$\Sigma_{1}\cap\Sigma_{2}\neq\emptyset\,.$

###### Proof.

Let $\mathsf{d}_{1}$ and $\mathsf{d}_{2}$ denote $\mathsf{d}_{\Sigma_{1}}$ and
$\mathsf{d}_{\Sigma_{2}}$ respectively and let
$\bar{\mathsf{d}}:=\mathsf{d}_{1}+\mathsf{d}_{2}$.
Assume by contradiction that $\Sigma_{1}\cap\Sigma_{2}=\emptyset$. Then it is
easily seen that $\bar{\mathsf{d}}$ attains one of its minima at a point $x\in
X\setminus(\Sigma_{1}\cup\Sigma_{2})$. Indeed it is sufficient to consider a
minimizing geodesic between $\Sigma_{1}$ and $\Sigma_{2}$ whose length is
$\mathsf{d}(\Sigma_{1},\Sigma_{2})>0$ and pick a point inside it.

By Theorem 5.1,

$\bm{\Delta}\mathsf{d}_{1}\leq-(N-1)\mathsf{d}_{1}\,,\quad\text{and}\quad\bm{\Delta}\mathsf{d}_{2}\leq-(N-1)\mathsf{d}_{2}\,,\quad\text{on
$X\setminus(\Sigma_{1}\cup\Sigma_{2})$}\,.$

Hence

(5.34) $\bm{\Delta}\bar{\mathsf{d}}\leq-(N-1)\bar{\mathsf{d}}\,,\quad\text{on
$X\setminus(\Sigma_{1}\cup\Sigma_{2})$}\,.$

In particular, there is a neighbourhood $U$ of $x$ such that
$\bar{\mathsf{d}}$ is superharmonic on $U$ and attains a minimum at the
interior point $x$. The strong maximum principle implies that
$\bar{\mathsf{d}}$ is constant in a neighbourhood of $x$, that contradicts the
strict superharmonicity of $\bar{\mathsf{d}}$ in (5.34), since
$\bar{\mathsf{d}}(x)>0$. ∎

######  Remark .

The assumptions of Theorem 5.3 cover in particular the classical case of
smooth minimal hypersurfaces in closed manifolds with positive Ricci
curvature. Indeed, as we already mentioned, smooth minimal hypersurfaces are,
locally, perimeter minimizing boundaries.

##  6\. Regularity theory

This section is dedicated to the partial regularity theory for minimal
boundaries on non collapsed $\operatorname{RCD}$ spaces. Our main result will
be that they are topologically regular away from sets of ambient codimension
three, and from the boundary of the space. Besides from a sharp Hausdorff
dimension estimate (see Theorem 6.5), we will obtain also a Minkowski estimate
for the quantitative singular set (see Theorem 6.7). Following a classical
pattern, these results will be achieved through two intermediate steps:

  * •

an $\varepsilon$-regularity result, Theorem 6.1 showing that under certain
assumptions at a given location and scale a minimal boundary is topologically
regular;

  * •

the analysis dedicated to guarantee that the assumptions of the
$\varepsilon$-regularity theorem are verified at many locations and scales
along the minimal boundary. This is pursued as follows:

    * –

in subsection 6.3, via dimension reduction arguments, we prove sharp Hausdorff
dimension estimates of the singular set (see Theorem 6.5). Here the arguments
depart from the classical ones: in the Euclidean (resp. smooth) setting,
minimal boundaries satisfy a very powerful monotonicity formula (resp. up to a
lower order term) which implies that every tangent space to a minimal boundary
is a cone. In the present non-smooth setting, it seems not possible to repeat
the Euclidean/smooth computations and it is not clear if such a (perturbed)
monotonicity formula holds;

    * –

in subsection 6.2 we prove sharp perimeter bounds for the equidistant sets
from locally minimal boundaries which will be used in subsection 6.4 to obtain
the quantitative regularity results (see Theorem 6.7) through a series of
covering arguments that control the regularity of the space and the regularity
of the minimal boundary together. The interpretation of minimality via
Laplacian bounds on the distance function obtained in subsection 5.1 will play
a key role here.

As some examples will show, the threshold dimension for the full regularity is
lower in this framework than in the Euclidean case: our Hausdorff codimension
three estimate for the singular set is sharp (see subsection 6.3), moreover,
already in ambient dimension $4$ there are examples of tangent cones with no
Euclidean splittings (see subsection 6.1) and of topologically irregular
minimal boundaries.

###  6.1. An $\varepsilon$-regularity theorem

The aim of this subsection is to establish an $\varepsilon$-regularity result
for minimal boundaries. This will provide a (weak) counterpart of the
classical statement for minimal boundaries in the Euclidean setting.

Usually, the outcome of an $\varepsilon$-regularity theorem is that if a
certain solution is close enough to a rigid model then it is regular. The
celebrated result for minimal boundaries in the Euclidean case from [54] says
that a minimal boundary contained in a sufficiently small strip around a
hyperplane is analytic.
Arguably, and as elementary examples show, this is too much to hope for in the
present setting. Our $\varepsilon$-regularity result will be more in the
spirit of Reifenberg’s original approach: we will show that a minimal boundary
which is close enough to the boundary of a half-space (in the Gromov-Haudorff
sense) is topologically regular.
This could be considered as the counterpart for minimal boundaries of the
celebrated $\varepsilon$-regularity result for manifolds with lower Ricci
curvature bounds obtained in [49, 41] and extended to $\operatorname{RCD}$
spaces in [89], see Theorem 2.2.

To avoid confusion let us clarify that in this subsection by local perimeter
minimizer in an open domain we intend that the perimeter is minimized among
all the competitors that are perturbations inside the domain. This is a much
stronger requirement than the one considered in subsection 5.1 to obtain mean
curvature bounds. For smooth hypersurfaces in smooth ambient spaces,
subsection 5.1 would correspond to minimality (i.e. vanishing mean curvature),
while here we will be concerned with locally area minimizers.
Moreover, this subsection will be independent of the theory of mean curvature
bounds that we have developed so far. Mean curvature bounds will enter into
play later on, when proving that the assumptions of the
$\varepsilon$-regularity theorem are in force at many locations and scales,
see subsection 6.2 and subsection 6.3.

Let us introduce some useful terminology, adapting the notion of flatness from
the Euclidean to the non smooth and non flat case. With respect to the
Euclidean realm, in the non flat framework there are many more rigid
situations to be considered. This is also due to the following result,
yielding existence of a large family of flat minimal boundaries.

######  Lemma .

Let $(Y,\mathsf{d}_{Y},\mathscr{H}^{N-1})$ be an $\operatorname{RCD}(0,N-1)$
metric measure space and let $X:=\mathbb{R}\times Y$ be endowed with the
canonical product metric measure structure. Let $E:=\\{t<0\\}$, where we
denoted by $t$ the coordinate of the Euclidean factor $\mathbb{R}$. Then $E$
is a perimeter minimizing set.

###### Proof.

The vector field $\nabla t$ is easily checked to be a calibration for $E$,
($t$ is harmonic, hence $\nabla t$ has vanishing divergence). The conclusion
follows from a classical calibration argument, exploiting Theorem 2.7 and
Theorem 2.9 as in the smooth setting. ∎

Recall that convergence in the $L^{1}$ strong sense of sets of finite
perimeter along pmGH converging sequences of metric measure spaces is
metrizable, see [6, Appendix A]. By the above, we are entitled to give the
following.

######  Definition  ($\varepsilon$-flat points).

Let $\varepsilon>0$. If $(X,\mathsf{d},\mathscr{H}^{N})$ is an
$\operatorname{RCD}(-\varepsilon,N)$ metric measure space and $E\subset X$ is
a set of finite perimeter, perimeter minimizing in $B_{2}(x)\subset X$ such
that:

  * •

there exists an $\operatorname{RCD}(0,N-1)$ metric measure space
$(Y,\mathsf{d}_{Y},\mathscr{H}^{N-1},y)$ such that the ball $B_{2}(x)\subset
X$ is $\varepsilon$-GH close to the ball $B_{2}((0,y))\subset\mathbb{R}\times
Y$;

  * •

$E$ is $\varepsilon$-close on $B_{2}(x)$ in the $L^{1}$ topology to
$\\{t<0\\}\subset\mathbb{R}\times Y$ and $\partial E\cap B_{2}(x)$ is
$\varepsilon$-GH close to $\\{t=0\\}\cap B_{2}(0,y)\subset\mathbb{R}\times Y$;

then we shall say that $E$ is $\varepsilon$-flat at $x$ in $B_{2}(x)$.

The notion of $\varepsilon$-flat set at $x$ in $B_{r}(x)$ can be introduced
analogously by scaling.

######  Definition  ($\varepsilon$-regular points).

Let $\varepsilon>0$. If $(X,\mathsf{d},\mathscr{H}^{N})$ is an
$\operatorname{RCD}(-\varepsilon,N)$ metric measure space and $E\subset X$ is
a set of finite perimeter, perimeter minimizing in $B_{2}(x)\subset X$, such
that:

  * •

the ball $B_{2}(x)\subset X$ is $\varepsilon$-GH close to the ball
$B_{2}(0^{N})\subset\mathbb{R}^{N}$;

  * •

$E$ is $\varepsilon$-close on $B_{2}(x)$ in the $L^{1}$ topology to
$\\{t<0\\}\subset\mathbb{R}^{N}$ and $\partial E\cap B_{2}(x)$ is
$\varepsilon$-GH close to $\\{t=0\\}\cap B_{2}(0^{N})\subset\mathbb{R}^{N}$,
where we denoted by $t$ one of the canonical coordinates on $\mathbb{R}^{N}$;

then we shall say that $E$ is $\varepsilon$-regular at $x$ in $B_{2}(x)$.

The notion of $\varepsilon$-regular set at $x$ in $B_{r}(x)$ can be introduced
analogously by scaling.

######  Remark .

Let $E\subset X$ be perimeter minimizing inside an open domain $\Omega\subset
X$. Let $x\in\partial E$ and assume that there exists an
$\operatorname{RCD}(0,N-1)$ metric measure space
$(Y,\mathsf{d}_{Y},\mathscr{H}^{N-1},y)$ such that, denoting by $t$ the
coordinate of the split factor $\mathbb{R}$ in the product $\mathbb{R}\times
Y$ with canonical product metric measure structure,

$\big{\\{}\big{(}\\{t<0\\},(0,y),\mathbb{R}\times
Y\big{)}\big{\\}}\in\operatorname{Tan}_{x}(E,X,\mathsf{d},\mathscr{H}^{N})\,.$

Then, for any $\varepsilon>0$ and any $r_{0}>0$, there exists $0<r<r_{0}$ such
that $E$ is $\varepsilon r$-flat in $B_{r}(x)$. This is a direct consequence
of Theorem 2.11, together with the very definition of tangent to a set of
finite perimeter.

Analogously, if

$\big{\\{}\big{(}\\{t<0\\},0^{N},\mathbb{R}^{N}\big{)}\big{\\}}\in\operatorname{Tan}_{x}(E,X,\mathsf{d},\mathscr{H}^{N})\,,$

then for any $\varepsilon>0$ and for any $r_{0}>0$ there exists $0<r<r_{0}$
such that $E$ is $\varepsilon r$-regular at $x$ on $B_{r}(x)$.

Below, we shall fix the scale $r=1$. As we already argued, the statements are
scale invariant, therefore this is not a loss of generality.

The stability of perimeter minimizers allows to get a measure bound out from
Gromov-Hausdorff closeness.

######  Lemma  (Perimeter density estimate for perimeter minimizers).

For any $\delta>0$ there exists $\varepsilon=\varepsilon(\delta,N)>0$ such
that the following holds. If $(X,\mathsf{d},\mathscr{H}^{N})$ is an
$\operatorname{RCD}(-\varepsilon,N)$ metric measure space, $E\subset X$ is
perimeter minimizing in $B_{4}(x)$, $x\in\partial E$ and $E$ is
$\varepsilon$-regular at $x$ in $B_{2}(x)$, then

(6.1) $1-\delta\leq\frac{\operatorname{Per}(E,B_{1}(x))}{\omega_{N-1}}\leq
1+\delta\,,$

where $\omega_{N-1}$ denotes the volume of the unit ball in
$\mathbb{R}^{N-1}$.

###### Proof.

The statement can be proved by a contradiction argument.

Consider a sequence of sets of finite perimeter $E_{n}\subset X_{n}$, where
$x_{n}\in\partial E_{n}$, $(X_{n},\mathsf{d}_{n},\mathscr{H}^{N})$ are
$\operatorname{RCD}(-1/n,N)$ metric measure spaces, $E_{n}$ is $1/n$-regular
in $B_{2}(x_{n})$ and perimeter minimizing in $B_{4}(x_{n})$. Then the
following holds: the balls
$B_{2}(x_{n})\subset(X_{n},\mathsf{d}_{n},\mathscr{H}^{N},x_{n})$ are
converging to
$B_{2}(0^{N})\subset(\mathbb{R}^{N},\mathsf{d}_{\mathrm{eucl}},\mathscr{H}^{N},0^{N})$
in the pmGH topology and the sets of finite perimeter $E_{n}$ are converging
to $\mathbb{H}^{N}$ on $B_{2}(0^{N})$ in the $L^{1}_{{\rm loc}}$-topology,
with boundaries $\partial E_{n}$ Hausdorff converging to the boundary
$\partial\mathbb{H}^{N}$ on $B_{2}(x_{n})$.
Then

$\operatorname{Per}(E_{n},B_{1}(x_{n}))\to\operatorname{Per}(\mathbb{H}^{N},B_{1}(0^{N}))=\omega_{N-1}\,,\quad\text{as
$n\to\infty$}\,,$

thanks to the weak convergence of perimeter measures in Theorem 2.11 and the
observation that $\operatorname{Per}(\mathbb{H}^{N},\partial B_{1}(0^{N}))=0$.
∎

######  Remark .

Let us recall that we can associate to any locally area minimizing cone
$C\subset\mathbb{R}^{N}$ with vertex at $0\subset\mathbb{R}^{N}$ its density

$\Theta_{0,C}:=\frac{\operatorname{Per}(C,B_{1}(0))}{\omega_{N-1}}=\frac{\operatorname{Per}(C,B_{r}(0))}{\omega_{N-1}r^{N-1}}\,,\quad\text{for
any $0<r<\infty$}\,.$

Then, among all the possible densities of minimal cones
$C\subset\mathbb{R}^{N}$, the halfspace attains the minimal one, and there is
a strictly positive gap between the density of the half-space and the
densities of all the other minimal cones.
This can be rephrased by saying that there exists $c_{N}>0$ such that, for any
minimal cone $C\subset\mathbb{R}^{N}$ with vertex at $0^{N}$ and different
from the half-space,

(6.2) $\Theta_{0,C}>1+c_{N}=\Theta_{0,\mathbb{H}^{N}}+c_{N}\,.$

The statement is classical, and it can be proved arguing by contradiction by
relying on the regularity theory for perimeter minimizers. More in detail, the
density at the vertex of a cone equals its density at infinity, which is
independent of the chosen base point. Namely

(6.3)
$\Theta_{0,C}=\lim_{r\to\infty}\frac{\operatorname{Per}(C,B_{r}(0))}{\omega_{N-1}r^{N-1}}=\lim_{r\to\infty}\frac{\operatorname{Per}(C,B_{r}(p))}{\omega_{N-1}r^{N-1}}\,,$

for any $p\in\partial C$. By the regularity theory, we can choose $p$ to be a
regular boundary point and apply the monotonicity formula to infer that

(6.4) $\Theta_{0,C}\geq\lim_{r\to
0}\frac{\operatorname{Per}(C,B_{r}(p))}{\omega_{N-1}r^{N-1}}=\Theta_{0,\mathbb{H}^{N}}\,.$

The argument above also shows that a cone with the same density of the half-
space must be the half-space.
In order to prove (6.2) we argue by contradiction. If there is a sequence of
cones $C_{n}$, all different from the half-space, and with densities
converging to the density of the half-space, by compactness and stability we
can extract a subsequence converging to a perimeter minimizer. The density at
infinity of this limit minimizer is easily seen to equal
$\Theta_{0,\mathbb{H}^{N}}$. By the above considerations, the limit is the
half-space. By the $\varepsilon$-regularity theorem $C_{n}$ is smooth on
$B_{1}(0)$ for any sufficiently large $n$. This is a contradiction to the
assumption that $C_{n}$ is a cone different from the half-space.

