# Half Space Property in $\mathsf{RCD}(0,N)$ spaces

Alessandro Cucinotta  Andrea Mondino

(February 2024)

###### Abstract

The goal of this note is to prove the Half Space Property for
$\mathsf{RCD}(0,N)$ spaces, namely that if
$(\mathsf{X},\mathsf{d},\mathfrak{m})$ is a parabolic $\mathsf{RCD}(0,N)$
space and $C\subset\mathsf{X}\times\mathbb{R}$ is locally the boundary of a
locally perimeter minimizing set and it is contained in a half space, then $C$
is a locally finite union of horizontal slices.

As a consequence, we obtain oscillation estimates and a Half Space Theorem for
minimal hypersurfaces in products $\mathsf{M}\times\mathbb{R}$, where
$\mathsf{M}$ is a parabolic smooth manifold (possibly weighted and with
boundary) with non-negative Ricci curvature.

On the way of proving the main result, we also obtain some properties of
Green’s functions on $\mathsf{RCD}(K,N)$ spaces that are of independent
interest.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
    1. 2.1 Sobolev spaces and the $\mathsf{RCD}$ condition
    2. 2.2 Poisson problem and regular sets
    3. 2.3 Sets of finite perimeter
  3. 3 Green’s functions in parabolic metric measure spaces
  4. 4 Proof of the Half Space Property
  5. 5 Applications

##  1 Introduction

In [45] Hoffman and Meeks proved the Half Space Theorem, stating that a
connected, proper, non-planar minimal surface in $\mathbb{R}^{3}$ cannot be
contained in a half space. Analogous results in the Riemannian setting were
then obtained by different authors ([43], [25], [24], [28], etc.) and in [62]
Rosenberg, Schulze and Spruck gave the following definition.

######  Definition.

A Riemannian manifold $\mathsf{M}$ has the Half Space Property if every proper
minimal hypersurface $S\subset\mathsf{M}\times\mathbb{R}$ not intersecting
$\mathsf{M}\times\\{0\\}$ equals a horizontal slice $\mathsf{M}\times\\{c\\}$.

The study of this property, which generalizes the Half Space Theorem, is
linked to the understanding of the following more general problem: what are
the conditions such that two minimal submanifolds $S_{1}$, $S_{2}$ of a
Riemannian manifold $\mathsf{M}$ must intersect? And if they do not intersect,
do they influence each other’s geometry?

Classical results in this direction, aside from the Half Space Theorem, are
the fact that the only positive solutions of the minimal surface equation in
Euclidean space are constant functions ([14]) and Frankel’s Theorem, stating
that two compact minimal hypersurfaces in a closed manifold with positive
Ricci curvature must intersect ([30]).

More recently, in the aforementioned work [62], it was proved that the Half
Space Property holds for parabolic manifolds with bounded sectional curvature
and the result was then extended to parabolic manifolds with a uniform lower
bound on the Ricci curvature in [23]. It was then proved in the very recent
work [26] that even the lower bound on the Ricci curvature can be removed, so
that the Half Space Property holds for general parabolic manifolds. We recall
that a manifold is parabolic if it does not admit a positive Green’s function
for the Laplacian and we note that without the parabolicity assumption the
Half Space Property fails in general, the catenoid in $\mathbb{R}^{4}$ being a
counterexample.

The validity of the Half Space Property for parabolic smooth Riemannian
manifolds with lower Ricci curvature bounds and the recent generalization of
Frankel’s Theorem to the setting of $\mathsf{RCD}$ spaces ([58]), i.e. non-
smooth spaces with a synthetic lower bound on the Ricci curvature, suggest
that also the Half Space Property might hold in the non-smooth setting.

We recall that an $\mathsf{RCD}(K,N)$ space is a metric measure space where
$K\in\mathbb{R}$ plays the role of a lower bound on the Ricci curvature, while
$N\in[1,+\infty)$ plays the role of an upper bound on the dimension; this
class includes measured Gromov-Hausdorff limits of smooth manifolds with
uniform Ricci curvature lower bounds and finite dimensional Alexandrov spaces
with sectional curvature bounded from below. Moreover, since a hypersurface in
a smooth manifold is minimal if and only if it is locally the boundary of a
locally perimeter minimizing set, it is natural, in the non-smooth setting, to
replace minimal hypersurfaces with local boundaries of local perimeter
minimizers. Recall that a set $C\subset\mathsf{X}$ is locally the boundary of
a locally perimeter minimizing set if, for every $x\in\mathsf{X}$ there exists
a (small) metric ball $B_{r}(x)$ and a set of finite perimeter
$E\subset\mathsf{X}$ minimizing the perimeter in $B_{r}(x)$ such that $C\cap
B_{r}(x)=\partial E\cap B_{r}(x)$.

We can now state the main result of this note.

######  Theorem 1.

Let $(\mathsf{X},\mathsf{d},\mathfrak{m})$ be a parabolic $\mathsf{RCD}(0,N)$
space and let $C\subset\mathsf{X}\times\mathbb{R}$. Then the following two
assertions are equivalent:

  1. 1.

$C$ is locally the boundary of a locally perimeter minimizing set, and $C$ is
contained in a half space, i.e. there exists $t_{0}\in\mathbb{R}$ such that
$C\subset\mathsf{X}\times[t_{0},\infty)$;

  2. 2.

$C$ is a locally finite union of horizontal slices, i.e. there exists a
discrete set $D\subset[t_{0},\infty)$ such that

$C=\bigcup_{d\in D}\mathsf{X}\times\\{d\\}.$

The previous theorem, in particular, is part of a wider class of recent
results that aim at generalizing to the non-smooth setting properties of
perimeter minimizing sets ([11], [29], [16], [58], etc.). It is interesting to
note that, while in the smooth category the Half Space Property holds for
parabolic manifolds even without the lower bound on the Ricci curvature, the
same is not true for metric measure spaces (see Example 4.29). In particular,
in the non-smooth setting, once we remove the $\mathsf{RCD}$ assumption, it is
not clear which conditions should be imposed (in addition to parabolicity) to
guarantee the validity of the Half Space Property.

Specializing Theorem 1 to the smooth category, we obtain that the Half Space
Property holds for certain free boundary minimal surfaces on weighted
manifolds with boundary. This is the content of Theorem 2. Given a manifold
$(\mathsf{M},g)$ we denote by $\mathfrak{m}_{g}$ its volume measure and by
$\mathsf{d}_{g}$ its distance. If $V:\mathsf{M}\to\mathbb{R}$ is a smooth
function, we say that the metric measure space
$(\mathsf{M}^{n},\mathsf{d}_{g},e^{-V}\mathfrak{m}_{g})$ is a weighted
manifold. A hypersurface $S$ in the weighted manifold $\mathsf{M}$ is minimal
if it is a critical point of the weighted area functional. We say that the
boundary of $\mathsf{M}$ is convex if its second fundamental form w.r.t. the
inward pointing unit normal is positive.

######  Theorem 2.

Let $(\mathsf{M}^{n},\mathsf{d}_{g},e^{-V}\mathfrak{m}_{g})$ be a parabolic
weighted manifold with convex boundary such that there exists $N>n$ satisfying

$\mathsf{Ric}_{\mathsf{M}}+\mathsf{Hess}_{V}-\frac{\nabla V\otimes\nabla
V}{N-n}\geq 0\quad\text{ on }\mathsf{M}\setminus\partial\mathsf{M}.$

If $S\subset\mathsf{M}\times(0,+\infty)$ is a properly embedded connected
minimal hypersurface intersecting $\partial\mathsf{M}\times\mathbb{R}$
orthogonally, then $S$ is a horizontal slice.

The previous result is new already in the boundaryless weighted setting and in
the framework of unweighted manifolds with boundary.

A second consequence of Theorem 1 is that the oscillation of area minimizing
hypersurfaces in an appropriate class of pointed manifolds grows with a
uniform rate as one moves away from the base point in each manifold. This is
stated precisely in Theorem 3 below.

We say that a function $P:[0,+\infty)\to(0,+\infty)$ is a modulus of
parabolicity if $\int_{1}^{+\infty}tP(t)^{-1}dt=+\infty$ and we say that a
pointed metric measure space $(\mathsf{X},\mathsf{d},\mathfrak{m},\mathsf{x})$
has modulus of parabolicity $P$ if
$\mathfrak{m}(B_{r}(\mathsf{x}))\leq\mathfrak{m}(B_{1}(\mathsf{x}))P(r)$ for
every $r\geq 0$. We say that a hypersurface $S\subset\mathsf{X}$ is an area
minimizing boundary in an open set $A\subset\mathsf{X}$ if the exists a set
$E\subset A$ minimizing the perimeter in $A$ such that $\partial E\cap A=S$.
If $S\subset\mathsf{X}\times\mathbb{R}$ and $(\mathsf{x},0)\in S$ we define
the oscillation of $S$ as

$\mathsf{Osc}_{\mathsf{x},r}(S):=\sup\\{|t|:(y,t)\in S\cap
B_{r}(\mathsf{x})\times(-r,r)\\}.$

######  Theorem 3.

Let $P$ be a modulus of parabolicity and $n\in\mathbb{N}$. For every $t,r,T>0$
there exists $R>0$ such that for every pointed manifold
$(\mathsf{M}^{n},g,\mathsf{x})$ with modulus of parabolicity $P$ and non-
negative Ricci curvature, and for every area minimizing boundary $S$ in
$B_{R}(\mathsf{x})\times(-R,R)$ containing $(\mathsf{x},0)$, if
$\mathsf{Osc}_{\mathsf{x},r}(S)\geq t$ then
$\mathsf{Osc}_{\mathsf{x},R}(S)\geq T$.

######  Remark.

In Section 5 we actually prove a more general result dealing with weighted
manifolds with boundary and with a stronger notion of oscillation (see
Corollary 5.9), but we preferred to state Theorem 3 in this form for the sake
of simplicity.

Now we comment on the proof of Theorem 1.
The proof of the implication $2\Rightarrow 1$ is a standard calibration
argument, we will thus only discuss the implication $1\Rightarrow 2$.

The proof that we give is totally different from the ones covering the
analogous results in the smooth setting. In the works [62] and [23] the
general principle is that if $S\subset\mathsf{M}\times\mathbb{R}$ is minimal
and contained in a half space then, given a compact set $D\neq\emptyset$, it
will be possible to construct a solution to the minimal surface equation
$u:\mathsf{M}\setminus D\to\mathbb{R}$ which is bounded from below and is not
constant if $S$ is not a slice. The next step is then to prove a suitable
generalization of De Giorgi’s a priori gradient estimate for solutions of the
minimal surface equation and to use the lower bound on $u$ to obtain a uniform
bound on its gradient. Finally one combines the parabolicity of $\mathsf{M}$
with the bound on the gradient of $u$ to deduce that also the graph of $u$ is
parabolic, and since $u$ is harmonic on its graph, by general properties of
parabolic manifolds, it has to be constant and $S$ has to be a slice.

There are several points of this strategy that it is not clear how to adapt to
the non-smooth framework. A key issue is that the a priori gradient estimate
on $u$ under a lower Ricci curvature bound is a consequence of the second
variation formula for minimal surfaces, and such formula relies crucially on
the smoothness of the ambient space. Moreover, even if such estimate were
available, the graph of a regular function on an $\mathsf{RCD}$ space is not
known, in general, to be an $\mathsf{RCD}$ space and so, losing even this mild
regularity, it would be difficult to adapt the previous technique, as it is
based on the parabolicity of the graph.

Our proof is instead closer to the one used in [58] to generalize Frankel’s
Theorem. The key idea taken from such proof is that if $M$ is a manifold with
non-negative Ricci curvature and $S$ is a minimal hypersurface contained in
$\mathsf{M}\times(0,+\infty)$, then the function
$\bar{\mathsf{d}}:=\mathsf{d}_{S}-\mathsf{d}_{\mathsf{M}\times\\{0\\}}$,
defined on $\mathsf{M}\times(-\infty,0)$, has negative Laplacian, is positive
and if it is constant then $S$ is a slice. Hence the goal is to use the
maximum principle for super-harmonic functions to show that $\bar{\mathsf{d}}$
is constant. This is immediate if $\mathsf{M}$ is compact, as in this case
$\bar{\mathsf{d}}$ achieves its minimum on all the points of the geodesic
realizing the distance between $\mathsf{M}\times\\{-1\\}$ and $S$.

If $\mathsf{M}$ is not compact, the argument is by contradiction. Exploiting
the geometric properties of distance functions and the super-harmonicity of
$\bar{\mathsf{d}}$, one shows that if $\bar{\mathsf{d}}$ is not constant, then
there exists an open bounded set $A\subset\mathsf{M}\times(-\infty,0)$ and
$\tau>0$ such that

$\Delta e^{-\bar{\mathsf{d}}}\geq 0\text{ on
}\mathsf{M}\times(-\infty,0),\quad\Delta e^{-\bar{\mathsf{d}}}\geq\tau\text{
on }A.$

If the product $\mathsf{M}\times\mathbb{R}$ is parabolic, then there exists a
continuous function $\phi:\mathsf{M}\times\mathbb{R}\to\mathbb{R}$ such that

$\Delta\phi\geq 0\text{ on
}\mathsf{M}\times(-\infty,0),\quad\Delta\phi\geq-\tau\text{ on }A$

which tends to $-\infty$ at infinity (as it happens on $\mathbb{R}$ and
$\mathbb{R}^{2}$ because of their parabolicity). As a consequence, the
function $e^{-\bar{\mathsf{d}}}+\phi$ is sub-harmonic and has a maximum in
$\mathsf{M}\times(-\infty,0]$. Moreover, since $e^{-\bar{\mathsf{d}}}$
increases as we move down a vertical line, by constructing $\phi$ in an
appropriate way we can assume that the maximum of $e^{-\bar{\mathsf{d}}}+\phi$
actually lies in $\mathsf{M}\times(-\infty,0)$. The maximum principle then
implies that $e^{-\bar{\mathsf{d}}}+\phi$ is constant, contradicting the
condition at infinity on $\phi$.

If $\mathsf{M}\times\mathbb{R}$ is not parabolic the previous argument has to
be modified, and working with metric measure spaces instead of Riemannian
manifolds becomes crucial. Consider on $\mathsf{M}\times\mathbb{R}$ the
distance $\mathsf{d}_{\times}$ induced by the product Riemannian metric and
the measure $\mathfrak{m}_{\times}$ which is the product volume measure. Using
the parabolicity assumption on $\mathsf{M}$ and the particular form of the
function $e^{-\bar{\mathsf{d}}}$, it is possible to replace the measure
$\mathfrak{m}_{\times}$ with a measure $\tilde{\mathfrak{m}}_{\times}$ such
that the product space
$(\mathsf{M}\times\mathbb{R},\mathsf{d}_{\times},\tilde{\mathfrak{m}}_{\times})$
is an $\mathsf{RCD}(K,N)$ space (possibly for $K<0$), it is parabolic and,
calling $\tilde{\Delta}$ the Laplacian in this modified space, we have that
there exists $\tau^{\prime}>0$ such that

$\tilde{\Delta}e^{-\bar{\mathsf{d}}}\geq 0\text{ on
}\mathsf{M}\times(-\infty,0),\quad\tilde{\Delta}e^{-\bar{\mathsf{d}}}\geq\tau^{\prime}\text{
on }A.$

To conclude, we can now repeat the previous argument involving the
construction of an auxiliary function $\phi$ and the maximum principle that,
in this case, is used on the modified space
$(\mathsf{M}\times\mathbb{R},\mathsf{d}_{\times},\tilde{\mathfrak{m}}_{\times})$.

This strategy works for $\mathsf{RCD}(0,N)$ spaces, but not on
$\mathsf{RCD}(K,N)$ spaces when $K<0$, as in this case it is not true in
general that the function $\bar{\mathsf{d}}$ is super-harmonic. In particular
the problem remains open for general $\mathsf{RCD}(K,N)$ spaces.

Finally, to implement the previously outlined strategy, we need to generalize
to $\mathsf{RCD}(K,N)$ spaces some classical properties of parabolic
manifolds. This is obtained by extending the work done in [18], [58] and [17]
on Green’s functions in the non smooth setting. Theorems 4 and 5 are the main
results in this regard. Given an $\mathsf{RCD}(K,N)$ space
$(\mathsf{X},\mathsf{d},\mathfrak{m})$ we denote by $p_{t}$ the corresponding
heat kernel and we refer to Section 3 for the precise definitions of Green’s
functions and regular sets.

######  Theorem 4.

Let $(\mathsf{X},\mathsf{d},\mathfrak{m},\mathsf{x})$ be a pointed
$\mathsf{RCD}(K,N)$ space with infinite diameter. The following are
equivalent:

  1. 1.

There is no positive Green’s function on $\mathsf{X}$ with pole $\mathsf{x}$.

  2. 2.

For every $(x,y)\in X\times X$ we have
$\int_{1}^{+\infty}p_{t}(x,y)\,dt=+\infty$.

  3. 3.

There exists $(x,y)\in X\times X$ such that
$\int_{1}^{+\infty}p_{t}(x,y)\,dt=+\infty$.

  4. 4.

Let $\\{B_{i}\\}_{i\in\mathbb{N}}$ be an exhaustion of regular sets containing
$\mathsf{x}$. Let $G_{\mathsf{x}}^{i}$ be the Green’s function on $B_{i}$ with
pole $\mathsf{x}$. Then for every $y\neq\mathsf{x}$ we have
$G_{\mathsf{x}}^{i}(y)\to+\infty$ as $i\to+\infty$.

######  Theorem 5.

Let $(\mathsf{X},\mathsf{d},\mathfrak{m})$ be an $\mathsf{RCD}(K,N)$ space. If
there exists $x\in X$ such that

$\int_{1}^{+\infty}\frac{t}{\mathfrak{m}(B_{t}(x))}\,dt=+\infty,$

then $\mathsf{X}$ is parabolic. This condition is also necessary if $K=0$.

The note is organized as follows. The first section contains preliminaries.
The second section concerns the theory of Green’s functions on $\mathsf{RCD}$
spaces and contains the proofs of Theorems 4 and 5. In Section 4 we rigorously
prove, following the previously outlined strategy, the Half Space Property for
$\mathsf{RCD}(0,N)$ spaces, while in the final section we deal with
applications in the smooth setting, proving Theorems 2 and 3.

Acknowledgements. A.M. is supported by the European Research Council (ERC),
under the European Union Horizon 2020 research and innovation programme, via
the ERC Starting Grant “CURVATURE”, grant agreement No. 802689.

##  2 Preliminaries

Throughout the note we will work on metric measure spaces
$(\mathsf{X},\mathsf{d},\mathfrak{m})$, where $(\mathsf{X},\mathsf{d})$ is a
separable complete metric space where balls are precompact and $\mathfrak{m}$
is a non-negative Borel measure on $\mathsf{X}$ which is finite on bounded
sets and whose support is the whole $\mathsf{X}$. Given a smooth manifold
$(\mathsf{M},g)$ we denote by $\mathsf{d}_{g}$ and $\mathfrak{m}_{g}$
respectively the distance induced by $g$ and the volume measure. Given an open
set $\Omega\subset\mathsf{X}$ we denote by $\mathsf{Lip}(\Omega)$,
$\mathsf{Lip}_{loc}(\Omega)$ and $\mathsf{Lip}_{c}(\Omega)$ respectively
Lipschitz functions, locally Lipschitz and Lipschitz functions with compact
support in $\Omega$. If $f\in\mathsf{Lip}_{loc}(\Omega)$ and $x\in\Omega$ we
define

$\mathsf{lip}(f)(x):=\limsup_{y\to x}\frac{|f(x)-f(y)|}{\mathsf{d}(x,y)}.$

Given a closed interval $I\subset\mathbb{R}$, we say that a curve
$\gamma:I\to\mathsf{X}$ is a geodesic if its length coincides with the
distance between its endpoints. Unless otherwise specified we assume that
geodesics have constant unit speed.

We will consider pointed measured Gromov Hausdorff convergence of metric
measure spaces, and we refer to [34] for the relevant background. We recall
that in the case of a sequence of uniformly locally doubling metric measure
spaces $(\mathsf{X}_{i},\mathsf{d}_{i},\mathfrak{m}_{i},\mathsf{x}_{i})$ (as
in the case of $\mathsf{RCD}(K,N)$ spaces), pointed measured Gromov-Hausdorff
convergence to $(\mathsf{X},\mathsf{d},\mathfrak{m},\mathsf{x})$ can be
equivalently characterized asking for the existence of a proper metric space
$(\mathsf{Z},\mathsf{d}_{z})$ such that all the metric spaces
$(\mathsf{X}_{i},\mathsf{d}_{i})$ are isometrically embedded into
$(\mathsf{Z},\mathsf{d}_{z})$, $\mathsf{x}_{i}\to\mathsf{x}$ and
$\mathfrak{m}_{i}\to\mathfrak{m}$ weakly in duality with continuous boundedly
supported functions in $\mathsf{Z}$. In this case we say that the convergence
is realized in the space $\mathsf{Z}$ (see [34]). Given a metric measure space
$(\mathsf{X},\mathsf{d},\mathfrak{m})$ and $x\in\mathsf{X}$, we denote by
$\mathsf{Tan}_{x}(\mathsf{X})$ the (possibly empty) collection of (isometry
classes of) metric measure spaces that are pointed measured Gromov Hausdorff
limits as $r\downarrow 0$ of the family
$(\mathsf{X},r^{-1}\mathsf{d},\mathfrak{m}(B_{r}(x))^{-1}\mathfrak{m},x)$.

###  2.1 Sobolev spaces and the $\mathsf{RCD}$ condition

We now recall the definition of Sobolev spaces in the setting of metric
measure spaces, the main references being [21], [6], [7] and [33].

