_{p}}(p)$, which shows that $\Sigma$ is a union of ordered $C^{2}$
minimal hypersurfaces around $p$. Then, the maximum principle implies that
these minimal hypersurfaces are actually disjoint as well. Moreover, by the
usual regularity estimates for the classical minimal surface equation, we can
bootstrap the $C^{2}$ regularity and deduce that $\Sigma$ is actually a union
of ordered, disjoint smooth ($C^{\infty}$) submanifolds in $B_{r_{p}}(p)$.
Since the choice of $p\neq p_{1},...,p_{l}$ was arbitrary, we conclude that
$\Sigma$ is smooth everywhere except possibly at the points $p_{1},...,p_{l}$.

Let now $K$ be a compact subset
$K\subset\Sigma\setminus\\{p_{1},...,p_{l}\\}$. By the above, the $E_{i}$
eventually satisfy $C^{2,\alpha}$ estimates around any point different from
$p_{1},...,p_{l}$; by compactness of $K$, covering it with finitely many balls
where such $C^{2,\alpha}$ estimates hold we deduce that, actually, the $E_{i}$
eventually satisfy uniform $C^{2,\alpha}$ estimates (depending on $K$) in a
whole neighbourhood of $K$. Since the $E_{i}$ are converging to $\Sigma$, we
conclude that we can eventually write the $E_{i}$, locally, as normal graphs
over $K$. This concludes the proof ∎

We are now ready to deduce the crucial result that the mass of our limit
varifold $V$ is the limit of the fractional perimeters of the $E_{i}$, which
we had not addressed up to now. We more generally prove the stronger result
that the variations of the varifold mass of $V$ of any order are the limits of
the corresponding variations of the fractional perimeters of the $E_{i}$. The
proof is delicate, since it needs to account for the potentially singular
convergence around the $p_{1},...,p_{l}$ as well as the risk (when localising
the fractional perimeter) of concentration of mass/classical perimeter close
to boundaries of domains but which the fractional perimeter would not be able
to capture.

######  Proposition 5.34.

In the setting of Theorem 5.31,

$\lim_{i}(1-s_{i})\textnormal{Per}_{s}(E_{i};M)=\gamma_{n}\|V\|_{M}\,,$

with $\gamma_{n}$ as in (3.7).
More generally, we have the following. Let $X$ be a vector field of class
$C^{\infty}$ on $M$, and let $\psi^{t}:=\psi_{X}^{t}$ denote its flow at time
$t$. Then, for every $k\in\mathbb{N}$,

$\frac{d^{k}}{dt^{k}}(1-s_{i})\textnormal{Per}_{s}(\psi^{t}(E_{i});M)\to\frac{d^{k}}{dt^{k}}\|\psi_{\\#}^{t}(V)\|_{M}\,\quad\mbox{as
}i\to\infty\,.$ (5.75)

###### Proof.

Step 1. We prove that the fractional perimeters of the $E_{i}$ converge to the
varifold mass of $V$.

Let $\delta>0$, which will be fixed until the end of the proof and then made
to go to $0$. Let $\Sigma$ be as in Proposition 5.33, so that the $E_{i}$
converge as uniform $C^{2}$ normal graphs over
$K:=\Sigma\setminus\\{B_{\delta}(p_{1})\cup...\cup B_{\delta}(p_{l})\\}$.
Define the compact manifold with boundary
$N:=M\setminus\\{B_{4\delta}(p_{1})\cup...\cup B_{4\delta}(p_{l})\\}$; by
slightly increasing the value of $\delta$ if necessary, we can moreover assume
that $\Sigma$ intersects $\partial N$ transversely and only in its smooth
part. Let now $\\{\mathcal{C}_{j}\\}$ be a decomposition of $N$ as in Lemma
4.8, with associated $(1+\delta)$-biLipschitz equivalences
$F_{j}:\mathcal{C}_{j}\to C_{j}\subset\mathbb{R}^{n}$ which are given by the
restriction of geodesic normal coordinate maps
$\widetilde{F}_{j}:B_{r_{j}}\to\mathcal{B}_{r_{j}}$ to the $C_{j}$. By the
proof of Lemma 4.8 and the assumption that $\Sigma$ is transverse to $\partial
N$, it is simple to see that we can moreover require that $\Sigma$ intersect
each of the $\mathcal{C}_{j}$ transversely and only in the smooth part of
$\mathcal{C}_{j}$.
It is clear then that, since the $E_{i}$ are converging (with multiplicity) as
$C^{2}$ submanifolds to $\Sigma$ in $N$, for all $i$ large enough the same
transversality properties are also true for the $E_{i}$, and in a quantifiable
way. We record in a claim the precise property that we will need, and which
follows directly from this discussion. We will use the notation
$\mathcal{N}_{r}F$ to denote the set (neighborhood) of points at distance at
most $r$ from a set $F$.

Claim. There exists a constant $C$ such that, for all $i$ large enough and $r$
small enough, $\partial
E_{i}\cap\bigcup_{j}\mathcal{N}_{3r}\partial\mathcal{C}_{j}$ can be covered
with $Cr^{2-n}$ (or less) balls $\\{B_{r/2}(q_{k})\\}_{k}$ of radius $r/2$.