In the Euclidean theory minimal boundaries are smooth, if the ambient
dimension is less or equal than $7$. Moreover, they are smooth in any
dimension in a region where they are sufficiently flat. These statements are
the outcome of the classification of minimal cones up to dimension $7$ and of
the already mentioned $\varepsilon$-regularity theorem in [54].

Notice that subsection 6.1 shows that there is no hope for such a statement in
our setting: consider a (possibly singular) Alexandrov space of dimension two
and its product with a line, then the Alexandrov space is a minimal boundary
inside the product. Hence the best regularity we can achieve for minimal
boundaries in ambient dimension three is the regularity of two dimensional
Alexandrov spaces.
Nevertheless one might hope that sufficiently flat minimal boundaries in the
sense of subsection 6.1 have flat tangents (i.e. $0$-flat). It turns that this
is not the case, at least when the ambient dimension is greater than $4$, due
to the following.

######  Remark .

Denote by $\mathbb{S}^{3}_{r}$ the three dimensional sphere of radius $r$
endowed with the canonical Riemannian metric, and by $\mathbb{H}^{3}_{r}$ the
upper hemisphere. Let also $0$ denote the tip of the cone. In [111] it is
shown that the cone $C(\mathbb{H}^{3}_{r})$ is perimeter minimizing in
$B_{1}(0)\subset C(\mathbb{S}^{3}_{r})$, for $r<1$ sufficiently close to $1$.

The effect of this remark is that in our framework there cannot be an
improvement of flatness, as it happens in the classical case, at least for
ambient dimension greater than $4$. The best we can hope for is that flatness
is preserved along scales.

######  Theorem 6.1 ($\varepsilon$-regularity).

Let $N>1$ be fixed. For any $\varepsilon>0$ there exists
$\delta=\delta(\varepsilon,N)>0$ such that the following holds. Let
$(X,\mathsf{d},\mathscr{H}^{N})$ be an $\operatorname{RCD}(-\delta,N)$ metric
measure space, $E\subset X$ be a set of locally finite perimeter,
$x\in\partial E$ be such that $E$ is perimeter minimizing on $B_{4}(x)$ and
$E$ is $\delta$-regular in $B_{2}(x)$; then for any $y\in\partial E\cap
B_{1}(x)$ and for any $0<r<1$, $E$ is $\varepsilon r$-regular in $B_{r}(y)$.

Moreover, for any $0<\alpha<1$, there exists $\delta=\delta(\alpha,N)>0$ such
that if $X$ and $E$ are as above (in particular, $x\in\partial E$ and $E$ is
$\delta$-regular at $x$ in $B_{2}(x)$), then $\partial E\cap B_{1}(x)$ is
$C^{\alpha}$-homeomorphic to the ball $B_{1}(0^{N-1})\subset\mathbb{R}^{N-1}$.

###### Proof.

We argue by contradiction. Let us suppose that the conclusion is not true.
Then we can find $\varepsilon>0$, a sequence of $\operatorname{RCD}(-1/n,N)$
metric measure spaces $(X_{n},\mathsf{d}_{n},\mathscr{H}^{N},x_{n})$ and sets
of locally minimal perimeter $E_{n}\subset X_{n}$ such that $x_{n}\in\partial
E_{n}$, $E_{n}$ is $1/n$-regular at $x_{n}$ in $B_{2}(x_{n})$ but there exist
$y_{n}\in B_{1}(x_{n})\cap\partial E_{n}$ and $r_{n}>0$ such that:

  * (i)

$E_{n}$ is $\varepsilon r$-regular at $y_{n}$ in $B_{r}(y_{n})$ for any
$r_{n}<r<1$ and for any $n\in\mathbb{N}$;

  * (ii)

$E_{n}$ is not $\varepsilon r_{n}/2$-regular at $y_{n}$ in
$B_{r_{n}/2}(y_{n})$.

It is easy to check that these assumptions force $r_{n}\to 0$. Moreover, we
can assume $\varepsilon>0$ small enough so that $\delta>0$ in (6.1) is smaller
than the density gap $c_{N}$ of (6.2).

Now let us rescale along the sequence in order to let the critical scales
$r_{n}$ become scale $1$. If we do so, letting
$\tilde{X}_{n}:=(X_{n},\mathsf{d}_{n}/r_{n},\mathscr{H}^{N},y_{n})$ and
looking at the sets $E_{n}$ in the rescaled metric measure spaces, by Theorem
2.2, $\tilde{X}_{n}$ converge in the pmGH topology to
$(\mathbb{R}^{N},\mathsf{d}_{\mathrm{eucl}},\mathscr{H}^{N},0^{N})$. Moreover,
thanks to Theorem 2.11 the sets $E_{n}$ converge in the $L^{1}_{{\rm loc}}$
topology to an entire minimizer of the perimeter $F\subset\mathbb{R}^{N}$.
Taking into account (i) and subsection 6.1, we can also infer that

(6.5)
$1-\delta\leq\frac{\operatorname{Per}(F,B_{r}(0^{N}))}{\omega_{N-1}r^{N-1}}\leq
1+\delta\,,\quad\text{for any $1<r<\infty$}\,.$

Since $F$ is an entire perimeter minimizer in $\mathbb{R}^{N}$, the standard
Euclidean monotonicity formula yields that

(6.6) $r\mapsto\frac{\operatorname{Per}(F,B_{r}(z))}{\omega_{N-1}r^{N-1}}$

is an increasing function, for any $z\in\partial F$. By (6.5), that guarantees
compactness of the sequence of scalings $F_{0,r}$ of $F$ for $r>1$, we are
allowed to consider a blow-down $G$ of $F$. A standard consequence of the
monotonicty formula is that $G$ is an entire minimal cone in $\mathbb{R}^{N}$.
Moreover, by (6.5) and our choice of $\delta>0$, we have that

$1-\delta\leq\Theta_{0,G}\leq 1+\delta\leq 1+c_{N}\,.$

Hence, by the Euclidean density gap subsection 6.1 and monotonicity, we infer
that $\Theta_{G}=1$. Therefore

(6.7)
$\lim_{r\to\infty}\frac{\operatorname{Per}(F,B_{r}(0))}{\omega_{N-1}r^{N-1}}=\Theta_{0,G}=1\,.$

Observe that the density at infinity of the entire minimal surface $F$ is
independent of the base point $z\in\partial F$, as one can easily verify.
Moreover, by De-Giorgi’s theorem, there exists $z_{0}\in F\cap B_{1}(0)$ such
that

(6.8) $\lim_{r\to
0}\frac{\operatorname{Per}(F,B_{r}(z_{0}))}{\omega_{N-1}r^{N-1}}=1\,.$

Relying again on the monotonicity formula, by (6.7) and (6.8) we infer that

$\frac{\operatorname{Per}(F,B_{r}(z_{0}))}{\omega_{N-1}r^{N-1}}=1\,,\quad\text{for
any $0<r<\infty$}\,.$

Then with a standard argument we obtain that $F$ is a half-space
$\mathbb{H}^{N}$ passing through $0$.

By condition (ii) above, the sets $E_{n}$, when considered in the scaled
metric measure spaces $\tilde{X}_{n}$, are not $\varepsilon/2$-regular at
$x_{n}$ in $B_{1/2}(x_{n})$. This clearly gives a contradiction, since their
limit is a half-space, as we just argued; in particular, they are
$\varepsilon/2$-regular at $x_{n}$ in $B_{1/2}(x_{n})$ as soon as $n$ is large
enough.

The second part of the statement follows from the previous one via
Reifenberg’s theorem for metric spaces, see for instance [41, Appendix 1]. ∎

######  Corollary .

Let $N>1$ be fixed. Then there exists $\delta=\delta(N)>0$ such that the
following holds. If $(M^{N},g)$ is a smooth $N$-dimensional Riemannian
manifold and $E\subset M$ is a set of locally finite perimeter such that, for
some $x\in M$ and $r>0$,

  * (i)

$\operatorname{Ric}_{M}\geq-\delta r^{-2}$ on $B_{4r}(x)$;

  * (ii)

$E$ is perimeter minimizing in $B_{4r}(x)$;

  * (iii)

$E$ is $\delta$-regular at $x$ on $B_{2r}(x)$.

Then $\partial E\cap B_{r}(x)$ is smooth.

###### Proof.

We only need to verify that all tangent cones at all points $x\in\partial
E\cap B_{r}(x)$ are Euclidean half-spaces. Then the classical regularity in
Geometric Measure Theory provides smoothness.
To this aim, observe that, by Theorem 6.1, all the tangent cones at any
$x\in\partial E\cap B_{r}(x)$ are entire perimeter minimizers in
$\mathbb{R}^{n}$ close to the Euclidean half-space at all scales. Then an
argument analogous to the one exploited in the proof of Theorem 6.1, relying
on the Euclidean density gap (see subsection 6.1), shows that the tangent
cones are half-spaces. ∎

######  Remark .

In subsection 6.1 there is no assumption on the injectivity radius of the
Riemannian manifold, nor on the full curvature tensor, which are the classical
assumptions for the $\varepsilon$-regularity theorems for minimal surfaces on
Riemannian manifolds, see for instance [47, 114].

######  Remark .

subsection 6.1 should be compared with some previous results obtained in [77]
and [80, Section 4]. Therein, uniform Reifenberg flatness was proved for
minimal bubbles w.r.t. families of smooth Riemannian metrics $g_{\varepsilon}$
uniformly converging to a background metric $g$ on a fixed manifold $M$. In
this regard subsection 6.1 is much stronger, since it deals with a weaker
notion of convergence of metrics. Moreover, Theorem 6.1 shows that ambient
regularity is not a key assumption for Reifenberg flatness, provided there is
a synthetic lower Ricci bound on the background.

###  6.2. Sharp perimeter bounds for the equidistant sets from minimal
boundaries

In this subsection we consider again local perimeter minimizers in the sense
of subsection 5.1. Our goal is to prove some sharp perimeter bounds for the
equidistant sets from minimal boundaries which will turn to be very useful to
establish the quantitative regularity results in subsection 6.4. The
interpretation of minimality via Laplacian bounds on the distance function
obtained in subsection 5.1 will play a key role here.

The following useful lemma is essentially taken from [29], see in particular
the proof of Theorem 7.4 therein. We omit the proof that can be obtained
relying on subsection 5.1, with arguments similar to those appearing in the
proofs of previous results in this note.

######  Lemma .

Let $(X,\mathsf{d},\mathscr{H}^{N})$ be an $\operatorname{RCD}(K,N)$ metric
measure space and let $E\subset X$ be a set of locally finite perimeter which
locally minimizes the perimeter in an open domain $\Omega\subset X$ according
to subsection 5.1. Then, for any Lipschitz function $\varphi:X\to\mathbb{R}$
with compact support in $\Omega$, it holds:

(6.9)
$\int\varphi\mathop{}\\!\mathrm{d}\operatorname{Per}(\\{\mathsf{d}_{\bar{E}}>r\\})=\int_{\\{0\leq\mathsf{d}_{\bar{E}}<r\\}}\mathop{}\\!\mathrm{d}\operatorname{div}(\varphi\nabla\mathsf{d}_{\bar{E}})\,,\quad\text{for
a.e. $r>0$}\,.$

######  Remark .

The local perimeter minimizing assumption above is used only to infer
regularity properties of the distance function, namely the fact that it has
measure valued Laplacian whose singular part on the boundary of the set is the
surfaces measure, rather than to obtain specific mean curvature bounds.
Indeed, the conclusion of subsection 6.2 holds for the boundary of any smooth
set on a smooth Riemannian manifold.

In order to ease the notation, let us denote by $E^{t}$ the open
$t$-enlargement of $E$, i.e.

(6.10) $E^{t}:=\\{x\in X\,:\,\mathsf{d}(x,\bar{E})<t\\}\,.$

We will need to compare the perimeter measure of the set $E$ and the measures
obtained by normalizing the restriction of the ambient volume measure to a
tubular neighbourhood of the set. Again, for all smooth hypersurfaces in the
smooth Riemannian setting, the perimeter and such a Minkowski-type measure
coincide, even though they do not for general sets. The next result states
that the perimeter minimality condition is robust enough to guarantee such an
extra regularity also in the $\operatorname{RCD}$ setting.

######  Proposition .

Let $(X,\mathsf{d},\mathscr{H}^{N})$ be an $\operatorname{RCD}(K,N)$ metric
measure space and let $E\subset X$ be a set of locally finite perimeter which
locally minimizes the perimeter in an open domain $\Omega\subset X$ according
to subsection 5.1. For any $0<\varepsilon<1$, let

$\mu_{\varepsilon}^{+}:=\frac{1}{\varepsilon}\mathscr{H}^{N}\mathop{\hbox{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt\vrule
height=0.5pt,width=6.0pt,depth=0.0pt}}\nolimits\\{0\leq\mathsf{d}_{{\bar{E}}}<\varepsilon\\}\quad\text{and}\quad\mu_{\varepsilon}^{-}:=\frac{1}{\varepsilon}\mathscr{H}^{N}\mathop{\hbox{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt\vrule
height=0.5pt,width=6.0pt,depth=0.0pt}}\nolimits\\{0\leq\mathsf{d}_{E^{c}}<\varepsilon\\}\,.$

Then both $\mu_{\varepsilon}^{+}$ and $\mu_{\varepsilon}^{-}$ weakly converge
to $\operatorname{Per}_{E}$ on $\Omega$ as $\varepsilon\to 0$.

###### Proof.

Let us prove the weak convergence to the perimeter of $\mu_{\varepsilon}^{+}$.
The weak convergence of $\mu_{\varepsilon}^{-}$ can be proved with an
analogous argument, replacing $\mathsf{d}_{\bar{E}}$ with
$\mathsf{d}_{E^{c}}$.

The family of measures $\mu_{\varepsilon}^{+}$ has locally uniformly bounded
mass, as it follows from subsubsection 2.4.6. We claim that for any weak limit
$\mu$ of the sequence of measures $\mu_{\varepsilon_{i}}^{+}$, where
$\varepsilon_{i}\downarrow 0$ as $i\to\infty$, it holds
$\mu=\operatorname{Per}_{E}$.