######  Definition 2.1.

Let $(\mathsf{X},\mathsf{d},\mathfrak{m})$ be a metric measure space,
$\Omega\subset\mathsf{X}$ an open set and let $p>1$. A function
$f\in\mathsf{L}^{p}(\Omega)$ is said to be in the Sobolev space
$\mathsf{W}^{1,p}(\Omega)$ if there exists a sequence of locally Lipschitz
functions $\\{f_{i}\\}_{i\in\mathbb{N}}\subset\mathsf{Lip}_{loc}(\Omega)$
converging to $f$ in $\mathsf{L}^{p}(\Omega)$ such that

$\limsup_{i\to+\infty}\int_{\Omega}\mathsf{lip}(f_{i})^{p}\,d\mathfrak{m}<+\infty.$

A function $f\in\mathsf{L}^{p}_{loc}(\Omega)$ is said to be in the Sobolev
space $\mathsf{W}^{1,p}_{loc}(\Omega)$ if for every
$\eta\in\mathsf{Lip}_{c}(\Omega)$ we have $f\eta\in\mathsf{W}^{1,p}(\Omega)$.

For any $f\in\mathsf{W}^{1,p}(\Omega)$ one can define an object $|\nabla f|$
(a priori depending on $p$, but independent of the exponent in the spaces that
we will work on) such that for every open set $A\subset\Omega$ we have

$\int_{A}|\nabla
f|^{p}\,d\mathfrak{m}=\inf\Big{\\{}\liminf_{n\to+\infty}\int_{A}\mathsf{lip}(f_{n})^{p}\,d\mathfrak{m}\Big{|}(f_{n})_{n}\subset\mathsf{L}^{p}(A)\cap\mathsf{Lip}_{loc}(A),\|f_{n}-f\|_{\mathsf{L}^{p}(A)}\to
0\Big{\\}}.$

The quantity in the previous expression will be called $p$-Cheeger energy and
denoted by $\mathsf{Ch}_{p}(f)$ while $|\nabla f|$ will be called relaxed
gradient. We define
$\|f\|_{\mathsf{W}^{1,p}(\Omega)}:=\|f\|_{\mathsf{L}^{p}(\Omega)}+\mathsf{Ch}_{p}(f)$.
One can check that with this norm the space $\mathsf{W}^{1,p}(\Omega)$ is
Banach. Later we will often write $\mathsf{Ch}$ in place of $\mathsf{Ch}_{2}$
for simplicity of notation. A deep result of Cheeger [21] implies that if
$f\in\mathsf{Lip}_{loc}(X)$ then $|\nabla
f|=\mathsf{lip}(f),\,\mathfrak{m}$-a.e., provided
$(\mathsf{X},\mathsf{d},\mathfrak{m})$ is a PI-space, so in particular this
holds for $\mathsf{RCD}(K,N)$-spaces, $N\in[1,\infty)$. We now introduce
functions of bounded variation following [56] (see also [3]).

######  Definition 2.2.

Let $(\mathsf{X},\mathsf{d},\mathfrak{m})$ be a metric measure space and let
$\Omega\subset\mathsf{X}$ be an open set. A function
$f\in\mathsf{L}^{1}(\Omega)$ is said to be of bounded variation if there
exists a sequence of locally Lipschitz functions
$\\{f_{i}\\}_{i\in\mathbb{N}}\subset\mathsf{Lip}_{loc}(\Omega)$ converging to
$f$ in $\mathsf{L}^{1}(\Omega)$ such that

$\limsup_{i\to+\infty}\int_{\Omega}\mathsf{lip}(f_{i})\,d\mathfrak{m}<+\infty.$

The space of such functions is denoted by $\mathsf{BV}(\Omega)$. A function
$f\in\mathsf{L}^{1}_{loc}(\Omega)$ is said to be in
$\mathsf{BV}_{loc}(\Omega)$ if for every $\eta\in\mathsf{Lip}_{c}(\Omega)$ we
have $f\eta\in\mathsf{BV}(\Omega)$.

For any $f\in\mathsf{BV}(\Omega)$ and any open set $A\subset\Omega$ we define

$|Df|(A)=\inf\Big{\\{}\liminf_{n\to+\infty}\int_{A}\mathsf{lip}(f_{n})\,d\mathfrak{m}\Big{|}(f_{n})_{n}\subset\mathsf{L}^{1}(A)\cap\mathsf{Lip}_{loc}(A),\|f_{n}-f\|_{\mathsf{L}^{1}(A)}\to
0\Big{\\}}.$

One can check that the quantity in the previous expression is the restriction
to the open subsets of $\Omega$ of a finite measure. We define
$\|f\|_{\mathsf{BV}(\Omega)}:=\|f\|_{\mathsf{L}^{1}(\Omega)}+|Df|(\Omega)$.
One can check that with this norm the space $\mathsf{BV}(\mathsf{X})$ is
Banach. A function $f$ belongs to $\mathsf{W}^{1,1}(\Omega)$ if
$f\in\mathsf{BV}(\Omega)$ and $|Df|\ll\mathfrak{m}$. In this case we denote by
$|\nabla f|$ the density of $|Df|$ with respect to $\mathfrak{m}$.

######  Definition 2.3.

Let $(\mathsf{X},\mathsf{d},\mathfrak{m})$ be a metric measure space and let
$\Omega\subset\mathsf{X}$ be an open set. For every $p\geq 1$ we denote by
$\mathsf{W}^{1,p}_{0}(\Omega)$ the closure in $\mathsf{W}^{1,p}(\Omega)$ of
$\mathsf{Lip}_{c}(\Omega)$.

The previous definition is particularly suitable for proper spaces, while
other definitions are to be preferred if one works without this assumption.
Since we will work in the setting of $\mathsf{RCD}(K,N)$ spaces with
$N\in[1,\infty)$ (and these are proper) we do not consider more general
definitions.

######  Definition 2.4.

We say that a metric measure space $(\mathsf{X},\mathsf{d},\mathfrak{m})$ is
infinitesimally Hilbertian if the space $\mathsf{W}^{1,2}(\mathsf{X})$ is a
Hilbert space.

If $(\mathsf{X},\mathsf{d},\mathfrak{m})$ is infinitesimally Hilbertian and
$\Omega\subset\mathsf{X}$ an open set, for every
$f,g\in\mathsf{W}^{1,1}_{loc}(\Omega)$ we define the measurable function
$\nabla f\cdot\nabla g:\Omega\to\mathbb{R}$ by

$\nabla f\cdot\nabla g:=\frac{|\nabla(f+g)|^{2}-|\nabla(f-g)|^{2}}{4}.$

As a consequence of the infinitesimal Hilbertianity assumption, the previously
defined product of gradients is bilinear in both entries. We then recall the
definition of Laplacian in the metric setting.

######  Definition 2.5.

Let $(\mathsf{X},\mathsf{d},\mathfrak{m})$ be infinitesimally Hilbertian and
let $\Omega\subset\mathsf{X}$ be an open set. Let
$f\in\mathsf{W}^{1,2}(\Omega)$. We say that $f\in D(\Delta,\Omega)$ if there
exists a function $h\in\mathsf{L}^{2}(\Omega)$ such that

$\int_{\Omega}gh\,d\mathfrak{m}=-\int_{\Omega}\nabla g\cdot\nabla
f\,d\mathfrak{m}\quad\text{for any }g\in\mathsf{W}^{1,2}_{0}(\Omega).$

In this case we say that $\Delta f=h$ in $\Omega$.

We also have the following more general definition.

######  Definition 2.6.

Let $(\mathsf{X},\mathsf{d},\mathfrak{m})$ be infinitesimally Hilbertian and
let $\Omega\subset\mathsf{X}$ be an open set. Let
$f\in\mathsf{W}^{1,1}_{loc}(\Omega)$ and let $\mu$ be a Radon measure on
$\Omega$. We say that $\Delta f=\mu$ in $\Omega$ in distributional sense if

$\int_{\Omega}h\,d\mu=-\int_{\Omega}\nabla g\cdot\nabla
f\,d\mathfrak{m}\quad\text{for any }g\in\mathsf{Lip}_{c}(\Omega).$

We now recall some properties of $\mathsf{RCD}(K,N)$ spaces, i.e.
infinitesimally Hilbertian metric measure spaces with Ricci curvature bounded
from below by $K\in\mathbb{R}$ and dimension bounded from above by
$N\in[1,+\infty)$ in synthetic sense.

The Riemannian Curvature Dimension condition $\mathsf{RCD}(K,\infty)$ was
introduced in [7] (see also [33, 5]) coupling the Curvature Dimension
condition $\mathsf{CD}(K,\infty)$, previously pioneered in [63, 64] and
independently in [54], with the infinitesimal Hilbertianity assumption. The
finite dimensional counterpart $\mathsf{RCD}(K,N)$ is then obtained coupling
the finite dimensional Curvature Dimension condition $\mathsf{CD}(K,N)$ with
the infinitesimal Hilbertianity assumption and was formalized in [33]. For a
thorough introduction to the topic we refer to the survey [1] and the
references therein. Let us also mention that in the literature one can find
also the (a priori weaker) $\mathsf{RCD}^{*}(K,N)$. It was proved in [27, 9],
that $\mathsf{RCD}^{*}(K,N)$ is equivalent to the dimensional Bochner
inequality. Moreover, [19] (see also [53]) proved that $\mathsf{RCD}^{*}(K,N)$
and $\mathsf{RCD}(K,N)$ coincide.

We now recall the properties that we will use later on in the note. The
$\mathsf{RCD}(K,N)$ condition implies that the measure is locally doubling
(see [63]) and the validity of a Poincaré inequality (see [61]). In particular
if $f$ is a locally Lipschitz function on a $\mathsf{RCD}(K,N)$ space, its
relaxed gradient coincides with its local Lipschitz constant $\mathsf{lip}(f)$
(see [44, Theorem $12.5.1$]).

We now recall a Bishop-Gromov type inequality for $\mathsf{RCD}(K,N)$ spaces
that can be found in [64]. We define
$V_{K,N}(r):=N\omega_{N}\int_{0}^{r}(s_{K,N}(t))^{N-1}\,dt$, where
$\omega_{N}$ is the volume of the $N$-dimensional Euclidean ball and

$s_{K,N}(t):=\begin{cases}\sqrt{\frac{N-1}{K}}\sin\Big{(}t\sqrt{\frac{K}{N-1}}\Big{)}&K>0\\\
r&K=0\\\
\sqrt{\frac{N-1}{-K}}\sinh\Big{(}t\sqrt{\frac{-K}{N-1}}\Big{)}&K<0.\end{cases}$

######  Proposition 2.7.

Let $(\mathsf{X},\mathsf{d},\mathfrak{m})$ be an $\mathsf{RCD}(K,N)$ space.
Then for every $x\in\mathsf{X}$ and $R>r>0$ we have

$\frac{\mathfrak{m}(B_{R}(x))}{\mathfrak{m}(B_{r}(x))}\leq\frac{V_{K,N}(R)}{V_{K,N}(r)}.$

######  Remark 2.8.

Let $(\mathsf{X},\mathsf{d},\mathfrak{m})$ be an $\mathsf{RCD}(K,N)$ space.
Proposition 2.7, together with the explicit expression of $V_{K,N}(r)$,
implies that for every $x\in\mathsf{X}$ and $R>0$ there exists a constant
$C>0$ such that $\mathfrak{m}(B_{r}(x))\geq Cr^{N}$ for every $r\in(0,R)$.

The next theorem follows from [57, 17]. We denote by $\mathsf{d}_{e}$ the
Euclidean distance.

######  Theorem 2.9.

Let $(\mathsf{X},\mathsf{d},\mathfrak{m})$ be an $\mathsf{RCD}(K,N)$ space.
There exists $k\in\mathbb{N}\cap[1,N]$, called essential dimension of
$\mathsf{X}$, such that for $\mathfrak{m}$-a.e. $x\in\mathsf{X}$ we have
$\mathsf{Tan}_{x}(\mathsf{X})=\\{(\mathbb{R}^{k},\mathsf{d}_{e})\\}$. Any such
point will be called a regular point for $\mathsf{X}$.

The next result follows from [48, 36] (see also [8, Theorem $4.1$]).

######  Proposition 2.10.

Let $(\mathsf{X},\mathsf{d},\mathfrak{m})$ be an $\mathsf{RCD}(K,N)$ space of
essential dimension $k>1$. Then for $\mathfrak{m}$-a.e. $x\in\mathsf{X}$ we
have

$\lim_{r\to 0^{+}}\frac{\mathfrak{m}(B_{r}(x))}{r^{k}}\in(0,+\infty).$

The next proposition is taken from [50, Theorem $1.1$].

######  Proposition 2.11.

Let $(\mathsf{X},\mathsf{d},\mathfrak{m})$ be an $\mathsf{RCD}(K,N)$ space of
essential dimension $1$. Then $\mathsf{X}$ is isometric to a closed connected
subset of $\mathbb{R}$ or to $S^{1}(r):=\\{x\in\mathbb{R}^{2}:|x|=r\\}$.

We now recall some properties of the heat flow in the $\mathsf{RCD}$ setting,
referring to [5, 7] for the proofs of these results. Given an
$\mathsf{RCD}(K,N)$ space $(\mathsf{X},\mathsf{d},\mathfrak{m})$, the heat
flow $P_{t}:\mathsf{L}^{2}(\mathsf{X})\to\mathsf{L}^{2}(\mathsf{X})$ is the
$\mathsf{L}^{2}(\mathsf{X})$-gradient flow of the Cheeger energy
$\mathsf{Ch}$.

It turns out that one can obtain a stochastically complete heat kernel
$p_{t}:\mathsf{X}\times\mathsf{X}\to[0,+\infty)$, so that the definition of
$P_{t}(f)$ can be then extended to $\mathsf{L}^{\infty}$ functions by setting

$P_{t}(f)(x):=\int_{\mathsf{X}}f(y)p_{t}(x,y)\,d\mathfrak{m}(y).$

The heat flow has good approximation properties, in particular if
$f\in\mathsf{W}^{1,2}(\mathsf{X})$, then $P_{t}(f)\to f$ in
$\mathsf{W}^{1,2}(\mathsf{X})$; while if
$f\in\mathsf{L}^{\infty}(\mathsf{X})$, then
$P_{t}f\in\mathsf{Lip}(\mathsf{X})$ for every $t>0$.

We conclude the section with two propositions on the heat kernel taken
respectively from [47, Theorem $1.1$] and [47, Lemma $3.3$] (see also [31]).

######  Proposition 2.12.

Let $(\mathsf{X},\mathsf{d},\mathfrak{m})$ be an $\mathsf{RCD}(K,N)$ space.
There exist constants $C>1$ and $c\geq 0$ such that for every
$x,y\in\mathsf{X}$ and every $t>0$

$\frac{1}{C\mathfrak{m}(B_{\sqrt{t}}(x))}\exp{\Big{\\{}\frac{-\mathsf{d}^{2}(x,y)}{3t}-ct\Big{\\}}}\leq
p_{t}(x,y)\leq\frac{C}{\mathfrak{m}(B_{\sqrt{t}}(x))}\exp{\Big{\\{}\frac{-\mathsf{d}^{2}(x,y)}{5t}+ct\Big{\\}}}.$

Moreover for every $x\in\mathsf{X}$ we have that $p_{t}(x,\cdot)$ is locally
Lipschitz in $\mathsf{X}\setminus\\{x\\}$ and for $\mathfrak{m}$-a.e.
$y\in\mathsf{X}$ and every $t>0$

$\mathsf{lip}(p_{t}(x,\cdot))(y)=|\nabla
p_{t}(x,\cdot)|(y)\leq\frac{C}{\sqrt{t}\mathfrak{m}(B_{\sqrt{t}}(x))}\exp{\Big{\\{}\frac{-\mathsf{d}^{2}(x,y)}{5t}+ct\Big{\\}}}.$

If $K=0$ we can take $c=0$ in the previous estimates.

######  Proposition 2.13.

Let $(\mathsf{X},\mathsf{d},\mathfrak{m})$ be an $\mathsf{RCD}(K,N)$ space
with $K<0$ and $N\in[1,+\infty)$. For any $0<s<s+1\leq t<+\infty$ and
$x,y,z\in\mathsf{X}$ it holds

$p_{t}(x,y)\geq
p_{s}(x,z)\exp{\Big{\\{}-\frac{\mathsf{d}(x,z)^{2}}{2e^{2K/3}}\Big{\\}}}\Big{(}\frac{1-e^{K/3}}{1-e^{2K/3}}\Big{)}^{N/2}\Big{(}\frac{1-e^{2Ks/3}}{1-e^{2K(t-1/2)/3}}\Big{)}^{N/2}.$

###  2.2 Poisson problem and regular sets

For the results of this subsection, unless otherwise specified,
$(\mathsf{X},\mathsf{d},\mathfrak{m})$ is an $\mathsf{RCD}(K,N)$ space and
$\Omega\subset\mathsf{X}$ is an open set. The next proposition is the maximum
principle on $\mathsf{RCD}(K,N)$ spaces and can be found in [37].

######  Proposition 2.14.

Let $\Omega\subset\mathsf{X}$ be open and connected and let
$f\in\mathsf{W}^{1,2}(\Omega)\cap C(\bar{\Omega})$ be a function whose
distributional Laplacian satisfies $\Delta f\geq 0$. If there exists
$x\in\Omega$ such that $f(x)=\max_{y\in\bar{\Omega}}f(y),$ then $f$ is
constant.

The next proposition is taken from [58, Theorem $2.58$].

######  Proposition 2.15.

Let $B\subset\subset\mathsf{X}$ be open with $\mathfrak{m}(\mathsf{X}\setminus
B)>0$, $f\in C(\bar{B})$, and $g\in\mathsf{W}^{1,2}(\mathsf{X})$. Then there
exists $h\in\mathsf{W}^{1,2}(B)$ satisfying

$\begin{cases}\Delta h=f\quad\in B\\\
h+g\in\mathsf{W}^{1,2}_{0}(B).\end{cases}$

The next two propositions are taken from [46].

######  Proposition 2.16.

Let $f\in D(\Delta,\Omega)$ be such that $\Delta f$ is continuous, then $f$
has a locally Lipschitz representative.

######  Proposition 2.17.

Let $r>0$ and $x\in\mathsf{X}$, then there exists a constant $c>0$ depending
on $r$ such that if $f\in D(\Delta,A)$ for every $A\subset\subset\mathsf{X}$
and $\Delta f$ is continuous, then

$\|\nabla f\|_{\mathsf{L}^{\infty}(B_{r}(x))}\leq
c(\|f\|_{{\mathsf{L}^{\infty}(B_{8r}(x))}}+\|\Delta
f\|_{{\mathsf{L}^{\infty}(B_{8r}(x))}}).$

######  Definition 2.18.

An open precompact subset $B\subset\bar{B}\subsetneq\mathsf{X}$ is said to be
regular if for every $f\in\mathsf{W}^{1,2}(B)$ admitting a representative
which is continuous on $\partial B$, there exists a function $u\in
D(\Delta,B)\cap C(\bar{B})$ such that $\Delta u=0$ on $B$,
$u-f\in\mathsf{W}^{1,2}_{0}(B)$ and $u=f$ on $\partial B$.

The next proposition concerns existence of regular sets and is taken from [13,
Theorem $14.1$]. In the aforementioned book the setting is the one of doubling
spaces supporting a Poincaré inequality and Newtonian Sobolev spaces, but the
statement that we are interested in, concerning precompact sets, holds also in
the $\mathsf{RCD}(K,N)$ setting (taking into account that Newtonian Sobolev
spaces coincide with the ones we are using by [6, Theorem $6.2$]).

######  Proposition 2.19.

Let $\mathsf{X}$ have infinite diameter. There exists a sequence
$\\{\Omega_{i}\\}_{i\in\mathbb{N}}$ of regular sets such that
$\Omega_{i}\subset\subset\Omega_{i+1}$ for every $i\in\mathbb{N}$ and
$\mathsf{X}=\cup_{i}\Omega_{i}$. Such a sequence of sets is called an
exhaustion of $\mathsf{X}$ with regular sets.

The next proposition follows from [12, Theorem $4.1$], noting that by the
maximum principle functions with negative Laplacian are super-harmonic in the
sense of [12]. The two subsequent corollaries are immediate consequences of
Proposition 2.20.

######  Proposition 2.20.

An open precompact set $B\subset\bar{B}\subsetneq\mathsf{X}$ is regular if for
every $x\in\partial B$ there exists a barrier i.e. a function $u_{x}\in
D(\Delta,B)\cap C(\bar{B})$, strictly positive on $B$, such that $\Delta
u_{x}\leq 0$ and

$\lim_{y\to x}u_{x}(y)=0.$

######  Corollary 2.21.

An open precompact set $B\subset\bar{B}\subsetneq\mathsf{X}$ is regular if and
only if for every $x\in\partial B$ there exists a function $u_{x}\in
D(\Delta,B)\cap C(\hbox to0.0pt{\hskip
1.5434pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$B$}})$,
strictly positive on $B$, such that $\Delta u_{x}=0$ and

$\lim_{y\to x}u_{x}(y)=0.$

######  Corollary 2.22.

Let $(\mathsf{X},\mathsf{d},\mathfrak{m})$ and
$(\mathsf{X}^{\prime},\mathsf{d}^{\prime},\mathfrak{m}^{\prime})$ be
$\mathsf{RCD}(K,N)$ spaces for $K\leq 0$. Let $B\subset\subset\mathsf{X}$ and
$B^{\prime}\subset\subset\mathsf{X}^{\prime}$ be regular sets. Then $B\times
B^{\prime}$ is regular in the product space.