This is to be read as “since $\Sigma$ is smooth and quantitatively transversal
to the $\mathcal{C}_{j}$, it accumulates area close to the boundaries of the
$\mathcal{C}_{j}$ comparably to how a hyperplane passing through the center of
a unit ball does so relative to the boundary of said ball. Since the $\partial
E_{i}$ converge in $C^{2}$ to $\Sigma$ in $K$, the same is true of the
$\partial E_{i}$”. In particular, if $\\{B_{r/2}(q_{k})\\}_{k}$ is a
collection of balls of radius $r$ as in the claim and $r$ is small enough, by
the classical perimeter bound in (5.71) for $i$ large enough we can bound

$\textnormal{Per}(E_{i};\mathcal{N}_{r}\partial\mathcal{C}_{j})\leq\sum_{k}\textnormal{Per}(E_{i};B_{r/2}(q_{k}))\leq
Cr^{2-n}r^{n-1}=Cr\,,$

which goes to $0$ as $r\to 0$ and thus shows that the $E_{i}$ do not
accumulate perimeter close to the boundary of the $\mathcal{C}_{j}$. This
simple technical assumption, which is satisfied by our $E_{i}$, is essential
to guarantee that the fractional and classical perimeters of our objects
inside the $\mathcal{C}_{j}$ are comparable.

We are now ready to begin the proof of Step 1. We will omit the constant
$\gamma_{n}$ in what follows, and every instance of a fractional perimeter
should be understood as incorporating this multiplicative constant in front.
Observe that, by transversality, we can split

$\textnormal{Per}(E_{i};M)=\textnormal{Per}(E_{i};M\setminus
N)+\textnormal{Per}(E_{i};N)=\textnormal{Per}(E_{i};M\setminus
N)+\sum_{j}\textnormal{Per}(E_{i};\mathcal{C}_{j})\,.$

An iterated application of the triangle inequality then leads to the bound

$\displaystyle\Big{|}\textnormal{Per}(E_{i};M)-(1-s_{i})\textnormal{Per}_{s_{i}}(E_{i};M)\Big{|}=$
(5.76) $\displaystyle=\Big{|}\textnormal{Per}(E_{i};M\setminus
N)+\sum_{j}\textnormal{Per}(E_{i};\mathcal{C}_{j})-(1-s_{i})\textnormal{Per}_{s_{i}}(E_{i};M)\Big{|}$
$\displaystyle\leq\Big{|}\textnormal{Per}(E_{i};M\setminus
N)-(1-s_{i})\textnormal{Per}_{s_{i}}(E_{i};M\setminus
N)\Big{|}+\sum_{j}\Big{|}\textnormal{Per}(E_{i};\mathcal{C}_{j})-(1-s_{i})\textnormal{Per}_{s_{i}}(E_{i};\mathcal{C}_{j})\Big{|}$
$\displaystyle\ \ \
+\Big{|}(1-s_{i})\textnormal{Per}_{s_{i}}(E_{i};M)-[(1-s_{i})\textnormal{Per}_{s_{i}}(E_{i};M\setminus
N)+\sum_{j}(1-s_{i})\textnormal{Per}_{s_{i}}(E_{i};\mathcal{C}_{j})]\Big{|}\,.$

Since $M\setminus N\subset\cup B_{4\delta}(p_{i})$, we immediately see that we
can bound the first term as

$\displaystyle\Big{|}\textnormal{Per}(E_{i};M\setminus
N)-(1-s_{i})\textnormal{Per}_{s_{i}}(E_{i};M\setminus N)\Big{|}\leq
C(\delta^{n-1}+\delta^{n-s})+\varepsilon(\sigma)\leq
C\delta^{n-1}+\varepsilon(\sigma)\,,$ (5.77)

simply by applying (5.71) and (3.30) on each of the balls $B_{4\delta}(p_{i})$
and bounding the remaining nonlocal interactions between different balls by
$\varepsilon(\sigma)$ thanks to disjointness (up to choosing $\delta$ small
enough) of the $B_{4\delta}(p_{i})$.
The second term is the main one. To obtain a bound for
$\Big{|}\textnormal{Per}(E_{i};\mathcal{C}_{j})-(1-s_{i})\textnormal{Per}_{s_{i}}(E_{i};\mathcal{C}_{j})\Big{|}$,
we can argue as in Proposition 5.26, using
$F_{j}^{-1}:C_{j}\subset\mathbb{R}^{n}\to\mathcal{C}_{j}$ as a chart
parametrisation and the fact that $F_{j}$ is a $(1+\delta)$-biLipschitz
equivalence. The only difference is that $C_{j}$ is not a cylinder, so that
the $F_{j}(E_{i})$ cannot be exactly written as a union of subgraphs inside
$C_{j}$; on the other hand, the $F_{j}$ are the restriction of geodesic normal
coordinate maps $\widetilde{F}_{j}:B_{r_{j}}\to\mathcal{B}_{r_{j}}$ to the
$C_{j}$, so that (up to choosing $r_{j}$ small enough) the
$\widetilde{F}_{j}(E_{i})$ are indeed a union of subgraphs $A_{k}$ inside
$\mathcal{B}_{r_{j}}$. We can then decompose $F_{j}(E_{i})=\cup_{k}(A_{k}\cap
C_{j})$, where (as argued at the beginning) the $A_{k}\cap C_{j}$ are
quantitatively transversal the $\partial C_{j}$. This allows us to apply [16,
Lemma 11] to each of the $A_{k}\cap C_{j}$, exactly as in the proof of
Proposition 5.26. Anyhow, we conclude that

$\displaystyle\sum_{j}\Big{|}\textnormal{Per}(E_{i};\mathcal{C}_{j})-(1-s_{i})\textnormal{Per}_{s_{i}}(E_{i};\mathcal{C}_{j})\Big{|}$