Let us start from the inequality $\mu\geq\operatorname{Per}_{E}$.
Letting $\varphi_{\varepsilon}^{+}:X\to\mathbb{R}$ be defined by
$\varphi_{\varepsilon}^{+}(x)=1$ on $\bar{E}$, $\varphi_{\varepsilon}^{+}=0$
on $X\setminus E^{\varepsilon}$ and

$\varphi_{\varepsilon}^{+}=\frac{1}{\varepsilon}(\varepsilon-\mathsf{d}(x,\bar{E}))\,,\quad\text{on
$E^{\varepsilon}\setminus E$}\,,$

it holds

$\mu_{\varepsilon}^{+}=\left\lvert\nabla\varphi_{\varepsilon}^{+}\right\rvert\mathscr{H}^{N}\,.$

Moreover, it is easy to check that $\varphi_{\varepsilon}^{+}$ converge
locally in $L^{1}$ to $\chi_{E}$. Hence, by the lower semicontinuity of the
total variation (in localized form), it is easy to infer that, for any open
set $A\subset\Omega$ such that $\mu(\partial A)=0$,

$\operatorname{Per}(E,A)\leq\liminf_{i\to\infty}\mu_{\varepsilon_{i}}^{+}(A)=\mu(A)\,.$

To prove the converse inequality, let us focus for simplicity on the case
$K=0$, the general case introduces only an additional error term of lower
order. Let us consider any non-negative Lipschitz function
$\varphi:X\to[0,\infty)$ with compact support in $\Omega$. We claim that

$\int\varphi\mathop{}\\!\mathrm{d}\mu\leq\int\varphi\mathop{}\\!\mathrm{d}\operatorname{Per}\,,$

which will imply the inequality $\mu\leq\operatorname{Per}_{E}$.
To prove this claim, we rely on subsection 6.2. Indeed, for a.e. $r>0$
sufficiently small, it holds that

$\int\varphi\mathop{}\\!\mathrm{d}\operatorname{Per}(\\{\mathsf{d}_{\bar{E}}>r\\})=\int_{\\{\mathsf{d}_{\bar{E}}<r\\}}\mathop{}\\!\mathrm{d}\operatorname{div}(\varphi\nabla\mathsf{d}_{\bar{E}})\,.$

Hence, for a.e. $r>0$, using the Leibniz rule for the divergence, Theorem 5.1
and subsection 5.1, we get

$\displaystyle\int\varphi\mathop{}\\!\mathrm{d}\operatorname{Per}(\\{\mathsf{d}_{\bar{E}}>r\\})=$
$\displaystyle\int_{E^{r}}\nabla\varphi\cdot\nabla\mathsf{d}_{\bar{E}}\mathop{}\\!\mathrm{d}\mathscr{H}^{N}+\int_{E^{r}}\varphi\bm{\Delta}\mathsf{d}_{\bar{E}}$
$\displaystyle\leq$
$\displaystyle\int_{E^{r}}\nabla\varphi\cdot\nabla\mathsf{d}_{\bar{E}}\mathop{}\\!\mathrm{d}\mathscr{H}^{N}+\int\varphi\mathop{}\\!\mathrm{d}\operatorname{Per}_{E}\,.$

Therefore, for any $s>0$ sufficiently small, by the coarea formula we get

$\displaystyle\int_{E^{s}}\varphi\mathop{}\\!\mathrm{d}\mathscr{H}^{N}=$
$\displaystyle\int_{0}^{s}\int\varphi\mathop{}\\!\mathrm{d}\operatorname{Per}(\\{\mathsf{d}_{\bar{E}}>r\\})$
$\displaystyle\leq$
$\displaystyle\int_{0}^{s}\left(\int_{E^{r}}\nabla\varphi\cdot\nabla\mathsf{d}_{\bar{E}}\mathop{}\\!\mathrm{d}\mathscr{H}^{N}+\int\varphi\mathop{}\\!\mathrm{d}\operatorname{Per}_{E}\right)$
$\displaystyle\leq$ $\displaystyle
s\operatorname{Lip}(\varphi)\mathscr{H}^{N}(E^{s}\cap\mathrm{spt}\varphi)+s\int\varphi\mathop{}\\!\mathrm{d}\operatorname{Per}_{E}\,.$

Hence

$\displaystyle\int\varphi\mathop{}\\!\mathrm{d}\mu=$
$\displaystyle\lim_{i\to\infty}\frac{1}{s_{i}}\int_{E^{s_{i}}}\varphi\mathop{}\\!\mathrm{d}\mathscr{H}^{N}$
$\displaystyle\leq$ $\displaystyle\limsup_{s\to
0}\frac{1}{s}\left(s\operatorname{Lip}(\varphi)\mathscr{H}^{N}(E^{s}\cap\mathrm{spt}\varphi)+s\int\varphi\mathop{}\\!\mathrm{d}\operatorname{Per}_{E}\right)$
$\displaystyle=$
$\displaystyle\int\varphi\mathop{}\\!\mathrm{d}\operatorname{Per}_{E}\,,$

where we used subsubsection 2.4.6 in the last inequality. This concludes the
proof of the inequality $\mu\leq\operatorname{Per}_{E}$ and hence the proof. ∎

Let us introduce the notation $\Sigma$ for the boundary $\partial E$ of a set
of finite perimeter $E$ which is locally perimeter minimizing in
$\Omega\subset X$ and let us denote, for any $h>0$,

$\Sigma^{h}:=\\{x\in\Omega\,:\,\mathsf{d}(\bar{E},x)=h\\}\,.$

The next result is a kind of monotonicity formula for equidistant sets from
minimal boundaries. Its proof is inspired by [33, Lemma 2], which deals with
the Euclidean case. The Laplacian bound for the distance from a locally
minimizing set of finite perimeter under lower Ricci curvature bounds
(obtained in Theorem 5.1) allows to extend it to the present framework.

######  Proposition .

Let $(X,\mathsf{d},\mathscr{H}^{N})$ be an $\operatorname{RCD}(K,N)$ metric
measure space and let $E\subset X$ be a set of locally finite perimeter which
locally minimizes the perimeter in an open domain $\Omega\subset X$ according
to subsection 5.1. Let $h>0$ be fixed. Let $\Gamma\subset\Sigma^{h}$ be any
compact set and denote

(6.11) $\displaystyle\Gamma_{\Sigma}$ $\displaystyle:=\\{y\in\Sigma\cap\Omega\
:\,\mathsf{d}(x,y)=h\quad\text{for some $x\in\Gamma$}\\}\,,$ (6.12)
$\displaystyle G$
$\displaystyle:=\\{x\in\Omega\,:\,\mathsf{d}_{\Gamma_{\Sigma}}(x)+\mathsf{d}_{\Gamma}(x)=h\\}\,.$

If $G\Subset\mathcal{K}$, where $\mathcal{K}$ has been defined in (5.2), then

(6.13)
$\operatorname{Per}(E^{h},\Gamma)\leq\operatorname{Per}(E,\Gamma_{\Sigma})+\int_{G}\mathrm{t}_{K,N}(\mathsf{d}_{E})\mathop{}\\!\mathrm{d}\mathscr{H}^{N}\
\,,$

where $\mathrm{t}_{K,N}$ was defined in (1.1), and

(6.14)
$\operatorname{Per}(E^{h},\Gamma)\leq\begin{cases}\operatorname{Per}(E,\Gamma_{\Sigma})\cos\left(\sqrt{\frac{K}{N-1}}h\right)^{N-1}\,&\quad\text{if
}K>0\\\ \quad\operatorname{Per}(E,\Gamma_{\Sigma})\,&\quad\text{if }K=0\\\
\operatorname{Per}(E,\Gamma_{\Sigma})\cosh\left(\sqrt{\frac{-K}{N-1}}h\right)^{N-1}\,&\quad\text{if
}K<0\,.\end{cases}$

######  Remark .

Note that $G$ is made by the union of minimizing geodesics connecting
$\Gamma_{\Sigma}$ with $\Sigma$ along which $\mathsf{d}_{E}$ is attained.

######  Remark .

The bounds obtained in subsection 6.2 are sharp. Indeed it is easily seen that
equality is achieved in the model spaces:

  * •

for $K>0$, let $(X,\mathsf{d},\mathscr{H}^{N})$ be the $N$-dimensional round
sphere of constant sectional curvature $K/(N-1)$ and $E$ be a half-sphere. It
is a standard fact that $E$ is locally perimeter minimizing inside a
sufficiently small open domain $\Omega$. It is immediate to see that
$\Sigma^{h}$ is (part of the boundary of) a spherical cap and one can check
that equality is attained in (6.14) by direct computations;

  * •

for $K=0$, let $(X,\mathsf{d},\mathscr{H}^{N})$ be the $N$-dimensional
Euclidean space and $E$ be a half-space. It is a standard fact that $E$ is
locally perimeter minimizing inside any open domain $\Omega$. It is immediate
to see that $\Sigma^{h}$ is (part of the boundary of) an equidistant half
space and that equality is attained in (6.14);

  * •

for $K<0$, let $(X,\mathsf{d},\mathscr{H}^{N})$ be the $N$-dimensional
hyperbolic space of constant sectional curvature $K/(N-1)$ and $E$ be a horo-
ball. It is a standard fact that $E$ is locally perimeter minimizing inside
any open domain $\Omega$. Also in this case, one can check that equality is
attained in (6.14) by direct computations.

###### Proof.

Notice that $G$ is the set spanned by those rays connecting $\Gamma_{\Sigma}$
to $\Gamma$. We would like to apply the Gauss-Green integration by parts
formula to the vector field $\nabla\mathsf{d}_{E}$ on $G$. Indeed, at an
heuristic level, the boundary of $G$ is made of three parts,
$\Gamma_{\Sigma}$, $\Gamma$ and some lateral faces whose unit normal we expect
to be orthogonal to $\nabla\mathsf{d}_{E}$. Then the conclusion would follow
from the fact that
$\operatorname{div}\nabla\mathsf{d}_{E}\leq\mathrm{t}_{K,N}\circ\mathsf{d}_{E}$,
by Theorem 5.1.

In order to make the argument rigorous, we are going to approximate the
characteristic function of the set $G$ (which in general may not be regular
enough), by suitable cut-off functions.
Let us introduce the shortened notation $\bar{\mathsf{d}}$ for the distance
from $\bar{E}$. Moreover, let us denote by $\mathsf{d}_{\Gamma}$ the distance
function from the compact set $\Gamma$ in the statement. Then, for any
$\varepsilon\in(0,\varepsilon_{0})$ let us set

$\varphi_{\varepsilon}:=\frac{1}{\varepsilon}\left(h+\varepsilon-(\bar{\mathsf{d}}+\mathsf{d}_{\Gamma})\right)_{+}\,,$

where we denoted by $(\cdot)_{+}$ the positive part. For any $\delta\in(0,h)$,
we introduce the monotone function $g_{\delta}$ satisfying:

$g_{\delta}(0)=g^{\prime}_{\delta}(0)=0\,,\quad
g^{\prime\prime}_{\delta}=\frac{1}{\delta}\left(\chi_{[0,\delta]}-\chi_{[h-\delta,h]}\right)\,.$

Observe that, in particular, $g^{\prime}_{\delta}(h)=0$. Recalling that
$\left\lvert\nabla\bar{\mathsf{d}}\right\rvert\leq 1$ a.e. and that
$g_{\delta}^{\prime}(\bar{\mathsf{d}})\bm{\Delta}\bar{\mathsf{d}}\leq
g_{\delta}^{\prime}(\bar{\mathsf{d}})\,\mathrm{t}_{K,N}(\bar{\mathsf{d}})$ by
Theorem 5.1, using chain rule we obtain:

(6.15) $\bm{\Delta}g_{\delta}(\bar{\mathsf{d}})\leq
g_{\delta}^{\prime\prime}(\bar{\mathsf{d}})+g_{\delta}^{\prime}(\bar{\mathsf{d}})\,\mathrm{t}_{K,N}(\bar{\mathsf{d}})\,.$

Now let $F\subset\mathcal{K}$ be an open neighbourhood of $G$ inside
$\mathcal{K}$. Relying on (6.15) and applying the Gauss Green integration by
parts formula (see Theorem 2.8), taking into account that there are no
boundary terms since either $\varphi_{\varepsilon}=0$ or
$g^{\prime}_{\delta}(\bar{\mathsf{d}})=0$ on the boundary of the domain for
$\varepsilon>0$ sufficiently small, we can compute:

$\displaystyle\int_{F}g^{\prime\prime}_{\delta}(\bar{\mathsf{d}})\varphi_{\varepsilon}\mathop{}\\!\mathrm{d}\mathscr{H}^{N}$
$\displaystyle\geq\int_{F}\varphi_{\varepsilon}\mathop{}\\!\mathrm{d}\bm{\Delta}g_{\delta}(\bar{\mathsf{d}})-\int_{F}\varphi_{\varepsilon}\,g_{\delta}^{\prime}(\bar{\mathsf{d}})\,\mathrm{t}_{K,N}(\bar{\mathsf{d}})\mathop{}\\!\mathrm{d}\mathscr{H}^{N}$
$\displaystyle=-\int_{F}\nabla(g_{\delta}(\bar{\mathsf{d}}))\cdot\nabla\varphi_{\varepsilon}\mathop{}\\!\mathrm{d}\mathscr{H}^{N}-\int_{F}\varphi_{\varepsilon}\,g_{\delta}^{\prime}(\bar{\mathsf{d}})\,\mathrm{t}_{K,N}(\bar{\mathsf{d}})\mathop{}\\!\mathrm{d}\mathscr{H}^{N}\,.$

Let us observe that

$\displaystyle\nabla(g_{\delta}(\bar{\mathsf{d}}))\cdot\nabla\varphi_{\varepsilon}=$
$\displaystyle
g^{\prime}_{\delta}(\bar{\mathsf{d}})\nabla\bar{\mathsf{d}}\cdot(-\nabla\bar{\mathsf{d}}-\nabla\mathsf{d}_{\Gamma})\varepsilon^{-1}$
$\displaystyle=$ $\displaystyle
g^{\prime}_{\delta}(\bar{\mathsf{d}})(-1-\nabla\bar{\mathsf{d}}\cdot\nabla\mathsf{d}_{\Gamma})\varepsilon^{-1}\leq
0\,,\quad\text{$\mathscr{H}^{N}$-a.e. on $F$}\,.$

Hence, for any $\varepsilon,\delta>0$ sufficiently small, it holds

$\int_{F}g^{\prime\prime}_{\delta}(\bar{\mathsf{d}})\varphi_{\varepsilon}\mathop{}\\!\mathrm{d}\mathscr{H}^{N}\geq-\int_{F}\varphi_{\varepsilon}\,g_{\delta}^{\prime}(\bar{\mathsf{d}})\,\mathrm{t}_{K,N}(\bar{\mathsf{d}})\mathop{}\\!\mathrm{d}\mathscr{H}^{N}\,.$

By the very definition of $g_{\delta}$, this implies that

$\displaystyle\frac{1}{\delta}\int_{F}\chi_{[0,\delta]}(\bar{\mathsf{d}})\varphi_{\varepsilon}\mathop{}\\!\mathrm{d}\mathscr{H}^{N}\geq$
$\displaystyle\frac{1}{\delta}\int_{F}\chi_{[h-\delta,h]}(\bar{\mathsf{d}})\varphi_{\varepsilon}\mathop{}\\!\mathrm{d}\mathscr{H}^{N}$
(6.16)
$\displaystyle-\int_{F}\varphi_{\varepsilon}\,g_{\delta}^{\prime}(\bar{\mathsf{d}})\,\mathrm{t}_{K,N}(\bar{\mathsf{d}})\mathop{}\\!\mathrm{d}\mathscr{H}^{N}\,.$

Relying on subsection 6.2, which guarantees the weak convergence of the
measures $\delta^{-1}\chi_{[0,\delta]}(\bar{\mathsf{d}})\mathscr{H}^{N}$ to
$\operatorname{Per}_{E}$ as $\delta\to 0$, we can pass to the limit in the
left hand side of (6.2). Moreover, by semicontinuity of the total variation,
for any weak limit $\nu$ of the sequence
$\delta^{-1}\chi_{[h-\delta,h]}(\bar{\mathsf{d}})\mathscr{H}^{N}$ (which is
easily seen to be pre-compact in the weak topology) as $\delta\to 0$, it holds
$\nu\geq\operatorname{Per}(E_{h})$. It is also easily seen that $0\leq
g_{\delta}^{\prime}(\bar{\mathsf{d}})\uparrow 1$ $\mathscr{H}^{N}$-a.e. on
$F$, as $\delta\downarrow 0$. Hence

(6.17)
$\int_{F}\varphi_{\varepsilon}\mathop{}\\!\mathrm{d}\operatorname{Per}_{E}\geq\int_{F}\varphi_{\varepsilon}\mathop{}\\!\mathrm{d}\operatorname{Per}_{E_{h}}-\int_{F}\varphi_{\varepsilon}\,\mathrm{t}_{K,N}(\bar{\mathsf{d}})\mathop{}\\!\mathrm{d}\mathscr{H}^{N}\,.$

Next, we pass to the limit as $\varepsilon\to 0$. Observe that

$\lim_{\varepsilon\to 0}\varphi_{\varepsilon}(x)=\begin{cases}1\,\quad\text{if
$x\in G$ }\\\ 0\,\quad\text{otherwise}\,.\end{cases}$

Therefore, passing to the limit in (6.17) as $\varepsilon\to 0$, we obtain
that

$\operatorname{Per}(E^{h},\Gamma)\leq\operatorname{Per}(E,\Gamma_{\Sigma})+\int_{G}\mathrm{t}_{K,N}(\bar{\mathsf{d}})\mathop{}\\!\mathrm{d}\mathscr{H}^{N}\
\,,$

as desired.