We now recall the definition of test functions in the $\mathsf{RCD}(K,N)$
setting, referring to [32] for the proofs of the results that we subsequently
list.

######  Definition 2.23.

We set

$\mathsf{Test}(\mathsf{X}):=\Big{\\{}\phi\in
D(\Delta,\mathsf{X})\cap\mathsf{L}^{\infty}(\mathsf{X}):|\nabla
f|\in\mathsf{L}^{\infty}(\mathsf{X}),\Delta
f\in\mathsf{L}^{\infty}(\mathsf{X})\cap\mathsf{W}^{1,2}(\mathsf{X})\Big{\\}}$

and

$\mathsf{Test}_{c}(\Omega):=\Big{\\{}\phi\in\mathsf{Test}(\mathsf{X}):\mathsf{supp}(\phi)\subset\subset\Omega\Big{\\}}.$

It has been proved that $\mathsf{Test}(\mathsf{X})$ is dense in
$\mathsf{W}^{1,2}(\mathsf{X})$, while $\mathsf{Test}_{c}(\Omega)$ is dense in
$\mathsf{W}^{1,2}_{0}(\Omega)$. Moreover if
$f\in\mathsf{W}^{1,2}(\mathsf{X})\cap\mathsf{L}^{\infty}(\mathsf{X})$ then
$P_{t}(f)\in\mathsf{Test}(\mathsf{X})$. Finally, it has been proved in [10]
that if $K\subset\Omega$ is compact then there exists
$\eta\in\mathsf{Test}_{c}(\Omega)$ such that $\eta=1$ on a neighbourhood of
$K$.

We then give a version of Weyl’s Lemma in $\mathsf{RCD}(K,N)$ spaces that
follows from [60, Theorem $1.3$].

######  Proposition 2.24.

Let $u\in\mathsf{L}^{1}_{loc}(\Omega)$ be a function such that

$\int_{\Omega}u\Delta\phi\,d\mathfrak{m}=0\quad\text{for every
}\phi\in\mathsf{Test}_{c}(\Omega),$

then $u\in\mathsf{W}^{1,2}_{loc}(\Omega)$.

###  2.3 Sets of finite perimeter

For the results of this section, unless otherwise specified, we will
implicitly assume that we are working on a fixed $\mathsf{RCD}(K,N)$ space
$(\mathsf{X},\mathsf{d},\mathfrak{m})$.

######  Definition 2.25.

Let $E\subset\mathsf{X}$. We say that $E$ has locally finite perimeter if
$1_{E}\in\mathsf{BV}_{loc}(\mathsf{X})$. For every Borel subset
$B\subset\mathsf{X}$ We denote $|D1_{E}|(B)$ by $P(E,B)$.

######  Definition 2.26.

Let $(\mathsf{X}_{i},\mathsf{d}_{i},\mathfrak{m}_{i},\mathsf{x}_{i})$ be a
sequence of $\mathsf{RCD}(K,N)$ spaces converging in pmGH sense to
$(\mathsf{Y},\mathsf{d},\mathfrak{m},\mathsf{y})$. We say that the Borel sets
$E_{i}\subset\mathsf{X}_{i}$ of finite measure converge in $\mathsf{L}^{1}$
sense to a set $E\subset\mathsf{Y}$ of finite measure if
$\mathfrak{m}_{i}(E_{i})\to\mathfrak{m}(E)$ and $1_{E_{i}}\mathfrak{m}_{i}\to
1_{E}\mathfrak{m}$ weakly in duality w.r.t. continuous compactly supported
functions in the space $(\mathsf{Z},\mathsf{d}_{z})$ realizing the pmGH
convergence.

We say that the Borel sets $E_{i}\subset\mathsf{X}_{i}$ converge in
$\mathsf{L}^{1}_{loc}$ sense to a set $E\subset\mathsf{Y}$ if $E_{i}\cap
B_{r}(x_{i})\to E\cap B_{r}(y)$ in $\mathsf{L}^{1}$ sense for every $r>0$.

The next proposition is taken from [2, Corollary $3.4$].

######  Proposition 2.27.

Let $(\mathsf{X}_{i},\mathsf{d}_{i},\mathfrak{m}_{i},\mathsf{x}_{i})$ be a
sequence of $\mathsf{RCD}(K,N)$ spaces converging in pmGH sense to
$(\mathsf{Y},\mathsf{d},\mathfrak{m},\mathsf{y})$. Let
$E_{i}\subset\mathsf{X}_{i}$ be Borel sets such that

$\sup_{i\in\mathbb{N}}P(E_{i},B_{r}(x_{i}))<+\infty\quad\text{for every }r>0.$

Then there exists a (non relabeled) subsequence and a Borel set
$F\subset\mathsf{Y}$ such that $E_{i}\to F$ in $\mathsf{L}^{1}_{loc}$.

######  Definition 2.28.

Let $E\subset\mathsf{X}$ be a set of finite perimeter. We say that a regular
point $x\in\mathsf{X}$ such that $x\in\partial E$ is in the reduced boundary
$\mathcal{F}E$ of $E$ if for every sequence
$\\{\epsilon_{i}\\}_{i\in\mathbb{N}}$ decreasing to zero the sets $E_{i}:=E$
in the rescaled spaces
$(\mathsf{X},\epsilon_{i}^{-1}\mathsf{d},\mathfrak{m}(B_{\epsilon_{i}}(x))^{-1}\mathfrak{m},x)$
converge in $\mathsf{L}^{1}_{loc}$ to a half space in $\mathbb{R}^{k}$.

The next proposition is taken from [16, Corollary $3.15$].

######  Proposition 2.29.

Let $E\subset\mathsf{X}$ be a set of finite perimeter. Then the perimeter
measure is concentrated on $\mathcal{F}E$.

The next proposition is taken from [15, Theorem $5.2$ and Proposition $6.1$].
We first introduce some notation. Let $A\subset\subset\mathsf{X}$ be a set of
finite perimeter and let $g\in\mathsf{Lip}_{loc}(\mathsf{X})$. Assume that $g$
has distributional Laplacian which is a finite Radon measure. Then there
exists measures $\mu_{1},\mu_{2}<<|D1_{A}|$ such that as $t\to 0$ we have

$1_{A}\nabla P_{t}(1_{A})\cdot\nabla g\to\mu_{1}\quad\text{and}\quad
1_{{{}^{c}}A}\nabla P_{t}(1_{A})\cdot\nabla g\to\mu_{2}$

in weak sense testing against functions in $\mathsf{Lip}_{c}(\mathsf{X})$. We
denote the density of $\mu_{1}$ and $\mu_{2}$ w.r.t. $|D1_{A}|$ respectively
by

$(\nabla g\cdot\nu_{E})_{int}\quad\text{and}\quad(\nabla
g\cdot\nu_{E})_{ext}.$

Moreover we denote by $A^{(1)}$ the set of points of $\mathsf{X}$ where $A$
has density $1$.

######  Proposition 2.30.

Let $A\subset\subset\mathsf{X}$ be a set of finite perimeter and let
$g\in\mathsf{Lip}_{loc}(\mathsf{X})$. Assume that $g$ has distributional
Laplacian which is a finite Radon measure. Then for any
$f\in\mathsf{Lip}_{c}(\mathsf{X})$ we have

$\int_{A^{(1)}}f\,d\Delta g+\int_{A}\nabla f\cdot\nabla
g\,d\mathfrak{m}=-\int_{\mathcal{F}A}f(\nabla
g\cdot\nu_{E})_{int}\,d\mathsf{Per}$

and

$\int_{A^{(1)}\cup\mathcal{F}A}f\,d\Delta g+\int_{A}\nabla f\cdot\nabla
g\,d\mathfrak{m}=-\int_{\mathcal{F}A}f(\nabla
g\cdot\nu_{E})_{ext}\,d\mathsf{Per}.$

We now turn our attention to minimal sets.

######  Definition 2.31.

Let $\Omega\subset\mathsf{X}$ be an open set. Let $E\subset\Omega$ be a set of
locally finite perimeter. We say that $E$ is perimeter minimizing in $\Omega$
if for every $x\in\Omega$, $r>0$ and $F\subset\Omega$ such that $F\Delta
E\subset\subset B_{r}(x)\cap\Omega$ we have that $P(E,B_{r}(x)\cap\Omega)\leq
P(F,B_{r}(x)\cap\Omega)$. If we say that $E$ is perimeter minimizing we
implicitly mean that $\Omega=\mathsf{X}$.

######  Definition 2.32.

Let $\Omega\subset\mathsf{X}$ be an open set. Let $E\subset\Omega$ be a set of
locally finite perimeter. We say that $E$ is locally perimeter minimizing in
$\Omega$ if for every $x\in\Omega$ there exists $r>0$ such that for every
$F\subset\Omega$ such that $F\Delta E\subset\subset B_{r}(x)\cap\Omega$ we
have $P(E,B_{r}(x)\cap\Omega)\leq P(F,B_{r}(x)\cap\Omega)$. If we say that $E$
is locally perimeter minimizing we implicitly mean that $\Omega=\mathsf{X}$.

Locally perimeter minimizing sets admit both a closed and an open
representative, and these have the same boundary which in addition is
$\mathfrak{m}$-negligible (see [49]). Whenever we consider the boundary of a
locally perimeter minimizing set, we will implicitly be referring to the
boundary of its closed (or open) representative.

######  Definition 2.33.

We say that a set $C\subset\mathsf{X}$ is locally the boundary of a locally
perimeter minimizing set if for every $x\in\mathsf{X}$ there exists an open
neighbourhood $U_{x}$ of $x$ and a set $E\subset U_{x}$ locally minimizing the
perimeter in $U_{x}$ such that $U_{x}\cap C=\partial E\cap U_{x}$.

It is immediate to check that if $C\subset\mathsf{X}$ is locally the boundary
of a locally perimeter minimizing set then it is closed. The next proposition
can be found in [58] (see also [35] for the extension to the collapsed case).

######  Proposition 2.34.

Let $(\mathsf{X},\mathsf{d},\mathfrak{m})$ be an $\mathsf{RCD}(0,N)$ space,
let $\Omega\subset\mathsf{X}$ be an open set and let $E\subset\Omega$ be the
relatively closed representative of a locally perimeter minimizing set in
$\Omega$. Let $\mathsf{d}_{E}:{{}^{c}}E\to\mathbb{R}$ be the distance function
from $E$. Then $\Delta\mathsf{d}_{E}\leq 0$ in distributional sense on every
open subset $\Omega^{\prime}\subset\subset{{}^{c}}E\cap\mathcal{K}$, where

$\mathcal{K}:=\\{x\in\mathsf{X}:\exists y\in\partial
E\cap\Omega:\mathsf{d}_{E}(x)=\mathsf{d}(x,y)\\}.$

The next proposition is taken from [58, Theorem $2.43$].

######  Proposition 2.35.

Let $(\mathsf{X}_{i},\mathsf{d}_{i},\mathfrak{m}_{i},\mathsf{x}_{i})$ be a
sequence of $\mathsf{RCD}(K,N)$ spaces converging in pmGH sense to
$(\mathsf{Y},\mathsf{d},\mathfrak{m},\mathsf{y})$. Let
$E_{i}\subset\mathsf{X}_{i}$ be a sequence of Borel sets converging in
$\mathsf{L}^{1}_{loc}$ sense to $E\subset\mathsf{Y}$. Assume that each $E_{i}$
is perimeter minimizing in $B_{r_{i}}(\mathsf{x}_{i})$ and that
$r_{i}\uparrow+\infty$. Then $E$ is perimeter minimizing and in the metric
space realizing the convergence we have that $\partial E_{i}\to\partial F$ in
Kuratowski sense.

We conclude the section by recalling the global Bernstein property in
Euclidean space. This is a classical result and the proof can be found in [38,
Theorem $17.4$].

######  Theorem 2.36.

Let $E\subset\mathbb{R}^{N}$ be a non empty perimeter minimizing set contained
in a half space. Then $E$ is itself a half space.

##  3 Green’s functions in parabolic metric measure spaces

The goal of this section is to prove Theorems 4 and 5. The techniques that we
use generally follow the lines of the arguments in [18, 58, 17], although we
work in higher generality since we do not assume any non collapsing condition
on the space $\mathsf{X}$. Throughout the section we assume that we are
working on a fixed $\mathsf{RCD}(K,N)$ space $(X,\mathsf{d},\mathfrak{m})$.

######  Definition 3.1.

The space $(\mathsf{X},\mathsf{d},\mathfrak{m})$ is said to be parabolic if
$\int_{1}^{+\infty}p_{t}(x,y)\,dt=+\infty$ for every
$(x,y)\in\mathsf{X}\times\mathsf{X}$.

In the smooth setting there are many possible equivalent definitions of
parabolicity. In Theorem 4 we show that also in the $\mathsf{RCD}(K,N)$
setting the previous definition is equivalent to the more classical one
involving non existence of Green’s functions. In the $\mathsf{RCD}(K,N)$
setting it is convenient to start from Definition 3.1 because it allows to
exploit the good properties of the heat flow.

######  Proposition 3.2.

The space $(\mathsf{X},\mathsf{d},\mathfrak{m})$ is parabolic if and only if
there exists $(x,y)\in\mathsf{X}\times\mathsf{X}$ such that
$\int_{1}^{+\infty}p_{t}(x,y)\,dt=+\infty$.

###### Proof.

It follows immediately by Proposition 2.13. ∎

We now define Green’s functions on $\mathsf{RCD}(K,N)$ spaces.

######  Definition 3.3.

Let $A\subset\subset\mathsf{X}$ be a regular set and let $x\in A$. We say that

$G_{x}\in\mathsf{W}^{1,1}_{0}(A)\cap C(\bar{A}\setminus\\{x\\})$

is a Green’s function on $A$ with pole $x$ if $\Delta G_{x}=-\delta_{x}$ in
distributional sense in $A$ and $G_{x}=0$ on $\partial A$. Similarly we say
that

$G_{x}\in\mathsf{W}^{1,1}_{loc}(\mathsf{X})\cap C(\mathsf{X}\setminus\\{x\\})$

is a Green’s function on $\mathsf{X}$ with pole $x$ if $\Delta
G_{x}=-\delta_{x}$ in distributional sense.

Note that our definition of Green’s functions on proper subsets of
$\mathsf{X}$ requires a pointwise boundary condition as this will be important
later on in the proof of Theorem 1. This is also the reason why we only define
Green’s functions on regular sets and not on arbitrary bounded precompact
sets. We will indeed show that with our definition if a bounded precompact set
admits a Green’s function then it is regular and viceversa.

The next proposition shows that a regular set admits at most one Green’s
function.

######  Proposition 3.4.

Let $A\subset\mathsf{X}$ be a regular set and let $x\in A$. Then there exists
at most one Green’s function with pole $x$ on $A$.

###### Proof.

Let $f_{1},f_{2}$ be two such functions. Then $f_{1}-f_{2}$ is harmonic in
distributional sense, so that by Proposition 2.24 it belongs to
$\mathsf{W}^{1,2}_{loc}(A)$. This implies that for every $\Omega\subset\subset
A$ we have $f_{1}-f_{2}\in D(\Delta,\Omega)$ and $\Delta(f_{1}-f_{2})=0$. In
particular, since we have that $f_{1}-f_{2}\in C(\bar{A}\setminus\\{x\\})$, we
obtain that $f_{1}-f_{2}\in C(\bar{A})$.

Suppose now that $f_{1}-f_{2}\neq 0$, so that there exists a point where such
difference is positive (or negative). Restricting to an appropriate precompact
set of $A$ and using the maximum principle we obtain a contradiction. ∎

Now we give a series of lemmas that are needed to prove existence of Green’s
functions on regular domains, which is the content of Proposition 3.10. The
proof of the next lemma is immediate.

######  Lemma 3.5.

Let $x\in\mathsf{X}$ and let $r>0$. If $f:\mathsf{X}\to\mathbb{R}$ satisfies
$\mathsf{lip}(f)(y)\leq c$ for every $y\in B_{r}(x)$, then $f$ is Lipschitz in
$B_{r/4}(x)$.

Let $T>0$ and define $G^{T}_{x}:\mathsf{X}\to\mathbb{R}$ by
$G^{T}_{x}(y):=\int_{0}^{T}p_{t}(x,y)\,dt$. The function $G^{T}_{x}$ is
summable as

$\int_{\mathsf{X}}G^{T}_{x}(y)\,d\mathfrak{m}(y)=\int_{0}^{T}\int_{\mathsf{X}}p_{t}(x,y)\,d\mathfrak{m}(y)\,dt=T.$

As a consequence, $G^{T}_{x}$ is real valued $\mathfrak{m}$-a.e. on
$\mathsf{X}$ and the definition is well posed.

######  Lemma 3.6.

The function $G^{T}_{x}$ is locally Lipschitz in $\mathsf{X}\setminus\\{x\\}$.

###### Proof.

Let $y\in\mathsf{X}\setminus\\{x\\}$. Thanks to Proposition 2.12 there exist
constants $c_{1},c_{2}>0$ such that for every $t\in(0,T)$ the function
$p_{t}(x,\cdot)$ is Lipschitz in $B_{\mathsf{d}(x,y)/8}(y)$ with local
Lipschitz constant bounded from above by

$\frac{c_{1}}{\sqrt{t}\mathfrak{m}(B_{\sqrt{t}}(x))}\exp{\Big{\\{}\frac{-\mathsf{d}^{2}(x,y)}{c_{2}t}\Big{\\}}}.$

As a consequence, for every $z\in B_{\mathsf{d}(x,y)/8}(y)$, using Fatou Lemma
we get

$\mathsf{lip}(G^{T}_{x})(z)=\limsup_{z_{1}\to
z}\int_{0}^{T}\frac{|p_{t}(x,z_{1})-p_{t}(x,z)|}{\mathsf{d}(z_{1},z)}\,dt\leq\int_{0}^{T}\frac{c_{1}}{\sqrt{t}\mathfrak{m}(B_{\sqrt{t}}(x))}\exp{\Big{\\{}\frac{-\mathsf{d}^{2}(x,y)}{c_{2}t}\Big{\\}}}\,dt.$
(1)

Moreover from Remark 2.8 we have that there exists a constant $C>0$ such that
$\mathfrak{m}(B_{r}(x))\geq Cr^{N}$ for every $r\leq\sqrt{T}$, so that the
r.h.s. of (1) is bounded. Hence $G^{T}_{x}$ is locally Lipschitz in a
neighbourhood of $y\in\mathsf{X}\setminus\\{x\\}$ because of Lemma 3.5. Since
$y$ was arbitrary $G^{T}_{x}$ is locally Lipschitz in
$\mathsf{X}\setminus\\{x\\}$. ∎

######  Lemma 3.7.

$G^{T}_{x}\in\mathsf{W}^{1,1}_{loc}(\mathsf{X})$.

