Nonlinear Isocapacitary Concepts of Mass in 3-Manifolds with Nonnegative Scalar Curvature
We deal with suitable nonlinear versions of Jauregui's isocapacitary mass in 3-manifolds with nonnegative scalar curvature and compact outermost minimal boundary. These masses, which depend on a parameter 1 < 𝑝 ≤ 2, interpolate between Jauregui's mass 𝑝 = 2 and Huisken's isoperime...
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| description | We deal with suitable nonlinear versions of Jauregui's isocapacitary mass in 3-manifolds with nonnegative scalar curvature and compact outermost minimal boundary. These masses, which depend on a parameter 1 < 𝑝 ≤ 2, interpolate between Jauregui's mass 𝑝 = 2 and Huisken's isoperimetric mass, as 𝑝 → 1⁺. We derive positive mass theorems for these masses under mild conditions at infinity, and we show that these masses do coincide with the ADM mass when the latter is defined. We finally work out a nonlinear potential theoretic proof of the Penrose inequality in the optimal asymptotic regime.
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 19 (2023), 091, 29 pages
Nonlinear Isocapacitary Concepts of Mass
in 3-Manifolds with Nonnegative Scalar Curvature
Luca BENATTI a, Mattia FOGAGNOLO b and Lorenzo MAZZIERI c
a) Università degli Studi di Pisa, Largo Bruno Pontecorvo 5, 56127 Pisa, Italy
E-mail: luca.benatti@unipi.it
URL: https://sites.google.com/view/luca-benatti
b) Università di Padova, via Trieste 63, 35121 Padova, Italy
E-mail: mattia.fogagnolo@unipd.it
URL: https://sites.google.com/view/mattiafogagnolo
c) Università degli Studi di Trento, via Sommarive 14, 38123 Povo (TN), Italy
E-mail: lorenzo.mazzieri@unitn.it
URL: https://sites.google.com/site/mazzierihome
Received May 03, 2023, in final form October 23, 2023; Published online November 10, 2023
https://doi.org/10.3842/SIGMA.2023.091
Abstract. We deal with suitable nonlinear versions of Jauregui’s isocapacitary mass in
3-manifolds with nonnegative scalar curvature and compact outermost minimal boundary.
These masses, which depend on a parameter 1 < p ≤ 2, interpolate between Jauregui’s
mass p = 2 and Huisken’s isoperimetric mass, as p→ 1+. We derive positive mass theorems
for these masses under mild conditions at infinity, and we show that these masses do coincide
with the ADM mass when the latter is defined. We finally work out a nonlinear potential
theoretic proof of the Penrose inequality in the optimal asymptotic regime.
Key words: Penrose inequality; positive mass theorem; isoperimetric mass; nonlinear poten-
tial theory; nonlinear potential theory
2020 Mathematics Subject Classification: 83C99; 35B40; 35A16; 31C15; 53C21
Dedicated to Jean–Pierre Bourguignon
on the occasion of his 75th birthday
1 Introduction
The isoperimetric concept of mass was introduced by Huisken [26] to study 3-manifolds with
nonnegative scalar curvature that are possibly nonsmooth, and not fulfilling the asymptotic
requirements needed to define the classic ADM mass [6]. Given a 3-manifold (M, g), the isoperi-
metric mass is indeed defined as
miso = sup
(Ωj)j∈N
lim sup
j→+∞
miso(Ωj), (1.1)
where the supremum is taken among all exhaustions (Ωj)j∈N consisting of domains with C 1-
boundary and the isoperimetric quasi-local mass miso(Ω) is
miso(Ω) =
2
|∂Ω|
(
|Ω| − |∂Ω|
3
2
6
√
π
)
.
This paper is a contribution to the Special Issue on Differential Geometry Inspired by Mathemati-
cal Physics in honor of Jean-Pierre Bourguignon for his 75th birthday. The full collection is available at
https://www.emis.de/journals/SIGMA/Bourguignon.html
mailto:luca.benatti@unipi.it
https://sites.google.com/view/luca-benatti
mailto:mattia.fogagnolo@unipd.it
https://sites.google.com/view/mattiafogagnolo
mailto:lorenzo.mazzieri@unitn.it
https://sites.google.com/site/mazzierihome
https://doi.org/10.3842/SIGMA.2023.091
https://www.emis.de/journals/SIGMA/Bourguignon.html
2 L. Benatti, M. Fogagnolo and L. Mazzieri
Notice that, without any further assumption on the 3-manifold, the isoperimetric mass might
in principle be any number in [−∞,+∞]. Unlike the ADM mass, which is defined on an
asymptotically flat chart x =
(
x1, x2, x3
)
at infinity as
mADM = lim
r→+∞
1
16π
�
{|x|=r}
(∂kgii − ∂igki)
xk
|x|
dσ,
the isoperimetric mass does not require passing to a chart to be defined. Rather, it is based
on the geometric concepts of volume and perimeter, making it well-defined even when there is
limited information on the asymptotic behaviour of the metric. Inspired by this observation,
in [11], we proved a Riemannian Penrose inequality [13, 27] for the isoperimetric mass in the class
of strongly 1-nonparabolic Riemannian manifolds with nonnegative scalar curvature. With the
locution strongly 1-nonparabolic manifolds, we denote manifolds on which any bounded Ω ⊂M ,
whose boundary is homologous to ∂M , admits a proper locally Lipschitz weak inverse mean
curvature flow (IMCF for short), that is a solution w1 to the problem
div
(
Dw1
|Dw1|
)
= |Dw1| on M ∖ Ω,
w1 = 0 on ∂Ω,
w1 → +∞ as d(x, ∂Ω) → +∞,
(1.2)
according to the definition introduced in [27]. The analysis leading to the isoperimetric Rieman-
nian Penrose inequality in [11] was carried out using a new asymptotic comparison between the
Hawking mass (see (2.11) below) and the isoperimetric mass along the level sets of the weak
IMCF.