The bounds in (6.14) follow from (6.13) thanks to the coarea formula and the
integral form of Grönwall’s Lemma if $K<0$. In the case $K=0$ they follow
directly from (6.13) since $\mathrm{t}_{0,N}=0$.
Let us deal with the remaining case $K>0$.
We introduce a function

$f_{K,N}:\left(0,\frac{\pi}{2}\sqrt{\frac{N-1}{K}}\right)\to\mathbb{R}\,,\quad\,f_{K,N}(r):=\int_{0}^{r}\cos\left(\sqrt{\frac{K}{N-1}}\,h\right)^{-(N-1)}\mathop{}\\!\mathrm{d}h\,.$

Notice that

(6.18)
$f^{\prime}_{K,N}(r)=\cos\left(\sqrt{\frac{K}{N-1}}\,r\right)^{-(N-1)}\,,$

for any $r>0$ and in particular $f^{\prime}_{K,N}(0)=1$. Moreover, the chain
rule for the Laplacian and a direct computation show that we can rephrase the
bound in Theorem 5.1 as

(6.19) $\Delta f_{K,N}\circ\mathsf{d}_{E}\leq 0\,.$

Then (6.14) in the case $K>0$ follows formally by applying the Gauss-Green
integration by parts formula to the vector field $\nabla
f_{K,N}\circ\mathsf{d}_{E}$ on the set $G$ introduced in (6.12). Indeed, the
contribution coming from the integration in the interior has a sign thanks to
(6.19), one of the two boundary terms is
$f^{\prime}_{K,N}(0)\operatorname{Per}(E,\Gamma_{\Sigma})=\operatorname{Per}(E,\Gamma_{\Sigma})$
and the other one can be estimated by

(6.20)
$f^{\prime}_{K,N}(h)\operatorname{Per}(E^{h},\Gamma)=\operatorname{Per}(E^{h},\Gamma)\cos\left(\sqrt{\frac{K}{N-1}}\,h\right)^{-(N-1)}\,.$

Therefore we obtain

(6.21)
$\operatorname{Per}(E^{h},\Gamma)\leq\operatorname{Per}(E,\Gamma_{\Sigma})\cos\left(\sqrt{\frac{K}{N-1}}\,h\right)^{(N-1)}\,,$

as we claimed. The rigorous justification of (6.21) can be obtained with an
approximation argument completely analogous to the one introduced in the first
part of the proof, approximating the characteristic function of $G$ with
suitable cut-off functions; we omit the details for the sake of brevity. ∎

A very useful result proved in Simons’ seminal paper on minimal varieties
[125] states that there are no two sided stable smooth minimal hypersurfaces
on closed manifolds with positive Ricci curvature. Thanks to the perimeter
monotonicity in subsection 6.2 we can partially generalize this fact to the
present framework.

######  Corollary  (Simons’ theorem in $\operatorname{RCD}$ spaces).

Let $(X,\mathsf{d},\mathscr{H}^{N})$ be an $\operatorname{RCD}(K,N)$ metric
measure space, for some $K>0$. Then, for any $r>0$, there is no non trivial
set of finite perimeter $E\subset X$ that minimizes the perimeter among all
the perturbations $F\subset X$ such that

$E\Delta F\subset B_{r}(\partial E)\,.$

###### Proof.

Let us argue by contradiction. If a set of finite perimeter as in the
statement exists, then it is locally perimeter minimizing according to
subsection 5.1. Hence it verifies the assumptions of subsection 6.2.
Therefore, for any $h>0$,

$\operatorname{Per}(E^{h})\leq\operatorname{Per}(E)+\int_{E^{h}\setminus
E}\mathrm{t}_{K,N}(\mathsf{d}_{E})\mathop{}\\!\mathrm{d}\mathscr{H}^{N}<\operatorname{Per}(E)\,.$

To conclude it is sufficient to observe that $E^{h}\Delta E\subset
B_{r}(\partial E)$ for any $h>0$ sufficiently small and we reach a
contradiction. ∎

###  6.3. Partial regularity of minimal boundaries away from sets of
codimension three

Our goal in this subsection is to prove that minimal boundaries have regular
blow-ups (and therefore are topologically regular) away from sets of ambient
codimension three (assuming for simplicity that the ambient space
$(X,\mathsf{d},\mathscr{H}^{N})$ is an $\operatorname{RCD}$ space without
boundary).

######  Definition  (Regular and singular sets on minimal boundaries).

Let $(X,\mathsf{d},\mathscr{H}^{N})$ be an $\operatorname{RCD}(K,N)$ metric
measure space and $E\subset X$ be locally perimeter minimizing inside a ball
$B_{2}(x)\subset X$. Suppose also that $\partial X\cap B_{2}(x)=\emptyset$.
The regular part $\mathcal{R}^{E}$ and the singular part $\mathcal{S}^{E}$ of
$\partial E$ are defined as

$\mathcal{R}^{E}:=\\{x\in\partial
E\,:\,(\mathbb{R}^{N},\mathsf{d}_{\mathrm{eucl}},\mathscr{H}^{N},0^{N},\\{x_{N}<0\\})\in\operatorname{Tan}_{x}(X,\mathsf{d},\mathscr{H}^{N},E)\\}\,,$
$\mathcal{S}^{E}:=\partial E\setminus\mathcal{R}^{E}\,.$

######  Remark .

If $E\subset X$ is locally perimeter minimizing and $x\in\partial X$, then for
any

(6.22)
$(Y,\mathsf{d}_{Y},\mathscr{H}^{N},F,y)\in\operatorname{Tan}_{x}(X,\mathsf{d},\mathscr{H}^{N},E)$

it holds that $F$ is an entire local perimeter minimizer in $Y$, as it follows
from the stability Theorem 2.11.

As a first regularity result, we establish topological regularity of the
regular set. This is indeed a direct consequence of the
$\varepsilon$-regularity Theorem 6.1.

######  Theorem 6.2 (Topological regularity of the regular set).

Let $(X,\mathsf{d},\mathscr{H}^{N})$ be an $\operatorname{RCD}(K,N)$ metric
measure space, assume that $B_{2}(x)\cap\partial X=\emptyset$ and let
$E\subset X$ be a set of locally finite perimeter that is locally perimeter
minimizing in $B_{2}(x)$. Then, for every $\alpha\in(0,1)$ there exists a
relatively open set $O_{\alpha}\subset\partial E\cap B_{1}(x)$ with
$\mathcal{R}^{E}\subset O_{\alpha}$ such that $O_{\alpha}$ is
$\alpha$-biHölder homeomorphic to an open, smooth $(N-1)$-dimensional
manifold.

######  Remark .

The $C^{0,\alpha}$ regularity of the manifold $O_{\alpha}$ containing the
regular set matches the (currently known) regularity of the regular part
$\mathcal{R}(X)$ of the ambient space $X$ (after Cheeger-Colding’s metric
Reifenberg Theorem [41, Appendix 1] and [89]). Higher regularity of
$\mathcal{R}^{E}$ (e.g. contained in a Lipschitz manifold), would require
first improving the structure theory of the ambient space.

The classical regularity result for perimeter minimizers in the Euclidean (or
smooth Riemannian) setting is that they are smooth away from sets of ambient
codimension $8$.

A key intermediate step is the fact that the blow-ups are flat Euclidean half-
spaces away from sets of ambient codimension $8$, see [62, 72].
The examples that we have already discussed in this note show that this
statement is false in the non smooth framework. Singular blow-ups already
appear in ambient dimension $3$, see the discussion after subsection 6.1.

As a first regularity result, below we prove that, if we restrict to regular
ambient points (or we consider $\operatorname{RCD}(K,N)$ metric measure spaces
$(X,\mathsf{d},\mathscr{H}^{N})$ such that the singular set is empty) then the
picture matches with the classical one and we can prove that the codimension
of the singular set of a perimeter minimizer is at least $8$.

######  Theorem 6.3.

Let $(X,\mathsf{d},\mathscr{H}^{N})$ be an $\operatorname{RCD}(K,N)$ metric
measure space, let $\Omega\subset X$ be an open domain and let $E\subset X$ be
a set of locally finite perimeter that is locally perimeter minimizing in
$\Omega$. Then

(6.23) $\dim_{H}\left(\mathcal{S}^{E}\cap\mathcal{R}\cap\Omega\right)\leq
N-8\,.$

In particular:

  * i)

if $\Omega\subset\mathcal{R}$, then

(6.24) $\dim_{H}\left(\mathcal{S}^{E}\cap\Omega\right)\leq N-8\,;$

  * ii)

if $\Omega\subset\mathcal{R}$ and $N\leq 7$, then
$\mathcal{S}^{E}\cap\Omega=\emptyset$.

###### Proof.

With the tools that we have developed so far, the proof reduces to a variant
of the classical dimension reduction technique to bound the dimension of
singular sets. See [61, 62] and [72, Chapter 11] for the case of perimeter
minimizers in the Euclidean setting, and [56] for the dimension bounds for the
singular strata on $\operatorname{RCD}(K,N)$ spaces
$(X,\mathsf{d},\mathscr{H}^{N})$.

We argue by contradiction. If (6.23) is not satisfied, then we construct a
(local) perimeter minimizer inside $\mathbb{R}^{N}$ whose singular set has
codimension less than $8$. This will lead to a contradiction, since the
singular set of a Euclidean (local) perimeter minimizer has codimension at
least $8$, by the classical regularity theory.

Let us suppose that (6.23) is not verified. Then there exists $\eta>0$ such
that

(6.25)
$\mathcal{H}^{N-8+\eta}\left(\mathcal{S}^{E}\cap\mathcal{R}\cap\Omega\right)>0\,.$

By [56, Lemma 3.6] (see also [72, Lemma 11.3] and [61, Theorem 2.10.17]),
there exists $x\in\mathcal{S}^{E}\cap\mathcal{R}\cap\Omega$ such that

(6.26) $\limsup_{r\to
0}\frac{\mathscr{H}^{N-8+\eta}_{\infty}\left(B_{r}(x)\cap\left(\mathcal{S}^{E}\cap\mathcal{R}\cap\Omega\right)\right)}{r^{N-8+\eta}}\geq
2^{N-8+\eta}\omega_{N-8+\eta}\,,$

where we denoted by $\mathscr{H}^{N-8+\eta}_{\infty}$ the pre-Hausdorff
measure of dimension $N-8+\eta$.

Now we claim that for any sequence $r_{i}\downarrow 0$ there exists a
subsequence, that we do not relabel, such that
$E\subset(X,\mathsf{d}/r_{i},\mathscr{H}^{N}/r_{i}^{N},x)$ converge in the
$L^{1}_{{\rm loc}}$ sense as sets of (locally) finite perimeter to an entire
(local) perimeter minimizer
$E_{\infty}\subset\left(\mathbb{R}^{N},\mathsf{d}_{eucl},\mathscr{H}^{N},0^{N}\right)$.
Here the compactness of the sequence follows from [6, Corollary 3.4] together
with the uniform perimeter bounds for perimeter minimizers inside the ball,
while the conclusion that $E$ is an entire (local) perimeter minimizer follows
from Theorem 2.11.

By scaling, denoting by
$E_{i}=E\subset(X,\mathsf{d}/r_{i},\mathscr{H}^{N}/r_{i}^{N},x)$ the set of
finite perimeter $E$ considered inside the rescaled metric measure space, we
can find a sequence $r_{i}\downarrow 0$ such that $E_{i}$ converge in
$L^{1}_{{\rm loc}}$ to an entire (locally) perimeter minimizer $E_{\infty}$ as
we discussed above and moreover

(6.27)
$\lim_{i\to\infty}\mathscr{H}^{N-8+\eta}_{\infty}\left(\mathcal{S}^{E_{i}}\cap
B_{1}^{i}(x)\cap\mathcal{R}(X_{i})\right)\geq
2^{N-8+\eta}\omega_{N-8+\eta}\,>\,0.$

We claim that (6.27) forces

(6.28) $\mathscr{H}^{N-8+\eta}_{\infty}\left(\mathcal{S}^{E_{\infty}}\cap
B_{1}(0^{N})\right)\,>\,0\,.$

In order to check (6.28) it is sufficient to prove that any limit point
$x_{\infty}$ of a sequence $(x_{i})$ such that
$x_{i}\in\mathcal{S}^{E_{i}}\cap\mathcal{R}(X_{i})\cap B_{1}^{i}(x)$ belongs
to $\mathcal{S}^{E_{\infty}}\cap B_{1}(0^{N})$. Once this statement has been
established, (6.28) will follow from (6.27) and the upper semicontinuity of
the pre-Hausdorff measure $\mathscr{H}^{N-8+\eta}_{\infty}$ under GH
convergence, see [56, Equation (3.36)].

Let us pass to the verification of the claim. Let us consider any $x_{\infty}$
as above. If we suppose by contradiction that it is a regular point of
$E_{\infty}$, then it follows from the $\varepsilon$-regularity Theorem 6.1
that for any $\varepsilon>0$ there exists $i\in\mathbb{N}$ such that, for any
$j\geq i$, $E_{j}\cap B_{1}^{j}(x)$ is $\varepsilon$-regular inside
$B_{1}^{j}(x)$. In particular, if $\varepsilon>0$ is sufficiently small, then
by the Euclidean density gap subsection 6.1, all the blow-ups of $E_{j}$
inside $B_{1}^{j}(x)\cap\mathcal{R}$ are flat half-spaces (see also the proof
of subsection 6.1). This leads to a contradiction since we are assuming that
$x_{\infty}$ is a limit of singular points $x_{k}\in\mathcal{S}^{E_{j}}\cap
B_{1}^{j}(x)\cap\mathcal{R}(X_{j})$.