###### Proof.

Consider for every $\epsilon>0$ the function
$G^{T}_{\epsilon,x}:\mathsf{X}\to\mathbb{R}$ given by

$G^{T}_{\epsilon,x}(y):=\int_{\epsilon}^{T}p_{t}(x,y)\,dt.$

It is easy to see that these converge in $\mathsf{L}^{1}(\mathsf{X})$ to
$G^{T}_{x}$ and, using Proposition 2.12 and Fatou Lemma, that for every
$y\in\mathsf{X}$

$\mathsf{lip}(G^{T}_{\epsilon,x})(y)\leq\int_{\epsilon}^{T}\frac{c_{1}}{\sqrt{t}\mathfrak{m}(B_{\sqrt{t}}(x))}\exp{\Big{\\{}\frac{-\mathsf{d}^{2}(x,y)}{c_{2}t}\Big{\\}}}\,dt\leq\int_{\epsilon}^{T}\frac{c_{3}}{\sqrt{t}\mathfrak{m}(B_{\sqrt{t}}(x))}\,dt<+\infty.$

In particular $G^{T}_{\epsilon,x}\in\mathsf{Lip}(\mathsf{X})\subset
W^{1,1}_{loc}(\mathsf{X})$. So if we prove that for every ball
$B_{r}(x)\subset\mathsf{X}$ with $r\geq T+1$, the functions $G^{T}_{\epsilon}$
restricted to $B_{r}(x)$ are a Cauchy sequence in
$\mathsf{W}^{1,1}(B_{r}(x))$, we obtain that the limit $G^{T}_{x}$ is in
$\mathsf{W}^{1,1}_{loc}(\mathsf{X})$ as well. So we compute

$\displaystyle\begin{split}\int_{B_{r}(x)}|\nabla(G^{T}_{\epsilon_{1},x}-G^{T}_{\epsilon_{2},x})|(y)d\mathfrak{m}(y)&=\int_{B_{r}(x)}\mathsf{lip}\Big{(}\int_{\epsilon_{1}}^{\epsilon_{2}}p_{t}(x,\cdot)dt\Big{)}(y)d\mathfrak{m}(y)\\\
&\leq\int_{B_{r}(x)}\int_{\epsilon_{1}}^{\epsilon_{2}}\frac{c_{4}}{\sqrt{t}\mathfrak{m}(B_{\sqrt{t}}(x))}\exp{\Big{\\{}\frac{-d^{2}(x,y)}{c_{5}t}\Big{\\}}}\,dt\,d\mathfrak{m}(y),\end{split}$
(2)

where for the last inequality we passed the Lipschitz constant inside the
integral using Fatou Lemma. Switching the integrals and applying a version of
Fubini’s Theorem (i.e. Cavalieri principle, see [4]) we then obtain that the
expression in (2) can be bounded as follows:

$\displaystyle\int_{\epsilon_{1}}^{\epsilon_{2}}\int_{B_{r}(x)}$
$\displaystyle\frac{c_{4}}{\sqrt{t}\mathfrak{m}(B_{\sqrt{t}}(x))}\exp{\Big{\\{}\frac{-d^{2}(x,y)}{c_{5}t}\Big{\\}}}\,d\mathfrak{m}(y)\,dt$
$\displaystyle=\int_{\epsilon_{1}}^{\epsilon_{2}}\frac{c_{4}}{\sqrt{t}\mathfrak{m}(B_{\sqrt{t}}(x))}\int_{0}^{1}\mathfrak{m}\Big{(}\Big{\\{}\exp{\Big{\\{}\frac{-\mathsf{d}^{2}(x,y)}{c_{5}t}\Big{\\}}}>s\Big{\\}}\cap
B_{r}(x)\Big{)}\,ds\,dt$
$\displaystyle\leq\int_{\epsilon_{1}}^{\epsilon_{2}}\frac{c_{6}}{\sqrt{t}}\sup_{z\in[0,T]}\int_{0}^{+\infty}e^{-c_{7}s}\frac{\mathfrak{m}(B_{\sqrt{sz}}(x)\cap
B_{r}(x))}{\mathfrak{m}(B_{\sqrt{z}}(x))}\,ds\,dt.$

We now prove that the supremum in the previous expression is finite, keeping
in mind that this is sufficient to prove that our sequence is Cauchy. We
compute:

$\displaystyle\sup_{z\in[0,T]}$
$\displaystyle\int_{0}^{+\infty}e^{-c_{7}s}\frac{\mathfrak{m}(B_{\sqrt{sz}(x)}\cap
B_{r}(x))}{\mathfrak{m}(B_{\sqrt{z}(x)})}\,ds$
$\displaystyle\leq\int_{0}^{1}e^{-c_{7}s}\,ds+\sup_{z\in[0,T]}\Big{(}\int_{1}^{r^{2}/z}e^{-c_{7}s}\frac{\mathfrak{m}(B_{\sqrt{sz}(x)})}{\mathfrak{m}(B_{\sqrt{z}(x)})}\,ds+\int_{r^{2}/z}^{+\infty}e^{-c_{7}s}\frac{\mathfrak{m}(B_{r}(x))}{\mathfrak{m}(B_{\sqrt{z}(x)})}\,ds\Big{)}$
$\displaystyle\leq
c_{8}+\sup_{z\in[0,T]}\Big{(}\int_{1}^{r^{2}/z}e^{-c_{7}s}\frac{V_{K,N}(B_{\sqrt{sz}(x)})}{V_{K,N}(B_{\sqrt{z}(x)})}\,ds+c_{9}e^{-c_{7}r^{2}/z}z^{-N/2}\Big{)}$

where the last inequality is a consequence of Remark 2.8. It is clear that the
finiteness of the last expression follows if we can prove the finiteness of

$\sup_{z\in[0,T]}\int_{1}^{r^{2}/z}e^{-c_{7}s}\frac{V_{K,N}(B_{\sqrt{sz}(x)})}{V_{K,N}(B_{\sqrt{z}(x)})}\,ds.$

Since when $s\in(1,r^{2}/z)$ we have $\sqrt{zs}\in(0,r)$, there exists a
constant $c_{10}$ depending on $r$ such that $V_{K,N}(B_{\sqrt{zs}(x)})\leq
c_{10}(sz)^{N/2}$ whenever $s\in(1,r^{2}/z)$. Hence combining with Remark 2.8
once again we get

$\sup_{z\in[0,T]}\int_{1}^{r^{2}/z}e^{-c_{7}s}\frac{V_{K,N}(B_{\sqrt{sz}(x)})}{V_{K,N}(B_{\sqrt{z}(x)})}\,ds\leq\sup_{z\in[0,T]}\int_{1}^{r^{2}/z}c_{11}e^{-c_{7}s}s^{N/2}\,ds\leq\int_{1}^{+\infty}c_{11}e^{-c_{7}s}s^{N/2}\,ds<+\infty.$

Concluding the proof that $G^{T}_{x}\in\mathsf{W}^{1,1}(B_{r}(x))$. ∎

######  Lemma 3.8.

Let $\Omega\subset\mathsf{X}$ be an open set and let $x\in\Omega$ and
$g\in\mathsf{W}^{1,1}_{0}(\Omega)$. If for every
$\phi\in\mathsf{Test}_{c}(\Omega,\mathsf{d},\mathfrak{m})$ we have

$\int_{\Omega}g(y)\Delta\phi(y)\,d\mathfrak{m}(y)=-\phi(x),$

then $\Delta g=-\delta_{x}$ in distributional sense in $\Omega$.

###### Proof.

Let $\phi\in\mathsf{Lip}_{c}(\Omega)$ and consider its extension to $0$ in
$\mathsf{X}$, which we still denote by $\phi$. Consider the heat flow
$P_{t}\phi\in\mathsf{Test}(\mathsf{X},\mathsf{d},\mathfrak{m})$. Let now
$\eta\in\mathsf{Test}_{c}(\Omega)$ be a positive function which is equal to
$1$ on an open set containing the support of $\phi$. Since the test functions
are an algebra, we get $\eta
P_{t}\phi\in\mathsf{Test}_{c}(\Omega,\mathsf{d},\mathfrak{m})$. Moreover $\eta
P_{t}\phi\to\phi$ in $\mathsf{W}^{1,2}(\mathsf{X})$ and uniformly, while the
functions $\eta P_{t}\phi$ are uniformly Lipschitz because $\phi$ is bounded.
Hence using the Dominated Convergence Theorem we get

$\int_{\Omega}\nabla g(y)\cdot\nabla\phi(y)\,d\mathfrak{m}(y)=\lim_{t\to
0}\int_{\Omega}\nabla g(y)\cdot\nabla(\eta
P_{t}\phi)\,d\mathfrak{m}(y)=-\lim_{t\to 0}\int_{\Omega}g(y)\Delta(\eta
P_{t}\phi)\,d\mathfrak{m}(y)=\lim_{t\to 0}\eta P_{t}\phi(x)=\phi(x),$

concluding the proof. ∎

######  Lemma 3.9.

Let $B\subset\mathsf{X}$ be a regular set, $x\in B$, $f\in C(\mathsf{X})$, and
$g\in\mathsf{Lip}_{loc}(\bar{B}\setminus\\{x\\})$. There exists
$h\in\mathsf{W}^{1,2}(B)\cap C(\bar{B})$ such that $\Delta h=f$ on $B$, $h+g$
is identically zero on $\partial B$ and for every positive
$\eta\in\mathsf{Lip}_{c}(B)$ equal to $1$ in a neighbourhood of $x$ and taking
value in $[0,1]$ we have $(1-\eta)(h+g)\in\mathsf{W}^{1,2}_{0}(B)$.

Moreover, if in addition we have that $g\in\mathsf{W}^{1,2}(B)$, then we can
ask that $h+g\in\mathsf{W}^{1,2}_{0}(B)$.

###### Proof.

Let $U\supset\supset\bar{B}$ be an open precompact subset of $\mathsf{X}$ and
consider the function $h_{1}\in\mathsf{W}^{1,2}_{0}(U)$ such that $\Delta
h_{1}=f$ on $U$. By Proposition 2.16 this function is Lipschitz on $\bar{B}$.

Let now $\eta\in\mathsf{Lip}_{c}(B)$ be as in the statement, then
$(1-\eta)g\in\mathsf{W}^{1,2}(B)\cap C(\bar{B})$ so that, $B$ being regular,
there exists $h_{2}\in\mathsf{W}^{1,2}(B)\cap C(\bar{B})$, harmonic on $B$,
such that $h_{2}+h_{1}+g$ is identically zero on $\partial B$ and
$h_{1}+h_{2}+(1-\eta)g\in\mathsf{W}^{1,2}_{0}(B)$. We then set
$h:=h_{2}+h_{1}$ and we note that this function has all the required
properties.

The case when $g\in\mathsf{W}^{1,2}(B)$ is analogous. ∎

We are finally in position to prove the existence of Green’s functions.

######  Proposition 3.10.

Let $B\subset\mathsf{X}$ be a regular set and let $x\in B$. For every $T>0$
there exists $g_{x}^{T}\in\mathsf{W}^{1,2}(B)\cap C(\bar{B})$ such that
$G_{x}:=G^{T}_{x}+g^{T}_{x}$ is a Green’s function on $B$ with pole $x$.

Moreover for every $\eta\in\mathsf{Lip}_{c}(B)$ equal to $1$ in a
neighbourhood of $x$ and taking value in $[0,1]$ we have
$(1-\eta)G_{x}\in\mathsf{W}^{1,2}_{0}(B)$.

###### Proof.

Let $g^{T}_{x}\in\mathsf{W}^{1,2}(B)\cap C(\bar{B})$ be the function given by
Lemma 3.9 such that

$\Delta g_{x}^{T}=-p_{T}(x,\cdot),\quad g_{x}^{T}+G_{x}^{T}=0\quad\text{on
}\partial B$

and such that for every $\eta$ as in the statement
$(1-\eta)(g_{x}^{T}+G_{x}^{T})\in\mathsf{W}^{1,2}_{0}(B).$ Now we define
$G_{x}:B\to\mathbb{R}$ by $G_{x}(y):=G^{T}_{x}(y)+g^{T}_{x}(y)$ and we note
that by construction $G_{x}\in\mathsf{W}^{1,1}_{0}(B)\cap
C(\bar{B}\setminus\\{x\\})$.

We now check that $G_{x}$ is a Green’s functions: let
$\phi\in\mathsf{Test}_{c}(B,\mathsf{d},\mathfrak{m})$ and note that

$\displaystyle\int_{B}$ $\displaystyle
G^{T}_{x}(y)\Delta\phi(y)\,d\mathfrak{m}(y)=\int_{0}^{T}\int_{\mathsf{X}}p_{t}(x,y)\Delta\phi(y)\,d\mathfrak{m}(y)\,dt=\int_{0}^{T}P_{t}\Delta\phi(x)\,dt$
$\displaystyle=\int_{0}^{T}\Delta
P_{t}\phi(x)\,dt=\int_{0}^{T}\frac{d}{dt}P_{t}\phi(x)\,dt=\int_{B}p_{T}(x,y)\phi(y)\,d\mathfrak{m}(y)-\phi(x),$

where the last inequality comes from the fact that the trajectories of the
heat flow at a fixed point are locally absolutely continuous. From the
previous computation and the definition of $G_{x}$ we deduce that for every
$\phi\in\mathsf{Test}_{c}(B,\mathsf{d},\mathfrak{m})$ we have

$\int_{B}G_{x}(y)\Delta\phi(y)\,d\mathfrak{m}(y)=-\phi(x),$

which by Lemma 3.8 implies that $G_{x}$ is a Green’s function for the
Laplacian. ∎

######  Corollary 3.11.

The function $G_{x}$ constructed in Proposition 3.10 is positive.

###### Proof.

We would like to use the maximum principle on $G_{x}$, but this function is
only $\mathsf{W}^{1,1}(B)\cap C(\bar{B}\setminus\\{x\\})$, while we would need
a function in $\mathsf{W}^{1,2}(B)\cap C(\bar{B})$ to apply such theorem.

For every $\epsilon>0$ consider the function
$G^{\epsilon}_{x}\in\mathsf{W}^{1,2}(B)\cap C(\bar{B})$ given by

$G_{x}^{\epsilon}(y):=\int_{\epsilon}^{T}p_{t}(x,y)\,dt+g_{x}^{T}(y).$

We have that $\Delta G_{x}^{\epsilon}=-p_{\epsilon}(x,\cdot)$ and
$G_{x}^{\epsilon}=-\int_{0}^{\epsilon}p_{t}(x,\cdot)\,dt$ on $\partial B$.

By the maximum principle these functions attain their minimum on the boundary,
so that on $B$ we have

$G_{x}^{\epsilon}\geq\min_{y\in\partial B}-\int_{0}^{\epsilon}p_{t}(x,y)\,dt$
(3)

At the same time it is easy to check that as $\epsilon$ decreases to zero also
the quantity in the right hand side of (3) goes to zero. Moreover
$G_{x}^{\epsilon}\to G_{x}$ pointwise in $B\setminus\\{x\\}$ as $\epsilon\to
0$, so that passing to the limit in (3) we conclude. ∎

As a consequence of this corollary we get the following proposition.

######  Proposition 3.12.

Let $B_{1}\subset\subset B_{2}$ be regular sets, let $x\in B_{1}$ and let
$G^{1}_{x}$ and $G^{2}_{x}$ be the corresponding Green’s functions. Then
$G^{2}_{x}\geq G^{1}_{x}$.

###### Proof.

By Proposition 2.24 the restriction to $\bar{B}_{1}$ of the difference
$G^{2}_{x}-G^{1}_{x}$ lies in $W^{1,2}_{loc}(B_{1})\cap C(\bar{B}_{1})$.

At the same time $G^{2}_{x}-G^{1}_{x}\geq 0$ on $\partial B_{1}$ by Corollary
3.11, so that using the maximum principle we conclude. ∎

The next proposition shows that we can use the Green’s function to construct
functions with prescribed Laplacian.

######  Proposition 3.13.

Let $B\subset\mathsf{X}$ be regular, and let $\phi\in\mathsf{Lip}_{c}(B)$.
Then

$f(x):=\int_{B}G_{x}(y)\phi(y)\,d\mathfrak{m}(y)$

is the unique function such that $f\in
D(\Delta,B)\cap\mathsf{W}^{1,2}_{0}(B)\cap C(\bar{B})$ with $\Delta f=-\phi$
and $f=0$ on $\partial B$.

###### Proof.

By Lemma 3.9 we have that there exists a function

$f\in D(\Delta,B)\cap W^{1,2}_{0}(B)\cap C(\bar{B})$

with $\Delta f=-\phi$ and $f=0$ on $\partial B$. By the maximum principle such
function is unique. Since $f\in\mathsf{W}^{1,2}_{0}(B)$ it is easy to check
that there exists a sequence
$\\{f_{i}\\}_{i\in\mathbb{N}}\subset\mathsf{Lip}_{c}(B)$ of functions such
that $\|f_{i}-f\|_{\mathsf{W}^{1,2}(B)}\to 0$ and such that for every
$K\subset\subset B$ we have $f_{i}=f$ on $K$ for $i$ sufficiently large. In
particular, since $f_{i}\in\mathsf{Lip}_{c}(B)$ and $G_{x}$ is a Green’s
function with pole $x$, we have

$f_{i}(x)=\int_{B}\nabla G_{x}(y)\cdot\nabla f_{i}(y)\,d\mathfrak{m}(y),$

so that to conclude it is sufficient to prove that as $i\to+\infty$ we have

$\int_{B}\nabla G_{x}(y)\cdot\nabla
f_{i}(y)\,d\mathfrak{m}(y)\to-\int_{B}G_{x}(y)\Delta f(y)\,d\mathfrak{m}(y).$
(4)

To this aim fix $\epsilon>0$ and let $\eta\in\mathsf{Lip}(B)$ be a function
taking value in $[0,1]$ which is equal to zero in a neighbourhood of $x$,
equal to $1$ out of a precompact open subset $U$ of $B$ and such that

$\Big{|}\int_{B}G_{x}(y)\Delta f(y)\,d\mathfrak{m}(y)-\int_{B}(\eta
G_{x})(y)\Delta f(y)\,d\mathfrak{m}(y)\Big{|}<\epsilon,$

and for every $i\in\mathbb{N}$ sufficiently large

$\Big{|}\int_{B}\nabla G_{x}(y)\cdot\nabla
f_{i}(y)\,d\mathfrak{m}(y)-\int_{B}\nabla(\eta G_{x})(y)\cdot\nabla
f_{i}(y)\,d\mathfrak{m}(y)\Big{|}<\epsilon.$

Note that such $\eta$ can be constructed because $\Delta
f\in\mathsf{L}^{\infty}(B)$ and, for $i$ sufficiently large, we have that

$|\nabla f_{i}|=|\nabla f|\text{ in }U.$

In particular, thanks to the arbitrariness of $\epsilon$, to obtain (4) it is
now sufficient to show that

$\int_{B}\nabla(\eta G_{x})(y)\cdot\nabla
f_{i}(y)\,d\mathfrak{m}(y)\to-\int_{B}(\eta G_{x})(y)\Delta
f(y)\,d\mathfrak{m}(y)\quad\text{as }i\to+\infty.$

Since $\eta G_{x}\in\mathsf{W}^{1,2}_{0}(B)$ by Proposition 3.10, we can
rewrite the difference of the previous quantities as

$\displaystyle\Big{|}-$ $\displaystyle\int_{B}\nabla(\eta G_{x})(y)\cdot\nabla
f_{i}(y)\,d\mathfrak{m}(y)-\int_{B}(\eta G_{x})(y)\Delta
f(y)\,d\mathfrak{m}(y)\Big{|}$ $\displaystyle=\Big{|}-\int_{B}\nabla(\eta
G_{x})\cdot\nabla f_{i}(y)\,d\mathfrak{m}(y)+\int_{B}\nabla(\eta
G_{x})\cdot\nabla f(y)\,d\mathfrak{m}(y)\Big{|}$ $\displaystyle\leq\|\eta
G_{x}\|_{\mathsf{W}^{1,2}(B)}\|f_{i}-f\|_{\mathsf{W}^{1,2}(B)}\to 0,$

concluding the proof. ∎

We now turn our attention to constructing a global Green’s function for
$\mathsf{X}$.

######  Lemma 3.14.

Let $(\mathsf{X},\mathsf{d},\mathfrak{m})$ be non parabolic. Let
$S\subset\subset\mathsf{X}$ and $x\in\mathsf{X}$. There exists a sequence
$t_{i}\to+\infty$ such that $p_{t_{i}}(x,\cdot)$ and
$\int_{1}^{t_{i}+1}p_{t}(x,\cdot)\,dt$ are uniformly bounded in $S$.

###### Proof.

The non parabolicity of $\mathsf{X}$ implies that there exists a sequence
$t_{i}\to+\infty$ such that $p_{t_{i}}(x,x)\leq 1$. By Proposition 2.13 there
exists a constant $C>0$ such that $p_{t}(x,y)\leq Cp_{t+1}(x,x)$ for every
$y\in S$ and $t>1$. Putting these facts together it is easy to check that the
sequence $\\{t_{i}\\}_{i\in\mathbb{N}}$ has the desired properties. ∎

######  Proposition 3.15.

Let $(\mathsf{X},\mathsf{d},\mathfrak{m})$ be non parabolic and let
$x\in\mathsf{X}$. Then the function
$G_{x}(y):=\int_{0}^{+\infty}p_{t}(x,y)\,dt$ is a Green’s function for
$\mathsf{X}$ with pole $x$.

###### Proof.

First of all we prove that $G_{x}\in\mathsf{L}^{1}_{loc}(\mathsf{X})$. Let
$S\subset\Omega$ be a compact set. By Proposition 2.13 we deduce that there
exists $C>0$ depending on $K,N,S$ such that $Cp_{t}(x,x)\geq p_{t-1}(x,y)$ for
every $t>1$ and every $y\in S$. As a consequence for every $y\in S$ we have

$\int_{1}^{+\infty}p_{t}(x,y)\,dt\leq
C\int_{2}^{+\infty}p_{t}(x,x)\,dt<+\infty,$ (5)

so that by Dominated Convergence Theorem and the Hölder continuity of the heat
kernel the function $\int_{1}^{+\infty}p_{t}(x,\cdot)\,dt$ is continuous in
$S$. Using such continuity and the stochastic completeness of the heat kernel
we then get

$\int_{S}\int_{0}^{+\infty}p_{t}(x,y)\,dt\,d\mathfrak{m}(y)\leq\int_{0}^{1}\int_{\mathsf{X}}p_{t}(x,y)\,d\mathfrak{m}(y)\,dt+\mathfrak{m}(S)\max_{y\in
S}\int_{1}^{+\infty}p_{t}(x,y)\,dt\leq 1+\mathfrak{m}(S)\max_{y\in
S}\int_{1}^{+\infty}p_{t}(x,y)\,dt,$

showing that $G_{x}\in\mathsf{L}^{1}_{loc}(\mathsf{X})$, as desired.

We now claim that for every
$\phi\in\mathsf{Test}(\mathsf{X},\mathsf{d},\mathfrak{m})\cap\mathsf{Lip}_{c}(\mathsf{X})$
we have

$\int_{\mathsf{X}}G_{x}(y)\Delta\phi(y)\,d\mathfrak{m}(y)=-\phi(x).$ (6)

To this aim fix such a $\phi$ and observe that

$\displaystyle\int_{\mathsf{X}}G_{x}(y)\Delta\phi(y)\,d\mathfrak{m}(y)$
$\displaystyle=\int_{0}^{+\infty}\int_{\mathsf{X}}p_{t}(x,y)\Delta\phi(y)\,d\mathfrak{m}(y)\,dt$
$\displaystyle=\int_{0}^{+\infty}P_{t}\Delta\phi(x)\,dt=\int_{0}^{+\infty}\Delta
P_{t}\phi(x)\,dt=\int_{0}^{+\infty}\frac{d}{dt}P_{t}\phi(x)\,dt$
$\displaystyle=\lim_{t\to+\infty}\int_{\mathsf{X}}p_{t}(x,y)\phi(y)\,d\mathfrak{m}(y)-\phi(x).$

We now claim that the limit appearing in the last expression is zero (we know
that the limit exists since the integral appearing at the beginning of the
computation is well defined). To prove the claim we note that since $\phi$ is
supported in a compact set $S$ we have

$\int_{1}^{+\infty}\Big{|}\int_{\mathsf{X}}p_{t}(x,y)\phi(y)\,d\mathfrak{m}(y)\Big{|}\,dt\leq
c_{1}\mathfrak{m}(S)\sup_{y\in S}\int_{1}^{+\infty}p_{t}(x,y)\,dt\leq
c_{2}\mathfrak{m}(S)\int_{1}^{+\infty}p_{t}(x,x)\,dt<+\infty,$ (7)

where in the last inequality we used Proposition 2.13. As a consequence there
exists a sequence $\\{t_{i}\\}_{i\in\mathbb{N}}$ increasing to $+\infty$ such
that $\int_{\mathsf{X}}p_{t_{i}}(x,y)\phi(y)\,d\mathfrak{m}(y)\to 0$, as
claimed.