In the present paper, we are going to develop a similar theory, in the case where the weak
IMCF (1.2) is replaced by the level set flow of weak solutions wp ∈ C 1,β
loc (M∖Ω) to the boundary
value problem
∆pwp = |Dwp|p on M ∖ IntΩ,
wp = 0 on ∂Ω,
wp → +∞ as d(x, ∂Ω) → +∞.
(1.3)
The link between the above problem and the weak IMCF (1.2) relies on the fact that wp → w1
as p → 1+ locally uniformly on M [30, 35, 36, 37], provided some natural global requirements
are met by the manifold (M, g). On the other hand, the solutions to problem (1.3) are di-
rectly connected to the notion of p-capacitary potential of a compact body Ω. In fact, setting
wp = −(p− 1) log up implies that up is p-harmonic, that is ∆pup = 0. These relationships have
been instrumental in demonstrating a series of geometric inequalities by means of monotonicity
formulas, holding along the level sets of solutions to equation (1.3). As p → 1+, these inequal-
ities become increasingly close to the desired result. This machinery, firstly introduced in the
case p = 2 for harmonic functions in [1, 4], has proven to be powerful enough to produce an
enhanced version of the Minkowski inequality [2, 21], later extended to Riemannian manifolds
with nonegative Ricci curvature [10] as well as to the anisotropic setting [47]. In [5] and subse-
quently in [3], the authors used this approach on 3-manifolds with nonnegative scalar curvature
to prove the Riemannian Penrose inequality for a single black hole, based on the monotonic be-
haviour of a suitable p-harmonic version of the Hawking mass (see (2.12) below). Throughout the
manuscript, Riemannian manifolds are assumed to be smooth, connected, metrically complete,
noncompact, with one single end.
The main object of interest in the present paper is the following nonlinear potential theoretic
version of Huisken’s isoperimetric mass (1.1), that we call the p-isocapacitary mass. It involves
Nonlinear Isocapacitary Concepts of Mass 3
the classical notion of p-capacity of a compact setK ⊂M , that, in dimension 3 and for 1 < p < 3,
is given by
cp(K) = inf
{
1
4π
(
p− 1
3− p
)p−1 �
M∖K
|Dv|p dµ
∣∣∣∣∣ v ∈ C∞
c (M), v ≥ 1 on K
}
.
When the boundary ofK is regular enough, such infimum is realized by the p-capacitary potential
of K, i.e., the unique p-harmonic function up, equal to 1 on ∂K and vanishing at infinity.
Definition 1.1 (p-isocapacitary mass). Let (M, g) be a Riemannian 3-manifold with compact
boundary, and let 1 < p < 3. Given a closed bounded subset Ω ⊂ M containing ∂M with
C 1-boundary the quasi-local p-isocapacitary mass of Ω is defined as
m(p)
iso(Ω) =
1
2pπcp(∂Ω)
2
3−p
(
|Ω| − 4π
3
cp(∂Ω)
3
3−p
)
.
The p-isocapacitary mass m(p)
iso of (M, g) is defined as
m(p)
iso = sup
(Ωj)j∈N
lim sup
j→+∞
m(p)
iso(∂Ωj), (1.4)
where the supremum is taken among all exhaustions {Ωj}j∈N.
As for the isoperimetric mass, the p-isocapacitary mass might be any number in [−∞,+∞].
The special and particularly relevant case of the 2-isocapacitary mass has been recently intro-
duced and studied by Jauregui [28].
A first natural and fundamental question about these newly introduced quantities is whether
they are nonnegative, on the class of 3-manifolds with nonnegative scalar curvature, where
a solution to (1.3) exists for any bounded Ω with regular boundary. The latter mentioned
property will be referred to as the strong p-nonparabolicity of the manifold (M, g), a terminology
that interpolates between the notion of strong nonparabolicity introduced by Ni [39] and the
notion of strong 1-nonparabolicity, which was employed in [11] and recalled above.
Our first main result is a nonlinear potential-theoretic version of the Riemannian Penrose
inequality, that, although not sharp, implies the positive mass theorem for the p-isocapacitary
mass, with its associated rigidity statement. We prove its validity under the following asymptotic
integral gradient estimate:
(†) Given any Ω ⊂M closed bounded subset with smooth and connected boundary homologous
to ∂M , the function wp ∈ C 1
loc
(
M ∖ Ω
)
that solves (1.3) satisfies
�
∂Ωt
|Dwp|2 dσ = o
(
et/(p−1)
)
as t→ +∞ where Ωt = {wp ≤ t}.