Given (6.28) we obtain a contradiction, since $E_{\infty}$ is an entire
Euclidean (local) perimeter minimizer and the classical dimension estimates
for the singular sets of perimeter minimizers give

(6.29) $\dim_{H}\left(\mathcal{S}^{E_{\infty}}\cap B_{1}(0^{N})\right)\leq
N-8\,.$

∎

######  Remark .

One of the key steps in the proof above is the fact that limits of singular
points of perimeter minimizers where the blow-up of the ambient is Euclidean
are singular points. In the smooth setting the second assumption is always
verified, but in the non smooth setting this is a non trivial requirement and
the example presented in subsection 6.1 shows that it is necessary in order
for the statement to hold. In particular, without further assumptions it is
not true that limits of singular boundary points of a perimeter minimizer are
singular boundary points of the limit.

We aim at obtaining a sharp dimension bound for the singular set of local
perimeter minimizers $\dim_{H}(\mathcal{S}^{E})\leq N-3$ within our framework.
To this aim, it will be necessary to consider also the intersection of the
minimal boundary with the ambient singular set $\partial E\cap\mathcal{S}(X)$.

In order to obtain the sharp dimension bound for the singular set in this
setting, there is a key additional difficulty with respect to the classical
case. Indeed it is not clear whether a monotonicity formula holds in this
generality, therefore we do not know if any blow-up of a local perimeter
minimizer is a cone. In order to circumvent this difficulty, following the
classical pattern of the dimension reduction, we will need first to iterate
blow-ups to reduce to the situation where the ambient is a cone of the form
$\mathbb{R}^{N-2}\times C(\mathbb{S}^{1}_{r})$, where $\mathbb{S}^{1}_{r}$ is
a circle, and then to perform the dimension reduction again in this simplified
setting (where a monotonicity formula holds).

We will rely on some classical tools of Geometric Measure Theory. The first
one is a monotonicity formula for perimeter minimizers inside cones, whose
proof can be obtained as in the classical case, see [110, Theorem 9.3], [61,
Theorem 5.4.3] and [111].

######  Theorem 6.4.

Let $(M,g)$ be a smooth Riemannian manifold of dimension $k\geq 1$ and with
$\operatorname{Ric}\geq k-1$. Let
$(X,\mathsf{d},\mathscr{H}^{N}):=C(M)\times\mathbb{R}^{N-k-1}$ be the product
of the metric measure cone $C(M)$ of tip $\\{o\\}$, with an
$(N-k-1)$-dimensional Euclidean factor. Let
$p\in\\{o\\}\times\mathbb{R}^{N-k-1}$ and let $E\subset X$ be perimeter
minimizing in $B_{2}(p)\subset X$. Then the ratio

(6.30) $(0,1)\ni
r\mapsto\frac{\operatorname{Per}_{E}(B_{r}(p))}{r^{N-1}}\quad\text{is
increasing.}$

Moreover, if the perimeter ratio is constant in $(0,2)$ then $E$ is a cone
with vertex $p$ inside $B_{1}(p)$.

The second tool is an elementary non existence result for entire local
perimeter minimizing cones passing through the tip inside non flat two
dimensional cones, whose proof is well known, see [111] and references
therein.

######  Proposition .

Let $N\geq 2$ be a given natural number. Let $0<r\leq 1$ and let $\mathcal{C}$
be the metric measure cone $C(\mathbb{S}^{1}_{r})\times\mathbb{R}^{N-2}$ with
canonical structure. Let $F:=C(I)\times\mathbb{R}^{N-2}\subset\mathcal{C}$,
where $I\subset\mathbb{S}^{1}_{r}$ is a set of finite perimeter. Then $F$ is a
local perimeter minimizer if and only if $r=1$ (i.e.
$\mathcal{C}=\mathbb{R}^{N}$) and $I$ is the half-circle
$[0,\pi]\subset\mathbb{S}^{1}_{1}$, up to isometry (i.e.
$F\subset\mathbb{R}^{N}$ is a half-space).

Notice that to pass from the case $N=2$ treated in [111] to the case of $N\geq
3$ it is sufficient to rely on a slight modification of [107, Lemma 28.13] to
drop the dimension.

######  Proposition .

Let $N\geq 2$ be a given natural number. Let $0<r\leq 1$ and let $\mathcal{C}$
be the metric measure cone $\mathbb{R}^{N-2}\times C(\mathbb{S}^{1}_{r})$ with
canonical structure and set of tips $\mathbb{R}^{N-2}\times\\{o\\}$. Let
$G\subset\mathcal{C}$ be any entire local perimeter minimizer. Then

(6.31) $\dim_{H}\left(\mathcal{S}^{G}\right)\leq N-3\,.$

###### Proof.

We argue via dimension reduction, reducing to the situation where subsection
6.3 can be applied.
Let us suppose without loss of generality that $r<1$ (i.e. the cone is
singular). If $r=1$ the statement follows from the classical Euclidean
regularity theory.

Step 1. We claim that any blow-up of $G\subset\mathcal{C}$ at a point
$x\in\mathcal{C}$ is either a minimal cone inside $\mathcal{C}$ if
$x\in\mathbb{R}^{N-2}\times\\{o\\}$ is an ambient singular point, or a minimal
cone in $\mathbb{R}^{N}$ if
$x\in\mathcal{C}\setminus\left(\mathbb{R}^{N-2}\times\\{o\\}\right)$ is an
ambient regular point.
In order to check this statement it is sufficient to observe that the
monotonicity formula

(6.32)
$r\mapsto\frac{\operatorname{Per}_{G}(B_{r}(x))}{r^{N-1}}\,\quad\text{is non
decreasing on $0<r<r_{x}$}$

holds for any $x\in\partial G$. Indeed, if $x\in\mathbb{R}^{N-2}\times\\{o\\}$
is a vertex, this follows from Theorem 6.4 (and we can take $r_{x}=\infty$
actually). If $x$ is a regular point of $\mathcal{C}$, the monotonicity
formula follows from the fact that $\mathcal{C}$ is isometric to a (flat)
Euclidean ball in a neighbourhood of $x$.
The fact that blow-ups are always cones follows then from a classical
argument, thanks to the uniform perimeter density bound (2.4.6) and the
rigidity in the monotonicity formula on $\mathcal{C}$ and $\mathbb{R}^{N}$.

Step 2. Let us assume by contradiction that (6.31) fails. Then, arguing as in
the proof of Theorem 6.3 (see in particular (6.26)) we can find $\eta>0$ such
that $\mathscr{H}^{N-3+\eta}(\partial G\cap\mathcal{S}(\mathcal{C}))>0$
(notice that
$\mathscr{H}^{N-3+\eta}(\mathcal{S}^{G}\cap\mathcal{R}(\mathcal{C}))=0$, by
Theorem 6.3). Therefore, there exist $x\in\partial
G\cap\mathcal{S}(\mathcal{C})=\partial G\cap\mathbb{R}^{N-2}\times\\{o\\}$ and
a sequence $r_{i}\downarrow 0$ such that

(6.33)
$\limsup_{i\to\infty}\frac{\mathscr{H}^{N-3+\eta}_{\infty}\left(\partial
G\cap\mathcal{S}(\mathcal{C})\cap B_{r_{i}}(x)\right)}{r_{i}^{N-3+\eta}}>0\,.$

By Step 1, we can find a subsequence of $(r_{i})$, that we do not relabel,
such that (6.33) holds and the blow-up of $G$ along the sequence $r_{i}$ is an
entire local perimeter minimizing cone $G^{1}$ inside a metric cone
$\mathcal{C}$ (which is the blow-up of $\mathcal{C}$ at any point
$x\in\mathcal{S}(\mathcal{C})$) with tip $o$. Moreover, it is easily seen that
any limit of points $x_{i}\in\partial G\cap\mathcal{S}(\mathcal{C})\cap
B_{r_{i}}(x)$ along this converging sequence belongs to $\partial
G^{1}\cap\mathcal{S}(\mathcal{C})\cap B_{1}(o)$. Hence, the upper
semicontinuity of the pre-Hausdorff measure implies

(6.34) $\mathscr{H}^{N-3+\eta}_{\infty}\left(\partial
G^{1}\cap\mathcal{S}(\mathcal{C})\cap B_{1}(o)\right)\,>\,0\,,$

that yields in turn

(6.35) $\mathscr{H}^{N-3+\eta}\left(\partial
G^{1}\cap\mathcal{S}(\mathcal{C})\cap B_{1}(o)\right)\,>\,0\,.$

Let us write $G^{1}=\mathbb{R}^{k}\times C(B^{1})$, where
$C(B^{1})\subset\mathbb{R}^{N-2-k}\times C(\mathbb{S}^{1}_{r})$ is an entire
local perimeter minimizing cone.
We claim that, after iterating a finite number of times the construction
above, it is possible to take $k=N-2$. Indeed, if we suppose that $k\leq N-3$,
then we obtain $\mathscr{H}^{N-3+\eta}\left(\left(\partial
G^{1}\setminus\mathbb{R}^{k}\times\\{o\\}\right)\cap\mathcal{S}(\mathcal{C})\right)>0$,
by (6.35).

In particular, there exist $z\in\left(\partial
G^{1}\cap\mathcal{S}(\mathcal{C})\right)\setminus\left(\mathbb{R}^{k}\times\\{o\\}\right)$
and a sequence $r_{j}\downarrow 0$ such that

(6.36)
$\limsup_{j\to\infty}\frac{\mathscr{H}^{N-3+\eta}_{\infty}\left(\partial
G^{1}\cap\mathcal{S}(\mathcal{C})\cap
B_{r_{j}}(z)\right)}{r_{j}^{N-3+\eta}}>0\,.$

Up to extraction of a subsequence, that we do not relabel, we find that a
blow-up of $G^{1}$ at $z$ along the sequence $r_{j}\downarrow 0$ is an entire
local perimeter minimizing cone of the form $G^{2}=\mathbb{R}^{k+1}\times
C(B^{2})$ such that

(6.37) $\dim_{H}\left(\partial
G^{2}\cap\mathcal{S}(\mathcal{C})\right)\,>\,N-3\,.$

This is due to Step 1 and to the fact that $G^{1}$ splits off a factor
$\mathbb{R}^{k}$, it is a cone, and we chose a point
$z\notin\mathbb{R}^{k}\times\\{o\\}$ as base point for the blow-up. The
additional splitting of $G^{2}$ can be justified with the very same arguments
of the Euclidean case, we refer for instance to [107, Theorem 28.11, Lemma
28.12, Lemma 28.13] whose statements and proofs work mutatis mutandis also in
our setting.

Step 3. The outcome of the previous two steps is that if (6.31) fails, then
there exists an entire local perimeter minimizing cone of the form
$G=\mathbb{R}^{N-2}\times C(B)\subset\mathcal{C}$. This is in contradiction
with subsection 6.3. ∎

######  Remark .

The Hausdorff dimension estimate (6.31) above is sharp. This is easily
verified by considering as entire local perimeter minimizer the set
$G:=\\{x>0\\}\subset\mathbb{R}\times C(\mathbb{S}^{1}_{r})$, where $0<r<1$ and
$x\in\mathbb{R}$ denotes the coordinate of the $\mathbb{R}$ factor. Then
$\partial G=\\{0\\}\times C(\mathbb{S}^{1}_{r})$ which has one singular point
$p=(0,o)$. Therefore $\mathscr{H}^{0}(\mathcal{S}^{G})=1$.

######  Theorem 6.5.

Let $(X,\mathsf{d},\mathscr{H}^{N})$ be an $\operatorname{RCD}(K,N)$ metric
measure space, let $\Omega\subset X$ be an open domain such that
$\Omega\cap\partial X=\emptyset$ and let $E\subset X$ be a set of locally
finite perimeter that is locally perimeter minimizing in $\Omega$. Then

(6.38) $\dim_{H}\left(\mathcal{S}^{E}\cap\Omega\right)\leq N-3\,.$

###### Proof.

The strategy of the proof is a refinement of the one of Theorem 6.3.
To simplify the notation, we assume throughout the proof that $E$ is an entire
local perimeter minimizer. Moreover, we assume that $N\geq 3$ and
$K\geq-(N-1)$. The case $K<-(N-1)$ can be reduced to $K=-(N-1)$ by scaling of
the distance. The case $N=1$ is elementary and the case $N=2$ can be treated
with a simpler variant of the argument presented below.

Step 1. Reduction to perimeter minimizers inside cones.
We aim to show via blow-up that if (6.38) fails for some local perimeter
minimizer on some $\operatorname{RCD}(K,N)$ metric measure space
$(X,\mathsf{d},\mathscr{H}^{N})$, then it fails also for an entire perimeter
minimizer inside a metric measure cone.
Notice that $\mathcal{S}^{E}\cap\mathcal{S}(X)=\partial E\cap\mathcal{S}(X)$
by the very definition of $\mathcal{S}^{E}$. Hence, if (6.38) fails, then
$\dim_{H}(\partial E\cap\mathcal{S}(X))>N-3$, by Theorem 6.3. In particular,
there exists $\varepsilon>0$ such that $\dim_{H}(\partial
E\cap\mathcal{S}_{\varepsilon}(X))>N-3$, where $\mathcal{S}_{\varepsilon}(X)$
is the quantitative $\varepsilon$-singular set of
$(X,\mathsf{d},\mathscr{H}^{N})$ defined by

$\mathcal{S}_{\varepsilon}(X):=\\{x\in
X\,:\,\mathsf{d}_{GH}(B_{r}(x),B_{r}(0^{N}))\geq\varepsilon r\,,\text{for all
$r\in(0,1)$}\\}\,.$

Indeed, by a well known argument (involving Bishop-Gromov volume monotonicity,
volume convergence and volume rigidity; see for instance the proof of [89,
Theorem 3.1] after [41]), it is easy to check that
$\mathcal{S}(X)=\bigcup_{\varepsilon>0}\mathcal{S}_{\varepsilon}(X)$.
Arguing as in the proof of Theorem 6.3 (see in particular (6.26)) we can find
$\eta>0$ such that $\mathscr{H}^{N-3+\eta}(\partial
E\cap\mathcal{S}_{\varepsilon}(X))>0$. Therefore, there exist $x\in\partial
E\cap\mathcal{S}_{\varepsilon}(X)$ and a sequence $r_{i}\downarrow 0$ such
that

(6.39)
$\limsup_{i\to\infty}\frac{\mathscr{H}^{N-3+\eta}_{\infty}\left(\partial
E\cap\mathcal{S}_{\varepsilon}(X)\cap
B_{r_{i}}(x)\right)}{r_{i}^{N-3+\eta}}>0\,.$

Applying Theorem 2.11, we can find a subsequence of $(r_{i})$, that we do not
relabel, such that (6.39) holds and the blow-up of $E$ along the sequence
$r_{i}$ is an entire local perimeter minimizer $F$ inside a metric cone $C(Z)$
with tip $p$, for some $\operatorname{RCD}(N-2,N-1)$ metric measure space
$(Z,\mathsf{d}_{Z},\mathscr{H}^{N-1})$. Moreover, it is easily seen that any
limit of points $x_{i}\in\partial E\cap\mathcal{S}_{\varepsilon}(X)\cap
B_{r_{i}}(x)$ along this converging sequence belongs to $\partial
F\cap\mathcal{S}_{\varepsilon}(C(Z))\cap B_{1}(p)$. Hence, the upper
semicontinuity of the pre-Hausdorff measure implies

(6.40) $\mathscr{H}^{N-3+\eta}_{\infty}\left(\partial
F\cap\mathcal{S}_{\varepsilon}(C(Z))\cap B_{1}(p)\right)\,>\,0\,,$

which yields $\dim_{H}(\mathcal{S}^{F})>N-3$, as we claimed.

