# Fill-ins with scalar curvature lower bounds and applications to positive
mass theorems

Stephen McCormick  Institutionen för teknikvetenskap och matematik
Luleå tekniska universitet
971 87 Luleå
Sweden<EMAIL_ADDRESS>
###### Abstract.

Given a constant $C$ and a smooth closed $(n-1)$-dimensional Riemannian
manifold $(\Sigma,g)$ equipped with a positive function $H$, a natural
question to ask is whether this manifold can be realised as the boundary of a
smooth $n$-dimensional Riemannian manifold with scalar curvature bounded below
by $C$. That is, does there exist a _fill-in_ of $(\Sigma,g,H)$ with scalar
curvature bounded below by $C$?

We use variations of an argument due to Miao and the author [21] to explicitly
construct fill-ins with different scalar curvature lower bounds, where we
permit the fill-in to contain another boundary component provided it is a
minimal surface. Our main focus is to illustrate the applications of such
fill-ins to geometric inequalities in the context of general relativity. By
filling in a manifold beyond a boundary, one is able to obtain lower bounds on
the mass in terms of the boundary geometry through positive mass theorems and
Penrose inequalities. We consider fill-ins with both positive and negative
scalar curvature lower bounds, which from the perspective of general
relativity corresponds to the sign of the cosmological constant, as well as a
fill-in suitable for the inclusion of electric charge.

###### Contents

  1. 1 Introduction
  2. 2 Brief Background
  3. 3 General Fill-In Construction
  4. 4 Fill-Ins With Scalar Curvature Lower Bounds
    1. 4.1 Negative scalar curvature lower bound
    2. 4.2 Positive scalar curvature lower bound
  5. 5 Applications
    1. 5.1 The asymptotically hyperbolic positive mass theorem with boundary
    2. 5.2 A Penrose-like inequality with electric charge
    3. 5.3 Engelhardt–Wall Outer Entropy and Bray’s Inner Bartnik Mass

##  1\. Introduction

Given a closed $(n-1)$-dimensional Riemannian manifold $(\Sigma,g)$ equipped
with a smooth function $H$, a natural question to ask is whether or not the
triple $(\Sigma,g,H)$ admits a non-negative scalar curvature fill-in. That is,
can $(\Sigma,g)$ be realised as the boundary of an $n$-dimensional manifold
$(\Omega,\gamma)$ with non-negative scalar curvature and outward-pointing mean
curvature $H$?. Recently the problem was highlighted by Gromov (Problem A in
[13]) and it has been the subject of several recent works (for example, [15,
28, 29]). Here we are predominantly interested in applications of such fill-
ins to mathematical general relativity, where the fill-in $\Omega$ is often
permitted to have another boundary component, provided that it is a minimal
surface. Indeed in many situations where the problem arises, such a minimal
surface inner boundary can itself be filled in and therefore causes no
problems.

In what follows, we will refer to a triple $(\Sigma,g,H)$ as Bartnik data
following the literature on Bartnik’s quasi-local mass, which concerns the
closely related problem of non-negative scalar curvature extensions – that is,
“filling out” to infinity – as opposed to fill-ins (see, for example, the
review articles [6] and [19]). Rather than focusing purely on “fill-ins” with
non-negative scalar curvature, it is interesting to ask if $(\Sigma,g,H)$ can
be realised as the boundary of a Riemannian manifold with any prescribed
scalar curvature lower bound.

This article serves to highlight an elementary approach to constructing fill-
ins used in [21] and demonstrate some new results using the same technique,
with an emphasis on the applications of such fill-ins. For example, these
fill-ins can lead to a kind of positive mass theorem with boundary, which we
illustrate in the case of asymptotically hyperbolic manifolds (Theorem 1.2) as
well as an example for asymptotically flat manifolds equipped with an electric
field (Theorem 5.2). Since the type of fill-ins constructed have a minimal
surface inner boundary, which means they also give a lower bound on the inner
Bartnik mass – a notion of quasi-local mass formulated by Bray [3] in the
spirit of the Bartnik mass, replacing the aforementioned extensions with fill-
ins. We state some of the main results more precisely below. The reader is
directed to Section 2 for more background on the problems considered here.

######  Theorem 1.1 (Fill-ins with negative scalar curvature lower bound).

Let $(\Sigma,g,H)$ be $(n-1)$-dimensional Bartnik data with $H>0$, and $C<0$
be a negative constant such that

$R_{g}-2H\Delta H^{-1}>0$

and

(1.1) $\max\left\\{\max_{\Sigma}\frac{n-2}{n-1}\,\frac{H^{2}}{R_{g}-2H\Delta
H^{-1}},\max_{\Sigma}\frac{H^{2}r_{o}^{2}}{(n-1)^{2}}\right\\}<1-\frac{C}{n(n-1)}r_{o}^{2}$

where $R_{g}$ is the scalar curvature of $g$ and $r_{o}$ is the area radius of
$(\Sigma,g)$. Then there exists a compact manifold $(\Omega,\gamma)$ with
boundary $\partial\Omega=\Sigma_{o}\cup\Sigma_{H}$ where $\Sigma_{o}$ is
isometric to $(\Sigma,g)$ with outward-pointing mean curvature $H$,
$\Sigma_{H}$ is minimal, and the scalar curvature $R_{\gamma}$ of $\gamma$ is
greater than $C$.

######  Remark 1.1.

The condition (1.1) is a technical condition that arises due to the
construction itself and can be thought of as a kind of “pointwise Hawking mass
positivity”. What we mean by this is that in the case that $R_{g}$ and $H$ are
constant, then the condition reduces to simply positivity of the a higher
dimensional Hawking mass tailored to the appropriate cosmological constant.
Note that while the Hawking mass is generally considered in dimension $3$ and
there are several inequivalent formulations of the Hawking mass in higher
dimensions, when $H$ is constant they all agree with the expected value for
spheres in Schwarzschild–anti-de Sitter manifolds. Namely, the expression

$m_{H}(\Sigma,g,H;C)=\frac{r_{o}}{2}\left(1-\frac{r_{o}^{2}}{(n-1)^{2}}\left(H^{2}+\frac{C(n-1)}{n}\right)\right).$

From this we are able to prove the following positive mass theorem for
asymptotically hyperbolic manifolds with boundary using the same method as was
used to obtain a Penrose-like inequality in [21]. Specifically, we obtain the
following.

######  Theorem 1.2 (Positive mass theorem for asymptotically hyperbolic
manifold with boundary).

Let $(M,\gamma)$ be an asymptotically hyperbolic $n$-manifold that is spin,
with scalar curvature $R_{\gamma}\geq C$ for some $C<0$, and inner boundary
$\Sigma=\partial M$. Assume that the Bartnik data $(\Sigma,g,H)$ induced on
the boundary satisfies

$R_{g}-2H\Delta H^{-1}>0$

and

$\max\left\\{\max_{\Sigma}\frac{n-2}{n-1}\,\frac{H^{2}}{R_{g}-2H\Delta
H^{-1}},\max_{\Sigma}\frac{H^{2}r_{o}^{2}}{(n-1)^{2}}\right\\}<1-\frac{C}{n(n-1)}r_{o}^{2}.$

Then $(M,\gamma)$ has positive asymptotically hyperbolic mass, in the sense of
Wang [32].

######  Remark 1.2.

The definition of mass in the asymptotically hyperbolic case is somewhat
subtle, however the precise definition will not be required here. This theorem
follows directly from applying a known positive mass theorem to a manifold
with corners, and for the sake of simplicity we apply the work of Bonini and
Qing directly [2], who prove a positive mass theorem for such manifolds with
corners using Wang’s definition of mass [32].

It is important to remark that the known positive mass theorem for
asymptotically hyperbolic manifolds in fact already applies for manifolds with
boundaries like those considered here provided that
$H\leq\sqrt{\frac{-C(n-1)}{n}}$. So from this perspective, the result provides
only a minor extension. The positive mass theorem with boundary presented here
is more interesting when thought of as leading to a Penrose-like inequality
assuming that the Riemannian Penrose inequality is established for
asymptotically hyperbolic manifolds. Details of this Penrose-like inequality
are given in Section 5, however in order to rigorously prove it we would first
require a proof of the Riemannian Penrose inequality in the asymptotically
hyperbolic case, which remains an open problem.

The next result demonstrates the existence of fill-ins with a non-negative
scalar curvature lower bound.

######  Theorem 1.3 (Fill-ins with non-negative scalar curvature lower bound).

Let $(\Sigma,g)$ be a closed $(n-1)$-dimensional manifold, $H$ be a positive
function. Suppose for some constant $C\geq 0$ we have

(1.2) $R_{g}-2H\Delta H^{-1}>\frac{n(n-2)H^{2}}{n(n-1)-Cr_{o}^{2}}$

where $R_{g}$ is the scalar curvature of $g$ and $r_{o}$ is the area radius of
$(\Sigma,g)$, which we further ask satisfies $r_{o}<\sqrt{\frac{n(n-1)}{C}}$
when $C>0$. Then there exists a compact manifold with scalar curvature bounded
below by $C$, whose boundary consists of two disconnected components, one
being a minimal surface and the other isometric to $(\Sigma,g)$ with outward-
pointing mean curvature $H$.

######  Remark 1.3.

Note that for coordinate $(n-1)$-spheres in an $n$-sphere, one would have
equality in (1.2). When $C=0$, this reduces precisely to the case established
by Miao and the author in [21].