To conclude the proof we only need to prove that

$G_{x}\in\mathsf{W}^{1,1}_{loc}(\mathsf{X})\cap
C(\mathsf{X}\setminus\\{x\\}).$ (8)

Indeed if this holds, we can combine equation (6) and Lemma 3.8 (which can be
used in our setting via a cut off argument) to obtain that $G_{x}$ is a
Green’s function for $\mathsf{X}$ with pole $x$.

To prove (8) it is sufficient to prove that
$\int_{1}^{+\infty}p_{t}(x,\cdot)\,dt\in\mathsf{W}^{1,1}_{loc}(\mathsf{X})\cap
C(\mathsf{X}\setminus\\{x\\})$ since $\int_{0}^{1}p_{t}(x,\cdot)\,dt$ lies in
such space by Lemmas 3.6 and 3.7. Consider the family of functions indexed by
$T$ given by $\int_{1}^{T}p_{t}(x,\cdot)\,dt$ and observe that these are in
$\mathsf{L}^{1}(\mathsf{X})\cap\mathsf{W}^{1,1}_{loc}(\mathsf{X})$ and
converge to $\int_{1}^{+\infty}p_{t}(x,\cdot)\,dt$ in
$\mathsf{L}^{1}_{loc}(\mathsf{X})$ by repeating the argument of (7), so that
if we prove that their Lipschitz constants are locally uniformly bounded we
are done.

To this aim we recall that for every $T>0$ we have the distributional equation

$\Delta\int_{1}^{T}p_{t}(x,\cdot)\,dt=p_{T}(x,\cdot)-p_{1}(x,\cdot).$

Fix then a compact set $S$, by Lemma 3.14 there exists a sequence
$\\{t_{i}\\}_{i\in\mathbb{N}}$ increasing to $+\infty$ such that

$\Delta\int_{1}^{t_{i}+1}p_{t}(x,\cdot)\,dt=p_{t_{i}+1}(x,\cdot)-p_{1}(x,\cdot)\quad\text{and}\quad\int_{1}^{t_{i}+1}p_{t}(x,\cdot)\,dt$

are uniformly bounded in the compact $S$. As a consequence, by the a priori
estimates on the gradient of solutions of the Poisson problem given in
Proposition 2.17, we deduce that the functions
$\int_{1}^{t_{i}+1}p_{t}(x,\cdot)\,dt$ have locally uniformly bounded
Lipschitz constants as desired. ∎

######  Remark 3.16.

The function $G_{x}$ that we constructed is not only continuous but also
locally Lipschitz in $\mathsf{X}\setminus\\{x\\}$.

We are now in position to prove Theorem 4, that we recall below.

######  Theorem 4.

Let $(\mathsf{X},\mathsf{d},\mathfrak{m},\mathsf{x})$ be an
$\mathsf{RCD}(K,N)$ metric measure space with infinite diameter. The following
are equivalent:

  1. 1.

There is no positive Green’s function on $\mathsf{X}$ with pole $\mathsf{x}$.

  2. 2.

For every $(x,y)\in\mathsf{X}\times\mathsf{X}$ we have
$\int_{1}^{+\infty}p_{t}(x,y)\,dt=+\infty$.

  3. 3.

There exists $(x,y)\in\mathsf{X}\times\mathsf{X}$ such that
$\int_{1}^{+\infty}p_{t}(x,y)\,dt=+\infty$.

  4. 4.

Let $\\{B_{i}\\}_{i\in\mathbb{N}}$ be an exhaustion of regular sets containing
$\mathsf{x}$. Let $G_{\mathsf{x}}^{i}$ be the Green’s function on $B_{i}$ with
pole $\mathsf{x}$. Then for every $y\neq\mathsf{x}$ we have
$G_{\mathsf{x}}^{i}(y)\to+\infty$ as $i\to+\infty$.

###### Proof.

The fact that $2$ and $3$ are equivalent has been shown in Proposition 3.2,
while $1$ implies $2$ by Proposition 3.15. So we only need to show that $2\to
4\to 1$.

$(2\to 4)$ By Proposition 3.10, for every $i\in\mathbb{N}$ the Green’s
function $G^{i}_{\mathsf{x}}$ can be written as
$G^{i}_{\mathsf{x}}(y)=\int_{0}^{T_{i}}p_{t}(\mathsf{x},y)dt+g^{T_{i}}_{\mathsf{x}}(y)$
where $g^{T_{i}}_{\mathsf{x}}\in\mathsf{W}^{1,2}(B_{i})\cap C(\bar{B_{i}})$
satisfies

$\Delta g^{T_{i}}_{\mathsf{x}}=-p_{T_{i}}(\mathsf{x},\cdot)\quad\text{on
}B_{i}\quad\text{and}\quad
g^{T_{i}}_{\mathsf{x}}=-\int_{0}^{T_{i}}p_{t}(\mathsf{x},\cdot)\quad\text{on
}\partial B_{i}.$

Taking into account the parabolicity assumption, it is sufficient to prove
that we can choose a sequence $T_{i}$ increasing to $+\infty$ in such a way
that $g^{T_{i}}_{\mathsf{x}}(y_{0})$ is bounded from below uniformly in $i$.

Since $\Delta g^{T_{i}}_{\mathsf{x}}\leq 0$, by the maximum principle it is
sufficient to consider the restriction of $g^{T_{i}}_{\mathsf{x}}$ to the
boundary of $B_{i}$, where the minimum is attained. So the problem reduces to
choosing $T_{i}$ increasing to $+\infty$ in such a way that
$\int_{0}^{T_{i}}p_{t}(\mathsf{x},\cdot)\,dt$ is uniformly bounded when
$y\in\partial B_{i}$.

This is clearly possible if we can show that for $T$ fixed, the quantity

$\sup_{y\in\partial B_{i}}\int_{0}^{T}p_{t}(\mathsf{x},y)\,dt$

converges to zero as $i\to+\infty$, and this follows from Proposition 2.12 and
the hypothesis on the diameter.

$(4\to 1)$ Suppose by contradiction that there exists a positive Green’s
function $G_{\mathsf{x}}\in\mathsf{W}^{1,1}_{loc}(\mathsf{X})\cap
C(\mathsf{X}\setminus\\{\mathsf{x}\\})$ with pole $\mathsf{x}$ in
$\mathsf{X}$. We claim that $G_{\mathsf{x}}\geq G^{i}_{\mathsf{x}}$ on $B_{i}$
and we note that, if the claim is true, then we contradict our hypothesis that
$G_{\mathsf{x}}^{i}(y)\to+\infty$ as $i\to+\infty$.

To prove the claim observe that the difference
$G_{\mathsf{x}}-G^{i}_{\mathsf{x}}$ is harmonic and belongs to
$\mathsf{W}^{1,2}_{loc}(B_{i})\cap C(\bar{B_{i}}\setminus\\{\mathsf{x}\\})$ by
Proposition 2.24. Moreover the difference $G_{\mathsf{x}}-G^{i}_{\mathsf{x}}$
is positive on $\partial B_{i}$, so that by the maximum principle it is
positive in $B_{i}$, as claimed. ∎

Now we turn our attention to proving Theorem 5. To this aim we need the
following technical lemma.

######  Lemma 3.17.

Let $(\mathsf{X},\mathsf{d},\mathfrak{m})$ be a non parabolic
$\mathsf{RCD}(K,N)$ space with infinite diameter and essential dimension
$k\geq 2$. Let $x\in\mathsf{X}$ be a regular point such that

$\lim_{r\to 0^{+}}\frac{\mathfrak{m}(B_{r}(x))}{r^{k}}\in(0,+\infty).$

Consider an exhaustion $\\{\Omega_{i}\\}_{i\in\mathbb{N}}$ of regular sets of
$\mathsf{X}$ containing $x$, and for each of these sets consider the
corresponding Green’s function $G^{i}_{x}$.

These functions admit continuous representatives with values in $[0,+\infty]$
such that $G_{x}^{i}(x)=+\infty$. Using these representatives, there exist
$T>1$ and a sequence $\\{c_{i}\\}_{i\in\mathbb{N}}\in(1,T)$ such that

$x\in\\{y\in\Omega_{i}:G^{i}_{x}(y)>c_{i}\\}\cap B_{1}(x)\subset\subset
B_{1}(x)\quad\text{ for every }i\in\mathbb{N}$

and each of the sets $\\{y\in\Omega_{i}:G^{i}_{x}(y)>c_{i}\\}$ is open, has
finite perimeter in $\mathsf{X}$ and satisfies

$\partial\\{y\in\Omega_{i}:G^{i}_{x}(y)>c_{i}\\}=\partial\\{y\in\Omega_{i}:G^{i}_{x}(y)<c_{i}\\}=\\{y\in\Omega_{i}:G^{i}_{x}(y)=c_{i}\\}.$

###### Proof.

We will write $\\{G^{i}_{x}>c_{i}\\}$ instead of
$\\{y\in\Omega_{i}:G^{i}_{x}(y)>c_{i}\\}$. Consider the Green’s function
$G_{x}$ given by Proposition 3.15. Since $x$ is a regular point and
$\mathsf{X}$ has essential dimension $k\geq 2$, using the bounds of
Proposition 2.12 and the explicit expressions for $G^{i}_{x}$ and $G_{x}$, we
get

$\lim_{y\to x}G_{x}(y)=\lim_{y\to x}G^{i}_{x}(y)=+\infty\quad\text{for every
}i\in\mathbb{N}.$ (9)

Hence, we have the desired continuous representatives. Moreover, since $G_{x}$
is bounded on compacts of $\mathsf{X}\setminus\\{x\\}$ we can pick $t>0$ such
that $\\{G_{x}>t\\}\cap B_{1}(x)$ contains $x$ and

$\\{G_{x}>t\\}\cap B_{1}(x)\subset\subset B_{1}(x).$

At the same time, using again (9) and the fact that $G_{x}\geq G_{x}^{i}$, we
get that for every $t^{\prime}\in[t,t+1]$ we have

$\\{G^{i}_{x}>t^{\prime}\\}\cap B_{1}(x)\subset\\{G_{x}>t\\}\cap
B_{1}(x)\subset\subset B_{1}(x),$

and that $\\{G^{i}_{x}>t^{\prime}\\}$ is open and contains $x$. Moreover,
since $G^{i}_{x}$ is locally Lipschitz on compact subsets of
$\mathsf{X}\setminus\\{x\\}$ (and tends to infinity in $x$, so that the level
sets stay away from $x$), for $\lambda^{1}$-a.e. $t^{\prime}\in[t,t+1]$ we
have that $\\{G^{i}_{x}>t^{\prime}\\}$ has finite perimeter and
$\mathfrak{m}(\\{G^{i}_{x}=t^{\prime}\\})=0$. We claim that this implies

$\partial\\{G^{i}_{x}>t^{\prime}\\}=\partial\\{G^{i}_{x}<t^{\prime}\\}=\\{G^{i}_{x}=t^{\prime}\\}.$
(10)

For the first equality we only show that
$\partial\\{G^{i}_{x}>t^{\prime}\\}\subset\partial\\{G^{i}_{x}<t^{\prime}\\}$,
the other case being analogous. Let $y\in\partial\\{G^{i}_{x}>t^{\prime}\\}$
and consider a small ball $B_{\epsilon}(y)$ which is precompact in
$X\setminus\\{x\\}$. Observe that $G_{x}^{i}(y)=t^{\prime}$ and that the ball
cannot be contained in $\\{G^{i}_{x}=t^{\prime}\\}$, because otherwise this
set would have positive measure. At the same time it cannot be contained in
$\\{G^{i}_{x}\geq t^{\prime}\\}$, otherwise the maximum principle would imply
that $G_{x}^{i}$ is constant in the ball. As a consequence such ball
intersects $\\{G^{i}_{x}<t^{\prime}\\}$, and since $\epsilon$ was arbitrarily
small, we have that $y\in\partial\\{G^{i}_{x}<t^{\prime}\\}$, as claimed.

The second equality in (10) follows because $\\{G^{i}_{x}=t^{\prime}\\}$
cannot contain any balls since it has zero measure, and so it is contained in
$\partial\\{G^{i}_{x}\neq t^{\prime}\\}$, but since
$\partial\\{G^{i}_{x}>t^{\prime}\\}=\partial\\{G^{i}_{x}<t^{\prime}\\}$ we get

$\partial\\{G^{i}_{x}\neq
t^{\prime}\\}=\partial\\{G^{i}_{x}>t^{\prime}\\}=\partial\\{G^{i}_{x}<t^{\prime}\\}.$

The fact that
$\partial\\{G^{i}_{x}<t^{\prime}\\}\subset\\{G^{i}_{x}=t^{\prime}\\}$ is
trivial. Then we define $T:=t+1$ and for every $i\in\mathbb{N}$ we pick
$c_{i}\in[t,t+1)$ to be one of the $t^{\prime}$ with the previously proved
properties. ∎

The next proposition is the key step to prove Theorem 5. The idea is adapted
from the proof of the analogous fact in the smooth setting given in [52], but
some crucial adjustments have been made to avoid integrating by parts on sets
that might not be regular enough. This is achieved by replacing balls in the
original proof with level sets of Green’s functions.

######  Proposition 3.18.

Let $(\mathsf{X},\mathsf{d},\mathfrak{m})$ be an $\mathsf{RCD}(K,N)$ space
with essential dimension $k$. If there exists $x\in\mathsf{X}$ such that

$\lim_{r\to
0^{+}}\frac{\mathfrak{m}(B_{r}(x))}{r^{k}}\in(0,+\infty)\quad\text{and}\quad\int_{1}^{+\infty}\frac{1}{P(B_{t}(x),\mathsf{X})}\,dt=+\infty,$

then $\mathsf{X}$ is parabolic.

###### Proof.

Consider first the case when $\mathsf{X}$ has finite diameter and assume by
contradiction that it is non parabolic. Note that the finiteness of the
diameter implies that $\mathsf{X}$ is compact. Consider then the functions
$g_{\epsilon}(y):=\int_{\epsilon}^{+\infty}p_{t}(x,y)\,dt$, which converge in
$\mathsf{L}^{1}(\mathsf{X})$ to the Green’s function $G_{x}$ given by
Proposition 3.15. Using ideas appearing in previous proofs one can show that
these function are Lipschitz (hence in $\mathsf{W}^{1,2}(\mathsf{X})$),
satisfy $\Delta g_{\epsilon}=-p_{\epsilon}(x,\cdot)\leq 0$ and have a minimum
since $\mathsf{X}$ is compact. Hence by the maximum principle they are all
constant, so that also $G_{x}$ is constant, a contradiction.

So we can suppose that $\mathsf{X}$ has infinite diameter. Suppose by
contradiction that $\mathsf{X}$ is non parabolic and assume first that the
essential dimension of $\mathsf{X}$ is not $1$. Pick an exhaustion
$\\{\Omega_{i}\\}_{i\in\mathbb{N}}$ of regular sets of $\mathsf{X}$ containing
$x$, and for each of these sets consider the corresponding Green’s function
$G^{i}_{x}$. By Lemma 3.17 there exist $T>1$ and a sequence
$\\{c_{i}\\}_{i\in\mathbb{N}}\in(1,T)$ such that

$x\in\\{G^{i}_{x}>c_{i}\\}\cap B_{1}(x)\subset\subset B_{1}(x)\quad\text{ for
every }i\in\mathbb{N}$

and each of the sets $\\{G^{i}_{x}>c_{i}\\}$ is open, has finite perimeter and
satisfies

$\partial\\{G^{i}_{x}>c_{i}\\}=\partial\\{G^{i}_{x}<c_{i}\\}=\\{G^{i}_{x}=c_{i}\\}.$

We can suppose without loss of generality that
$\bar{B}_{1}(x)\subset\Omega_{i}$ for every $i\in\mathbb{N}$.

We then define

$l_{i}:=\sup_{z\in B_{1}(x)}\\{\mathsf{d}(x,z):G^{i}_{x}(z)>c_{i}\\}$

and claim that we have

$T\geq\int_{l_{i}}^{\mathsf{d}(\partial\Omega_{i},x)}\frac{1}{P(B_{r}(x),\mathsf{X})}\,dr.$

To prove the claim let $0<\epsilon<<1$ such that

$\partial\\{G^{i}_{x}>\epsilon\\}=\partial\\{G^{i}_{x}<\epsilon\\}=\\{G^{i}_{x}=\epsilon\\}$

and such that, setting

$s_{i}^{\epsilon}:=\inf_{z\in\Omega_{i}}\\{\mathsf{d}(x,z):G^{i}_{x}(z)<\epsilon\\},$

it holds $s_{i}^{\epsilon}>l_{i}$. The first condition can be achieved by the
same argument of Lemma 3.17, while the second can be obtained using that
$G_{x}^{i}>0$ on $\Omega_{i}$ and $G_{x}^{i}=0$ on $\partial\Omega_{i}$.

Consider the function $g_{i}:\mathsf{X}\to\mathbb{R}$ such that $g(y)$ is
equal to $c_{i}$ in $\bar{B}_{l_{i}}(x)$, is equal to $\epsilon$ in
${{}^{c}}B_{s_{i}^{\epsilon}}(x)$, and in $B_{s_{i}^{\epsilon}}(x)\setminus
B_{l_{i}}(x)$ is defined by

$g_{i}(y):=(c_{i}-\epsilon)\frac{\int_{\mathsf{d}(x,y)}^{s_{i}^{\epsilon}}{P(B_{r}(x),\mathsf{X})}^{-1}\,dr}{\int_{l_{i}}^{s_{i}^{\epsilon}}{P(B_{r}(x),\mathsf{X})}^{-1}\,dr}+\epsilon.$

Let

$A_{i}^{\epsilon}:=\\{y\in
B_{1}(x):G_{x}^{i}(y)<c_{i}\\}\cup\\{y\in\Omega_{i}\cap{{}^{c}}B_{1}(0):G_{x}^{i}(y)>\epsilon\\}.$

Thanks to our conditions on $\\{G_{x}^{i}>c_{i}\\}$ and
$\\{G_{x}^{i}>\epsilon\\}$ we have that the set $A_{i}^{\epsilon}$ is open,
contains $B_{s_{i}^{\epsilon}}(x)\setminus\bar{B}_{l_{i}}(x)$ and its boundary
is made of disjoint connected sets where $G_{x}^{i}$ takes either the value
$c_{i}$ or $\epsilon$. As a consequence, modulo translations and inversions,
we can use $G_{x}^{i}$ as a barrier in Proposition 2.20 to show that
$A_{i}^{\epsilon}$ is a regular set.

Hence there exists a unique function $h\in D(\Delta,A_{i}^{\epsilon})\cap
C(\bar{A}_{i}^{\epsilon})$ such that
$h-g_{i}\in\mathsf{W}^{1,2}_{0}(A_{i}^{\epsilon})$, $\Delta h=0$ and
$h-g_{i}=0$ on $\partial A_{i}^{\epsilon}$. From this it follows that
$h=G_{x}^{i}$, which in particular gives
$G_{x}^{i}-g_{i}\in\mathsf{W}^{1,2}_{0}(A_{i}^{\epsilon})$. Hence, using the
fact that harmonic functions minimize the Dirichlet energy, we get

$\int_{A_{i}^{\epsilon}}|\nabla
G_{x}^{i}|^{2}\,d\mathfrak{m}\leq\int_{A_{i}^{\epsilon}}|\nabla
g_{i}|^{2}\,d\mathfrak{m}.$

At the same time, using the coarea formula, one can check that

$\int_{A_{i}^{\epsilon}}|\nabla
g_{i}|^{2}\,d\mathfrak{m}=(c_{i}-\epsilon)^{2}\Big{(}\int_{l_{i}}^{s_{i}^{\epsilon}}{P(B_{r}(x),\mathsf{X})}^{-1}\,dr\Big{)}^{-1},$

so that

$\int_{A_{i}^{\epsilon}}|\nabla
G_{x}^{i}|^{2}\,d\mathfrak{m}\leq(c_{i}-\epsilon)^{2}\Big{(}\int_{l_{i}}^{s_{i}^{\epsilon}}{P(B_{r}(x),\mathsf{X})}^{-1}\,dr\Big{)}^{-1}.$
(11)

Moreover by construction we have that the vector field $\nabla G_{x}^{i}$ is
defined in a neighbourhood of $\bar{A}_{i}^{\epsilon}$, where it is bounded
and admits bounded divergence, so that we can apply Proposition 2.30 to get

$\int_{A_{i}^{\epsilon}}|\nabla
G_{x}^{i}|^{2}\,d\mathfrak{m}=-c_{i}\int_{\partial\\{G_{i}^{\epsilon}<c_{i}\\}\cap
B_{1}(x)}(\nabla
G_{x}^{\epsilon}\cdot\nu_{A_{i}^{\epsilon}})_{ext}\,d\mathsf{Per}-\epsilon\int_{\partial\\{G_{i}^{\epsilon}>\epsilon\\}}(\nabla
G_{x}^{\epsilon}\cdot\nu_{A_{i}^{\epsilon}})_{ext}\,d\mathsf{Per}.$

Using the definitions it is then easy to check that

$\int_{\partial\\{G_{i}^{\epsilon}<c_{i}\\}\cap B_{1}(x)}(\nabla
G_{x}^{\epsilon}\cdot\nu_{A_{i}^{\epsilon}})_{ext}\,d\mathsf{Per}=-\int_{\partial\\{G_{i}^{\epsilon}>c_{i}\\}\cap
B_{1}(x)}(\nabla
G_{x}^{\epsilon}\cdot\nu_{({{}^{c}}A_{i}^{\epsilon})})_{int}\,d\mathsf{Per},$

so that, taking into account that $\Delta G_{i}=-\delta_{x}$ and using again
Proposition 2.30, we obtain

$\int_{A_{i}^{\epsilon}}|\nabla
G_{x}^{i}|^{2}\,d\mathfrak{m}=c_{i}\int_{\partial\\{G_{i}^{\epsilon}>c_{i}\\}\cap
B_{1}(x)}(\nabla
G_{x}^{\epsilon}\cdot\nu_{({{}^{c}}A_{i}^{\epsilon})})_{int}\,d\mathsf{Per}-\epsilon\int_{\partial\\{G_{i}^{\epsilon}>\epsilon\\}}(\nabla
G_{x}^{\epsilon}\cdot\nu_{A_{i}^{\epsilon}})_{ext}\,d\mathsf{Per}=c_{i}-\epsilon.$

Putting together this with (11) we obtain

$c_{i}-\epsilon\geq\int_{l_{i}}^{s_{i}^{\epsilon}}{P(B_{r}(x),\mathsf{X})}^{-1}\,dr,$

and letting $\epsilon\to 0$ we deduce that

$T\geq\int_{l_{i}}^{\mathsf{d}(\partial\Omega_{i},x)}\frac{1}{P(B_{r}(x),\mathsf{X})}\,dr$

for every $i\in\mathbb{N}$, as claimed.