Step 2. Dimension reduction.
In Step 1, we found an entire local perimeter minimizer $F\subset C(Z)$, where
$C(Z)$ is a metric measure cone. Let us consider the maximal Euclidean factor
$\mathbb{R}^{k}$ split off by $C(Z)$ and write $C(Z)=\mathbb{R}^{k}\times
C(W)$ for some $0\leq k\leq N-2$ and some $\operatorname{RCD}(N-k-2,N-k-1)$
metric measure space $(W,\mathsf{d}_{W},\mathscr{H}^{N-k-1})$. Arguing
inductively, we wish to prove that it is possible to assume that $k=N-2$
iterating the construction of Step 1.
Indeed, let us suppose that $k\leq N-3$. Then by (6.40) there exists a set of
singular points of $F$ with positive $\mathscr{H}^{N-3+\eta}_{\infty}$ pre-
Hausdorff measure not contained $\mathbb{R}^{k}\times\\{p\\}$. Iterating the
construction of Step 1 with a base point
$y\notin\left(\mathbb{R}^{k}\times\\{p\\}\right)$ such that

(6.41)
$\limsup_{i\to\infty}\frac{\mathscr{H}^{N-3+\eta}_{\infty}\left(\partial
F\cap\mathcal{S}_{\varepsilon}(C(Z))\cap
B_{r_{i}}(y)\right)}{r_{i}^{N-3+\eta}}>0\,,$

we obtain that, up to extraction of a subsequence that we do not relabel, the
blow-up of $F$ at $y$ is an entire local perimeter minimizer
$G\subset\mathbb{R}^{k+1}\times C(V)$, where
$(V,\mathsf{d}_{V},\mathscr{H}^{N-k-1})$ is an
$\operatorname{RCD}(N-k-2,N-k-1)$ metric measure space. Indeed, the blow-up
$G$ is an entire local perimeter minimizer by the usual stability Theorem
2.11. Moreover, the fact that the ambient space splits an additional Euclidean
factor follows by the choice of base point
$y\notin\left(\mathbb{R}^{k}\times\\{p\\}\right)$ (and the fact that $C(Z)$ is
a cone), via the splitting theorem [64].

Step 3. Conclusion.
The outcome of the previous two steps is that, if (6.38) fails for a local
perimeter minimizer $E\subset X$, where $(X,\mathsf{d},\mathscr{H}^{N})$ is an
$\operatorname{RCD}(K,N)$ m.m.s. (without boundary), then it fails for an
entire perimeter minimizer $F\subset\mathbb{R}^{N-2}\times
C(\mathbb{S}^{1}_{r})$, where $0<r\leq 1$. However, this would contradict
subsection 6.3 and the proof is complete.

∎

In the next statement, obtained combining Theorem 6.2, Theorem 6.3 and Theorem
6.5, we summarize the main regularity results of the present section.

######  Theorem 6.6.

Let $(X,\mathsf{d},\mathscr{H}^{N})$ be an $\operatorname{RCD}(K,N)$ metric
measure space. Let $E\subset X$ be a set of locally finite perimeter. Assume
that $E$ is perimeter minimizing in $B_{2}(x)\subset X$ and
$B_{2}(x)\cap\partial X=\emptyset$. Then for any $\alpha\in(0,1)$ there exists
a relatively open set $O_{\alpha}\subset\partial E\cap B_{1}(x)$ such that:

  * •

$O_{\alpha}$ is $\alpha$-bi-Hölder homeomorphic to a smooth open
$(N-1)$-dimensional manifold;

  * •

$\mathcal{R}^{E}\subset O_{\alpha}$ and

(6.42) $\displaystyle\dim_{H}\big{(}(\partial
E\setminus\mathcal{R}^{E})\cap\mathcal{R}(X)\big{)}$ $\displaystyle\leq
N-8\,,$ (6.43) $\displaystyle\dim_{H}\big{(}(\partial
E\setminus\mathcal{R}^{E})\cap\mathcal{S}(X)\big{)}$ $\displaystyle\leq
N-3\,.$

######  Remark  (Sharpness of Theorem 6.6 and a conjecture).

Both the Hausdorff codimension bounds (6.42) and (6.43) are sharp:

  * •

(6.42) is sharp already in $\mathbb{R}^{N}$, by the classical example of
Simons’ cone $C_{S}\subset\mathbb{R}^{8}$;

  * •

the sharpness of (6.43) was discussed in subsection 6.3.

Since $\mathcal{R}^{E}\subset O_{\alpha}$, the bounds (6.42)-(6.43) of course
imply

(6.44) $\displaystyle\dim_{H}\big{(}(\partial E\setminus
O_{\alpha})\cap\mathcal{R}(X)\big{)}$ $\displaystyle\leq N-8\,,$ (6.45)
$\displaystyle\dim_{H}\big{(}(\partial E\setminus
O_{\alpha})\cap\mathcal{S}(X)\big{)}$ $\displaystyle\leq N-3\,.$

Note that (6.44) is sharp already in $\mathbb{R}^{N}$, by the example of the
Simons’ cone $C_{S}\subset\mathbb{R}^{8}$: indeed, for any $\alpha\in(0,1)$,
it holds that $O_{\alpha}=C_{S}\setminus\\{0^{8}\\}$, so that
$\dim_{H}\big{(}C_{S}\setminus O_{\alpha}\big{)}=0$.
Instead, we conjecture that the optimal dimension bound for the topologically
regular part of $\partial E$ contained in the ambient singular set is

$\dim_{H}\big{(}(\partial E\setminus O_{\alpha})\cap\mathcal{S}(X)\big{)}\leq
N-4\,.$

Note that ambient Hausdorff co-dimension 4 would be sharp, from the example
given by $E:=C(\mathbb{RP}^{2})\times[0,\infty)\subset
C(\mathbb{RP}^{2})\times\mathbb{R}=:X$.

###  6.4. Quantitative estimates for singular sets of minimal boundaries

Our goal is to obtain Minkowski content estimates for the singular sets of
boundaries of locally perimeter minimizing sets in our context, in analogy
with the Euclidean theory [47, 114] and with the Minkowski estimates for the
quantitative singular sets of non collapsed Ricci limit spaces [46, 45] and
$\operatorname{RCD}$ spaces [19].

The strategy that we adopt has been partly inspired by [33], which proposed an
alternative approach to the regularity theory of locally perimeter minimizing
boundaries in the Euclidean framework. A key additional difficulty in our
setting, besides the fact that the spaces are not smooth, is that they are
curved (and we aim to an effective regularity theory, i.e. without the
dependence on flatness parameters such as the injectivity radius). Therefore
we will need to control at the same time the regularity of the space (with
constants only depending on the Ricci curvature and volume lower bounds) and
the regularity of the minimal boundary inside it.

######  Definition .

Let $(X,\mathsf{d},\mathscr{H}^{N})$ be an $\operatorname{RCD}(K,N)$ space.
Let $\eta>0$ and $r\in(0,1)$ be fixed. The quantitative regular set
$\mathcal{R}_{\eta,r}\subset X$ is defined by

$\mathcal{R}_{\eta,r}:=\\{x\in
X\,:\,\mathsf{d}_{GH}(B_{s}(x),B_{s}(0^{N}))\leq\eta s\,\quad\text{for any
$0<s<r$}\\}\,,$

where we indicated by $B_{r}(0^{N})\subset\mathbb{R}^{N}$ the Euclidean ball
of radius $r$.

######  Definition .

Let $(X,\mathsf{d},\mathscr{H}^{N})$ be an $\operatorname{RCD}(K,N)$ space.
Let $\eta>0$ and $r\in(0,1)$ be fixed. For any $0\leq k\leq N$, we shall
denote

$\displaystyle\mathcal{S}^{k}_{\eta,r}:=\\{x\in X$
$\displaystyle\,:\,\mathsf{d}_{GH}(B_{s}(x),B_{s}(0^{k+1},z^{*}))\geq\eta
s\,,$ $\displaystyle\quad\text{for any $\mathbb{R}^{k+1}\times C(Z)$ and all
$r<s<1$}\\}\,,$

where $B_{s}(0^{k+1},z^{*})$ denotes the ball centred at the tip of a cone
$\mathbb{R}^{k+1}\times C(Z)$.

In an analogous way we can deal with boundary points of local perimeter
minimizers.

######  Definition  (Quantitative singular sets for minimizing boundaries).

Let $(X,\mathsf{d},\mathscr{H}^{N})$ be an $\operatorname{RCD}(K,N)$ metric
measure space and let $E\subset X$ be a set of locally finite perimeter. Let
us suppose that $E$ is locally perimeter minimizing inside a ball
$B_{2}(x)\subset X$ and that $\partial X\cap B_{2}(x)=\emptyset$.
For any $\delta>0$, let $\mathcal{S}^{E}_{\delta}\subset\partial E$ be the
quantitative singular set defined by

$\displaystyle\mathcal{S}^{E}_{\delta}:=\\{x\in\partial E\,$
$\displaystyle:\,\text{there exists no $r\in(0,1)$}$ $\displaystyle\text{for
which $E\cap B_{r}(x)$ is $\delta$-regular at $x$}\\}\,.$

Moreover, for any $r>0$ we shall denote

$\displaystyle\mathcal{S}^{E}_{\delta,r}:=\\{x\in\partial E\,$
$\displaystyle:\,\text{there exists no $s\in(r,1)$}$ $\displaystyle\text{for
which $E\cap B_{s}(x)$ is $\delta$-regular at $x$}\\}\,$

and

$\overline{\mathcal{S}}^{E}_{\delta,r}:=\\{x\in\partial E\,:\,\text{$E\cap
B_{r}(x)$ is not $\delta$-regular at $x$}\\}\,.$

######  Remark .

A direct consequence of the definitions is that

$\mathcal{S}^{E}_{\delta}=\cap_{r>0}\mathcal{S}^{E}_{\delta,r}=\cap_{i\in\mathbb{N}}\mathcal{S}^{E}_{\delta,r_{i}}\,\quad\text{and
}\,\quad\mathcal{S}^{E}_{\delta,r}=\cap_{s>r}\overline{\mathcal{S}}^{E}_{\delta,s}\,,$

for any $\delta>0$ and for any sequence $r_{i}\downarrow 0$.

######  Definition  (Quantitative regular sets for minimal boundaries).

Let $(X,\mathsf{d},\mathscr{H}^{N})$ and $E\subset X$ be as in subsection 6.4.
Given $\eta>0$ and $r>0$ we shall denote by

$\displaystyle\mathcal{R}^{E}_{\eta,r}$ $\displaystyle:=\\{x\in\partial
E\,:\,E\cap B_{s}(x)\quad\text{is}\quad\text{ $\eta$-regular for any
$s\in(0,r)$}\\}\,,$ $\displaystyle\mathcal{R}^{E}_{\eta}$
$\displaystyle:=\bigcup_{r>0}\mathcal{R}^{E}_{\eta,r}=\\{x\in\partial
E\,:\,\exists\,r>0\,\text{ s.t. }E\cap B_{r}(x)\quad\text{is
$\eta$-regular}\\}\,,$

the quantitative regular sets of the minimal boundary $\partial E$.

######  Remark .

Let us notice that

(6.46) $\mathcal{R}^{E}=\bigcap_{\eta>0}\mathcal{R}^{E}_{\eta}.$

This is a consequence of the very definitions and of the
$\varepsilon$-regularity Theorem 6.1. Also, $\mathcal{R}^{E}_{\eta}$ is open
as soon as $\eta<\eta(N)$. Moreover,

$\mathcal{R}^{E}_{\eta}=\partial
E\setminus\mathcal{S}^{E}_{\eta}\,,\quad\text{for any $\eta>0$}\,.$

######  Remark .

An inspection of the proof of Theorem 6.1 shows that, if $\eta<\eta(N)$ and

$x\in\mathcal{R}\cap\mathcal{R}^{E}_{\eta}\,,$

then $x\in\mathcal{R}^{E}$ (cf. also with subsection 6.1).

######  Theorem 6.7.

For every $K\in\mathbb{R}$ and $N\in[1,\infty)$ there exists $\delta_{K,N}>0$
with the following property. Let $(X,\mathsf{d},\mathscr{H}^{N})$ be an
$\operatorname{RCD}(K,N)$ metric measure space, $x\in X$ such that $\partial
X\cap B_{2}(x)=\emptyset$, and $E\subset X$ be a set of locally finite
perimeter such that $E$ is perimeter minimizing in $B_{2}(x)$. Then for any
$0<\delta\in(0,\delta_{K,N})$ and for any $\gamma\in(0,1)$ there exist
$C=C(K,N,\delta,\operatorname{Per}\big{(}E,B_{2}(x)),\gamma\big{)}>0$ and
$r_{0}=r_{0}\big{(}K,N,\operatorname{Per}(E,B_{2}(x))\big{)}>0$ so that the
following Minkowski content-type estimate on the quantitative singular set
$\mathcal{S}^{E}_{\delta}\subset\partial E$ holds:

(6.47) $\mathscr{H}^{N}\big{(}T_{r}(\mathcal{S}^{E}_{\delta})\cap
B_{1}(x)\big{)}\leq C\,r^{2-\gamma}\,\quad\text{for any $r\in(0,r_{0})$ },$

where $T_{r}$ denotes the tubular neighbourhood of radius $r>0$.

When $(X,\mathsf{d},\mathscr{H}^{N})$ is a non collapsed Ricci limit space or
a finite dimensional Alexandrov space with curvature bounded below, the bounds
(6.47) can be strengthened to

(6.48) $\mathscr{H}^{N}(T_{r}(\mathcal{S}^{E}_{\delta})\cap B_{1}(x))\leq
C\,r^{2}\,,\quad\text{for any $r\in(0,r_{0})$}\,.$

######  Remark .

There is no direct implication between (6.47) and the Hausdorff dimension
estimate in Theorem 6.5. Indeed, while it is easily seen that (6.47) is much
stronger than the Hausdorff dimension estimate $\dim_{H}(\mathcal{S}^{E})\leq
N-2$, it does not imply the sharp estimate $\dim_{H}(\mathcal{S}^{E})\leq
N-3$. On the other hand, the Minkowski type estimate (6.47) is not implied by
any Hausdorff dimension estimate. As an elementary example just to fix the
ideas, note for instance that $\mathbb{Q}^{N}\subset\mathbb{R}^{N}$ has
Hausdorff dimension $0$, but no Minkowski content-type estimate holds since
any tubular neighbourhood of $\mathbb{Q}^{N}$ is the whole space
$\mathbb{R}^{N}$.

######  Remark .

While the proof of the Hausdorff dimension bound
$\dim_{H}(\mathcal{S}^{E})\leq N-3$ for local perimeter minimizers is
independent of the mean curvature bounds proved in section 5, these play a key
role in the proof of Theorem 6.7.

Theorem 6.7 will be proved at the end of the section. Below, we first
establish a series of auxiliary results.

Thanks to subsubsection 2.4.6, there exist constants
$C_{K,N},\bar{\delta}_{K,N}>0$ such that, if $(X,\mathsf{d},\mathscr{H}^{N})$
is an $\operatorname{RCD}(K,N)$ metric measure space and $E\subset X$ is a set
of finite perimeter minimizing the perimeter in $B_{2}(x)\subset X$, then

(6.49) $\mathscr{H}^{N}\big{(}(E^{\delta}\setminus\bar{E})\cap
B_{1}(x)\big{)}\leq
C_{K,N}\,\operatorname{Per}(E,B_{2}(x))\,\delta\,,\quad\text{for any
}\delta\in(0,\bar{\delta}_{K,N})\,,$

where we keep the notation $E^{\delta}$ for the $\delta$-enlargement of the
set $E$, see (6.10).