######  Remark 1.4.

There is no positive mass theorem or Penrose-like inequality in this case, as
the counterexamples of Brendle, Marques and Neves, to Min-Oo’s Conjecture [4]
demonstrate that a positive mass theorem in the usual sense does not hold
here. This can be seen explicitly in Proposition 4.1, which demonstrates that
there exist complete fill-ins of Bartnik data corresponding to the
cosmological horizon in a negative mass Schwarzschild–de Sitter manifold.

As another application of the fill-in construction, Section 5.2 establishes a
charged Penrose-like inequality. That is, a lower bound on the total mass of
an asymptotically flat manifold with boundary, equipped with an electric
field, satisfying dominant energy conditions. This is stated precisely in
Theorem 5.1.

This article is organised as follows. Section 2 provides a brief background
and overview of some related results before Section 3 gives the general fill-
in construction used here. Section 4 constructs the fill-ins required for
positive and negative scalar curvature lower bounds, and finally Section 5
provides all of the applications to positive mass-type theorems.

##  2\. Brief Background

A Riemannian manifold with scalar curvature bounded below by a constant $C$
may be interpreted as time-symmetric initial data for the Einstein equations
satisfying the dominant energy condition with cosmological constant
$\Lambda=C/3$. In the case where $C\leq 0$ the notion of total mass of an
isolated system in the context of general relativity is well-understood,
corresponding to the mass of an asymptotically hyperbolic manifold ($C<0$) or
an asymptotically flat manifold ($C=0$). It is therefore no surprise that
notions of mass in general relativity are connected with the problem of fill-
ins with scalar curvature lower bounds.

More specifically, for $C=0$ the positive mass theorem [25, 33] is a well-
known foundational result in mathematical relativity, and for $C<0$ analogous
positive mass theorems have been established for asymptotically hyperbolic
manifolds [9, 32]. That is, a complete asymptotically flat (resp.
asymptotically hyperbolic) Riemannian manifold with scalar curvature bounded
below by $0$ (resp. $C$, where $C<0$) has non-negative mass. The case where
$C>0$, corresponding to a positive cosmological constant, does not have a
standard notion of mass nor appear to have any prospect for a positive mass
theorem to hold in general (see Remark 1.4 and Proposition 4.1).

From the known positive mass theorems, it follows that if Bartnik data
$(\Sigma,g,H)$ can be realised as the boundary of an asymptotically flat or
asymptotically hyperbolic manifold with scalar curvature lower bound of $C$
and negative mass (defined appropriately for the value of $C$), then it cannot
admit a fill-in with scalar curvature bounded below by $C$. This is due to
that fact that if such a fill-in exists then through a gluing procedure one
could obtain an asymptotically flat or asymptotically hyperbolic manifold with
minimimal surface boundary (if the boundary is non-empty) and negative mass,
which would contradict the relevant positive mass theorem. Similarly, quasi-
local positive mass theorems can also provide obstructions to the existence of
fill-ins. For example, Shi and Tam’s proof of the positivity of the Brown–York
mass can be rephrased as a result on fill-ins as follows.

######  Theorem 2.1 (Shi–Tam [26]).

Let $(\Sigma,g,H)$ be $(n-1)$-dimensional Bartnik data where $3\leq n\leq 7$
and with $H>0$, and such that $g$ is isometric to strictly convex closed
hypersurface in $\mathbb{R}^{n}$. Then if

(2.1) $\int_{\Sigma}Hd\mu_{g}>\int_{\Sigma}H_{o}d\mu_{g},$

where $H_{o}$ is the mean curvature of $(\Sigma,g)$ isometrically embedded
$\mathbb{R}^{n}$, there exists no fill-in of $(\Sigma,g,H)$ with non-negative
scalar curvature.

######  Remark 2.1.

When $n=3$, the existence of the required isometric embedding is well-known to
be equivalent to the condition that $g$ has positive Gauss curvature [23, 24].
Furthermore, the dimensional restriction here is simply required to apply the
positive mass theorem.

Shi and Tam also proved a positivity statement for an asymptotically
hyperbolic quasi-local mass in dimension $3$, which similarly gives the non-
existence of fill-ins with negative scalar curvature lower bounds as follows.

######  Theorem 2.2 (Shi–Tam [27], see also Wang–Yau [30]).

Let $(\Sigma,g,H)$ be $2$-dimensional topologically spherical Bartnik data
with $H>0$ and Gauss curvature $K_{g}>\frac{C}{6}$ for some $C<0$ that
satisfies

$\int_{\Sigma}(H_{o}-H)\cosh(-\sqrt{\frac{C}{6}}\,r)d\mu_{g}<0$

where $H_{o}$ is the mean curvature of $(\Sigma,g)$ isometrically embedded in
$3$-dimensional hyperbolic space with constant scalar curvature equal to $C$,
and $r$ is the geodesic distance function from a fixed point in $\Sigma$. Then
$(\Sigma,g,H)$ admits no fill-in with scalar curvature bounded below by $C$.

######  Remark 2.2.

The isometric embedding into hyperbolic space required for Theorem 2.2 not
only exists, but is unique up to an isometry of hyperbolic space [7, 24]. It
seems likely that a version of Theorem 2.2 would also hold in higher
dimensions, however some care should be taken in checking the details with
particular attention given to the existence of required isometric embedding.

It will be a recurring theme that the size of $H$ governs whether or not a
fill-in exists. Jauregui [15] shows this in a clear way with the following
theorem.

######  Theorem 2.3 (Jauregui [15]).

Let $(\Sigma,g,H)$ be $2$-dimensional Bartnik data data with positive Gauss
curvature and $H>0$. Then there exists $\lambda_{o}>0$ such that
$(\Sigma,g,\lambda H)$ admits a fill-in with non-negative scalar curvature for
all $\lambda<\lambda_{o}$ and there exists no such fill-in for
$\lambda>\lambda_{o}$.

######  Remark 2.3.

Theorem 2.3 also likely can be extended in a straightforward manner to higher
dimension than $3$, although it relies on Theorem 2.1 so some extra hypotheses
regarding the isometric embedding are likely required.

In a similar spirit to Jauregui’s result [15], Shi, Wang, Wei and Zhu [28]
establish the following result.

######  Theorem 2.4 (Shi–Wang–Wei–Zhu [28]).

Let $g$ be a metric on the sphere $\Sigma=\mathbb{S}^{n-1}$ ($3\leq n\leq 7$)
such that there exists a continuous path in the space of smooth positive
scalar curvature metrics on $\mathbb{S}^{n-1}$ from $g$ to the standard round
metric. Then there exists a constant $h_{o}$ depending on $g$ such that
$(\Sigma,g,H)$ admits no fill-in with non-negative scalar curvature for all
$H>0$ satisfying

(2.2) $\int_{\Sigma}Hd\mu_{g}>h_{o}.$

While there are several results demonstrating that if $H$ is too large in some
sense then no fill-in can exist, it is difficult to explicitly quantify how
large $H$ may be. It would be interesting to obtain an explicit, computable
lower bound on $H$ in terms of $g$ for an obstruction of the existence of a
fill-in, such as an explicit value for $h_{o}$ in (2.2).

##  3\. General Fill-In Construction

We construct fill-ins of Bartnik data $(\Sigma,g,H)$ by constructing a metric
$\gamma$ on the cylinder $\Sigma\times I$ for some interval $I$, such that one
boundary component induces the Bartnik data while the other is a minimal
surface. This construction originates with [18] and has been used successfully
in several related problems (see [6] for a survey).

We now give the general construction of these collars so we may refer to it
later. Consider a metric of the form

(3.1) $\gamma=A(x)^{2}dt^{2}+E(t)^{2}g$

where $A$ is a positive function on $\Sigma$, $E$ is a positive function of
$t$, and $g$ is a fixed metric on $\Sigma$.

Computing the scalar curvature of the metric $\gamma$ we obtain

(3.2)
$\displaystyle\begin{split}E(t)^{2}R_{\gamma}=&\,R_{g}-2A^{-1}\Delta_{\Sigma}A\\\
&-(n-1)A^{-2}\left((n-2)E^{\prime}(t)^{2}+2E(t)E^{\prime\prime}(t).\right)\end{split}$

We will be interested in choices of $E$ and $A$ that allow us to prescribe the
mean curvature of some constant $t$ boundary surface, and ensure a prescribed
lower bound on $R_{\gamma}$. The mean curvature of each slice with constant
$t$ can computed directly as

(3.3) $H_{t}(x)=\frac{(n-1)E^{\prime}(t)}{E(t)A(x)}.$

For convenience, we will choose the parameter $t$ such that $t=0$ is the
surface with prescribed mean curvature, and we use $t<0$ for the fill-in, so
that $t$ is increasing towards the outer boundary. We also must prescribe the
induced metric on the outer boundary surface. To this end, we will always
scale $E$ such that $E(0)=1$. Letting $H=H(x)$ be the mean curvature we wish
to prescribe, we find

(3.4) $A(x)=\frac{(n-1)E^{\prime}(0)}{H(x)}$

and then (3.2) becomes

(3.5)
$\displaystyle\begin{split}E(t)^{2}R_{\gamma}=&\,R_{g}-2H\Delta_{\Sigma}(H^{-1})\\\
&-\frac{H^{2}}{(n-1)(E^{\prime}(0))^{2}}\left((n-2)E^{\prime}(t)^{2}+2E(t)E^{\prime\prime}(t).\right)\end{split}$

In what follows, we will use this construction with different choices of $E$,
coming from profile functions of model spherically symmetric metrics. We now
demonstrate that the idea used to prove the main result of [21] can be used to
construct fill-ins with general scalar curvature lower bounds. Let
$\left(\Sigma,g,H\right)$ be Bartnik data where $\Sigma$ is an $(n-1)$-sphere
and $H$ is a positive function. We consider a metric of the form