Theorem 1.2 (p-isocapacitary Riemannian Penrose inequality). Let (M, g) be a strongly p-
nonparabolic Riemannian 3-manifold, 1 < p < 3, with nonnegative scalar curvature and with
smooth, compact, connected, minimal, possibly empty boundary. Assume also that (M, g) satis-
fies (†) and H2(M,∂M ;Z) = {0}. Then
cp(∂M)
1
3−p ≤ 5− p
2
m(p)
iso. (1.5)
In particular, m(p)
iso ≥ 0 and it vanishes if and only if (M, g) is isometric to the flat 3-dimensional
Euclidean space.
4 L. Benatti, M. Fogagnolo and L. Mazzieri
As we are going to show below in Theorem 1.3, under natural assumptions of asymptotic
flatness, the p-capacitary masses coincide with one another, and they are all equal to the classical
ADM mass. Then, in the limit as p→ 1+, (1.5) yields the Riemannian Penrose inequality. In [12]
and its extension [48], the version of (1.5) with the sharp exponent is derived employing the
weak inverse mean curvature flow. The above result on the other hand just involves nonlinear
potential theoretic concepts, such as the notions of p-nonparabolicity and of p-isocapacitary
mass.
The condition (†) is actually very mild. As we are going to detail in Remark 3.4, it is
always fulfilled on manifolds that are merely C 0-asymptotically flat, provided a suitable Ricci
lower bound is also satisfied. C 1-asymptotically flat Riemannian manifolds are also fulfilling
condition (†) for 1 < p ≤ 2, as proved in Lemma 2.5. This latter class of manifolds is particularly
natural in the framework of mathematical general relativity, as the works of Bartnik [7] and
Chruściel [16] showed that the ADM mass is well defined on C 1
τ -asymptotically flat Riemannian
manifolds, with τ > 1/2. In fact, our second main result shows that on C 1
τ -asymptotically
Flat Riemannian 3-manifolds with nonnegative scalar curvature the p-isocapacitary masses do
coincide with the ADM mass for any 1 ≤ p ≤ 2. This fact was previously known only for p = 2
and only for harmonically flat manifolds, as proven by Jauregui in the insightful paper [28,
Corollary 8]. Most of the authors assume that the scalar curvature belongs to L1(M) in the
definition of the ADM mass so that the latter is not only a geometric invariant but also a finite
number. Here, we do not assume any integrability of the scalar curvature, hence, our ADM
mass is not finite a priori. The statement below, can be understood in the sense that all masses
are infinite as soon as one of them is, and they all coincide with one another as soon as one of
them is finite.
Theorem 1.3. Let (M, g) be a C 1
τ -asymptotically flat Riemannian 3-manifold, τ > 1/2, with
nonnegative scalar curvature and with smooth, compact, minimal, possibly empty boundary. As-
sume that H2(M,∂M ;Z) = {0}. Then
m(p)
iso = miso = mADM
for all 1 < p ≤ 2.
What we will actually prove in this paper is the chain of inequalities mADM ≤ m(p)
iso ≤ miso.
This will in turn give the identities above due to the equality miso = mADM, obtained in the
setting of Theorem 1.3 in [11, Theorem 4.13]. In the proof of Theorem 1.3, the inequality
m(p)
iso ≥ miso is deduced substantially arguing as in [28, Theorem 5], combining some of the com-
putations in [19] together with an extension of the main estimate in [12, 48] for the isoperimetric
mass (see Proposition 2.13).
The reverse inequality, for which the harmonically flat condition was invoked in [28], is
instead proven by integrating the sharp isoperimetric inequality in [15, Corollary C.3] to obtain
a sharp p-isocapacitary inequality in terms of the isoperimetric mass, Theorem 5.5. This last
step is inspired by the classical derivation of the sharp p-isocapacitary inequality from the sharp
isoperimetric inequality, as in [10, Theorem 4.1]. The identification with the ADM mass finally
follows from [11, Theorem 4.13], where it was showed to coincide with the isoperimetric mass in
the above optimal regime, sharpening [29, Theorem 3].
It is natural to conjecture that the equivalence among p-isocapacitary masses also holds under
weaker asymptotic assumptions, where the ADM mass may not even be well defined. In this
direction, in the generality of C 0-asymptotically flat Riemannian manifold satisfying (†) for
1 < p ≤ 2, we will prove the following two-sided estimate (see Lemma 5.1 and Theorem 5.6):
m(p)
iso ≤ miso ≤
22p−1π
p−1
2 ppC
3
2
(p−1)
S
(3− p)(p− 1)p−1
1
3−p
m(p)
iso,
Nonlinear Isocapacitary Concepts of Mass 5
where CS is the global Sobolev constant on (M, g). The lower bound is optimal, while the upper
bound sharpens as p → 1+. As a consequence, in this generality miso can at least be recovered
as the limit of its p-capacitary versions as p→ 1+.
In concluding the paper, we propose an alternative proof of the Riemannian Penrose inequality
in the sharp asymptotic regime given in [11], that is for C 1
τ -asymptotically flat Riemannian 3-
manifolds, with τ > 1/2. In this previous work, we exploited the better asymptotic behaviour
of harmonic functions and the monotonicity of the 2-Hawking mass discovered in [5] to improve
the original argument by Huisken and Ilmanen [27] based on the IMCF, as far as the asymptotic
analysis at infinity is concerned.