Taking into account that $l_{i}\leq 1$ and letting $i\to+\infty$ we obtain
that

$T\geq\int_{1}^{+\infty}\frac{1}{P(B_{r}(x),\mathsf{X})}\,dr,$

a contradiction.

We now need to consider the case when $\mathsf{X}$ has essential dimension
$1$. A space with infinite diameter and essential dimension $1$ must be a
weighted version of either $\mathbb{R}$ or of a closed half line by
Proposition 2.11. In this case we can repeat an analogous proof, replacing the
sets $\\{G_{x}^{i}>c_{i}\\}$ with a fixed interval containing $x$. This is
sufficient because on each endpoint of this interval each $G_{x}^{i}$ will
trivially have a singleton as image, allowing to construct the analogue of the
function $g_{i}$ in the previous case. Similarly the computation to estimate
$\int_{A_{i}^{\epsilon}}|\nabla G_{x}^{i}|^{2}\,d\mathfrak{m}$ can be carried
out in the same way, replacing $A_{i}^{\epsilon}$ with an appropriate union of
two intervals and exploiting their regularity. ∎

######  Corollary 3.19.

Let $(\mathsf{X},\mathsf{d},\mathfrak{m})$ be an $\mathsf{RCD}(K,N)$ space. If
there exists $x\in\mathsf{X}$ such that

$\int_{1}^{+\infty}\frac{t}{\mathfrak{m}(B_{t}(x))}\,dt=+\infty,$

then $\mathsf{X}$ is parabolic.

###### Proof.

It is clear that if the condition of the statement holds for $x\in\mathsf{X}$,
then, thanks to Proposition 2.10, it holds also for a regular point
$x^{\prime}\in\mathsf{X}$ satisfying

$\lim_{r\to
0^{+}}\frac{\mathfrak{m}(B_{r}(x^{\prime}))}{r^{k}}\in(0,+\infty),$

where $k$ is the essential dimension of $\mathsf{X}$. The statement then
follows from the previous proposition repeating the strategy of [40, Corollary
$2$]. ∎

Putting this together with the heat kernel estimates of Proposition 2.12 we
obtain Theorem 5.

######  Theorem 5.

Let $(\mathsf{X},\mathsf{d},\mathfrak{m})$ be an $\mathsf{RCD}(K,N)$ space. If
there exists $x\in\mathsf{X}$ such that

$\int_{1}^{+\infty}\frac{t}{\mathfrak{m}(B_{t}(x))}\,dt=+\infty,$

then $\mathsf{X}$ is parabolic. This condition is also necessary if $K=0$.

###### Proof.

Thanks to Corollary 3.19 we only need to prove that if $K=0$ the condition of
the statement is also sufficient. This follows immediately since by
Proposition 2.12 we have that

$\int_{1}^{+\infty}p_{t}(x,x)\,dt\geq\int_{1}^{+\infty}\frac{c_{1}}{\mathfrak{m}(B_{\sqrt{t}}(x))}\,dt=+\infty,$

concluding the proof ∎

We conclude the section with a proposition taken from [41, Example $3.19$]
concerning spaces with finite measure.

######  Proposition 3.20.

Let $(\mathsf{M},\mathsf{d}_{g},e^{-V}\mathfrak{m}_{g})$ be a geodesically
complete weighted manifold with finite measure. Then for every
$(x,y)\in\mathsf{M}\times\mathsf{M}$, setting
$\mathfrak{m}:=e^{-V}\mathfrak{m}_{g}$, we have

$p_{t}(x,y)\to\frac{1}{\mathfrak{m}(\mathsf{M})}\text{ as }t\to+\infty.$

######  Corollary 3.21.

If $(\mathsf{X},\mathsf{d},\mathfrak{m})$ is parabolic and
$(\mathsf{M},\mathsf{d}_{g},e^{-V}\mathfrak{m}_{g})$ is a geodesically
complete weighted Riemannian manifold with finite mass then the product space
is parabolic.

###### Proof.

It follows immediately from the previous proposition and the definition of
parabolicity. ∎

##  4 Proof of the Half Space Property

In this section $(\mathsf{X},\mathsf{d},\mathfrak{m})$ will be a fixed
parabolic $\mathsf{RCD}(0,N)$ space and $C\subset\mathsf{X}\times\mathbb{R}$
will be locally the boundary of a locally perimeter minimizing set. We assume
moreover that $C\subset\mathsf{X}\times\mathbb{R}_{+}$. We will denote by
$\mathsf{d}_{C}:{{}^{c}}C\to\mathbb{R}$ the distance from $C$, by
$\mathsf{d}_{0}$ the signed distance from $\mathsf{X}\times\\{0\\}$ with
positive sign in the negative half space and negative sign otherwise, and by
$\bar{\mathsf{d}}:\mathsf{X}\times(-\infty,0)\to\mathbb{R}$ the difference
$\mathsf{d}_{C}-\mathsf{d}_{0}$. Observe that $\bar{\mathsf{d}}\geq 0$ since
$C$ is contained in the upper half space.

Generic points in $\mathsf{X}\times\mathbb{R}$ will be usually denoted by
$\bar{x}=(x,t_{x})$, the canonical distance and the canonical measure in the
poduct space will be $\mathsf{d}_{\times}$ and $\mathfrak{m}_{\times}$
respectively.

We now develop the simple machinery needed to prove Propositions 4.3 and 4.8,
which are crucial steps in the previously outlined argument to prove Theorem
1.

######  Definition 4.1.

Given $\bar{x}\in\mathsf{X}\times\mathbb{R}$ we say that a geodesic
$\gamma:[0,\mathsf{d}_{C}(\bar{x})]\to\mathsf{X}\times\mathbb{R}$ is a
_connector_ if it realizes the distance between $\bar{x}$ and $C$. We say that
such connector is _vertical_ if its image is contained in a set of the form
$\\{x\\}\times\mathbb{R}\subset\mathsf{X}\times\mathbb{R}$.

Connectors will be useful because their behavior is related to the gradient of
the function $\bar{\mathsf{d}}$. The next lemma is taken from [37, Lemma
$2.6$].

######  Lemma 4.2.

For $\mathfrak{m}_{\times}$-a.e. $\bar{x}\in\mathsf{X}\times(-\infty,0)$ there
exists a unique connector.

######  Proposition 4.3.

$\mathsf{d}_{C}:{{}^{c}}C\to\mathbb{R}$ admits negative distributional
Laplacian.

###### Proof.

From [20, Lemma $4.16$] we get that $\mathsf{d}_{C}$ admits distributional
Laplacian in its domain and that the singular part of such distributional
Laplacian is a negative Radon measure. Hence we only need to show that the
regular part of this measure is negative as well. To this aim let
$\bar{x}\notin C$ be a point with a single connector, and let $\bar{y}\in C$
be such that $\mathsf{d}_{C}(\bar{x})=\mathsf{d}(\bar{x},\bar{y})$. Let
$U_{\bar{y}}\ni\bar{y}$ and $E\subset U_{\bar{y}}$ be the sets as in
Definition 2.33. We claim that there exists a neighbourhood $V_{\bar{x}}$ of
$\bar{x}$ such that

$V_{\bar{x}}\subset\\{\bar{z}\in{{}^{c}}C:\exists\bar{z}_{1}\in
U_{\bar{y}}:\mathsf{d}_{C}(\bar{z})=\mathsf{d}(\bar{z},\bar{z}_{1})\\}.$

If this were not the case we could find a sequence $\bar{z}_{n}\to\bar{x}$
whose connectors’ endpoints lie outside of $U_{\bar{y}}$. Modulo passing to a
subsequence we then have that the endpoints of these connectors converge to a
point $\bar{z}\in{{}^{c}}U_{\bar{y}}$, and the geodesic connecting $\bar{x}$
and $\bar{z}$ will be a second connector for $\bar{x}$, a contradiction.

Hence we can apply Proposition 2.34 to obtain that the distributional
Laplacian of $\mathsf{d}_{C}$ is negative in $V_{\bar{x}}$. Repeating the same
argument for every point with a single connector and taking the union of the
resulting open sets we get an open set of full measure where the
distributional Laplacian of $\mathsf{d}_{C}$ is negative. This implies that
the regular part of the distributional Laplacian of $\mathsf{d}_{C}$ is
negative on its domain, concluding the proof. ∎

The next Lemma shows the first link between connectors and the quantity
$|\nabla\bar{\mathsf{d}}|$.

######  Lemma 4.4.

For $\mathfrak{m}_{\times}$-a.e. $\bar{x}\in\mathsf{X}\times(-\infty,0)$ with
a vertical connector we have $|\nabla\bar{\mathsf{d}}|(\bar{x})=0$.

###### Proof.

For $\mathfrak{m}_{\times}$-a.e. point in
$\bar{x}\in\mathsf{X}\times(-\infty,0)$ we have

$|\nabla\bar{\mathsf{d}}|(\bar{x})^{2}=2-2\nabla\mathsf{d}_{C}\cdot\nabla\mathsf{d}_{0}(\bar{x})$

so that we only need to prove that for $\mathfrak{m}_{\times}$-a.e. point
$\bar{x}$ with a unique vertical connector we have
$\nabla\mathsf{d}_{C}\cdot\nabla\mathsf{d}_{0}(\bar{x})=1$. Since distance
functions have local Lipschitz constant $1$ everywhere, for
$\mathfrak{m}_{\times}$-a.e. point $\bar{x}$ we get

$\nabla\mathsf{d}_{C}\cdot\nabla\mathsf{d}_{0}(\bar{x})\leq|\nabla\mathsf{d}_{C}|(\bar{x})|\nabla\mathsf{d}_{0}|(\bar{x})=1.$

Moreover one can easily check that for $\mathfrak{m}_{\times}$-a.e. $\bar{x}$
we have

$\nabla\mathsf{d}_{C}\cdot\nabla\mathsf{d}_{0}(\bar{x})=\lim_{\epsilon\to
0}\frac{\mathsf{lip}(\mathsf{d}_{C}+\epsilon\mathsf{d}_{0})^{2}(\bar{x})-\mathsf{lip}(\mathsf{d}_{C})^{2}(\bar{x})}{2\epsilon}.$

Then, moving along the vertical connector, we deduce that
$\mathsf{lip}(\mathsf{d}_{C}+\epsilon\mathsf{d}_{0})\geq 1+\epsilon$, so that
for $\mathfrak{m}_{\times}$-a.e. $\bar{x}$

$\nabla\mathsf{d}_{C}\cdot\nabla\mathsf{d}_{0}(\bar{x})\geq\lim_{\epsilon\to
0}\frac{(1+\epsilon)^{2}-1}{2\epsilon}=1,$

concluding the proof. ∎

######  Definition 4.5.

Let $\bar{x}=(x,t_{x})\in\mathsf{X}\times(-\infty,0)$, let
$\gamma:[0,\mathsf{d}_{C}(\bar{x})]\to\mathsf{X}\times\mathbb{R}$ be a
connector and let $\bar{y}=(y,t_{y}):=\gamma(\mathsf{d}_{C}(\bar{x}))$. We
define the slope $\mathsf{s}(\bar{x},\gamma)>0$ of $\gamma$ by

$\mathsf{s}(\bar{x},\gamma):=1-\frac{|t_{x}-t_{y}|}{\mathsf{d}_{C}(\bar{x})}.$

The role played by the slope of a connector is clarified in the next lemma.

######  Lemma 4.6.

Let $\bar{x}\in\mathsf{X}\times(-\infty,0)$ and $\gamma$ be a connector for
$\bar{x}$. Then

$\mathsf{lip}(\bar{\mathsf{d}})(\bar{x})\geq\mathsf{s}(\bar{x},\gamma).$

###### Proof.

The projection of $\gamma$ to $\mathbb{R}$ is a geodesic of constant speed
$1-\mathsf{s}(\bar{x},\gamma)$ so that

$\mathsf{d}_{0}(\gamma(s))=\mathsf{d}_{0}(\bar{x})-(1-\mathsf{s}(\bar{x},\gamma))s,$

while, since $\gamma$ is a connector, we have

$\mathsf{d}_{C}(\gamma(s))=\mathsf{d}_{C}(\bar{x})-s.$

Using these facts we obtain

$\bar{\mathsf{d}}(\gamma(s))-\bar{\mathsf{d}}(\bar{x})=\mathsf{d}_{C}(\gamma(s))-\mathsf{d}_{0}(\gamma(s))-\mathsf{d}_{C}(\bar{x})+\mathsf{d}_{0}(\bar{x})=-s+(1-\mathsf{s}(\bar{x},\gamma))s=-s\mathsf{s}(\bar{x},\gamma).$

Dividing both sides of the previous equation by $s$ and taking the limit as
$s$ goes to zero we immediately obtain that
$\mathsf{lip}(\bar{\mathsf{d}})(\bar{x})\geq\mathsf{s}(\bar{x},\gamma)$. ∎

The next lemma shows that the slope is continuous in points with a single
connector. It is easy to check that the statement may fail for general points.

######  Lemma 4.7.

Let $\bar{x}\in\mathsf{X}\times(-\infty,0)$ be a point admitting a unique
connector $\gamma$. Let
$\\{\bar{x}_{n}\\}_{n\in\mathbb{N}}\subset\mathsf{X}\times(-\infty,0)$ be a
sequence such that $\bar{x}_{n}\to\bar{x}$, and
$\\{\gamma_{n}\\}_{n\in\mathbb{N}}$ a corresponding sequence of connectors.
Then

$\lim_{n\to+\infty}\mathsf{s}(\bar{x}_{n},\gamma_{n})=\mathsf{s}(\bar{x},\gamma).$

###### Proof.

For every $n\in\mathbb{N}$ we define

$\bar{y}_{n}:=\gamma_{n}(\mathsf{d}_{C}(\bar{x}_{n}))\in C$

and we note that these points have uniformly bounded distance from $\bar{x}$
because of the triangular inequality. We claim that the sequence $\bar{y}_{n}$
converges to the endpoint of the connector $\gamma$. If we are able to prove
the claim, the statement of the lemma immediately follows by noting that the
slope of a connector depends continuously on the positions of the starting
point and the endpoint.

To prove the claim we observe that, modulo passing to a (non relabeled)
subsequence, we have that $\bar{y}_{n}\to\bar{y}\in C$. Moreover, since
$\mathsf{d}_{C}(\bar{x}_{n})=\mathsf{d}_{\times}(\bar{x}_{n},\bar{y}_{n})$,
passing to the limit and using the continuity of distance functions we obtain
that

$\mathsf{d}_{C}(\bar{x})=\mathsf{d}_{\times}(\bar{x},\bar{y}).$

This implies, by the uniqueness of the connector in $\bar{x}$, that $\bar{y}$
is the endpoint of $\gamma$. In particular the limit point does not depend on
the subsequence that we chose, so that the full sequence
$\\{\bar{y}_{n}\\}_{n\in\mathbb{N}}$ converges to $\bar{y}$, as claimed. ∎

######  Proposition 4.8.

If $\bar{\mathsf{d}}:\mathsf{X}\times(-\infty,0)\to\mathbb{R}$ is not
constant, then there exists a ball $B\subset\mathsf{X}\times(-\infty,0)$ and a
real number $\tau>0$ such that $|\nabla\bar{\mathsf{d}}(\bar{x})|\geq\tau$ for
$\mathfrak{m}_{\times}$-a.e. $\bar{x}\in B$.

###### Proof.

Since $\bar{\mathsf{d}}$ is not constant there exists a set of positive
measure in $\mathsf{X}\times(-\infty,0)$ where $|\nabla\bar{\mathsf{d}}|>0$.
By Lemmas 4.2 and 4.4, this implies that there exists a point
$\bar{x}\in\mathsf{X}\times(-\infty,0)$ with a single non vertical connector
$\gamma$ such that $\mathsf{s}(\bar{x},\gamma)>0$. By lemma 4.7 and a standard
argument by contradiction, this implies that there exists $\tau>0$ such that
for every $\bar{y}$ in a sufficiently small ball centered in $\bar{x}$, and
every connector $\gamma_{\bar{y}}$ of $\bar{y}$, we have
$p(\bar{y},\gamma_{\bar{y}})>\tau.$ The conclusion immediately follows by
Lemma 4.6. ∎

######  Remark 4.9.

A direct computation shows that $\Delta e^{-\bar{\mathsf{d}}}\geq
e^{-\bar{\mathsf{d}}}|\nabla\bar{\mathsf{d}}|^{2}.$ Combining this with the
previous proposition we deduce that there exists a ball
$B\subset\mathsf{X}\times(-\infty,0)$ and $c>0$ such that

$\Delta e^{-\bar{\mathsf{d}}}\geq c1_{B}.$

The next proposition follows from a standard computation and gives the
explicit dependence of the Laplacian on the weight in a weighted
$\mathsf{RCD}$ space.

######  Proposition 4.10.

Let $(\mathsf{Y},\mathsf{d}_{y},\mathfrak{m}_{y})$ be an $\mathsf{RCD}(K,N)$
space and let $V\in\mathsf{W}^{1,2}_{loc}(\mathsf{X})$. Suppose that the space
$(\mathsf{Y},\mathsf{d}_{y},\tilde{\mathfrak{m}}_{y})$ with
$\tilde{\mathfrak{m}}=e^{-V}\mathfrak{m}_{y}$ is
$\mathsf{RCD}(K^{\prime},N^{\prime})$ for a suitable couple
$(K^{\prime},N^{\prime})\in\mathbb{R}\times[1,+\infty)$. Then for every
$u\in\mathsf{Lip}_{loc}(\mathsf{Y})$, denoting by $\tilde{\Delta}$ the
Laplacian in $(\mathsf{Y},\mathsf{d}_{y},\tilde{\mathfrak{m}}_{y})$, we have
that $\tilde{\Delta}u=\Delta u-\nabla u\cdot\nabla V$ in distributional sense.

The next step to prove Theorem 1 is to replace the product measure
$\mathfrak{m}_{\times}$ on $\mathsf{X}\times\mathbb{R}$ with a measure
$\tilde{\mathfrak{m}}_{\times}$, such that the resulting space is parabolic
and the Laplacian bounds given by Remark 4.9 are preserved (modulo reducing
$c$) w.r.t. the new measure. To this aim consider the modified metric measure
space $(\mathbb{R},\mathsf{d}_{e},e^{-V}\lambda^{1})$ where $\mathsf{d}_{e}$
is the Euclidean distance and the function $V:\mathbb{R}\to\mathbb{R}_{+}$ is
defined as follows.

Let $\bar{x}=(p_{x},t_{x})\in\mathsf{X}\times(-\infty,0)$ and
$\tau,\epsilon>0$ be such that $|\nabla\bar{d}|\geq\tau$ on
$B_{\epsilon}(\bar{x})\subset\mathsf{X}\times(-\infty,0)$. Note that this is
possible thanks to Proposition 4.8. Let $f:\mathbb{R}\to\mathbb{R}$ be an odd
increasing function such that $f(0)=0$, $f(t)=1$ for every $t\geq 1$ and
$f(t)=-1$ for every $t\leq-1$. Define

$V(s):=\int_{t_{x}}^{s}f(t-t_{x})\,dt.$

Note that $V$ is symmetric w.r.t. $t_{x}$ and that $V^{\prime}(s)=1$ if
$s\in[t_{x}+1,+\infty)$, $V^{\prime}(s)\in(-1,1)$ if $s\in(t_{x}-1,t_{x}+1)$
and $V^{\prime}(s)=-1$ if $s\in(-\infty,t_{x}-1]$. In particular
$\int_{\mathbb{R}}e^{-V(s)}\,ds<+\infty$.

As anticipated, we use such function to modify the measure in the product
space $\mathsf{X}\times\mathbb{R}$.

######  Proposition 4.11.

There exists a couple $(K,N)$ such that

$(\mathsf{X}\times\mathbb{R},\mathsf{d}_{\times},e^{-V(-\mathsf{d}_{0})}\mathfrak{m}_{\times}),$

is an $\mathsf{RCD}(K,N)$ space.