(3.6) $\gamma=A(x)^{2}dt^{2}+\frac{u_{m}(t)^{2}}{r_{o}^{2}}g$

on $\Sigma\times[t_{o},0]$ where $r_{o}$ is the area radius of $g$ and
$t_{o}<0$ will be determined later such that $\Sigma\times\\{t_{o}\\}$
corresponds to a minimal surface boundary. Note that this is simply (3.1) with
$E(t)=\frac{u_{m}(t)}{r_{o}}$ and $g(t)=g$ a constant path. In [21] the
function $u_{m}$ was chosen to be a Schwarzschild profile function, however
here we would like to include Schwarzschild–de Sitter and Schwarzschild–anti-
de Sitter profiles too. Specifically $u_{m}$ is taken to be such that
$u_{m}(0)=r_{o}$ and satisfies

$u_{m}^{\prime}(t)=\sqrt{1+\epsilon u(t)^{2}-\frac{2m}{u(t)^{n-2}}}$

where $\epsilon\in\mathbb{R}$ depends on the desired scalar curvature lower
bound and $m>0$ is some parameter. Specifically, for some $C\in\mathbb{R}$, we
seek fill-ins with scalar curvature bounded below by $C$, and to do so we will
choose $\epsilon=-\frac{C}{n(n-1)}$. From (3.3), the mean curvature of each
constant $t$ slice is given by

(3.7) $H_{t}(x)=\frac{n-1}{A(x)u_{m}(t)}\sqrt{1+\epsilon
u_{m}(t)^{2}-\frac{2m}{u_{m}(t)^{n-2}}}.$

It is important to note that for $\epsilon\geq 0$, or $\epsilon<0$ and $m$ not
too large, one may solve find a value of $u_{m}=r_{H}$ such that $1+\epsilon
u_{m}(t)^{2}-\frac{2m}{u_{m}(t)^{n-2}}=0$, corresponding to an apparent
horizon in the model manifold. In fact, if $\epsilon<0$ there are two such
values of $u_{m}$ where this quantity vanishes, in which case $r_{H}$ is taken
to be the smaller of the two, as the larger radius corresponds to a
cosmological horizon in the Schwarzschild–de Sitter manifold. In particular,
if $r_{o}>r_{H}$ then there is always a $t_{o}<0$ such that $H_{t_{o}}=0$. We
will always ensure this is true so that the interior boundary of our collar is
a minimal surface.

Similarly, equation (3.5) for the scalar curvature gives

(3.8)
$\displaystyle\begin{split}R_{\gamma}-C=\frac{r_{o}^{2}}{u_{m}^{2}}&\left(R_{g}-2H\Delta
H^{-1}-\frac{n-2}{n-1}H^{2}(1+\epsilon
r_{o}^{2}-\frac{2m}{r_{o}^{n-2}})^{-1}\right.\\\
&\left.-\left(\frac{C}{r_{o}^{2}}+\frac{n}{n-1}H^{2}\epsilon(1+\epsilon
r_{o}^{2}-\frac{2m}{r_{o}^{n-2}})^{-1}\right)u_{m}^{2}\right).\end{split}$

When $C=0$ this is exactly what was considered in [21]. We consider the cases
of $C<0$ and $C\geq 0$ separately, however the idea is the same in both cases.
We choose $m>0$ such that the manifold
$\left(\Sigma\times[t_{o},0],\gamma\right)$ has a minimal surface at the
surface $t=t_{o}$, scalar curvature bounded below by $C$ and mean curvature of
the surface $t=0$ prescribed as above.

##  4\. Fill-Ins With Scalar Curvature Lower Bounds

###  4.1. Negative scalar curvature lower bound

We consider the case where the scalar curvature lower bound $C$ is negative,
and first establish Theorem 1.1. Although Theorem 1.2 is essentially a
corollary of this, we reserve the proof of that until Section 5 to discuss
with other applications of the fill-ins satisfying scalar curvature bounds.

###### Proof of Theorem 1.1.

Consider the fill-in constructed above in Section 3, whose metric is given by
(3.6). In this case, we choose $\epsilon=-\frac{C}{n(n-1)}>0$ and the function
$u_{m}$ is the radial profile function for an Schwarzschild–anti-de Sitter
manifold with scalar curvature equal to $C$. By (3.7), the fill-in constructed
above has a minimal surface at some $t=t_{o}$ provided $m>0$. So we simply
seek to choose $m>0$ such that the right-hand side of (3.8) is non-negative. A
straightforward albeit non-optimal way to ensure that, is to impose

(4.1) $R_{g}-2H\Delta H^{-1}-\frac{n-2}{n-1}H^{2}(1+\epsilon
r_{o}^{2}-\frac{2m}{r_{o}^{n-2}})^{-1}\geq 0$

and

(4.2) $\frac{C}{r_{o}^{2}}+\frac{n}{n-1}H^{2}\epsilon(1+\epsilon
r_{o}^{2}-\frac{2m}{r_{o}^{n-2}})^{-1}\leq 0.$

With our choice of $\epsilon$, (4.2) becomes

(4.3) $m\leq\frac{r_{o}^{n-2}}{2}\left(1+\epsilon
r_{o}^{2}-\frac{H^{2}r_{o}^{2}}{(n-1)^{2}}\right),$

and (4.1) can be expressed similarly as

(4.4) $m\leq\frac{r_{o}^{n-2}}{2}\left(1+\epsilon
r_{o}^{2}-\frac{n-2}{n-1}\frac{H^{2}}{R_{g}-2H\Delta H^{-1}}\right).$

In order for the fill-in to have a minimal surface inner boundary, rather than
a cusp, we require $m>0$ so we obtain a fill-in provided

(4.5) $\max\left\\{\max_{\Sigma}\frac{n-2}{n-1}\,\frac{H^{2}}{R_{g}-2H\Delta
H^{-1}},\max_{\Sigma}\frac{H^{2}r_{o}^{2}}{(n-1)^{2}}\right\\}<1+\epsilon
r_{o}^{2}.$

∎

Note that in the case where $C=0$, (4.2) is trivially satisfied and this
reduces to what was shown in [21].

###  4.2. Positive scalar curvature lower bound

We next turn to consider a positive lower bound on the scalar curvature and
prove Theorem 1.3.

###### Proof of Theorem 1.3.

We again use the same fill-in metric (3.6) from Section 3, however in this
case with $C>0$ and $\epsilon=-\frac{C}{n(n-1)}<0$. In this case, the model
space is the Schwarzschild–de Sitter family of manifolds. In this case, a
minimal surface is again located where $u^{\prime}_{m}=0$, however here we
must take a little more care with the roots of

(4.6) $1+\epsilon x^{2}-\frac{2m}{x^{n-2}}.$

When $\epsilon\geq 0$ we find that there is only one root, and therefore one
minimal surface in the model space, which represents a black hole’s horizon.
However, when $\epsilon<0$ and provided that $m$ is not too large, (4.6) has
two real positive roots, $0<r_{+}<r_{-}$. These correspond to a black hole
horizon at $r_{+}$ and a cosmological horizon at $r_{-}$ in the model
Schwarzschild–de Sitter manifold. This model is a compact manifold with two
connected minimal surface boundary components, one sphere at $r_{+}$ and
another sphere at $r_{-}$. We will choose $m$ arbitrarily small but positive
and require that the area radius of $g$, $r_{o}$ satisfy $r\in(r_{+},r_{-})$.
Since the roots $r_{+}$ and $r_{-}$ tend to $0$ and
$\frac{1}{\sqrt{-\epsilon}}$ respectively, as $m\to 0$, the condition
$r_{o}^{2}<\frac{n(n-1)}{C}$ guarantees $r_{+}>2m>0$ for sufficiently small
$m$. In particular, we have $r_{o}\geq u_{m}(t)\geq r_{+}>2m>0$. Turning back
to the scalar curvature equation, (3.8) implies that $R_{\gamma}\geq C$ is
equivalent to

$R_{g}-2H\Delta H^{-1}-\frac{H^{2}}{n-1}(1+\epsilon
r_{o}^{2}-\frac{2m}{r_{o}^{n-2}})^{-1}\left(n-2-n\epsilon
u_{m}^{2}\right)+\frac{n(n-1)\epsilon u_{m}^{2}}{r_{o}^{2}}\geq 0.$

As we do not expect to obtain anything optimal by this method, it will suffice
to make a crude estimate using the fact that $u_{m}$ is positive.
Specifically, we ask that

$R_{g}-2H\Delta H^{-1}-\frac{n-2}{n-1}H^{2}(1+\epsilon
r_{o}^{2}-\frac{2m}{r_{o}^{n-2}})^{-1}\geq 0,$

which is ensured by choosing

$0<m\leq\frac{r^{n-2}}{2}\left(1+\epsilon
r_{o}^{2}-\frac{n-2}{n-1}\frac{H^{2}}{R_{g}-2H\Delta H^{-1}}\right).$

That is, there exists a fill-in with minimal surface boundary and scalar
curvature bounded below by $C$, provided that

(4.7) $\frac{n-1}{n-2}\frac{R_{g}-2H\Delta
H^{-1}}{H^{2}}<\left(1-\frac{C}{n(n-1)}r_{o}^{2}\right)^{-1}.$

∎

Note that equation (4.7) amounts to a strong, point-wise quasi-local positive
mass assumption, which is somewhat natural in the context of non-positive
scalar curvature lower bounds given the positive mass theorem. However, as
mentioned above, the analogous positive mass theorem does not hold for
positive scalar curvature lower bounds. That is, fill-ins with positive scalar
curvature bounds should exist under far weaker assumptions than in the case of
non-positive scalar curvature bounds.