Replacing the IMCF with the level sets flow of the solutions wp to (1.3), we obtain a nonsharp
family of p-Penrose Inequalities in terms of the p-capacity of the horizon. These then provide
the optimal and classical Riemannian Penrose inequality in the limit as p→ 1+. We recall that
a minimal boundary ∂M is outermost if no other closed minimal surface homologous to ∂M
is contained in M . Moreover, we understand that, in case such boundary is empty, then no
minimal closed hypersurfaces exists in (M, g). We can now state the last main result of the
paper.
Theorem 1.4. Let (M, g) be a C 1
τ -asymptotically flat Riemannian 3-manifold, τ > 1/2, with
nonnegative scalar curvature and with smooth, compact, minimal, connected and outermost pos-
sibly empty boundary. Then
cp(∂M)
1
3−p ≤ 2mADM (1.6)
for any 1 < p ≤ 2. Letting p→ 1+, we get√
|∂M |
16π
≤ mADM. (1.7)
Both in the above result and in Theorem 1.3, the restriction to p ≤ 2 is due to technical
reasons, that can be devised in the proof of Theorem 5.6.
Outline of the paper
In Section 2, we gather some basic facts about p-harmonic potentials. The content of this
section is substantially well known. In Section 3, we work out the main asymptotic comparison
at infinity between the p-Hawking mass and the quasi-local p-isocapacitary mass, see Lemma 3.2.
We deduce the nonsharp Riemannian Penrose inequality Theorem 1.2 for the p-isocapacitary
mass. In Section 5, we show relations among the p-isocapacitary masses for the various values
of p, in turn obtaining Theorem 1.3. Finally, in Appendix A, we include a proof of the full
monotonicity of the p-Hawking mass, since the original [3, Theorem 1.1] actually yields such
result only along regular values. We will also relate such quantity with a similar one considered
in [14] that will naturally appear in the asymptotic comparison argument ruling our main results.
2 Preliminaries in nonlinear potential theory
We first remind that the operator ∆p is defined by ∆pf = div
(
|∇f |p−2∇f
)
, for f ∈ C 2 with
nonvanishing gradient. As far as basic principles and regularity for weakly p-harmonic functions
are concerned, we just refer the reader to [17, 18, 22, 31, 32, 33, 41, 42, 43, 44, 45], and [24, 25] for
the theory of p-capacitary potentials in exterior domains (see also [8, Chapter 1] and reference
therein), including existence issues. The function wp that solves (1.3) can be written as wp =
6 L. Benatti, M. Fogagnolo and L. Mazzieri
−(p− 1) log up, where up ∈ C 1,β
loc (M ∖ IntΩ) is the solution to
∆(p)
g up = 0 on M ∖ IntΩ,
up = 1 on ∂Ω,
up → 0 as d(x, ∂Ω) → +∞,
(2.1)
the following definition of strongly p-nonparabolic Riemannian manifold is consistent with that
of strong nonparabolicity [39] and with the limit case of strong 1-nonparabolicity [11].
Definition 2.1 (strongly p-nonparabolic). We say that a 3-dimensional Riemannian mani-
fold (M, g) with compact possibly empty boundary is strongly p-nonparabolic, 1 < p < 3, if
there exists a solution to (1.3) for some Ω ⊆ M closed bounded with smooth boundary homol-
ogous to ∂M .
Remark 2.2. By the maximum principle, in a strongly p-nonparabolic manifold every Ω with
C 1-boundary homologous to ∂M admits a solution to (1.3).
This definition naturally comes with the notion of the p-capacity of a compact subsetK ⊂M ,
which we recall to be
cp(K) = inf
{
1
4π
(
p− 1
3− p
)p−1 �
M∖K
|Dv|p dµ
∣∣∣∣∣ v ∈ C∞
c (M), v ≥ 1 on K
}
.
When the boundary of K is sufficiently regular, the above infimum is realized by the function up
solving (2.1). A fundamental property of the p-capacity is that it is monotone with respect to
the standard inclusion of sets. More specifically, if we consider sublevel sets Ωt = {wp ≤ t} of
solutions to (1.3), we have that their p-capacities grow exponentially with respect to the arrival
time parameter t. This is completely analogous to the exponential growth of the area along the
IMCF. We recall this useful property in the following lemma, proved in [24, Lemma 3.8].
Lemma 2.3. Let (M, g) be a 3-dimensional Riemannian manifold with compact possibly empty
boundary ∂M . Let Ω ⊆M be a closed bounded subset homologous to ∂M with C 1-boundary, wp
the solution to (1.3) starting at Ω and Ωt = {wp ≤ t}, 1 < p < 3. We have
cp(∂Ωt) = et cp(∂Ω) =
1
4π
�
∂Ωt
(
|Dwp|
3− p
)p−1
dσ.
2.1 Estimates on asymptotically flat Riemannian manifolds
In [9], we prove the asymptotic behaviour of p-harmonic potential (2.1) assuming a lower bound
on the Ricci curvature other than asymptotic assumptions on the metric. Here we want to
remove the additional assumption on Ricci curvature. We will prove that it is superfluous if we
assume the metric is C 1-asymptotically flat.
We start by giving the precise definition of asymptotically flat 3-manifolds.
Definition 2.4 (asymptotically flat Riemannian manifolds). A 3-dimensional Riemannian man-
ifold (M, g) with compact possibly empty boundary is C k
τ -asymptotically flat
(
resp. C k-asymp-
totically flat
)
, with order k ∈ N and rate τ > 0, if the following conditions are satisfied.