###### Proof.

Note that $(\mathbb{R},\mathsf{d}_{e},e^{-V(-\mathsf{d}_{0})}\lambda^{1})$ is
an $\mathsf{RCD}(K_{0},N_{0})$ space for some $K_{0}\in(-\infty,0)$ and
$N_{0}\in(1,+\infty)$. Since

$(\mathsf{X}\times\mathbb{R},\mathsf{d}_{\times},e^{-V(-\mathsf{d}_{0})}\mathfrak{m}_{\times})=(\mathsf{X},\mathsf{d},\mathfrak{m})\otimes(\mathbb{R},\mathsf{d}_{e},e^{-V}\lambda^{1}),$

we deduce that also
$(\mathsf{X}\times\mathbb{R},\mathsf{d}_{\times},e^{-V(-d_{0})}\mathfrak{m}_{\times})$
is an $\mathsf{RCD}(K_{0},N+N_{0})$ space, as desired. ∎

######  Proposition 4.12.

The space

$(\mathsf{X}\times\mathbb{R},\mathsf{d}_{\times},e^{-V(-\mathsf{d}_{0})}\mathfrak{m}_{\times})$

is parabolic.

###### Proof.

As we noted earlier the factor $(\mathbb{R},\mathsf{d}_{e},e^{-V}\lambda^{1})$
has finite mass. The statement follows applying Corollary 3.21. ∎

In what follows we denote by $\tilde{\mathfrak{m}}_{\times}$ the measure
$e^{-V(-\mathsf{d}_{0})}\mathfrak{m}_{\times}$ and by $\tilde{\Delta}$ the
laplace operator in
$(\mathsf{X}\times\mathbb{R},\mathsf{d}_{\times},\tilde{\mathfrak{m}}_{\times})$.
We recall that both in
$(\mathsf{X}\times\mathbb{R},\mathsf{d}_{\times},\tilde{\mathfrak{m}}_{\times})$
and $(\mathsf{X}\times\mathbb{R},\mathsf{d}_{\times},\mathfrak{m}_{\times})$
the relaxed gradient of a Lipschitz function coincides with the local
Lipschitz constant, so that in particular it does not change depending on the
space that we consider. The next theorem contains the crucial computation
showing that the Laplacian bounds of Proposition 4.8 are preserved in the
modified product space.

######  Theorem 4.13.

The function $e^{-\bar{\mathsf{d}}}$ satisfies
$\tilde{\Delta}e^{-\bar{\mathsf{d}}}\geq 0$ in $\mathsf{X}\times(-\infty,0)$.
Moreover there exist $\tau>0$ and
$B_{\epsilon}(\bar{x})\subset\mathsf{X}\times(-\infty,0)$ such that
$\tilde{\Delta}e^{-\bar{\mathsf{d}}}\geq\tau>0$ on $B_{\epsilon}(\bar{x})$.

###### Proof.

Using the chain rule for Laplacians and taking into account Proposition 4.10
we have that

$\tilde{\Delta}e^{-\bar{\mathsf{d}}}=e^{-\bar{\mathsf{d}}}(-\tilde{\Delta}\bar{\mathsf{d}}+|\nabla\bar{\mathsf{d}}|^{2})=e^{-\bar{\mathsf{d}}}(-\Delta\bar{\mathsf{d}}+\nabla\bar{\mathsf{d}}\cdot\nabla
V(-\mathsf{d}_{0})+|\nabla\bar{\mathsf{d}}|^{2})=e^{-\bar{\mathsf{d}}}(-\Delta\bar{\mathsf{d}}-V^{\prime}(-d_{0})\nabla\bar{\mathsf{d}}\cdot\nabla\mathsf{d}_{0}+|\nabla\bar{\mathsf{d}}|^{2}).$
(12)

By the fact that $\Delta\mathsf{d}_{0}=0$ and Proposition 4.3 we have that
$-\Delta\bar{\mathsf{d}}=-\Delta\mathsf{d}_{C}\geq 0.$ Moreover

$|\nabla\bar{\mathsf{d}}|^{2}=2-2\nabla\mathsf{d}_{C}\cdot\nabla
d_{0}\quad\text{and}\quad\nabla\bar{\mathsf{d}}\cdot\nabla\mathsf{d}_{0}=\nabla\mathsf{d}_{C}\cdot\nabla\mathsf{d}_{0}-1,$

so that (12) is bounded below by

$\displaystyle
e^{-\bar{\mathsf{d}}}(V^{\prime}(-\mathsf{d}_{0})-V^{\prime}(-\mathsf{d}_{0})\nabla\mathsf{d}_{C}\cdot\nabla\mathsf{d}_{0}+2-2\nabla\mathsf{d}_{C}\cdot\nabla\mathsf{d}_{0})$
$\displaystyle=e^{-\bar{\mathsf{d}}}(2+V^{\prime}(-\mathsf{d}_{0}))(1-\nabla\mathsf{d}_{C}\cdot\nabla\mathsf{d}_{0})=e^{-\bar{\mathsf{d}}}(2+V^{\prime}(-\mathsf{d}_{0}))\frac{|\nabla\bar{\mathsf{d}}|^{2}}{2}.$

Summing up we obtain

$\tilde{\Delta}e^{-\bar{\mathsf{d}}}\geq
e^{-\bar{\mathsf{d}}}(2+V^{\prime}(-\mathsf{d}_{0}))\frac{|\nabla\bar{\mathsf{d}}|^{2}}{2}.$
(13)

Since by construction $|V^{\prime}(-\mathsf{d}_{0})|\leq 1$ the first part of
the statement follows. For the second part it is sufficient to apply
Proposition 4.8 to equation (13). ∎

The next proposition concerns regularity of sets in the modified product space
and is an immediate consequence of Corollary 2.22.

######  Proposition 4.14.

Let $B\subset\mathsf{X}$ be a regular set in
$(\mathsf{X},\mathsf{d},\mathfrak{m})$ and let
$I_{t}:=(t_{x}-t,t_{x}+t)\subset(\mathbb{R},\mathsf{d}_{e},e^{-V}\lambda^{1})$.
Then $B\times I_{t}$ is regular in
$(\mathsf{X}\times\mathbb{R},\mathsf{d}_{\times},\tilde{\mathfrak{m}}_{\times})$.

The next two lemmas concern some simple geometric properties of
$\bar{\mathsf{d}}$.

######  Lemma 4.15.

Let $y\in\mathsf{X}$ and $t\in(-\infty,0)$. Then
$\bar{\mathsf{d}}(y,0)\geq\bar{\mathsf{d}}(y,t)$.

###### Proof.

By triangular inequality we have
$\mathsf{d}_{C}(y,t)\leq\mathsf{d}_{C}(y,0)+t,$ so that

$\bar{\mathsf{d}}(y,t)=\mathsf{d}_{C}(y,t)-t\leq\mathsf{d}_{C}(y,0)=\bar{\mathsf{d}}(y,0),$

as claimed. ∎

The next lemma follows easily by the definition of product distance.

######  Lemma 4.16.

Suppose that $C$ and $\mathsf{X}\times\\{0\\}$ have positive distance and that
there exists a geodesic
$\gamma:[0,\mathsf{d}_{\times}(C,\mathsf{X}\times\\{0\\})]\to\mathsf{X}\times\mathbb{R}$
realizing such distance. Then $\gamma$ is vertical, i.e. its image is
contained in a set of the form $\\{x\\}\times\mathbb{R}$.

The next theorem is the key step to prove the Half Space Property. In the
proof we follow the argument outlined in the Introduction.

######  Theorem 4.17.

The function $\bar{\mathsf{d}}:\mathsf{X}\times(-\infty,0)\to\mathbb{R}$ is
constant.

###### Proof.

Suppose that the statement does not hold.

Assume first that $\mathsf{X}$ has finite diameter. In this case $\mathsf{X}$
is compact, so that the restriction
$\mathsf{d}_{C}:\mathsf{X}\times\\{-1\\}\to\mathbb{R}$ has a minimum on a
point $x\in\mathsf{X}\times\\{-1\\}$. Let $\gamma$ be the connector of $x$ to
$C$ and note that it has to be vertical by Lemma 4.16. In particular
$\bar{\mathsf{d}}$ attains its minimum on all the points in its domain that
are in the image of $\gamma$. By Proposition 4.3 and the maximum principle
this implies that such function is constant.

Assume now that $\mathsf{X}$ has infinite diameter. Then by Theorem 4.13 there
exist $\bar{x}=(x,t_{x})$ and $\epsilon,\tau>0$ such that
$\tilde{\Delta}e^{-\bar{\mathsf{d}}}\geq\tau>0$ on $B_{\epsilon}(\bar{x})$.
Modulo replacing $x$ with a sufficiently close point we may also suppose that
$x$ is regular for $\mathsf{X}$ and that

$\lim_{r\to 0^{+}}\frac{\mathfrak{m}(B_{r}(x))}{r^{k}}\in(0,+\infty),$

where $k\in\mathbb{N}$ is the essential dimension of $\mathsf{X}$.

Let $\phi\in\mathsf{Lip}_{c}(B_{\epsilon}(\bar{x}))$ be a positive function,
which is equal to $\tau$ on $B_{\epsilon/2}(\bar{x})$, takes values in
$[0,\tau]$, and is symmetric with respect to $\mathsf{X}\times\\{t_{x}\\}$. It
is clear that such a function exists. Let also $\\{B_{i}\\}_{i\in\mathbb{N}}$
be an exhaustion of $(\mathsf{X},\mathsf{d},\mathfrak{m})$ with regular sets
containing $x$.

Consider now for every $i\in\mathbb{N}$ and $\bar{z}\in
B_{i}\times[t_{x}-i,t_{x}+i]$ the Green’s function $G_{\bar{z}}^{i}$ with pole
$\bar{z}$ relative to the set

$B_{i}\times[t_{x}-i,t_{x}+i]\subset(\mathsf{X}\times\mathbb{R},\mathsf{d}_{\times},\tilde{\mathfrak{m}}_{\times}),$

and note that such function exists thanks to Propositions 4.14 and 3.10.

In addition, since the space
$(\mathsf{X}\times\mathbb{R},\mathsf{d}_{\times},\tilde{\mathfrak{m}}_{\times})$
is parabolic by Proposition 4.12, then for every $\bar{y}\neq\bar{x}$ (and $i$
sufficiently large) we get by Theorem 4 that

$G_{\bar{x}}^{i}(\bar{y})\to\infty\quad\text{as }i\uparrow+\infty.$

We claim that this implies that for every
$S\subset\subset\mathsf{X}\times\mathbb{R}$ we have

$\inf_{\bar{y}\in S}G_{\bar{x}}^{i}(\bar{y})\to\infty\quad\text{as
}i\uparrow+\infty.$ (14)

This follows because for every $\bar{y}\neq\bar{x}$ the sequence
$\\{G_{\bar{x}}^{i}(\bar{y})\\}_{i\in\mathbb{N}}$ increases as $i$ increases
by Proposition 3.12 while, considering the $[0,+\infty]$-valued continuous
representative given by Lemma 3.17, we have that
$G_{\bar{x}}^{i}(\bar{x})=+\infty$ (here we are using our particular choice of
the point $x\in\mathsf{X}$ and the fact that the essential dimension of the
product space $\mathsf{X}\times\mathbb{R}$ is greater than or equal to $2$).

Consider now for every $i\in\mathbb{N}$ the function
$g_{i}:B_{i}\times[t_{x}-i,t_{x}+i]\to\mathbb{R}$ defined by

$g_{i}(\bar{z}):=\int_{\mathsf{X}}G_{\bar{z}}^{i}(\bar{y})\phi(\bar{y})\,d\tilde{\mathfrak{m}}_{\times}(\bar{y}).$

By Proposition 3.13 each of these functions, in its domain of definition,
satisfies $\tilde{\Delta}g_{i}=-\phi$, is locally Lipschitz and continuous up
to the boundary, and takes value zero on the boundary itself.

Moreover using (14) and the fact that $\phi=\tau$ on $B_{\epsilon/2}(\bar{x})$
it is easy to check that

$g_{i}(\bar{x})\to+\infty\quad\text{as }i\to+\infty.$

In particular we can pick $i_{0}$ large enough so that $g_{i_{0}}(\bar{x})>2$
and $i_{0}>|t_{x}|$. Consider now the function
$e^{-\bar{\mathsf{d}}}+g_{i_{0}}.$ This function must have a maximum in

$B_{i_{0}}\times[t_{x}-i_{0},t_{x}+i_{0}]\cap\mathsf{X}\times(-\infty,0]$

since since it is continuous and
$(e^{-\bar{\mathsf{d}}}+g_{i_{0}})(\bar{x})\geq 2$ while
$(e^{-\bar{\mathsf{d}}}+g_{i_{0}})(\bar{y})\leq 1$ for every

$\bar{y}\in\partial(B_{i_{0}}\times[t_{x}-i_{0},t_{x}+i_{0}]\cap\mathsf{X}\times(-\infty,0])\setminus\mathsf{X}\times\\{0\\}.$
(15)

Moreover by Theorem 4.13 we have

$\tilde{\Delta}(e^{-\bar{\mathsf{d}}}+g_{i_{0}})\geq 0\quad\text{in
}B_{i_{0}}\times[t_{x}-i_{0},t_{x}+i_{0}]\cap\mathsf{X}\times(-\infty,0),$

so that if we can prove that the maximum lies in

$B_{i_{0}}\times[t_{x}-i_{0},t_{x}+i_{0}]\cap\mathsf{X}\times(-\infty,0),$

then by the maximum principle we would obtain that
$e^{-\bar{\mathsf{d}}}+g_{i_{0}}$ is constant, which contradicts the fact that
$(e^{-\bar{\mathsf{d}}}+g_{i_{0}})(\bar{x})\geq 2$ while
$(e^{-\bar{\mathsf{d}}}+g_{i_{0}})(\bar{y})\leq 1$ on the set in (15).

To prove that a maximum lies in the desired set we suppose by contradiction
that all the maximum points lie in $\mathsf{X}\times\\{0\\}$. Let $(y_{0},0)$
be such maximum point, and observe that by Lemma 4.15 we have

$e^{-\bar{\mathsf{d}}}(y_{0},2t_{x})\geq e^{-\bar{\mathsf{d}}}(y_{0},0),$

so that if we can prove that

$g_{i_{0}}(y_{0},2t_{x})=g_{i_{0}}(y_{0},0)$ (16)

we would obtain that also $(y_{0},2t_{x})$ is a maximum point, a
contradiction.

To this aim observe that the function $g_{i_{0}}$ is the only function in

$\mathsf{W}^{1,2}_{0}(B_{i_{0}}\times(t_{x}-i_{0},t_{x}+i_{0}))\cap
C(\bar{B}_{i_{0}}\times[t_{x}-i_{0},t_{x}+i_{0}])$ (17)

taking value zero on the boundary and such that
$\tilde{\Delta}g_{i_{0}}=-\phi$. On the other hand, given that $\phi$ and
$\tilde{\mathfrak{m}}_{\times}$ are symmetric w.r.t.
$\mathsf{X}\times\\{t_{x}\\}$, also the reflected function
$g^{\prime}_{i_{0}}(a,t_{a}):=g_{i_{0}}(a,2t_{x}-t_{a})$ belongs to the set in
(17), is zero on the boundary, and satisfies
$\tilde{\Delta}g^{\prime}_{i_{0}}=-\phi$. In particular
$g^{\prime}_{i_{0}}=g_{i_{0}}$, implying (16). ∎

######  Corollary 4.18.

There exists $a\in(0,+\infty)$ such that $\mathsf{X}\times\\{a\\}\subset C$
and

$C\cap\mathsf{X}\times(-\infty,a)=\emptyset.$

###### Proof.

By the previous proposition the function
$\mathsf{d}_{C}:\mathsf{X}\to\mathbb{R}$ is constant. Let $a>0$ be its
constant value. It is clear that $C\cap\mathsf{X}\times(-\infty,a)=\emptyset$.
To show that $\mathsf{X}\times\\{a\\}\subset C$ suppose by contradiction that
there exists $x\in\mathsf{X}$ such that $(x,a)\notin C$. Then there exists a
ball $B_{\epsilon}(x,a)\subset\mathsf{X}\times\mathbb{R}$ such that
$B_{\epsilon}(x,a)\cap C=\emptyset$ and this implies that
$\mathsf{d}_{C}(x)>a$, which is a contradiction. ∎

The remaining part of the section is dedicated to showing that $C$ is actually
a union of horizontal slices. The idea is that, using the previous corollary,
we can prove that $C\setminus\mathsf{X}\times\\{a\\}$ is locally a boundary of
a locally perimeter minimizing set. Then, reapplying Corollary 4.18 to the
aforementioned set, we obtain that $C\setminus\mathsf{X}\times\\{a\\}$
contains a set of the form $\mathsf{X}\times\\{b\\}$ for some $b>a$ and
$\mathsf{X}\times(-\infty,b)\cap C=\mathsf{X}\times\\{a\\}$. Finally,
iterating this procedure, we obtain that $C$ is made of horizontal slices.

To follow such a plan we first need a few intuitive results whose proofs are
quite technical. These are presented in the next propositions. We will
repeatedly use the fact that
$P(\mathsf{X}\times(-\infty,c),A\times\mathbb{R})=\mathfrak{m}(A)$ for every
$c\in\mathbb{R}$.

######  Proposition 4.19.

Let $x\in\mathsf{X}$ be a regular point for $\mathsf{X}$ and let $U_{(x,a)}$
and $E\subset U_{(x,a)}$ be the sets given by Definition 2.33 applied to $C$
at $x$. Modulo passing to the complement of $E$ we can suppose that
$E\subset\mathsf{X}\times(a,+\infty)\cap U_{(x,a)}$. Then there exists
$\epsilon_{0}>0$ such that for every $\epsilon\in(0,\epsilon_{0})$

$B_{\epsilon}(x)\times(a+\epsilon,a+2\epsilon)\subset E.$

###### Proof.

We first claim that there exists $\epsilon_{0}>0$ and an open set

$B_{2\epsilon_{0}}(x)\times(a-2\epsilon_{0},a+2\epsilon_{0})\subset\mathsf{X}\times\mathbb{R}$

such that for every $(y,t)\in\mathsf{X}\times(a,+\infty)\cap\partial E\cap
B_{2\epsilon_{0}}(x)\times(a-2\epsilon_{0},a+2\epsilon_{0})$ we have

$\frac{|t-a|}{\mathsf{d}(x,y)}<1.$

To prove the claim we suppose by contradiction that there exists a sequence
$\\{(x_{n},t_{n})\\}\subset\mathsf{X}\times(a,+\infty)\cap\partial E$
converging to $(x,a)$ and satisfying

$\frac{|t_{n}-a|}{\mathsf{d}(x,x_{n})}\geq 1\quad\text{for every
}n\in\mathbb{N}.$ (18)

Let $k$ be the essential dimension of $\mathsf{X}$. Let
$(\mathsf{Z},\mathsf{d}_{Z})$ be the space realizing the blow up of
$\mathsf{X}$ at $x$ along the scales $\\{|t_{n}-a|^{-1}\\}_{n\in\mathbb{N}}$,
and observe that
$(\mathsf{Z}\times\mathbb{R},\mathsf{d}_{e}\times\mathsf{d}_{Z})$ naturally
realizes the blow up of $\mathsf{X}\times\mathbb{R}$ at $(x,a)$ along the same
scales. When doing such blow up the set $E$ converges in
$\mathsf{L}^{1}_{loc}$ to a perimeter minimizing set
$E^{\prime}\subset\mathbb{R}^{k}\times\mathbb{R}$ such that

$\mathbb{R}^{k}\times\\{0\\}\subset\partial E^{\prime},\quad
E^{\prime}\subset\mathbb{R}^{k}\times(0,+\infty).$

Moreover, because of (18) and Proposition 2.35, there exists $a^{\prime}>0$
such that $(0,a^{\prime})\in\partial E^{\prime}$. This is a contradiction
because of the global Bernstein Property on $\mathbb{R}^{k+1}$. This proves
our initial claim.

Hence the set

$\Big{\\{}(y,t)\in
B_{2\epsilon_{0}}(x)\times(a-2\epsilon_{0},a+2\epsilon_{0})\cap\mathsf{X}\times(a,+\infty):\frac{|t-a|}{\mathsf{d}(x,y)}>1\Big{\\}}$

is either contained in ${{}^{c}}E$ or in $E$. If we manage to prove that it is
contained in $E$ we are done.

So suppose by contradiction that the previous set is contained in ${{}^{c}}E$.
Then repeating the same blow up argument that we used at the beginning of the
proof we obtain a perimeter minimizing set
$E^{\prime}\subset\mathbb{R}^{k}\times\mathbb{R}$ such that

$\mathbb{R}^{k}\times\\{0\\}\subset\partial E^{\prime},\quad
E^{\prime}\subset\mathbb{R}^{k}\times(0,+\infty)$

and

$\\{(x,t)\in\mathbb{R}^{k}\times\mathbb{R}:|t|>|x|\\}\subset{{}^{c}}E^{\prime}.$

This is again a contradiction by the global Bernstein Property in Euclidean
space. ∎

######  Proposition 4.20.

Let $x\in\mathsf{X}$ and let $U_{(x,a)}$ and $E\subset U_{(x,a)}$ be the sets
given by Definition 2.33 applied to $C$ at $x$. We can assume w.l.o.g. that
$U_{(x,a)}=A\times I$, where $A\subset\mathsf{X}$ is open and $I$ is an
interval, and that $E$ lies above the slice $\mathsf{X}\times\\{a\\}$. Then
for every $\delta>0$ there exists $0<\epsilon<\delta/2$ and an open set
$A^{\prime}\subset A$ such that

$\mathfrak{m}(A\setminus A^{\prime})<\delta,\quad\text{and
}A^{\prime}\times(a+\epsilon,a+2\epsilon)\subset E.$

###### Proof.

For every regular $x\in A$ consider the neighbourhood $V_{x}\subset A$ and the
number $\epsilon_{x}>0$ given by the previous proposition. Define then

$A_{t}:=\\{x\in A:\epsilon_{x}>t\\}$

and, choosing $\epsilon$ small enough, we have that $\mathfrak{m}(A\setminus
A_{\epsilon})<\delta$. Setting

$A^{\prime}:=\bigcup_{x\in A_{\epsilon}}V_{x}$

we conclude. ∎

In the next propositions the symbols $U_{(x,a)}$, $A$, $I$ and $E$ will denote
the objects introduced in the statement of the previous proposition.