One can see this from the counterexample to Min-Oo’s conjecture constructed by
Brendle, Marques and Neves in [4]. Therein they establish the existence of
compact manifolds with scalar curvature bounded below by $n(n-1)$, with a
neighbourhood of the boundary isometric to a neighbourhood of the boundary of
the $n$-dimensional hemisphere, and strictly positive scalar curvature
somewhere on the interior. From a perturbation of this counterexample one can
obtain a manifold with the same scalar curvature lower bound with a
neighbourhood of the boundary isometric to a neighbourhood of the cosmological
horizon (unstable minimal surface) in a negative mass Schwarzschild–de Sitter
manifold. Although this is a fairly obvious consequence of the main results of
[4], as it does not appear to be recorded explicitly, we state it here for
completeness.

######  Proposition 4.1.

There exists a compact Riemannian manifold $(M,g)$ with boundary, having
scalar curvature $R_{g}\geq 6$, with the inequality strict ($R_{g}>6$) on an
open subset, such that there exists a subset $\Omega\supset\partial M$
isometric to a neighbourhood of the boundary of a Schwarzschild–de Sitter
manifold with negative mass.

###### Proof.

By Theorem 4 of [4] there exists a metric $g_{o}$ on the hemisphere
$S_{+}^{n}$ with scalar curvature $R_{g_{o}}>n(n-1)$ everywhere, $g_{o}$ is
exactly equal to the standard round metric on $\partial S_{+}^{n}=S^{n-1}$,
and the outward-pointing mean curvature $H$ of $\partial S_{+}^{n}$ with
respect to $g_{o}$ is strictly negative. By a small rescaling, one can also
obtain a metric $g_{\varepsilon}$ that also satisfies
$R_{g_{\varepsilon}}>n(n-1)$ with negative mean curvature on the boundary,
such that $g_{\varepsilon}$ restricted to $\partial S_{+}^{n}$ is round with
area $A_{\varepsilon}>4\pi$. Note that the cosmological horizon boundary of a
Schwarzschild–anti-de Sitter manifold of mass $m$ has area $\widetilde{A}_{m}$
satisfying

$m=\frac{1}{2}\left(\frac{\widetilde{A}_{m}}{\omega_{n-1}}\right)\left(1-\frac{\widetilde{A}_{m}}{\omega_{n-1}}\right).$

That is, the metric $g_{\varepsilon}$ restricted to the boundary $\partial
S_{+}^{n}$ is equal to the boundary metric for a negative mass
Schwarzschild–anti-de Sitter manifold. One can therefore apply Theorem 5 of
[4] to obtain the result. Note that although the negative mass
Schwarzschild–anti-de Sitter manifolds are singular at a point, this does not
affect the application of Theorem 5 of [4] since it is purely a local
construction. ∎

######  Remark 4.1.

The above is simply a perturbative construction, so the manifolds obtained
correspond to a (negative) mass parameter very close to zero. It would be an
interesting question to ask for Bartnik data $(S^{n-1},g_{0},H=0)$, with
$g_{o}$ round, how large can $|S^{n-1}|_{g_{o}}$ be and still admit a fill-in
with scalar curvature bounded below by $n(n-1)$.

##  5\. Applications

###  5.1. The asymptotically hyperbolic positive mass theorem with boundary

As mentioned above, in [21], the fill-ins constructed were used to prove a
“Penrose-like” inequality. In particular, it was shown that for an
asymptotically flat manifold with boundary $\Sigma$ satisfying the hypotheses
of Theorem 1.3 with $C=0$, there exists a fill-in with minimal surface
boundary whose area $A$ satisfies

(5.1)
$\left(\frac{A}{\omega_{n-1}}\right)^{\frac{n-2}{n-1}}=\left(\frac{|\Sigma|}{\omega_{n-1}}\right)^{\frac{n-2}{n-1}}\left(1-\frac{n-2}{n-1}\frac{H^{2}}{R_{g}-2H\Delta
H^{-1}}\right),$

and then via the Riemannian Penrose inequality we obtain that the ADM mass is
bounded below by the right-hand side of (5.1).

An analogous result follows by the same reasoning for asymptotically
hyperbolic manifolds with boundary using the fill-in constructed in Section
4.1 if one assumes the Riemannian Penrose inequality holds in this case. While
this version of the Riemannian Penrose inequality is only conjectured, and not
yet established, it is known that the positive mass theorem holds for
asymptotically hyperbolic manifolds with minimal surface boundary [9]. So we
obtain the following positive mass theorem for asymptotically hyperbolic
manifolds with boundary.

###### Proof of Theorem 1.2.

Let $(M,\gamma)$ satisfy the hypotheses of Theorem 1.2 with boundary Bartnik
data $(\Sigma,g,H)$, and let $\gamma_{\Omega}$ be the fill-in metric on
$\Omega=\Sigma\times[t_{o},1]$ constructed for the proof of Theorem 1.1,
filling in this Bartnik data. We can attach $(\Omega,\gamma_{\Omega})$ to
$(M,\gamma)$ along their matching Bartnik data to form a manifold with corner
à la Miao [22] and then double it along the minimal surface to form a complete
asymptotically hyperbolic manifold with corners and two ends. The conclusion
follows now from the positive mass theorem with corners, which has been
established in the asymptotically hyperbolic case by Bonini and Qing [2]. ∎

######  Remark 5.1.

If the conjectured Riemannian Penrose inequality for asymptotically hyperbolic
manifolds were established, then one could use a suitable version of it for
manifolds with corners to obtain an improved lower bound on the mass of an
asymptotically hyperbolic manifold with boundary. In particular, we have the
following.

Let $(M,\gamma)$ be an asymptotically hyperbolic $n$-manifold with scalar
curvature bounded below by $C=-\epsilon n(n-1)$ and interior boundary
$\Sigma$, and define the quantity

(5.2)
$\chi=\frac{1}{n-1}\max\left\\{\max_{\Sigma}\frac{(n-2)H^{2}}{R_{g}-2H\Delta
H^{-1}},\max_{\Sigma}\frac{H^{2}r_{o}^{2}}{(n-1)}\right\\}.$

If $\chi<1+\epsilon r_{o}^{2}$, where $r_{o}$ is the area radius of $\Sigma$,
then assuming the asymptoticaly hyperbolic Penrose inequality holds (on a
manifold with corners), we would conclude that the total mass of $(M,\gamma)$
is bounded below by $\frac{1}{2}r_{o}^{n-2}\left(1+\epsilon
r_{o}^{2}-\chi\right)$.

We state this as a remark and omit a formal proof of this statement, as we
require a precise statement of the appropriate Riemannian Penrose inequality,
which remains an open problem. However, a sketch is provided as we will refer
back to at the end of Section 5.3.

###### Sketch of proof.

Choose $m=\frac{r_{o}^{n-2}}{2}\left(1+\epsilon r_{o}^{2}-\chi\right)>0$ and
obtain a fill-in with minimal surface boundary as in Section 4.1. One can
quickly check from the form of the metric (3.6) that the area radius $r_{H}$
of this minimal surface satisfies

$1+\epsilon r_{H}^{2}-\frac{2m}{r_{H}^{n-2}}=0.$

From our choice of $m$, we therefore obtain the relationship

(5.3) $\frac{1}{2}r_{H}^{n-2}\left(1+\epsilon
r_{H}^{2}\right)=\frac{1}{2}r_{o}^{n-2}\left(1+\epsilon
r_{o}^{2}-\chi\right),$

where the left-hand side of the equation is exactly the lower bound on the
total mass of an asymptotically hyperbolic manifold conjectured by the
Riemannian Penrose inequality. This fill-in can then be glued to $(M,\gamma)$
and the conclusion would follow from the Riemannian Penrose inequality. ∎

######  Remark 5.2.

There are two different conjectured versions of the asymptotically hyperbolic
Penrose inequality corresponding roughly to whether the asymptotically
hyperbolic manifold is being viewed as time-symmetric initial data for a
spacetime with negative cosmological constant, or as an asymptotically
hyperbolic slide of an asymptotically flat spacetime. The former considers the
horizon to be a minimal surface, whereas the latter considers a surface of
constant mean curvature equal to $n-1$. Here we consider the former version,
and while it seems that many suspect it to hold there is some reason to
suspect that perhaps only the latter is true in general. We do not wish to
speculate on the conjecture here, however it is worthwhile noting that the
construction given above could in principle be used to construct a
counterexample if one exists. That is, if one can find an asymptotically
hyperbolic manifold (perhaps only defined near infinity) containing a surface
on which the right-hand side of 5.3 is larger than the total mass, then after
gluing, a counterexample to the asymptotically hyperbolic Penrose inequality
would be constructed. However, the author has attempted this with no success.