1. There exists a compact set K ⊆ M such that M ∖K is differmorphic to R3 ∖ {|x| ≤ R},
through a map
(
x1, x2, x3
)
whose components are called asymptotically flat coordinates.
Nonlinear Isocapacitary Concepts of Mass 7
2. In the chart
(
M ∖K,
(
x1, x2, x3
))
the metric tensor is expressed as
g = gijdx
i ⊗ dxj = (δij + ηij)dx
i ⊗ dxj
with
3∑
i,j=1
k∑
|β|=0
|x||β|+τ |∂βηij | = O(1) (resp. = o(1)), as |x| → +∞.
C 0-asymptotically flat Riemannian manifolds are in particular strongly p-nonparabolic. This
is a consequence of [35, Theorem 3.6], implying it under the mere existence of a global (even
weighted) Sobolev inequality.
We first point out a decay estimate for the gradient of up holding on C 1-asymptotically
flat Riemannian manifolds. It is immediately obtained as a consequence of the Cheng–Yau
inequality for p-harmonic functions [46] when the Ricci curvature is quadratically asymptotically
nonnegative (see also [9, Proposition 2.27]). However, as the proof presented in [46] is purely
integral, integrating by parts the term containing the Ricci curvature and exploiting the C 1
decay of the metric leads to the following.
Lemma 2.5. Let (M, g) be a C 1-asymptotically flat Riemannian 3-manifold with compact pos-
sibly empty boundary. Let Ω ⊂ M be a closed bounded subset with C 1-homologous to ∂M , and
let up be the solution to (2.1), 1 < p < 3. Then, there exist C > 0 and R > 0 such that
|Dup|(x) ≤ C
up(x)
|x|
(2.2)
on {|x| ≥ R}.
Proof. We drop the subscript p. Let |x| ≥ R for some R large enough so that |Γk
ij | ≤ C/|x|
|gij − δij | ≤ C on {|x| ≥ R/3} for some C > 0 and every i, j, k = 1, 2, 3 and B|x|/2(x) ⊆
{|x| ≥ R/3} for all x ∈ {|x| ≥ R}. The constant C may change during the proof, but its
value depends only on R, g and p, not on u or x. We explain how to modify the proof of the
Cheng–Yau inequality in the ball B = B|x|/2(x) presented in [46] to replace their lower bound
on the Ricci curvature with the C 1-decay of the metric coefficients. The proof begins with an
integral version of the Bochner formula
�
B
L (f)ψ dµ = 2
�
B
f
p
2
−1|DDu|2ψ dµ+
(
p
2
− 1
) �
B
|Df |2f
p
2
−2ψ dµ
+
�
B
f
p
2
−1Ric(Du,Du)ψ dµ (2.3)
for any ψ ∈ C∞
c (B), where f = |Du|2 and
L (f) = div
[
fp/2−1
(
Df + (p− 2)⟨Du |Df⟩ Du
|Du|2
)]
− pfp/2−1⟨Du |Df⟩.
The only term containing the second derivatives of the metric is the one involving the Ricci
curvature tensor. It can be written as
�
B
f
p
2
−1Ric(Du,Du)ψ dµ =
�
B
(
∂kΓ
k
ij − ∂iΓ
k
kj + Γk
ijΓ
m
km − Γm
ikΓ
k
jm
)
DiuDju f
p
2
−1ψ dµ. (2.4)
8 L. Benatti, M. Fogagnolo and L. Mazzieri
The terms containing products of Christoffel symbols can be estimated using the C 1-asymptotic
behaviour of the metric, using the fact we are on {|x| ≥ R}. Indeed,
�
B
(
Γk
ijΓ
m
km − Γm
ikΓ
k
jm
)
DiuDju f
p
2
−1ψ dµ ≥ − C
|x|2
�
B
f
p
2ψ dµ. (2.5)
On the other hand, using integration by parts, we have
�
B
∂kΓ
k
ijD
iuDju f
p
2
−1ψ dµ = −
�
B
Γk
ij∂k
(
DiuDju f
p
2
−1ψ
√
det g
)
dµδ
≥ −C
�
B
1
|x|
|DDu||Du|f
p
2
−1ψ + f
p
2
(
ψ
|x|2
+
|Dψ|
|x|
)
dµ
≥ −C
�
B
ε2|DDu|2f
p
2
−1ψ + f
p
2
((
1 + ε2
)
ψ
ε2|x|2
+
|Dψ|
|x|
)
dµ (2.6)
for any ε > 0, where we employed Young’s inequality in the last step to the product between(
|x|−1|DDu|
)
and |Du|, recalling that f = |Du|2. Observe that we used the C 1-asymptotic
behaviour of the metric not only to control |∂g|, but also to estimate |∂∂u|2 in terms of |DDu|2
and |x|−2|Du|2. We can deal with the remaining term in the same way. Combining (2.4)–(2.6),
we finally get
�
B
f
p
2
−1Ric(Du,Du)ψ dµ ≥ −C
�
B
ε2|DDu|2f
p
2
−1ψ + f
p
2
((
1 + ε2
)
ψ
ε2|x|2
+
|Dψ|
|x|
)
dµ (2.7)
Following the argument in [46, Theorem 1.1], one can chose ψ = f bη2 for some b > 1 and with
|Dη| ≤ C η/|x|. With this specification, the integrand of the last term in (2.7) is pointwise
estimated by
|Dψ|
|x|
f
p
2 ≤ Cψ
(
ε2|DDu|2f
p
2
−1 +
(
1 + ε2
)
ε2|x|2
f
p
2ψ
)
.