######  Proposition 4.21.

We have

$P(E,A\times\\{a\\})\geq\mathfrak{m}(A).$

###### Proof.

Fix $\delta>0$ and consider the open set $A^{\prime}\subset A$ and the number
$0<\epsilon<\delta/2$ such that

$\mathfrak{m}(A\setminus A^{\prime})<\delta,\quad\text{and
}A^{\prime}\times(a+\epsilon,a+2\epsilon)\subset E.$

We claim that

$P(E,A^{\prime}\times(a-\delta,a+\delta))\geq\mathfrak{m}(A^{\prime}).$ (19)

If the claim holds we then obtain that

$P(E,A\times(a-\delta,a+\delta))\geq\mathfrak{m}(A)-\delta,$

and passing to the limit on both sides as $\delta$ goes to zero we conclude.

To prove (19) consider a sequence
$\\{f_{i}\\}_{i\in\mathbb{N}}\in\mathsf{Lip}_{loc}(A^{\prime}\times(a-\delta,a+\delta))$
such that

$f_{i}\to 1_{E}\quad\text{in
}\mathsf{L}^{1}(A^{\prime}\times(a-\delta,a+\delta))$

and

$P(E,A^{\prime}\times(a-\delta,a+\delta))=\lim_{i\to+\infty}\int_{A^{\prime}\times(a-\delta,a+\delta)}\mathsf{lip}(f_{i})\,d\mathfrak{m}_{\times}.$
(20)

We claim that we can replace each $f_{i}$ with functions $g_{i}$ mantaining
the same properties and such that $g_{i}=0$ on
$A^{\prime}\times\\{a-3\epsilon/2\\}$ and $g_{i}=1$ on
$A^{\prime}\times\\{a+3\epsilon/2\\}$. This is possible because (20) implies,
though a lower semi-continuity argument and the fact that
$A^{\prime}\times(a+\epsilon,a+2\epsilon)\subset E$ and
$A^{\prime}\times(a-2\epsilon,a-\epsilon)\subset{{}^{c}}E$, that

$\lim_{i\to+\infty}\int_{A^{\prime}\times(a+5\epsilon/4,a+5\epsilon/3)}\mathsf{lip}(f_{i})\,d\mathfrak{m}_{\times}=\lim_{i\to+\infty}\int_{A^{\prime}\times(a-5\epsilon/3,a-5\epsilon/4)}\mathsf{lip}(f_{i})\,d\mathfrak{m}_{\times}=0.$

These identities allow, though a cut off argument, to construct
$g_{i}\in\mathsf{Lip}_{loc}(A^{\prime}\times(a-\delta,a+\delta))$ such that

$\displaystyle g_{i}\to 1_{E}$ $\displaystyle\quad\text{in
}\mathsf{L}^{1}(A^{\prime}\times(a-\delta,a+\delta)),$ $\displaystyle
P(E,A^{\prime}\times(a-\delta,a+\delta))$
$\displaystyle=\lim_{i\to+\infty}\int_{A^{\prime}\times(a-\delta,a+\delta)}\mathsf{lip}(g_{i})\,d\mathfrak{m}_{\times}$

and

$g_{i}=0\text{ on }A^{\prime}\times\\{-3\epsilon/2\\};\quad g_{i}=1\text{ on
}A^{\prime}\times\\{3\epsilon/2\\}.$

In particular, denoting by $\mathsf{lip}_{t}$ the local Lipschitz constant of
a function restricted to the vertical component of the product space, we
obtain

$\displaystyle\int_{A^{\prime}\times(a-\delta,a+\delta)}\mathsf{lip}(g_{i})\,d\mathfrak{m}_{\times}\geq\int_{A^{\prime}\times(a-\delta,a+\delta)}\mathsf{lip}_{t}(g_{i})\,d\mathfrak{m}_{\times}=\int_{A^{\prime}}\int_{(a-\delta,a+\delta)}\mathsf{lip}_{t}(g_{i})\,ds\,d\mathfrak{m},$

and by the condition on the values of $g_{i}$ on the horizontal slices the
quantity $\int_{(a-\delta,a+\delta)}\mathsf{lip}_{t}(g_{i})\,ds$ is greater
than $1$ for every $x\in A^{\prime}$, so that the last integral is greater
than or equal to $\mathfrak{m}(A^{\prime})$. Summing up we obtained

$P(E,A^{\prime}\times(a-\delta,a+\delta))=\lim_{i\to+\infty}\int_{A^{\prime}\times(a-\delta,a+\delta)}\mathsf{lip}(g_{i})\,d\mathfrak{m}_{\times}\geq\mathfrak{m}(A^{\prime}),$

as desired. ∎

######  Proposition 4.22.

For every $\delta_{1},\delta_{2},\delta_{3}>0$ there exists
$0<\epsilon<\delta_{1}$ such that

$P(E\cap\mathsf{X}\times(-\infty,a+\epsilon),A\times(a,a+\delta_{2}))\geq\mathfrak{m}(A)-\delta_{3}.$

###### Proof.

Let $\delta:=\min\\{\delta_{1},\delta_{2},\delta_{3}\\}$. By Proposition 4.20
there exist $0<\epsilon^{\prime}<\delta/2$ and an open set $A^{\prime}\subset
A$ such that

$\mathfrak{m}(A\setminus A^{\prime})<\delta,\quad\text{and
}A^{\prime}\times(a+\epsilon^{\prime},a+2\epsilon^{\prime})\subset E.$

Setting $\epsilon:=3\epsilon^{\prime}/2$, and taking into account that we have

$E\cap\mathsf{X}\times(-\infty,a+\epsilon)\cap
A^{\prime}\times(a+\epsilon^{\prime},a+2\epsilon^{\prime})=\mathsf{X}\times(-\infty,a+\epsilon)\cap
A^{\prime}\times(a+\epsilon^{\prime},a+2\epsilon^{\prime}),$

we get that

$P(E\cap\mathsf{X}\times(-\infty,a+\epsilon),A\times(a,a+\delta))\geq\mathfrak{m}(A^{\prime})\geq\mathfrak{m}(A)-\delta,$

which immediately implies the statement. ∎

In what follows we will use the notation
$E^{\prime}:=E\cup\mathsf{X}\times(-\infty,a)$. The next lemma is the key step
to show that $C\setminus\mathsf{X}\times\\{a\\}$ is locally a boundary of a
locally perimeter minimizing set.

######  Lemma 4.23.

We have that

$P(E,A\times I)-P(\mathsf{X}\times(-\infty,a),A\times I)\geq
P(E^{\prime},A\times I).$

###### Proof.

By Proposition 4.21 we have that

$P(E,A\times I)-P(\mathsf{X}\times(-\infty,a),A\times I)\geq
P(E,A\times(a,b))$

and this last quantity can be rewritten as

$P(E,A\times(a,b))=P(E^{\prime},A\times(a,b))=P(E^{\prime},A\times
I)-P(E^{\prime},A\times\\{a\\}).$

In particular to conclude we only need to show that

$P(E^{\prime},A\times\\{a\\})=0.$ (21)

To this aim fix $\delta>0$ and, using Proposition 4.22, pick a sequence
decreasing to zero $\\{\epsilon_{i}\\}_{i\in\mathbb{N}}$ such that

$P(E\cap\mathsf{X}\times(-\infty,a+\epsilon_{i}),A\times(a,a+\delta))\geq\mathfrak{m}(A)-1/i.$
(22)

So, by the lower semi-continuity of perimeters,

$\displaystyle P(E^{\prime},A\times(a-\delta,a+\delta))$
$\displaystyle\leq\liminf_{i\to+\infty}P(E\cup\mathsf{X}\times(-\infty,a+\epsilon_{i}),A\times(a-\delta,a+\delta))$
$\displaystyle=\liminf_{i\to+\infty}P(E\cup\mathsf{X}\times(-\infty,a+\epsilon_{i}),A\times(a,a+\delta)).$

Using (22) we get

$\displaystyle P(E\cup\mathsf{X}$
$\displaystyle\times(-\infty,a+\epsilon_{i}),A\times(a,a+\delta))$
$\displaystyle\leq
P(E,A\times(a,a+\delta))+\mathfrak{m}(A)-P(E\cap\mathsf{X}\times(-\infty,a+\epsilon_{i}),A\times(a,a+\delta))$
$\displaystyle\leq P(E,A\times(a,a+\delta))+1/i.$

Summing up we obtained that

$P(E^{\prime},A\times(a-\delta,a+\delta))\leq P(E,A\times(a,a+\delta)),$

which implies (21). ∎

######  Proposition 4.24.

$E^{\prime}$ is locally perimeter minimizing in $U_{(x,a)}$.

###### Proof.

Let $B\subset\mathsf{X}\times\mathbb{R}$ be a Borel set such that $B\Delta
E^{\prime}\subset\subset U_{(x,a)}$ and note that
$B\cap\mathsf{X}\times(a,+\infty)$ is a competitor for $E$ in $U_{(x,a)}$. In
particular, using subadditivity of perimeters first and the fact that $E$
minimizes the perimeter in $U_{(x,a)}$ later, we get

$P(B,U_{(x,a)})\geq
P(B\cap\mathsf{X}\times(a,+\infty),U_{(x,a)})-P(\mathsf{X}\times(a,+\infty),U_{(x,a)})$
$\geq P(E,U_{(x,a)})-P(\mathsf{X}\times(a,+\infty),U_{(x,a)}).$

By Proposition 4.23 this last quantity is greater than or equal to
$P(E^{\prime},U_{(x,a)})$, so that summing up we obtained

$P(B,U_{(x,a)})\geq P(E^{\prime},U_{(x,a)}),$

concluding the proof. ∎

######  Lemma 4.25.

Let $y\in A$ be a regular point of $\mathsf{X}$, then $(y,a)\notin\partial
E^{\prime}$. In particular there exists a small ball
$B\subset\mathsf{X}\times\mathbb{R}$ containing $(y,a)$ such that
$B\cap\partial E^{\prime}=\emptyset$.

###### Proof.

Suppose by contradiction that $y\in\partial E^{\prime}$. Now let
$(\mathsf{Z},\mathsf{d}_{Z})$ be the space realizing the blow up of
$\mathsf{X}$ at $y$ and observe that
$(\mathsf{Z}\times\mathbb{R},\mathsf{d}_{e}\times\mathsf{d}_{Z})$ naturally
realizes the blow up of $\mathsf{X}\times\mathbb{R}$ at $(y,a)$ along the same
scales. When doing such blow up the set $E^{\prime}$ converges in
$\mathsf{L}^{1}_{loc}$ to a perimeter minimizing set
$E^{\prime\prime}\subset\mathbb{R}^{k}\times\mathbb{R}$ such that

$\mathbb{R}^{k}\times(-\infty,0)\subset E^{\prime\prime}\quad\text{and}\quad
0\in\partial E^{\prime\prime}.$

Moreover $0\in\partial(E^{\prime\prime}\cap\mathbb{R}^{k}\times(0,+\infty))$
since $E^{\prime\prime}\cap\mathbb{R}^{k}\times(0,+\infty)$ is the blow up of
$E$ at $(y,a)$. In particular the complement ${{}^{c}}E^{\prime\prime}$ cannot
be a half space, contradicting the global Bernstein Property in Euclidean
space. ∎

######  Corollary 4.26.

$\partial E^{\prime}\cap\mathsf{X}\times\\{a\\}\cap U_{(x,a)}=\emptyset$.

###### Proof.

Suppose by contradiction that $(y,a)\in\partial
E^{\prime}\cap\mathsf{X}\times\\{a\\}\cap U_{(x,a)}$. Consider then the
function $\mathsf{d}_{\partial E^{\prime}}+|\mathsf{d}_{0}|$ defined in a
small neighbourhood of $(y,a)$ in $\mathsf{X}\times(-\infty,a)$. If the
neighbourhood is small enough this function is super-harmonic (since
$E^{\prime}$ is locally perimeter minizing) and attains its minimum on each
point of the vertical line connecting $(y,a)$ and $(y,0)$ (since
$(y,a)\in\partial E^{\prime}$). Hence the aforementioned function has to be
constant, contradicting the previous lemma. ∎

######  Lemma 4.27.

$\overline{(C\setminus\mathsf{X}\times\\{a\\})}\cap\mathsf{X}\times\\{a\\}=\emptyset$.
In particular $C\setminus\mathsf{X}\times\\{a\\}$ is still locally a boundary
of a locally perimeter minizing set.

###### Proof.

It is immediate to check that for every $x\in\mathsf{X}$ we have

$U_{(x,a)}\cap\overline{(C\setminus\mathsf{X}\times\\{a\\})}\cap\mathsf{X}\times\\{a\\}\subset
U_{(x,a)}\cap\partial E^{\prime}\cap\mathsf{X}\times\\{a\\}$

and this last set is empty by the previous lemma. ∎

The next theorem gives the explicit description of $C$ and this immediately
implies the Half Space Property for $\mathsf{RCD}(0,N)$ spaces, i.e. Theorem
1.

######  Theorem 4.28.

There exists a sequence $\\{a_{i}\\}_{i\in\mathbb{N}}\subset(0,+\infty]$
increasing to $+\infty$ and $N\in\mathbb{N}\cup\\{+\infty\\}$ such that

$C=\bigcup_{i\in\mathbb{N},\,i\leq N}\mathsf{X}\times\\{a_{i}\\}.$

###### Proof.

We know that there exists $a_{1}:=a>0$ such that

$\mathsf{X}\times(-\infty,a_{1})\cap
C=\emptyset\quad\text{and}\quad\mathsf{X}\times\\{a_{1}\\}\subset C.$

If $C=\mathsf{X}\times\\{a_{1}\\}$ we have nothing left to prove, so suppose
that this is not the case. Then the set
$C_{1}:=C\setminus\mathsf{X}\times\\{a_{1}\\}$ is still locally a boundary of
a perimeter minimizing set, so that there exists $a_{2}>a_{1}$ such that

$\mathsf{X}\times(-\infty,a_{2})\cap
C_{1}=\emptyset\quad\text{and}\quad\mathsf{X}\times\\{a_{2}\\}\subset C_{1}.$

If $C_{2}:=C_{1}\setminus\mathsf{X}\times\\{a_{2}\\}=\emptyset$ the proof is
concluded, otherwise we apply the same argument to
$C_{2}:=C_{1}\setminus\mathsf{X}\times\\{a_{2}\\}$ and we keep iterating this
process as long as $C_{k}\neq\emptyset$. If there exists $N\in\mathbb{N}$ such
that $C_{N}=\emptyset$ we set $a_{i}=+\infty$ for every $i\geq N+1$ and the
proof is concluded. If this is not the case, and $a_{i}\to+\infty$ we set
$N:=+\infty$ and the proof is again concluded.

We claim that the previous two cases are the only possible. Indeed if
$C_{k}\neq\emptyset$ for every $k$ and there exists $b\in\mathbb{R}$ such that
$a_{i}\uparrow b$, then in any neighbourhood of a point in
$\mathsf{X}\times\\{b\\}$ the set $E$ whose boundary locally coincides with
$C$ would have infinite perimeter, a contradiction. ∎

We conclude the section with a counterexample showing that there exists an
infinitesimally Hilbertian metric measure space which is parabolic (meaning
that it admits a point $x$ with no positive Green’s function with pole $x$)
and does not have the Half Space Property.

######  Example 4.29.

Consider the metric measure space $(\mathsf{X},\mathsf{d},\mathfrak{m})$
obtained by gluing $([0,+\infty),\mathsf{d}_{e},(x\wedge 1)\lambda^{1})$ and
$(\mathbb{R},\mathsf{d}_{e},\lambda^{1})$ in the respective origins. Using the
fact that weighted manifolds are infinitesimally Hilbertian (see [55]), and
that such space is a weighted manifold outside of the gluing point, we obtain
that $\mathsf{X}$ is infinitesimally Hilbertian as well. We now show that
$\mathsf{X}$ is not parabolic in $2$ steps.

Step 1: Denote by $(a_{\mathbb{R}},b_{\mathbb{R}})$ intervals in
$(\mathbb{R},\mathsf{d}_{e},\lambda^{1})$ and by
$(a_{\mathbb{\mathsf{Y}}},b_{\mathbb{\mathsf{Y}}})$ intervals in
$([0,+\infty),\mathsf{d}_{e},(x\wedge 1)\lambda^{1})$. We claim that, given
$c_{\mathbb{R}}\in(0,1)$, if a function
$u\in\mathsf{W}^{1,1}((-c_{\mathbb{R}},c_{\mathbb{R}})\cup(0_{\mathsf{Y}},c_{\mathsf{Y}}))$
is harmonic in its domain, then the restriction of $u$ to
$(-c_{\mathbb{R}},c_{\mathbb{R}})$ is affine.
To this aim note that the restriction of $u$ to
$(-c_{\mathbb{R}},c_{\mathbb{R}})$ belongs to
$\mathsf{W}^{1,1}((-c_{\mathbb{R}},c\textbf{}))$ and in particular it is
continuous.

Moreover by restricting $u$ to open sets of its domain that do not contain the
gluing point we obtain, by harmonicity, that $u$ is affine in
$(-c_{\mathbb{R}},0_{\mathbb{R}})$ and $(0_{\mathbb{R}},c_{\mathbb{R}})$
(possibly with different slopes) and that $u$ is smooth in
$(0_{\mathsf{Y}},c_{\mathsf{Y}})$. Hence, if we manage to prove that the
restriction of $u$ to $(-c_{\mathbb{R}},c_{\mathbb{R}})$ is $C^{1}$ we
conclude the proof of Step $1$.

To this aim let
$\phi\in\mathsf{Lip}_{c}((-c_{\mathbb{R}},c_{\mathbb{R}})\cup(0_{\mathsf{Y}},c_{\mathsf{Y}}))$
and consider the harmonicity condition (we are using that relaxed gradients
coincide with local Lipschitz constants in our space and we are denoting by
$u^{\prime}$ and $\phi^{\prime}$ standard derivatives):

$0=\int_{(-c_{\mathbb{R}},c_{\mathbb{R}})\cup(0_{\mathsf{Y}},c_{\mathsf{Y}})}\nabla
u\cdot\nabla\phi\,d\mathfrak{m}=\int_{-c_{\mathbb{R}}}^{0}u^{\prime}(x)\phi^{\prime}(x)\,d\lambda^{1}+\int_{0}^{c_{\mathbb{R}}}u^{\prime}(x)\phi^{\prime}(x)\,d\lambda^{1}+\int_{0_{\mathsf{Y}}}^{c_{\mathsf{Y}}}u^{\prime}(x)\phi^{\prime}(x)x\,d\lambda^{1}(x).$

Integrating by parts and using the previously found regularity we obtain

$0=\phi(0)\Big{(}\lim_{\begin{subarray}{c}x\in(-c_{\mathbb{R}},0)\\\ x\to
0\end{subarray}}u^{\prime}(x)-\lim_{\begin{subarray}{c}x\in(0,c_{\mathbb{R}})\\\
x\to 0\end{subarray}}u^{\prime}(x)\Big{)},$

which shows that the restriction of $u$ to $(-c_{\mathbb{R}},c_{\mathbb{R}})$
is $C^{1}$ and concludes the proof of Step $1$.

Step 2: We now prove that $\mathsf{X}$ is not parabolic.
By step $1$ we deduce that if $x\in\mathbb{R}\setminus\\{0\\}$ and $G_{x}$ is
the corresponding Green’s function on $\mathsf{X}$, then the restriction of
$G_{x}$ to $\mathbb{R}$ is a Green’s function on the standard real line. This
in particular implies that $G_{x}$ cannot be positive, proving that
$\mathsf{X}$ is parabolic.

We finally show that the Half Space Property fails on $\mathsf{X}$. This is
done again in two steps.

Step 1. Consider the function $1_{\mathbb{R}}:\mathsf{X}\to\mathbb{R}$ defined
as $1_{\mathbb{R}}\equiv 1$ on $\mathbb{R}$ and $1_{\mathbb{R}}\equiv 0$ on
$\mathsf{X}\setminus\mathbb{R}$. We claim that its epigraph
$\mathsf{Epi}(1_{\mathbb{R}})\subset\mathsf{X}\times\mathbb{R}$ satisfy
$P(\mathsf{Epi}(1_{\mathbb{R}}),\\{0\\}\times\mathbb{R})=0$.
Since $P(\mathsf{Epi}(1_{\mathbb{R}}),\\{0\\}\times\\{0,1\\})=0$, to prove the
above claim it is sufficient to show that
$P(\mathsf{Epi}(1_{\mathbb{R}}),\\{0\\}\times[\epsilon,1-\epsilon])=0$ for
every $\epsilon>0$ small enough. This would follow if we can show that for
$\delta>0$ fixed we have

$P(\mathsf{Epi}(1_{\mathbb{R}}),B^{\mathsf{X}}_{\delta}(0)\times[\epsilon,1-\epsilon])=0.$

So let $\delta>0$ be fixed and consider the sequence $f_{n}$ given by