###  5.2. A Penrose-like inequality with electric charge

In the framework of general relativity, it is common to consider the Einstein
equations coupled to other equations governing the matter content of the
universe to model. Considering $(M,\gamma)$ as initial data from the
perspective of general relativity, we can add an electric field $E$ – a vector
field on $M$ – to describe initial data for the coupled Einstein–Maxwell
system, gravity coupled to the electric field. In this section we will only
consider the case of vanishing cosmological constant, which in the preceding
sections equated to manifolds with non-negative scalar curvature. This
restriction to only considering vanishing cosmological constant is not
required to construct the fill-ins, however the estimates become much messier
and these “charged” fill-ins are not of particular interest independent of a
Penrose-like inequality. However, such an inequality cannot hold in the
positive cosmological constant case and remains out of reach for the negative
cosmological constant case, for the reasons discussed above. On the other
hand, the scalar curvature lower bound considered here is not zero and in fact
depends on the electric field $E$. Specifically, we will require

(5.4) $R_{g}\geq(n-1)(n-2)|E|_{g}^{2}.$

The divergence of the electric field corresponds to the charge of any matter
source terms, and in order to later apply a charged Riemannian Penrose
inequality [14, 16, 17, 20] we will ask that this vanishes. It is not strictly
required that the $\nabla\cdot E=0$ for the charged Riemannian Penrose
inequality to hold (see [20]), however it will nevertheless be fruitful to
impose this for the fill-ins constructed. In the preceding sections, a fill-in
is taken as a manifold with boundary metric and mean curvature prescribed
since this is the appropriate boundary condition to glue the fill-in to an
exterior manifold while preserving the scalar curvature condition. However,
when an electric field is present we would also like to preserve the sign of
$\text{div}(E)$ in a distributional sense when performing the gluing, which
amounts to matching $\phi=E\cdot n$ the normal component of the electric field
on $\Sigma$. Therefore the appropriate Bartnik data for including electric
charge is the triple $(\Sigma,g,H,\phi)$ (see, for example, Section 3 of [8]).

We again consider a metric of the form (3.6) except now use a
Reissner–Nordström manifold as our model and profile curve

(5.5) $\gamma=A(x)^{2}dt^{2}+\frac{v_{m,Q}(t)^{2}}{r_{o}^{2}}g,$

where $v_{m,Q}$ satisfies $v_{m,Q}(0)=r_{o}$ and

$v^{\prime}_{m,Q}(t)=\sqrt{1+\frac{Q^{2}}{v(t)^{2(n-2)}}-\frac{2m}{v(t)^{n-2}}}.$

In what follows we will use the shorthand $v=v_{m,Q}$ for the sake of
presentation. We will also assume $H$ is constant here, as it simplifies the
computations considerably and does not change the qualitative properties of
the estimate we obtain. We again choose $A(x)$ according to (3.4) and compute
the scalar curvature of $\gamma$ similarly to (3.8) to obtain

(5.6)
$R_{\gamma}=\frac{r_{o}^{2}}{v^{2}}\left(R_{g}-\frac{n-2}{n-1}H^{2}\left(1+\frac{Q^{2}}{r_{o}^{2(n-2)}}-\frac{2m}{r_{o}^{n-2}}\right)^{-1}\left(1-\frac{Q^{2}}{v^{2(n-2)}}\right)\right).$

We then set the electric field as

$E=\frac{r_{o}^{n-1}\phi}{v^{n-1}A}\partial_{t},$

which is easily checked to be divergence-free. Then we see that the
appropriate energy condition, $R_{g}\geq(n-1)(n-2)|E|_{g}^{2}$, is equivalent
to

$R_{g}-\frac{n-2}{n-1}H^{2}\left(1+\frac{Q^{2}}{r_{o}^{2(n-2)}}-\frac{2m}{r_{o}^{n-2}}\right)^{-1}\left(1-\frac{Q^{2}}{v^{2(n-2)}}\right)-\frac{(n-1)(n-2)r_{o}^{2(n-2)}\phi^{2}}{v^{2(n-2)}}\geq
0.$

Assuming $H>(n-1)\phi>0$, we may choose $Q$ in such a way to cancel out the
$v^{2}$ terms. Namely, we set

$\displaystyle Q^{2}$
$\displaystyle=\frac{(n-1)^{2}r_{o}^{2(n-2)}\hat{\phi}^{2}}{H^{2}}\left(1+\frac{Q^{2}}{r_{o}^{2(n-2)}}-\frac{2m}{r_{o}^{n-2}}\right)$
$\displaystyle Q^{2}$
$\displaystyle=(n-1)^{2}r_{o}^{2(n-2)}\hat{\phi}^{2}\left(1-\frac{2m}{r_{o}^{2(n-2)}}\right)\left(H^{2}-(n-1)^{2}\hat{\phi}^{2}\right)^{-1},$

where we write $\hat{\phi}=\max_{\Sigma}(\phi)$ for the sake of presentation.
This leaves $m<\frac{r_{o}^{2(n-2)}}{2}$ as a free parameter, which we will
choose to ensure

$R_{g}-\frac{n-2}{n-1}H^{2}(1+\frac{Q^{2}}{r_{o}^{2(n-2)}}-\frac{2m}{r_{o}^{n-2}})^{-1}\geq
0.$

With our choice of $Q$, this is equivalent to

$R_{g}-\frac{n-2}{n-1}\,\frac{\left(H^{2}-(n-1)^{2}\hat{\phi}^{2}\right)}{\left(1-\frac{2m}{r_{o}^{n-2}}\right)}\geq
0,$

so we choose

$m=\frac{r_{o}^{n-2}}{2}\left(1-\frac{n-2}{n-1}\,\frac{H^{2}-(n-1)^{2}\hat{\phi}^{2}}{\min_{\Sigma}(R_{g})}\right).$

Our fill-in now satisfies the appropriate energy condition for the charged
Riemannian Penrose inequality (5.4), however we have yet to check that the
metric is non-singular. In fact, we require that $m>Q$ to ensure that
$v^{\prime}$ vanishes somewhere and therefore providing us with a minimal
surface boundary. Note that with our choice of $m$, we have

$\displaystyle Q^{2}$
$\displaystyle=r_{o}^{2(n-2)}\left(\frac{n-2}{n-1}\,\frac{H^{2}-(n-1)^{2}\hat{\phi}^{2}}{\min_{\Sigma}(R_{g})}\right)\left(\frac{H^{2}}{(n-1)^{2}\hat{\phi}^{2}}-1\right)^{-1}$
$\displaystyle Q^{2}$
$\displaystyle=\frac{(n-1)(n-2)\hat{\phi}^{2}r_{o}^{2(n-2)}}{\min_{\Sigma}(R_{g})}.$

In order to ensure $m>Q$, we calculate the difference $m-Q$ from the above
expressions to obtain

(5.7)
$\displaystyle\begin{split}\frac{2R_{g}}{r_{o}^{(n-2)}}(m-Q)=&\,\min_{\Sigma}(R_{g})-\frac{n-2}{n-1}H^{2}+(n-2)(n-1)\hat{\phi}^{2}\\\
&-2\hat{\phi}\sqrt{(n-1)(n-2)\min_{\Sigma}(R_{g})}.\end{split}$

The right-hand side of (5.7) is a quadratic expression in
$\sqrt{\min_{\Sigma}(R_{g})}$, so we can directly check that this is positive
when

$\min_{\Sigma}(R_{g})>\frac{n-2}{n-1}\left((n-1)\hat{\phi}+H\right)^{2}.$

Note that when $\phi\equiv 0$ this reduces to the condition for the existence
of fill-in for the uncharged case [21]. By construction, we have proven the
following.

######  Theorem 5.1.

Let $(\Sigma,g,H,\phi)$ be charged Bartnik data with constant $H$ satisfying
$H>(n-1)\phi>0$ and satisfying

$\min_{\Sigma}(R_{g})>\frac{n-2}{n-1}\left((n-1)\max_{\Sigma}(\phi)+H\right)^{2}$

then there exists a metric $\gamma$ and divergence-free vector field $E$ on
$M=\Sigma\times[0,1]$ satisfying $R_{g}\geq(n-1)(n-2)|E|_{\gamma}^{2}$, such
that one boundary component is a minimal surface and on the other the induced
metric is $g$, outward-pointing mean curvature is $H$, and the outward normal
component of $E$ is $\phi$.

From this fill-in, we are able to prove an electrically charged Penrose-like
inequality (and charged positive mass theorem for manifold with boundary).

######  Theorem 5.2.

Let $(M,\gamma,E)$ be an asymptotically flat manifold with charge of dimension
$3\leq n\leq 7$, and boundary $\Sigma$ with charged Bartnik data
$(\Sigma,g,H,\phi)$, satisfying $\nabla\cdot E\geq 0$ and

$\min\limits_{\Sigma}R_{g}>\frac{n-2}{n-1}\left((n-1)\max\limits_{\Sigma}\phi+\max\limits_{\Sigma}H\right)^{2}.$

Then

$\mathfrak{m}_{ADM}\geq m+\frac{Q_{\Sigma}^{2}-Q^{2}}{m+\sqrt{m^{2}-Q^{2}}},$

where $Q_{\Sigma}=\frac{1}{\omega_{n-1}}\int_{S}\phi\,d\mu_{g}$ is the
electric charge on $\Sigma$ in $M$, and the parameters $m$ and $Q$ are given
by

(5.8)
$\displaystyle\begin{split}m&=\frac{r_{o}^{n-2}}{2}\left(1-\frac{n-2}{n-1}\,\frac{H^{2}-(n-1)^{2}\max_{\Sigma}\phi^{2}}{\min_{\Sigma}(R_{g})}\right)\\\
Q^{2}&=\frac{(n-1)(n-2)\max_{\Sigma}\phi^{2}r_{o}^{2(n-2)}}{\min_{\Sigma}(R_{g})}.\end{split}$

Furthermore, if $\nabla\cdot E\equiv 0$ then we have

(5.9) $\mathfrak{m}_{ADM}\geq|Q_{\Sigma}|.$

######  Remark 5.3.

The expression for $m$ given by (5.8) can be compared to the charged Hawking
mass, while the expression $Q_{\Sigma}^{2}-Q^{2}$ can be seen to vanish when
$(S,g)$ is a round sphere and $\phi$ is constant.