######  Corollary .

There exist constants $C_{K,N},\bar{\delta}_{K,N}>0$ with the following
property. Let $(X,\mathsf{d},\mathscr{H}^{N})$ be an $\operatorname{RCD}(K,N)$
metric measure space and $E\subset X$ is a set of finite perimeter minimizing
the perimeter in $B_{2}(x)\subset X$. Then, for any
$\delta\in(0,\bar{\delta}_{K,N})$, there exists $\rho\in(1/2,1)$ such that

$\operatorname{Per}(B_{\rho}(x),E^{\delta}\setminus\bar{E})\leq
C_{K,N}\,\operatorname{Per}(E,B_{2}(x))\,\delta\,.$

###### Proof.

The conclusion follows from the estimate (6.49). Indeed, by the coarea formula
Theorem 2.4 applied to the distance function from $x$ we can bound

$\displaystyle\int_{1/2}^{1}\operatorname{Per}\big{(}B_{s}(x),E^{\delta}\setminus\bar{E}\big{)}\mathop{}\\!\mathrm{d}s$
$\displaystyle\leq\mathscr{H}^{N}\big{(}(E^{\delta}\setminus\bar{E})\cap(B_{1}(x)\setminus
B_{1/2}(x))\big{)}$
$\displaystyle\leq\mathscr{H}^{N}\big{(}(E^{\delta}\setminus\bar{E})\cap
B_{1}(x)\big{)}\,$ $\displaystyle\leq
C_{K,N}\,\operatorname{Per}(E,B_{2}(x))\,\delta\,.$

∎

The so-called interior/exterior touching ball condition is a regularity
property for domains $\Omega\subset X$. The interior one amounts to ask that
at any given point $x\in\partial\Omega$, there exists a point $y\in\Omega$ and
$r>0$ such that $\mathsf{d}(x,y)=r$ and $B_{r}(y)\subset\Omega$.
When it holds uniformly, on a smooth Riemannian manifold, it yields a control
on the second fundamental form of the boundary of the domain.

Our next goal is to prove that minimal boundaries verify a weak
interior/exterior touching ball condition in our setting.

######  Definition  (Set of touching points).

Let $(X,\mathsf{d},\mathfrak{m})$ be an $\operatorname{RCD}(K,N)$ metric
measure space and let $E\subset X$ be a local perimeter minimizer. For any
$\delta\in(0,\delta_{0})$, we let $\mathcal{C}_{\delta}\subset\partial E$ be
the set of interior and exterior touching points of balls of radius $\delta$,
i.e.

$\displaystyle\mathcal{C}_{\delta}:=\\{x\in\partial E\,$
$\displaystyle:\,\text{there exist $B_{\delta}(x_{1})\subset E$}$
$\displaystyle\text{and $B_{\delta}(x_{2})\subset E^{c}$ such that
$x\in\partial B_{\delta}(x_{1})\cap\partial B_{\delta}(x_{2})$}\\}\,.$

######  Proposition .

There exist constants $\delta_{K,N},\,C_{K,N}>0$ such that, for any
$\operatorname{RCD}(K,N)$ metric measure space
$(X,\mathsf{d},\mathscr{H}^{N})$, for any $x\in X$ and for any set of finite
perimeter $E\subset X$ such that $E$ is perimeter minimizing in
$B_{2}(x)\subset X$ the following holds:

(6.50) $\operatorname{Per}(E,B_{1/2}(x)\setminus\mathcal{C}_{\delta})\leq
C_{K,N}\,\operatorname{Per}(E,B_{2}(x))\,\delta\,,\quad\text{for any
}\delta\in(0,\delta_{K,N})\,.$

###### Proof.

It is sufficient to estimate the size of the set $\mathcal{C}^{e}_{\delta}$ of
touching points of exterior tangent balls as in (6.50). A similar argument
will give the estimate for the size of the set of touching points of interior
balls $\mathcal{C}^{i}_{\delta}$. Then the estimate for $\mathcal{C}_{\delta}$
will follow, since
$\mathcal{C}_{\delta}=\mathcal{C}^{i}_{\delta}\cap\mathcal{C}^{e}_{\delta}$.

Let us fix $\delta\in(0,\bar{\delta}_{K,N})$ and choose $\rho\in(1/2,1)$ given
by subsection 6.4 above. We can also assume that
$\operatorname{Per}(E^{\delta},\partial B_{\rho}(x))=0$ up to slightly perturb
$\rho$. Observe that $E\cup(E^{\delta}\cap B_{\rho}(x))$ is a compactly
supported perturbation of $E$ in $B_{2}(x)$. Hence, by perimeter minimality,
it holds:

$\operatorname{Per}(E,B_{2}(x))\leq\operatorname{Per}(E\cup(E^{\delta}\cap
B_{\rho}(x)),B_{2}(x))\,.$

Therefore

$\displaystyle\operatorname{Per}(E,B_{\rho}(x))\leq$
$\displaystyle\operatorname{Per}(E^{\delta},B_{\rho}(x))+\operatorname{Per}(B_{\rho}(x),E^{\delta}\setminus\bar{E})$
(6.51) $\displaystyle\leq$
$\displaystyle\operatorname{Per}(E^{\delta},B_{\rho}(x))+C_{K,N}\,\operatorname{Per}(E,B_{2}(x))\,\delta\,.$

Letting $\Gamma_{\delta}:=\partial E^{\delta}\cap\overline{B}_{\rho}(x)$ and
$\Gamma_{\delta,\Sigma}$ be the set of touching points of minimizing geodesics
from $\Gamma_{\delta}$ to $\Sigma$, we can estimate by subsection 6.2

(6.52)
$\operatorname{Per}(E^{\delta},B_{\rho}(x))\leq\operatorname{Per}(E,\Gamma_{\delta,\Sigma})+C_{K,N}\operatorname{Per}(E,B_{2}(x))\,\delta\,.$

Notice that all the points in $\Gamma_{\delta,\Sigma}$ are touching points of
exterior balls of radius $\delta$ on $\partial E$. Hence
$\Gamma_{\delta,\Sigma}\subset\mathcal{C}_{\delta}^{e}$. Taking into account
(6.4) and (6.52), then we can estimate

(6.53) $\operatorname{Per}(E,B_{1/2}(x)\setminus\mathcal{C}^{e}_{\delta})\leq
C_{K,N}\,\operatorname{Per}(E,B_{2}(x))\,\delta\,.$

Combining (6.53) with the analogous estimate valid for the set of touching
points of interior balls, we get (6.50). ∎

######  Remark .

It is worth pointing out the following nontrivial consequence of subsection
6.4: if $E$ is locally perimeter minimizing, then
$\operatorname{Per}_{E}$-a.e. point $x\in\partial E$ is an intermediate point
of a minimizing geodesic along which the signed distance function from $E$ is
realized (that would correspond to a perpendicular geodesic on a smooth
Riemannian manifold).

Given a set of finite perimeter $E\subset\mathbb{R}^{n}$, locally perimeter
minimizing in an open domain, the existence of an interior and of an exterior
touching balls at a given point $x\in\partial E$ are enough to guarantee the
regularity of the boundary near to the touching point.
One way to verify this conclusion is to argue that the presence of both an
interior and an exterior touching ball forces the tangent cone at the point to
be flat and this is enough to guarantee regularity in a neighbourhood, as we
already pointed out.

There is also a more quantitative approach, whose starting point is given by
the following observation: there exists $C=C_{n}>0$ such that if $x\in\partial
B_{C_{n}\lambda/\delta}(x_{1})\cap\partial B_{C_{n}\lambda/\delta}(x_{2})$,
where $\delta,\lambda>0$ and $B_{C_{n}\lambda/\delta}(x_{1})$ and
$B_{C_{n}\lambda/\delta}(x_{2})$ are an interior and an exterior touching ball
respectively, then

(6.54) $E\cap B_{\lambda}(y)\,\quad\text{is $\delta$-flat at $y$, for every
$y$ such that $\left\lvert x-y\right\rvert<\lambda$}\,.$

As subsection 6.1 clearly illustrates, the existence of an interior and an
exterior touching ball at a boundary point of a perimeter minimizing set is
not enough to guarantee that the tangent is flat, nor that the boundary is
regular in a neighbourhood of the point.

Even on a smooth Riemannian manifold, in order to guarantee
$\delta$-regularity, the existence of interior/exterior touching balls needs
to be combined with closeness (at the given scale) of the ball to the
Euclidean ball, as shown in the next lemma.

######  Lemma .

There exists a constant $C=C_{N}>0$ such that the following holds. Let
$(X,\mathsf{d},\mathscr{H}^{N})$ be an $\operatorname{RCD}(-(N-1),N)$ metric
measure space and let $E\subset X$ be a set of finite perimeter that locally
minimizes the perimeter in $B_{2}(x)\subset X$. Let $\lambda\in(0,1/2)$,
$\delta>0$ and assume that:

  * (i)

$x\in\partial E$ is a touching point of an interior and an exterior ball of
radius $C_{N}\lambda/\delta$;

  * (ii)

$B_{\lambda}(x)$ is $\delta\lambda$-GH close to
$B_{\lambda}(0)\subset\mathbb{R}^{N}$.

Then, for any $y\in\partial E\cap B_{\lambda/2}(x)$, $E\cap B_{\lambda}(y)$ is
$2\delta$-regular in $B_{\lambda}(y)$.

###### Proof.

Condition (ii) guarantees scale invariant $\delta$-closeness, in GH sense, of
$B_{\lambda}(x)$ to $B_{\lambda}(0)\subset\mathbb{R}^{N}$ and of
$B_{\lambda}(y)$ to $B_{\lambda}(0)\subset\mathbb{R}^{N}$ for any
$y\in\partial E\cap B_{\lambda/2}(x)$. The proof is then reduced to the
Euclidean setting, where the existence of interior/exterior touching balls
with radii $C_{N}\lambda/\delta$ guarantees $\delta$-flatness, as we remarked
in (6.54). ∎

By (6.54), we can bound in an effective way the perimeter of the set where
there are no interior/exterior touching balls of a given size. In order to
guarantee that regularity of the ambient balls is in force at many locations
and scales along $\partial E$, we will rely on the quantitative bounds for the
singular strata of noncollapsed $\operatorname{RCD}$ spaces, obtained in [19]
following the strategy of the previous [46].

We will be focusing on codimension two singularities. With this aim, let us
state an $\varepsilon$-regularity result that follows from [29].

######  Theorem 6.8 (Boundary $\varepsilon$-regularity).

Let $K\in\mathbb{R}$ and $1\leq N<\infty$ be fixed. Then there exists
$\varepsilon=\varepsilon(K,N)>0$ such that the following holds. If
$(X,\mathsf{d},\mathscr{H}^{N})$ is an $\operatorname{RCD}(K,N)$ space, $x\in
X$ and $s\in(0,1)$ are such that

$\mathsf{d}_{GH}(B_{s}(x),B_{s}(0^{N-1},z^{*}))<\varepsilon s\,,$

for some $B_{s}(0^{N-1},z^{*})\subset\mathbb{R}^{N-1}\times C(Z)$ and
$\partial X\cap B_{s}(x)=\emptyset$, then

$\mathsf{d}_{GH}(B_{s}(x),B_{s}(0^{N}))<2\varepsilon s\,,$

where $B_{s}(0^{N})\subset\mathbb{R}^{N}$ is the Euclidean ball of dimension
$N$.

###### Proof.

There are only two possibilities for the cone $C(Z)$. Either
$C(Z)=\mathbb{R}^{+}$ or $C(Z)=\mathbb{R}$, with the canonical metric measure
structure, in both cases. The possibility that $C(Z)=\mathbb{R}^{+}$ can be
excluded thanks to [29, Theorem 1.6]. Hence $C(Z)=\mathbb{R}$ the ball is
$2\varepsilon$-regular, as we claimed.

∎

Thanks to Theorem 6.8, we can easily check that if
$(X,\mathsf{d},\mathscr{H}^{N})$ is an $\operatorname{RCD}(-(N-1),N)$ space
and $B_{s}(x)\cap\partial X=\emptyset$ for some ball $B_{s}(x)\subset X$, then

(6.55) $B_{s}(x)\setminus\mathcal{S}^{N-2}_{\eta,r}=\\{y\in
B_{s}(x)\,:\,\mathsf{d}_{GH}(B_{t}(y),B_{t}(0^{N}))<\eta t\,,\text{ for any
$t\in(0,r)$}\\}\,,$

for any $\eta\in(0,\eta(N))$.

Let us recall the volume estimate for the quantitative singular stratum
obtained in [19] (see [19, Theorem 2.4] and the discussion below it) after
[46].

######  Theorem 6.9.

Let $K\in\mathbb{R}$, $2\leq N<\infty$, $1\leq k\leq N$, $v,\eta,\gamma>0$ be
fixed. Then there exists a constant $c=c(K,N,k,\eta,v)>0$ such that the
following holds. If $(X,\mathsf{d},\mathscr{H}^{N})$ is an
$\operatorname{RCD}(K,N)$ space and

$\frac{\mathscr{H}^{N}(B_{1}(x))}{v_{K,N}(1)}\geq v\,,$

then for any $r\in(0,1/2)$ it holds

(6.56) $\mathscr{H}^{N}(B_{1/2}(x)\cap\mathcal{S}^{k}_{\eta,r})\leq
c(K,N,k,\eta,v,\gamma)r^{N-k-\gamma}\,.$

######  Remark .

In [45, Theorem 1.7] it has been shown that for non collapsed Ricci limit
spaces it is possible to replace $(N-k-\gamma)$ with $(N-k)$ at the exponent
in (6.56) to obtain a much stronger estimate

(6.57) $\mathscr{H}^{N}(B_{1/2}(x)\cap\mathcal{S}^{k}_{\eta,r})\leq
c(N,K,k,\eta,v)r^{N-k}\,,\quad\text{for any $r\in(0,1/2)$ }.$

The very same estimate (6.57) was established in [101, Corollary 1.5] for
$N$-dimensional Alexandrov spaces with curvature bounded below by $K$.

Relying on Theorem 6.9, let us estimate the size of the intersection of the
quantitative singular stratum $\mathcal{S}^{k}_{\eta,r}$ with the boundary of
a locally perimeter minimizing set of finite perimeter.