The last term in (2.7) can be absorbed in the others. Plugging this into (2.3), we deduce
�
B
L (f)ψ dµ ≥
(
2− ε2
)�
B
f
p
2
−1|DDu|2ψ dµ+
(p
2
− 1
)�
B
|Df |2f
p
2
−2ψ dµ
−
C
(
1 + ε2
)
ε2|x|2
�
B
f
p
2ψ dµ (2.8)
for any ε > 0. Plugging now the last displayed identity at the bottom of [46, p. 763] into (2.8),
we get, choosing ε > 0 small enough (depending only on p), the inequality [46, formula (2.3)],
with κ given by a suitable uniform constant multiplying |x|−2. From this point on, the proof
can be followed line by line, and yields the Cheng–Yau inequality of [46, Theorem 1.1] in terms
of κ above in the ball B|x|/4(x). This is exactly the claimed (2.2). ■
Remark 2.6. We can rewrite the above estimate in terms of wp = −(p− 1) log up. It reads
|Dwp(x)| ≤
C
|x|
on {|x| ≥ R}, for R large enough and some positive constant C > 0 depending on p.
The following is a double-sided control on the solution up to (2.1) of a bounded Ω ⊂M with
smooth boundary with respect to the Euclidean distance.
Nonlinear Isocapacitary Concepts of Mass 9
Lemma 2.7. Let (M, g) be a C 1-asymptotically flat Riemannian 3-manifold with compact pos-
sibly empty boundary. Let Ω ⊂ M be a closed bounded subset with C 1-homologous to ∂M , and
let up be the solution to (2.1), 1 < p < 3. Then, there exist C > 0 and R > 0 such that
C−1|x|−
3−p
p−1 ≤ up(x) ≤ C|x|−
3−p
p−1
on {|x| ≥ R}.
Proof. We drop subscript p. The rightmost inequality follows by [35, Theorem 3.6], since
having a positive isoperimetric constant is equivalent to having a global Sobolev inequality.
We get the leftmost inequality adapting the argument used for [25, Proposition 5.9]. Inte-
grating Lemma 2.5 as in [46] we have a Harnack inequality holding on large coordinate spheres
max
{|x|=r}
u ≤ C min
{|x|=r}
up,
where C does not depend on r. We are now committed to proving that
max
{|x|=r}
u ≥ Cr
− 3−p
p−1 ,
which concludes the proof. Let m = max{u(x) | |x| = r}. Then
cp({|x| ≤ r}) ≥ cp({u ≥ m}) = m−(p−1)cp(∂Ω).
Using [23, Theorem 2.6], we have
m cp(∂Ω)
− 1
p−1 ≥ cp({|x| ≤ r})−
1
p−1 ≥
+∞∑
j=0
(
cp
({
|x| ≤ 2jr
}
,
{
|x| < 2j+1r
}))− 1
p−1 , (2.9)
where we point out that the p-capacity of a condenser (K,A) where A is a open subset of M
and K is a compact subset of A, is just defined as
cp(K,A) = inf
{
1
4π
(
p− 1
3− p
)p−1 �
A∖K
|Dv|p dµ
∣∣∣∣∣ v ∈ C∞
c (A), v ≥ 1 on K
}
.
Picking now a test function ψr ∈ C∞
c
({
|x| ≤ 2j+1r
})
taking the value 1 on
{
|x| ≤ 2jr
}
and such
that |∇ψr| ≤ C/r, we directly estimate the right-hand side of (2.9) with
+∞∑
j=0
(
cp
({
|x| ≤ 2jr
}
,
{
|x| < 2j+1r
}))− 1
p−1
≥ p− 1
3− p
(4π)
− 1
p−1C
+∞∑
j=0
( �
{|x|<2j+1r}
|∇ψr|p dµ
)− 1
p−1
≥ p− 1
3− p
(4π)
− 1
p−1C
+∞∑
j=0
(
(2jr)p
|{|x| ≤ 2j+1r}|
) 1
p−1
≥ p− 1
3− p
(4π)
− 1
p−1C
� +∞
2r
(
t
|{|x| ≤ t}|
) 1
p−1
dt. (2.10)
Finally, since |{|x| ≤ t}| = t3(4π/3+ o(1)) as t→ +∞, we can choose R such that |{|x| ≤ r}| ≤
Cr3 for every r ≥ R, so that, by (2.9) and (2.10), we get
m ≥ C
� +∞
2r
(
t
|{|x| ≤ t}|
) 1
p−1
dt ≥ Cr
− 3−p
p−1 ,
which concludes the proof. ■
10 L. Benatti, M. Fogagnolo and L. Mazzieri
Corollary 2.8. We can rewrite the above estimates in terms of wp = −(p−1) log up. They read
(3− p) log |x| − C−1 ≤ wp ≤ (3− p) log |x|+C
on {|x| ≥ R}, for R large enough and some positive constant C > 0 depending on p.