###### Proof.

We apply Theorem 5.1 to construct a fill-in of the Bartnik data
$(\Sigma,g,H_{o},\phi)$, where $H_{o}=\max_{\Sigma}(H)$. We then obtain a
(charged) manifold with corner by attaching the fill-in to $M$ and can apply
the charged Riemannian Penrose inequality for manifolds with corners
established in [8]. Note that the results in [8] are stated only in dimension
$3$ only, however it is clear from the proof that the charged Riemannian
Penrose inequality with corners holds in dimension up to $7$ (see remark 5.4
below). Specifically, from Theorem 1.3 of [8] we have

(5.10)
$\mathfrak{m}_{ADM}(M,\gamma)\geq\frac{r_{H}^{n-2}}{2}\left(1+\frac{Q_{\Sigma}^{2}}{r_{H}^{2(n-2)}}\right),$

where $r_{H}$ is the area radius of the minimal surface boundary.

From the definition of the profile function $v$ used in the fill-in, we have
that the minimal surface occurs when $v^{\prime}=0$, which implies

$1+\frac{Q^{2}}{r_{H}^{2(n-2)}}-\frac{2m}{r_{H}^{n-2}}=0,$

and then from (5.10), we have

(5.11) $\mathfrak{m}_{ADM}(M,\gamma)\geq
m+\frac{Q_{\Sigma}^{2}-Q^{2}}{r_{H}^{n-2}},$

with

$r_{H}^{n-2}=m+\sqrt{m^{2}-Q^{2}}.$

Finally note that (5.9) follows from applying the charged positive mass
theorem [12, 10] instead of the charged Riemannian Penrose inequality. ∎

######  Remark 5.4.

The charged Riemannian Penrose inequality with corners established in [8] is
presented in dimension 3, following [22]. However, as illustrated in the
appendix of [21], the argument holds up to dimension $7$ (where the standard
problem concerning the regularity of minimal surfaces prevents it immediately
being generalised to higher dimensions than that). Naturally, there are
dimensional constants in the inequality used, which we are careful to
correctly include here.

######  Remark 5.5.

As with the other inequalities established by this method, it is expected that
there is room slightly improve the inequality, however comparing it to the
lower bound on ADM mass in terms of the charged Hawking mass in dimension 3,
one sees that the method is unlikely to achieve an optimal inequality.

###  5.3. Engelhardt–Wall Outer Entropy and Bray’s Inner Bartnik Mass

Recently, Wang [31] noted that the concept of outer entropy due to Engelhardt
and Wall [11] in the context of the AdS/CFT correspondence is essentially the
same concept as Bray’s inner Bartnik mass [3]. The former was formulated from
the perspective of the AdS/CFT correspondence for asymptotically hyperbolic
manifolds while the latter was formulated from a purely geometric perspective
for asymptotically flat manifolds. In particular, the outer entropy is
equivalent to an asymptotically hyperbolic analogue of the inner mass, rather
than the standard one. Nevertheless, at the heart of both is the problem of
constructing fill-ins of Bartnik data with a minimal surface boundary, and
taking the supremum of the minimal surface area over an appropriate class of
fill-ins111Technically, the inner mass is defined using fill-ins that extend
out to another asymptotic end, but this distinction is minor..

The Penrose-like inequality obtained in [21] in fact was first motivated by
considerations of the Bartnik–Bray inner mass. Since this mass is taken as a
supremum, we immediately conclude that the inner Bartnik mass of Bartnik data
$(\Sigma,g,H)$ is bounded below by the right-hand side of (5.1).

While an asymptotically hyperbolic analogue of the standard Bartnik mass has
been recently investigated [5], to the best of the author’s knowledge an
asymptotically hyperbolic inner Bartnik mass has not been considered in the
literature. Nevertheless, there is an obvious analogue one could consider,
which is the one equivalent to the outer entropy. Namely, we define the
asymptotically hyperbolic inner Bartnik mass of given Bartnik data
$(\Sigma,g,H)$ as the supremum, taken over the set of all fill-ins with scalar
curvature bounded below by $-\epsilon n(n-1)$ with no closed minimal surfaces
except for a minimal surface boundary, of the quantity

$\frac{1}{2}r^{n-2}\left(1+\epsilon r^{2}\right)$

where here $r$ is the area radius of the minimal surface boundary. It follows
from (5.3) that this asymptotically hyperbolic inner Bartnik mass of some data
$(\Sigma,g,H)$ is bounded below by

(5.12) $\frac{1}{2}r_{o}^{n-2}\left(1+\epsilon r_{o}^{2}-\chi\right)$

where $\chi$ is given by (5.2) and $r_{o}$ is the area radius of $g$. Then
observation of Wang connects this to the Engelhardt–Wall outer entropy, which
is simply the supremum of the area of the minimal surface over the same set of
fill-ins. That is the lower bound for the outer entropy is
$\omega_{n-1}r_{H}^{n-1}$ where $r_{H}$ satisfies (5.3).

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Optimizing Packet Reception Rates for Low Duty-Cycle BLE Relay Nodes
This work is financed by the ERDF – European Regional Development Fund through the Operational Programme for Competitiveness and Internationalisation - COMPETE 2020 Programme within project WHERE.IS (POCI-01-0247-FEDER-024191).

Nuno Paulino, Luís M. Pessoa
INESC TEC and Faculty of Engineering,
University of Porto, Porto, Portugal

{nuno.m.paulino<EMAIL_ADDRESS>André Branquinho, Rafael Tavares, Igor Ferreira
Wavecom, Aveiro, Portugal

{abranquinho, rtavares<EMAIL_ADDRESS>
In order to achieve the full potential of the Internet-of-Things, connectivity between devices should be ubiquitous and efficient. Wireless mesh networks are a critical component to achieve this ubiquitous connectivity for a wide range of services, and are composed of terminal devices (i.e., nodes), such as sensors of various types, and wall powered gateway devices, which provide further internet connectivity (e..g, via WiFi).

When considering large indoor areas, such as hospitals or industrial scenarios, the mesh must cover a large area, which introduces concerns regarding range and the number of gateways needed and respective wall cabling infrastructure. Solutions for mesh networks implemented over different wireless protocols exist, like the recent BLE 5.1. Besides range concerns, choosing which nodes forward data through the mesh has a large impact on performance and power consumption.

We address the area coverage issue via a battery powered BLE relay device of our own design, which acts as a range extender by forwarding packets from end nodes to gateways. We present the relay's design and experimentally determine the packet forwarding efficiency for several scenarios and configurations. In the best case, up to 35 of the packets transmitted by 11 nodes can be forwarded to a gateway by a single relay under continuous operation. A battery lifetime of 1 year can be achieved with a relay duty cycle of 20.

BLE, Bluetooth, low-energy, wireless sensor networks, mesh networks

§ INTRODUCTION

Wireless mesh networks can be the platform for many applications. A common use case are sensor networks <cit.>, but others include domotics <cit.>, automated inventory tracking or localization <cit.>. Specific scenarios include healthcare <cit.>, security <cit.> and warehouses and industrial facilities <cit.>.

Depending on the application, mesh networks can be built with WiFi devices, for example, but WiFi end-points or routers typically require wall power.  On the other hand, BLE devices benefit from a comparatively lower cost, power efficiency, and smaller device sizes. The lower power consumption alone negates the need for installation of a wired infrastructure in favor of battery power, reducing costs further. A BLE mesh network with battery powered end and relay nodes can be deployed in legacy locations without such an infrastructure. However, the range and data-rate for BLE devices is lesser than that of protocols such as WiFi <cit.>, meaning that denser networks may be required, which introduces the need for efficient are coverage and packet relaying.

We address the specific case where end nodes periodically transmit data on advertising channels only. The range of the network is restricted by the range of the edge nodes to the gateways, and on the available wall power for the gateways.

To address this, intermediate nodes acting as relays are designed to extend the advertising range of the nodes. These nodes should also be battery powered, since otherwise they could be easily replaced with gateways. However, continuous operation by the relays to listen for sporadic transmissions from the nodes would result in an unsuitably short battery life.

By configuring their listening period with a low duty-cycle (i.e., configuring the network nodes to a low-power operating mode for a given listening period), the battery life can be considerably extended. Consequently, since BLE packet transmission is sporadic, especially as the network nodes operate asynchronously, idling some (or all) the relay nodes inevitably leads to packet losses. However, some applications may not consider that all data is high priority, and some degree of data loss and/or end-to-end delay may be acceptable.

To characterize the efficiency of a system reliant on battery-powered relay nodes, we present an in-house design for such a BLE relay, and characterize the system's packet loss in different conditions. Specifically, we vary the number of client nodes, the listening time spent on each BLE channel, and apply two different forwarding policies, one of which has additional configuration parameters. Additionally, we subject the system to noise from other Bluetooth devices external to the network.

We validate the operation of our BLE relay design by manufacture and assembly, employing some of the units as beacons (so we may configure transmission periods), while another unit performs the relay function under several software configurations which implement our operating policies.

This paper is organized as follows: <Ref> reviews related work, <Ref> describes the network topology we addressed, <Ref> presents the design characteristics of the BLE relay node, and the configurable operating parameters, like the duty cycle and forwarding policy. <Ref> presents experimental evaluation of packet reception rates for different scenarios. <Ref> concludes the paper.