######  Proposition .

Let $K\in\mathbb{R}$, $2\leq N<\infty$, $1\leq k\leq N$, $v,\eta,\gamma>0$ be
fixed. Then there exists a constant $c=c(K,N,k,\eta,v)>0$ such that the
following holds. If $(X,\mathsf{d},\mathscr{H}^{N})$ is an
$\operatorname{RCD}(K,N)$ space such that

$\frac{\mathscr{H}^{N}(B_{1}(x))}{v_{K,N}(1)}\geq v\,$

and $E\subset X$ is a set of finite perimeter which is perimeter minimizing in
$B_{2}(x)$, then there exists $r_{0}=r_{0}(K,N)>0$ independent of $k,\,\eta$
and $\gamma$ such that

(6.58)
$\displaystyle\operatorname{Per}(E,B_{1/2}(x)\cap\mathcal{S}^{k}_{\eta,r})$
$\displaystyle\leq c(K,N,k,\eta,v)\,r^{N-k-1-\gamma}\,,\quad\text{for any
$r\in(0,r_{0})$}\,,$ (6.59)
$\displaystyle\operatorname{Per}(E,B_{1/2}(x)\setminus\mathcal{R}_{\eta,r})$
$\displaystyle\leq c(K,N,\eta,v)\,r^{1-\gamma}\,,\quad\qquad\quad\text{ for
any $r\in(0,r_{0})$}\,.$

###### Proof.

Let us consider a covering of $\partial E\cap
B_{1/2}(x)\cap\mathcal{S}^{k}_{\eta,r}$ with balls $B_{r\eta}(x_{i})$ such
that the balls $B_{r\eta/5}(x_{i})$ are disjoint, via a Vitali covering
argument.
As shown for instance in [19, equation (2.5)], unwinding the definitions, one
can check that

(6.60) $T_{\eta
r}(\mathcal{S}^{k}_{2\eta,r})\subset\mathcal{S}^{k}_{\eta,r}\,,$

where $T_{\eta r}$ denotes the tubular neighbourhood of radius $r\eta$. Thus,
we can estimate:

(6.61)
$\operatorname{Per}(E,B_{1/2}(x)\cap\mathcal{S}^{k}_{\eta,r})\leq\sum_{i}\operatorname{Per}(E,B_{\eta
r}(x_{i}))\leq C\sum_{i}\frac{\mathscr{H}^{N}(B_{\eta r}(x_{i}))}{\eta r}\,,$

where the constant $C$ is given by subsubsection 2.4.6.
Relying on (6.60), Theorem 6.9 and the Vitali covering condition, we obtain

$\sum_{i}\frac{\mathscr{H}^{N}(B_{\eta r}(x_{i}))}{\eta r}\leq\frac{C}{\eta
r}\mathscr{H}^{N}(T_{\eta r}(\mathcal{S}^{k}_{2\eta,r}\cap
B_{1/2}(x)))\leq\frac{c}{\eta}r^{N-k-1-\gamma}\,,$

which gives (6.58) when combined with (6.61).
The estimate (6.59) follows from (6.58), thanks to (6.55). ∎

######  Remark .

Relying on the observation in subsection 6.4, in the case of non collapsed
Ricci limit spaces and finite dimensional Alexandrov spaces with curvature
bounded below, it is possible to strengthen (6.58) and (6.59) to

(6.62) $\operatorname{Per}(E,B_{1/2}(x)\cap\mathcal{S}^{k}_{\eta,r})\leq
c(K,N,k,\eta,v)\,r^{N-k-1}\,,$

for any $r\in(0,r_{0})$ and

(6.63) $\operatorname{Per}(E,B_{1/2}(x)\setminus\mathcal{R}_{\eta,r})\leq
c(K,N,\eta,v)\,r\,,$

for any $r\in(0,r_{0})$.

###### Proof of Theorem 6.7.

We claim that for any $\delta\in(0,\delta_{K,N})$ there exists a constant

$C=C(K,N,\delta,\gamma,\operatorname{Per}(E,B_{2}(x))>0$

such that for any $r\in(0,r_{0})$ and any Vitali covering of
$\overline{\mathcal{S}}^{E}_{\delta,r}\cap B_{1}(x)$ with balls $B_{r}(x_{i})$
such that $x_{i}\in\overline{\mathcal{S}}^{E}_{\delta,r}$ and $B_{r/5}(x_{i})$
are pairwise disjoint for $i=1,\dots,N(r)$, it holds

(6.64) $N(r)\leq C\,r^{2-N-\gamma}\,.$

Indeed, for any ball $B_{r}(x_{i})$ as above, it holds

$B_{r}(x_{i})\cap\mathcal{R}_{\delta/2,2r}\cap\mathcal{C}_{C_{K,N}\frac{r}{\delta}}=\emptyset\,,$

where $\mathcal{C}_{C_{K,N}\frac{r}{\delta}}$ is the set of contact points of
touching balls as in subsection 6.4. This is a consequence of subsection 6.4:
if by contradiction $y$ belongs to the intersection above, then $E$ is
$\delta$-regular on $B_{r}(z)$ for any $z\in B_{r}(y)$. Hence $E$ is
$\delta$-regular on $B_{r}(x_{i})$, a contradiction.
Since the balls $B_{r/5}(x_{i})$ are disjoint, we can bound

$\displaystyle\sum_{i\leq
N(r)}\operatorname{Per}\big{(}E,B_{r/5}(x_{i})\big{)}\leq$
$\displaystyle\operatorname{Per}\bigg{(}E,\bigcup_{i\leq
N(r)}B_{r}(x_{i})\bigg{)}$ $\displaystyle\leq$
$\displaystyle\operatorname{Per}\big{(}E,B_{1}(x)\setminus(\mathcal{R}_{\delta/2,2r}\cap{\mathcal{C}}_{C_{K,N}\frac{r}{\delta}})\big{)}\leq
C\,r^{1-\gamma}\,,$

for some $C=C(K,N,\delta,\gamma,\operatorname{Per}(E,B_{2}(x))>0$, where the
last inequality follows from subsection 6.4 and (6.59). By the Ahlfors
regularity of the perimeter measure subsubsection 2.4.6, we easily get (6.64).

Notice that, for non collapsed Ricci limit spaces and finite dimensional
Alexandrov spaces, the estimate (6.64) can be strengthened into

(6.65) $N(r)\leq C\,r^{2-N}\,.$

This is a consequence of the better Minkowski bounds obtained in [45, 101] in
such a setting, arguing as we did above, using subsection 6.4 and subsection
6.4.

To conclude the proof, in all the cases of $\operatorname{RCD}(K,N)$ spaces,
non collapsed Ricci limit spaces and finite dimensional Alexandrov spaces, it
is sufficient to rely on the Ahlfors regularity bound for $\mathscr{H}^{N}$
and to recall that

(6.66)
$\mathcal{S}^{E}_{\delta,r}=\bigcap_{s>r}\overline{\mathcal{S}}^{E}_{\delta,s}\,$

and

(6.67) $\mathcal{S}^{E}_{\delta}=\bigcap_{r>0}\mathcal{S}^{E}_{\delta,r}\,.$

∎

######  Remark .

If $(X,\mathsf{d},\mathscr{H}^{N})$ is a smooth Riemannian manifold equipped
with its volume measure, then (6.47) can be strengthened into

(6.68) $\mathscr{H}^{N}\big{(}T_{r}(\mathcal{S}^{E}_{\delta})\cap
B_{1}(x)\big{)}\leq C\,r^{8}\,\quad\text{for any $r\in(0,r_{0})$ }\,,$

if we allow the constants $C$ and $r_{0}$ to depend on the norm of the full
Riemann curvature tensor on $B_{2}(x)$ and on a lower bound on the injectivity
radius on $B_{2}(x)$, as proved in [114, Theorem 1.6].
Since the constants in (6.47) only depend on the dimension, the lower Ricci
curvature bound and on the perimeter of $E$ on $B_{1}(x)$, our estimates are
not encompassed by those in [114] even in the case of smooth manifolds.

##  Appendix A Laplacian bounds Vs mean curvature bounds:
a comparison with the classical literature

The aim of this subsection is to put Theorem 5.1 and Theorem 5.2 into
perspective. In particular, we wish to clarify why Laplacian bounds on the
distance function can be understood as mean curvature bounds. For this reason,
we are going to present some mostly well known results about the distance
function from minimal hypersurfaces on smooth Riemannian manifolds, focusing
for simplicity on the non-negative Ricci curvature case.

As we already remarked, the fact that the distance from a smooth minimal
hypersurface is subharmonic in a manifold with non-negative Ricci curvature is
classical. To the best of our knowledge, the first reference where this result
is explicitly stated, even though without proof, is [134]. Therein, the
Laplacian bound was understood in the viscosity sense. In subsequent
contributions, such as [116] and [48], superharmonicity of the distance was
understood in the sense of barriers, following the seminal [34, 44].

######  Theorem A.1.

Let $(M^{n},g)$ be a smooth Riemannian manifold with non-negative Ricci
curvature and let $\Sigma\subset M$ be a smooth hypersurface. Then
$\bm{\Delta}\mathsf{d}_{\Sigma}\leq 0$ on $M\setminus\Sigma$ if and only if
$\Sigma$ is minimal, in the sense that it has vanishing mean curvature.

###### Proof.

We only give an indication of the argument, a complete proof of the
implication from minimality to subharmonicity of the distance can be found for
instance in [48].

Notice that the Laplacian of the distance from a smooth hypersurface coincides
with its mean curvature along the hypersurface, thanks to a classical
computation in Riemannian Geometry. One possible strategy to check
subharmonicity of the distance is to observe that the singular part of the
Laplacian has negative sign, in great generality. Then we can consider
minimizing geodesics along which the distance to the hypersurface is realized.
Along these rays, the vanishing mean curvature condition at the starting point
propagates to nonnegativity of the Laplacian of the distance, thanks to the
non-negative Ricci curvature condition.

The converse implication, from subharmonicity of the distance to minimality,
relies on the same principle, combined with the fact that
$\mathsf{d}_{\Sigma}$ is smooth on any side of $\Sigma$ locally in a
neighbourhood of any point. In order to check that the mean curvature
$H_{\Sigma}$ vanishes at a given $p\in\Sigma$, let us consider the minimizing
geodesic $\gamma:(-\varepsilon,\varepsilon)\to M$ such that $\gamma(0)=p$ and
$\gamma^{\prime}(0)$ is perpendicular to $T_{p}\Sigma$. Then observe that,
combining the superharmonicity of $\mathsf{d}_{\Sigma}$ with the already
mentioned connection between mean curvature and Laplacian of the distance,

$0\leq-\lim_{t\uparrow
0}\Delta\mathsf{d}_{\Sigma}(\gamma(t))=H_{\Sigma}(p)=\lim_{t\downarrow
0}\Delta\mathsf{d}_{\Sigma}(\gamma(t))\leq 0\,,$

hence $H_{\Sigma}(p)=0$. ∎

On smooth Riemannian manifolds with non-negative Ricci curvature, the distance
from a minimal hypersurface is subharmonic even for certain minimal
hypersurfaces that are not globally smooth. This is a key point for the sake
of the applications, since minimal hypersurfaces that are built through
variational arguments might be non smooth in ambient dimension greater than
$8$.

Notice that Theorem 5.1 already gives a substantial contribution in this
direction. Indeed, we can cover at least all the minimal hypersurfaces that
are locally boundaries of sets of locally minimal perimeter.222In particular
it provides a different proof of the first implication in Theorem A.1 since,
as we already mentioned, all smooth minimal hypersurfaces are locally
boundaries of perimeter minimizing sets.

Actually, the principle “minimality implies subharmonicity of the distance
function” extends even to minimal hypersurfaces that are not necessarily
locally boundaries.

Let us introduce some terminology, following [138] for this presentation.

######  Definition .

Given a smooth Riemannian manifold $(M^{n},g)$, a singular hypersurface with
singular set of codimension no less than $k$ ($k<n-1$, $k\in\mathbb{N}$) is a
closed set $\overline{\Sigma}\subset M$ such that
$\mathscr{H}^{n-1}(\overline{\Sigma})<\infty$, where the regular part
$\mathcal{R}(\Sigma)$ is defined by

$\displaystyle\mathcal{R}(\Sigma):=\\{$ $\displaystyle
x\in\overline{\Sigma}\,:\,\overline{\Sigma}$ $\displaystyle\quad\text{is a
smooth embedded hypersurface in a neighbourhood of $x$}\\}$

and $\mathcal{S}(\Sigma):=\overline{\Sigma}\setminus\mathcal{R}(\Sigma)$ is
the singular part which we assume to satisfy
$\dim_{H}(\mathcal{S}(\Sigma))\leq n+1-k$.

Given such a singular hypersurface, it represents an integral varifold, that
we denote as $[\Sigma]$. We will say that $\Sigma$ is minimal if $[\Sigma]$ is
a stationary varifold and the tangent cones of $[\Sigma]$ have all
multiplicity one.

######  Remark .

We recall that the minimality condition above is equivalent to the requirement
that the mean curvature vanishes on $\mathcal{R}(\Sigma)$ and the density of
$[\Sigma]$ is finite everywhere. Moreover, as shown in [138, Lemma 6.3],
minimal hypersurfaces produced through min-max are minimal according to
Appendix A above.

The next statement originates from an argument due to Gromov in his proof of
the isoperimetric inequality [78].

######  Theorem A.2.

Let $(M^{n},g)$ be a smooth Riemannian manifold with non-negative Ricci
curvature. Let $\overline{\Sigma}\subset X$ be minimal in the sense of
Appendix A. Then $\mathsf{d}_{\overline{\Sigma}}$ is subharmonic on
$M\setminus\overline{\Sigma}$.

###### Proof.

The proof is divided in two steps. The first is about controlling the mean
curvature at footpoints of minimizing geodesics on the hypersurface. The
second deals with the propagation of the mean curvature bound to obtain a
Laplacian bound, as in previous arguments in this note.

Step 1. As proved for instance in [138, Lemma 2.1] along the original argument
due to Gromov, the following holds. For any $p\in
M\setminus\overline{\Sigma}$, let
$\gamma:[0,\mathsf{d}(p,\overline{\Sigma})]\to M$ be a minimizing geodesic
connecting $p$ to $\overline{\Sigma}$, and let $\gamma(0)=q$ be the footpoint
of the geodesic on $\overline{\Sigma}$, then $q\in\mathcal{R}(\Sigma)$.

Indeed, the geodesic sphere of radius $\mathsf{d}(p,q)/2$ centred at
$\gamma(\mathsf{d}(p,q)/2)$ is a smooth hypersurface near to $q$ and
$\overline{\Sigma}$ lies on one side of it. Since all tangent cones have
multiplicity one, the tangent cone to $[\Sigma]$ at $q$ is unique and it is a
hyperplane. Hence, by Allard’s regularity theorem [1], $\overline{\Sigma}$ is
regular at $q$. Therefore, the mean curvature of $\Sigma$ is vanishing in a
neighbourhood of $q$.

Step 2. Let us propagate the information that the mean curvature is vanishing
in the classical sense near to footpoints of minimizing geodesics to prove
that $\mathsf{d}_{\overline{\Sigma}}$ is subharmonic on
$M\setminus\overline{\Sigma}$.

We can rely for instance on the localization technique to argue that it is
sufficient to control the regular part of the Laplacian of
$\mathsf{d}_{\overline{\Sigma}}$ (see for instance [37, Theorem 1.3, Corollary
4.16] and Step 2 in the proof of Theorem 5.1). Then, to control the regular
part, it is enough to observe that $\mathsf{d}_{\overline{\Sigma}}$ is smooth
near to initial points of rays in the localization (thanks to the smoothness
of $\Sigma$ obtained in Step 1). Moreover the Laplacian of the distance is
vanishing there, therefore it remains non-negative along the rays by the non-
negative Ricci curvature assumption. ∎

######  Remark .

The proof of Theorem A.2 above works in particular for hypersurfaces that are
locally boundaries of locally perimeter minimizing sets, once we appeal to the
classical Euclidean regularity theory for local perimeter minimizers. In
particular it provides a different proof of Theorem 5.1 for smooth Riemannian
manifolds. However, the use of deep regularity theorems in Geometric Measure
Theory, makes the extension of this strategy to non smooth ambient spaces
unlikely, as already pointed out in [119]. The interest towards proofs of mean
curvature bounds and regularity results for area minimizing surfaces not
heavily relying on GMT tools was pointed out also in [79, 80].

######  Remark .

As remarked in [123], if $E\subset\mathbb{R}^{n}$ is an open set and
$\Delta\mathsf{d}_{\partial E}\leq 0$ locally in a neighbourhood of $\partial
E$ and away from $\partial E$, then $\partial E$ satisfies the minimal
surfaces equation in the viscosity sense. Indeed the signed distance from a
smooth boundary is smooth in a neighbourhood of any point along the boundary,
where its Laplacian corresponds to the mean curvature, as we pointed out in
the proof of Theorem A.1 above. See also [133] for some arguments in the same
spirit in the Riemannian framework.

## References

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