We conclude by resuming some basic asymptotic expansions for wp, substantially worked out
in [9]. We specialise in the case of C 1-asymptotically flat Riemannian 3-manifolds and take
advantage of the above observations in order to get rid of any Ricci curvature assumption.
Lemma 2.9. Let (M, g) be a C 1-asymptotically flat Riemannian 3-manifold with compact, pos-
sibly empty boundary. Fix 1 < p < 3. Let Ω ⊆ M be a closed bounded subset with C 1-boundary
homologous to ∂M , wp the solution to (1.3) starting at Ω and Ωt = {wp ≤ t}. Then, for every
1 < q < 3
(1) wp = (3− p) log |x| − log(cp(∂Ω)) + o(1) as |x| → +∞,
(2) Diwp = (3− p) xi
|x|2 (1 + o(1)) as |x| → +∞,
(3) limt→+∞ e
− 3−q
3−p
t
cq(∂Ωt) = 1,
(4) limt→+∞ e
− 2
3−p
t |∂Ωt| = 4π.
Proof. Items (1) and (2) follow with the same strategy of [9, Theorem 3.1], replacing [9,
Corollary 2.25 and Proposition 2.27] with Lemmas 2.7 and 2.5, respectively. Having (1), item (3)
follows at once. Indeed, for every ε > 0
{(3− p) log |x| ≤ t− ε} ⊂ Ωt ⊂ {(3− p) log |x| ≤ t+ ε},
for sufficiently large t. Hence, by monotonicity of the q-capacity we obtain
cq({(3− p) log |x| = t− ε}) ≤ cq(∂Ωt) ≤ cq({(3− p) log |x| = t+ ε}).
Dividing both sides by e−(3−q)t/(3−p) and passing to the limit as t → +∞, in virtue of [9,
Lemma 2.21] we get
e
− 3−q
3−p
ε ≤ lim
t→+∞
e
− 3−q
3−p cq(∂Ωt) ≤ e
3−q
3−p
ε
,
from which we infer item (3) sending ε→ 0+. Item (4) follows as [9, Proposition 3.4] replacing [9,
Theorems 1.1 and 3.1] with items (1) and (2), respectively. ■
Remark 2.10. Up to this point, the content of this Section can be extended in any dimen-
sion n ≥ 3 with obvious modifications. From now on we focus on dimension 3 since the mono-
tonicity formulas introduced are peculiar to this dimension.
2.2 Concepts of mass in nonlinear potential theory
The classical Hawking mass mH , that is
mH(∂Ω) =
|∂Ω|
1
2
16π
3
2
(
4π −
�
∂Ω
H2
4
dσ
)
, (2.11)
for Ω ⊂M , monotonically increases along the level sets of the weak IMCF [27]. Such a property
is clearly not preserved in general when one replaces the weak IMCF with solutions wp to (1.3).
Nonlinear Isocapacitary Concepts of Mass 11
For this reason, we consider a different family of quasi-local masses. We will call p-Hawking
mass the quantity
m(p)
H (∂Ω) =
cp(∂Ω)
1
3−p
8π
[
4π +
�
∂Ω
|Dwp|2
(3− p)2
dσ −
�
∂Ω
|Dwp|
(3− p)
H dσ
]
(2.12)
for ∂Ω ∈ C 1 with weak second fundamental form in L2(∂Ω). This should be thought of as
a p-version of the classical Hawking mass. In fact, it is immediately seen that the p-Hawking
mass formally converges to the Hawking mass as p → 1+, having in mind that along the weak
IMCF w1 we have |Dw1| = H and that the p-capacity of an outward minimizing set recovers
the perimeter in such limit [20, Theorem 1.2]. Crucially, as the Hawking mass is monotone
along the weak IMCF [27, Geroch Monotonicity Formula 5.8], so the function t 7→ m(p)
H (∂Ωt),
for Ωt = {wp ≤ t}, is monotone nondecreasing, as proven in [3] (see actually Appendix A for the
full monotonicity result).
Theorem 2.11. Let (M, g) be a strongly p-nonparabolic Riemannian 3-manifold, 1 < p < 3, with
nonnegative scalar curvature and with connected, compact, possibly empty boundary. Assume
that H2(M,∂M ;Z) = {0}. Let Ω ⊆ M be a closed bounded subset with connected C 1-boundary
homologous to ∂M and with second fundamental form h ∈ L2(∂Ω), wp the solution to (1.3) start-
ing at Ω and Ωt = {wp ≤ t}. The function t 7→ m(p)
H (∂Ωt) defined in (2.12) admits a monotone
nondecreasing BVloc(0,+∞) representative and
d
dt
m(p)
H (∂Ωt) =
cp(∂Ωt)
1
3−p
(3− p)8π
(
4π −
�
∂Ωt
R⊤
2
dσ +
�
∂Ωt
|̊h|2
2
+
R
2
+
|D⊤|Dwp||
2
|Dwp|2
dσ
+
�
∂Ωt
5− p
p− 1
(
|Dwp|
3− p
− H
2
)2
dσ
)
(2.13)
holds at every t regular for wp.
The p-Hawking mass has the very useful feature of dominating the Hawking mass times
a constant involving the global Sobolev constant of the underlying Riemannian manifold.