§ RELATED WORK

A comprehensive survey on the research efforts in BLE mesh topologies is presented by Darroudi et al. <cit.>. The survey categorizes and compares nearly 30 approaches to BLE network designs, including standardization solutions proposed by the Bluetooth SIG and the IETF, academic solutions, and proprietary solutions.

A major distinction between mesh approaches is whether data is transmitted by flooding (e.g., using the BLE advertising channels), or by through end-to-end connections through specific nodes. A comparison is presented in <cit.>, where the authors compare the Trickle flooding protocol <cit.> with the FruityMesh connection based protocol <cit.>. Both are evaluated regarding their multi-hop efficiency, for a network of nine intermediate nodes placed between two source and sink nodes. The packet delivery ratio and the end-to-end delay are measured. Both approaches are comparable in this scenario, with a packet delivery rate of close to 40 when 10 packets are generated per second by the source node. FruityMesh suffers an end-to-end delay which is approximately 9x higher compared to Trickle, but in turn requires 3x less power.

Kim et al. <cit.> present BLEMesh. A packet forwarding protocol is proposed to transmit batches of packets. Less transmissions are required in total to transport data end-to-end, through intermediate nodes, relative to naive flooding or routing based approaches. The packets include priority tables used by intermediate nodes to determine if a received packet should be re-transmitted, based on whether or not that packet was already forwarded by a node of higher priority. A downside is that the payload capability of the BLE packet diminishes as the number of nodes and batch size increases. A simulated evaluation for a mesh with 5 nodes, and assuming only one advertising channel, achieves a reduction of 54.5 in the required number of transmissions, relative to flood routing.

Brandão et al. <cit.> propose the Drypp protocol, based on the Trickle flooding protocol <cit.>. Trickle is a mesh network protocol for BLE where each node captures and attempts to re-transmit data at a later time, unless it meanwhile listens to redundant transmissions sent by other nodes. Drypp introduces a load balancing method which relies on dynamic adaptation of the protocol parameters based on each node's battery level. For three test nodes implementing the Drypp protocol, an 11 increase in network lifetime was achieved relative to Trickle, in exchange for a 7.5 decrease in throughput.

A BLE mesh network relying on a routing protocol is evaluated in <cit.>. The proposed mesh network is designed for environmental monitoring and disaster scenarios, and both the edge (sensor nodes) and the Wi-Fi capable gateway nodes are battery powered. Information if propagated based on Trickle routing <cit.>. To extend battery life, the sensor nodes are periodically shut off, and modifications to the trickle algorithm are introduced to prevent packet loss due to these power-down periods. Given the periods for listening and transmission time, the authors estimated a lifetime of 589 days for a sensor node, and 511 days for a gateway, when equipped with 6000 and 8000 lithium polymer batteries, respectively.

The work in <cit.> specifically addresses optimization of the use of Bluetooth relays in mesh networks. Connection-less mesh networks propagate data by controlled flooding between nodes, until the destination node of a particular data packet is reached. However, this leaves the network vulnerable to excessive flooding as a function of the number of nodes used as relays and/or selected to be relays. The authors employ state-of-the-art relay selection algorithms to a BLE mesh network, and evaluate the effect of six different relay selection algorithms to a Connected Dominating Set (CDS) representation of the mesh. Using an in-house simulator, different relays densities were tested with two end nodes exchanging 1000 messages one-way. The lowest packet loss can be achieved by computing the routing with the fewest hops, but the lowest power consumption is possible for a genetic algorithm which find the minimum CDS of the network, at the cost of suffering the highest packet loss (as high as 80).

In <cit.>, a method for relay node management is proposed based on a tree representation for the mesh network, together with an integer linear programming formulation which minimizes the number of relay nodes required to ensure connectivity between all nodes. The algorithm requires that the number of nodes and network topology be know to determine the relay routing. Using an in-house simulator, the authors evaluate the routing efficiency and energy consumption of a system composed of up to 100 nodes in an indoor configuration where line-of-sight is not possible for all pairs of nodes. A power consumption reduction of up to 12x is claimed over the conventional case where any relay node can be used as a relay during forwarding (i.e., flooding).

In general, the choice of protocol and network topology is application dependant. <Ref> summarizes the results (or a subset of results) from the experimental evaluations shown in this section. The values reported are our best effort at a comparison of the presented approaches, as well as our own. Depending on the respective experiments, some columns show either scalar, ranges of values, or lists (correspondence between list values is kept column to column). Node power reports the power consumption of each node of the tested mesh, taking into account the entire operating time, including any sleep periods of the nodes (i.e., the average power consumption throughout the experiment lifetime.

The experiments we conducted can be categorized as controlled flooding mechanism, but where we rely on details specific to a class of applications to determine forwarding behaviour. We consider end nodes with a constant packet rate, and envision a tree topology for the network where a relay is responsible for the end nodes within its range, and where relays are out of range amongst themselves. Additionally, we are not concerned with end-to-end delay, as data is non-critical given equal importance. We also conduct experiments while introducing real-world noise due to other wireless devices external to the network, which we have not observed in other works we have identified.

§ NETWORK TOPOLOGY

The use-case network topology for the evaluation of our relay, and respective forwarding policies, is shown in <Ref>. We target use cases where the end nodes are battery powered, and periodically transmit information about the environment (e.g., sensor data). The gateways are BLE/Wi-Fi devices which synchronize the status of the network with the centralized system. The network was designed and tested according to the features/constraints of the Bluetooth 4.1 specification <cit.>.

BLE mesh topology, with battery-powered end nodes and intermediate relay nodes, and wall-powered BLE/Wi-Fi gateways interfacing with an upstream server system

One of the characteristics of BLE is the transmission range (approximately 20). This means that either all nodes placed throughout the site have to be within this range of a wall-powered gateway in order for data to be retrieved by those nodes, or that data is forwarded through nodes. The former is a potentially expensive solution, and the later is the object of study on mesh network routing protocols.

However, if the end-nodes are simple sensors and cannot move data to and from each other (or if they are physically placed in such a way that a sequence of hops from end node to gateway cannot be established), more sophisticated battery-powered intermediate nodes are required which do not gather data themselves, but serve as range extenders to the gateways.

This paper focuses presents a design of the relay node, which functions as a packet receiver, gatherer, and re-transmitter. This makes it possible to extend the network range in situations where the indoor configuration or cost do not allow for a more ubiquitous distribution of wall-powered gateways. It also provides a cheaper solution relative to fully-fledged gateways, since it may replace them where Wi-Fi capabilities are not needed. Additionally, since the relays are battery powered, they are easy to relocate according to changes in the application requirements, or simply to tune the quality of the sensed data. The relays are compatible with any off-the-shelf end node which is BLE/Bluetooth 4.1 compliant.

§ BLE RELAY NODE

The purpose of the BLE relay device is to serve as a packet forwarder. It discards (i.e, does not forward) packets originating from devices which are not part of the its own network. Currently this is done by MAC address filtering. The only payload sent is the identification of each node.

We implemented two relay designs, both based on a single Nordic Semiconductor nRF52832 micro-controller <cit.>, which performs the packet reception and re-transmission, and idles the relay by going into a low-power mode. The configuration parameters listed earlier, such as listening intervals and periodicity, are controlled by the firmware residing on the non-volatile program memory of the nRF52832 chip. All relay implementations are composed by one single-layer, dual-sided, FR-4 PCB with a 1 thickness.

BLE relay prototype A, powered by a 3.3 button cell

BLE relay prototype B, with multiple power sources; the experimental evaluations in this paper consider only this relay variant

Two variants of the relay prototype; functionally identical with different power sources

The first prototype relay is shown in <Ref>. It contains the nRF52832 chip, a J-Link type programming header, and a single 3.3 CR2032 button cell battery. The relay is considerably small, with with a 23x38x10 profile. The antenna for reception and transmission of Bluetooth packets is a co-planar IFA, tuned for 2.4.

A second prototype designed for longer lifespan is shown in <Ref>. A series of four 3.3 AA batteries powers the relay when deployed in a location where wall power is unavailable, which is the primary use-case of the device. Alternatively, a mini-USB connector accepts a 5 input. An LTC4419 chip <cit.> is used as a power selector, which prioritizes the USB power input. A TPS62125 <cit.> regulates the chosen input to 3.3 for the nRF52832. Finally, the J-Link programming header powers the device in the absence of other power sources. The antenna design is identical to that of prototype A (albeit with a longer trace to the PCB edge, of 2.1), and the device is 74x64x25.

The relay's software can accept a number of configuration parameters which will be the focus of the experimental evaluation. <Ref> shows the cyclical operation mode of the relay during scanning. The relay stays in a given channel during a scan interval, and listens on that channel during the length of the scan window. In our tests we vary the length of the scan interval and set the scan window to an equal value. We evaluate the effects of two forwarding policies and estimate lifetime of the devices as a function of the sleep time (for the best performing scan interval and policy). Only advertising channels are used, and paired connections are not established, which is typical for one-way sensor meshes.

§ EXPERIMENTAL EVALUATION

We evaluate the relay's performance regarding packet reception and forwarding, for different scan interval lengths, policies, and sleep time. We employed the experimental setup show in <Ref>. In addition to the elements of the system shown, additional BLE nodes were placed in the environment, to act as noise, thus subjecting the system to a realistic operating condition. For all our tests, the scan window occupies the entire duration of the scan interval, in order to evaluate only the effects of the listening time, forwarding policy and sleep time. Exploring the effects of the length of sleep time (i.e., device duty cycle), in conjunction with non-equal scan window and interval lengths, , on power savings and performance is out of the scope of this paper.

Given this, we evaluated the following characteristics:

* the rate of packets received by the relay while subject to noise, for different scan intervals (i.e., advertising channel switching periods);

* the forwarding efficiency between the relay and a gateway using an immediate forwarding policy, first with two client nodes, and then with 11 client nodes;

* forwarding efficiency for 11 nodes, under a policy which buffers received packets and forwards replicas to the gateway, to reduce the overhead of switching between radio modes;

* power consumption as a function of device duty cycle (i.e., sleep time).

Experimental setup for relay efficiency evaluation

In order to account for all transmitted and received packets, the relay and the terminal gateway communicate every packet received via serial connection. Each packet is annotated with the originating node. Since the transmission period of the nodes is known, we know the total transmitted packets for a given run time. We can then compute the packet losses in different conditions, between the nodes and the relay, and between the relay and the gateway.

§.§ Relay Reception Efficiency for 2 Nodes

In this test, the relay's packet reception rate under noise was tested for two client nodes,  set to transmit advertising packets with period of 1. The test environment contained another 15 BLE nodes, external to the network, advertising at different intervals and thus acting as noise.

We varied the relay's scan interval between 50 and 1150. The scan window occupies the entire period.
What is measured in this case is the packet reception rate under noise, and due to the intrinsic loss of packets due to the randomness of the selected transmission and reception channels. The Bluetooth specification outlines a total of 40 channels, three of which (37, 38 and 39) are used for advertising packets.

<Ref> shows the measured reception rates of the relay. Three runs were performed per configuration. Per run, each of the two nodes transmitted 600 packets. For a transmission rate of 1 packet per second, this totals an experimental time of 90 per configuration. For all experiments the average reception rate is 88 ($\sigma = 1.02\%$).

The scan interval does not affect the reception rate significantly. Even so, it is marginally more efficient for the relay to stay tuned into a single channel for as long as possible, i.e., longer scan intervals. This might contribute to a slightly reduced packet loss since less time is spent switching radio channels, which contributes to idle time.

Also, since the Bluetooth protocol also dictates that an advertising event must be sent by a node on all three channels, the likelihood of the relay capturing a packet is higher by staying on a single channel for a period of time which is greater than the node's transmission period.

Note that in this scenario the relay's radio never transmits, and we evaluated the best case reception rate in a noisy scenario. Since the radio is half-duplex, once the relay begins forwarding packets, its reception rate will consequently decrease, as we present next.

§.§ Relay Forwarding Efficiency for 2 Nodes and 11 Nodes

<Ref> shows the reception efficiencies between the nodes and the relay, and between the relay and gateway. <Ref> shows the case with 2 client nodes, and 15 nodes acting as noise, and <Ref> shows the case with 11 client nodes, and 6 nodes acting as noise.

In these experiments, the sleep time is zero, as we wish to evaluate the performance, for a long period of operation, only as a function of the network size, scan interval, and noise introduced by other devices. The relay has an immediate forwarding policy for every packet received.

<Ref> shows that the relay experiences a greater packet loss relative to the data in <Ref>, since it was configured to interrupt the scan interval and re-transmit immediately. This policy intended to reduce the travel time of the packets to the gateway. However this means that only one packet is relayed per scan interval, which explains the loss of packets from the nodes to the relay. Consequently, the number of packets forwarded to the gateway diminishes as the scan interval increases.

For scan intervals greater than 350, the number of packets received by the gateways actually exceeds those forwarded. This is due to two factors. Firstly, for forwarding the relay must be switched to advertising mode for a duration such that only one packet is sent. However, non-deterministic behaviour during channel switching and switching between reception and transmission sometimes produces duplicate packets. Secondly, the gateway may receive packets directly from the nodes, depending on transmission power. This leads to an apparent increase in system performance for lengthier scan intervals, despite the relay's losses.

<Ref> shows the same metrics when 11 nodes are introduced into the system. For the same reason as before, the reception rate (for both the relay and gateway) decreases with the relay's scan interval. However, this case shows how the relay effectively acts as an intermediate buffer to hold packets. The shorter the scan intervals, the quicker the relay echoes packets, decreasing the likelihood that packets are missed while the gateway is occupied, either by being in a non-listening state, e.g., switching between channels, or by being busy processing beacons received either directly from the nodes or by the relay.

However, even in the best case, only approximately 16 of the total packets sent arrive at the gateway, which implies significant energy expenditure by the beacons without benefit. The next section improves this with a different forwarding policy.

§.§ Relay Forwarding Efficiency for 11 Nodes & Batching Policy

We programmed the relay with a forwarding policy based on a listening period, and a forwarding period. During the listening period, the relay accumulates the captured packets, e.g., 4 packets from node #1, 10 from node #2, and one from node #3. During forwarding, the relay echoes up to $N$ repetitions of a packet per node, regardless of how many packets were received per node. For instance, for 10 packets received for node #1, five echoes will be transmitted. This reduces the total traffic, and also normalizes the amount of packets sent upstream to the gateways, potentially boosting reception of packets sent by nodes under noiser conditions.

<Ref> shows the reception rates, this time including also the rate of successful transfer between the relay and gateway. The three scenarios employ a listening period of 10, and a different number of packet repetitions each, e.g., five packet repetitions for <Ref>. For each case, the interval between repetitions is also varied. Once again, the sleep time is zero, and the scan interval is 50 for all cases.

The listening time is also shown, which represents the amount of time during each listen-and-forward cycle that the relay is listening. The relay first listens during the scan time ($S_{Time}$) (switching between channels every scan interval), and buffers the packets. Then it enters forwarding mode where each packet is re-sent a given number of times ($Nr_{Repeats}$) at a set interval ($R_{Interval}$). Given that there are 11 nodes, the ratio between listening and forwarding time can be estimated as:

\begin{equation}
L(\%) = \frac{S_{Time}}{S_{Time} + R_{Interval} \times Nr_{Repeats} \times N_{Nodes} }
\end{equation}

In the best case, with 5 repetitions at 10 interval, up to 35 of packets are now successfully forwarded to the gateway, which is 2.2x increase in performance relative to immediate forwarding. Although the relay captures less packets directly from the nodes, due to the lengthier forwarding period, the overall forwarding efficiency is higher.

In a multi-relay scenario, a superior performance should be expected, although the best strategy regarding scan interval, repeat interval, and repeat count would have to be determined. However, a possible approach would be to have each relay in the system forward only a subset of all nodes, thus reducing its own load and preventing excessive in-system noise.

§.§ Estimated Power Consumption vs. Sleep Time

This section explores the power consumption in continued operation as a function of sleep time, given that the forwarding rates during uptime are indicated by the previous experiments. To retrieve power consumption, we utilized a power profiler kit from Nordic Semiconductors <cit.>.

We first use the power profiler to measure the current draw during radio operation (i.e., during scan window periods). Regardless of configuration values, the relay draws 7.5.

We then evaluate the power consumption for different duty cycles defined by the scan and sleep times. The scan interval and window remain equal at 50, and adopt a batching policy with 5 repetitions and a 10 repeat time. The efficiency for this case was 35.

The average current draw and efficiency as a function of the duty cycle can be calculated by the product of the cycle and these baseline values of 7.5 and 35, respectively. The battery life is computed based on the relay's four AA batteries totaling 12000.

<Ref> shows the resulting efficiencies and battery life. The efficiency is shown based on the experimental runs with 11 nodes transmitting at a 1 interval. In this case, a duty cycle of 100 leads to the 35 efficiency, but a battery life of only approximately 2.16 months. To attain a battery life of a year, a duty cycle of 20 is required, with an estimated efficiency of 7. Note that the effective efficiency of forwarding remains 35, since a duty cycle of 20 implies that, in the best case, 20 of all packets would be forwarded.

Additionally, note that for all experiments, the efficiency is dependant on the total amount of packets sent by the nodes. These experimental runs impose a 1 period per node. For some applications like sensor networks for temperature or light intensity readings with periods of in the order of minutes, longer update periods would be tolerable, especially since a long battery life is also desired for the nodes.

We can extrapolate that for a node transmission period of 2.5, the relay could forward 87 of the packets, given the same up-time and fewer packets, a behaviour similar to the one observed for <cit.> (see <Ref>). For a duty cycle of 20 to ensure close to a year of battery life, the estimated efficiency would increase by 10 percentage points.

The efficiency and power consumption are still subject to additional parameters such as multiple relays, tweaks to the batching policy, different values for scan and sleep times which resulting the same duty cycle, node transmission period and number of nodes. This exploration is out of the scope of this paper and left as future work.

§ CONCLUSION

We have presented an evaluation of a Bluetooth device and packet forwarding policies in mesh networks. The objective of the relay device is to extend the range of transmission between end devices, such as Bluetooth nodes, and the gateway devices, which are wall-powered and communicate with a central server. The relays allow for more area coverage without additional gateways, which are more costly, and without the necessary additional wall-power infrastructure.

We first evaluated the relay's packet reception with 2 nodes, under noise generated by 17 nodes which were not part of the system, for values of the scan window between 50 and 1150, and found that the relay can receive up to 90 of the node transmissions, for a node transmission period of 1.

We then evaluated the forwarding efficiency, measured as the number of packets received by the gateway versus the total number of packets sent by the nodes. For a policy where the relay immediately forwards a received packet, only 16 of packets sent by 11 nodes are received by the gateway. By employing a policy of deferred forwarding, and multiple packet repetitions per listened node, this increases to 35.

Finally, we measured the power draw of the device using a power analyzer, and estimated the lifetime of the four AA batteries (12000) for different duty cycles and node transmission periods.


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