Lemma 2.12. Let (M, g) be a strongly p-nonparabolic Riemannian 3-manifold, 1 < p < 3, with
compact possibly empty boundary. Assume that the Sobolev constant CS of (M, g) is positive.
Then for every outward minimising Ω ⊂ M with C 1-boundary homologous to ∂M with second
fundamental form h ∈ L2(∂Ω), we have
m(p)
H (∂Ω) ≥
(
(3− p)(p− 1)p−1
22p−1π
p−1
2 ppC
3
2
(p−1)
S
) 1
3−p
mH(∂Ω).
Proof. Observe that
�
∂Ω
|Dwp|
(3− p)
H dσ −
�
∂Ω
|Dwp|2
(3− p)2
dσ =
�
∂Ω
H2
4
dσ −
�
∂Ω
(
H
2
− |Dwp|
3− p
)2
dσ ≤
�
∂Ω
H2
4
dσ.
It is then enough to proceed as in the proof of [20, Theorem 1.3] to prove that
cp(∂Ω)
1
3−p ≥
(
(3− p)(p− 1)p−1
4πppC
3
2
(p−1)
S
) 1
3−p
|∂Ω|
1
2 . ■
12 L. Benatti, M. Fogagnolo and L. Mazzieri
In [12, Theorem 2] (see also the new proof proposed in [40]) then extended to all 1 < p < 3 [48],
the authors prove an upper bound for the capacity in terms of the area and the Willmore
deficit. Observe that in their proof the asymptotically flat condition is assumed only to grant
the existence of an IMCF starting at some Ω. Here we assume the existence of the IMCF
by requiring that (M, g) is strongly 1-nonparabolic (see [11]). We report here the statement,
referring the reader to [12, 48] for the proof.
Proposition 2.13 (Nonlinear version of Bray–Miao’s estimate). Let (M, g) be a strongly 1-non-
parabolic Riemannian 3-manifold with nonnegative scalar curvature and with connected, compact,
possibly empty boundary. Assume that H2(M,∂M ;Z) = {0}. Let Ω ⊂ M be closed, bounded
with connected C 1-boundary homologous to ∂M with h ∈ L2(∂Ω). Then
cp(∂Ω) ≤
(
|∂Ω|
4π
) 3−p
2
2F1
(
1
2
,
3− p
p− 1
,
2
p− 1
; 1− 1
16π
�
∂Ω
H2 dσ
)−(p−1)
,
where 2F1 is the hypergeometric function.
We recall that the hypergeometric function satisfies the following useful relation
2F1
(
1
2
,
3− p
p− 1
,
2
p− 1
;
2m
t
)
=
3− p
p− 1
t
3−p
p−1
� +∞
t
s
2
p−1
(
1− 2m
s
)− 1
2
ds, (2.14)
where 1 < p < 3 and m ∈ R. Here, the integrand on right-hand side is, up to a scaling factor,
the radial derivative of the rotationally symmetric p-capacitary potential of the horizon of the
Schwarzschild of mass m.
Combining the previous proposition, the minimality of ∂M and the isoperimetric Riemannian
Penrose inequality [11, Theorem 1.3], we obtain a sharp Penrose-type inequality for the p-
capacity of the boundary, for every 1 < p < 3.
Theorem 2.14. Let (M, g) be a strongly 1-nonparabolic Riemannian 3-manifold with nonnega-
tive scalar curvature and with smooth, compact, connected, minimal outermost boundary. Then,
for every 1 < p < 3 it holds
cp(∂M)
1
3−p ≤ 2
(
√
π
Γ
(
2
p−1
)
Γ
(
2
p−1 − 1
2
))− p−1
3−p
miso, (2.15)
where Γ is the gamma function. Moreover, the equality holds in (2.15) if and only if (M, g) is
isometric to(
Rn ∖ {|x| < 2miso},
(
1 +
miso
2|x|
)4(
δijdx
i ⊗ dxj
))
.
Proof. By [11, Lemma 2.8], under the above assumptions we have H2(M,∂M ;Z) = {0}. By
Proposition 2.13, we have
cp(∂M)
1
3−p ≤ 2
(
√
π
Γ
(
2
p−1
)
Γ
(
2
p−1 − 1
2
))− p−1
3−p
√
|∂M |
16π
.
Then, (2.15) follows from [11, Theorem 1.3]. The equality in (2.15) implies the equality in [11,
Theorem 1.3] yielding the rigidity statement. ■
This result has been provided in [12, Theorem 4] for p = 2 and [48, Theorem 1.1] for every
1 < p < 3, in terms of the ADM mass and in the asymptotic flat regime. One can recover such
formulation applying Theorem 2.14 in conjunction with Theorem 1.3, proved below.
Nonlinear Isocapacitary Concepts of Mass 13
3 p-isocapacitary Riemannian Penrose inequality
In establishing the asymptotic comparison between the p-Hawking mass (2.12) and the p-iso-
capacitary mass, the quantity
m̃(p)
H (∂Ω) =
cp(∂Ω)
1
3−p
4π(3− p)
(
4π −
�
∂Ω
|Dwp|2
(3− p)2
dσ
)
(3.1)
will naturally appear. This quantity is closely related to the ones studied in [14, 38] (see
Theorem A.2 and Remark A.3 for details). In the following lemma, we discuss the monotonicity
properties of (3.1) along the level sets of the solution wp to (1.3) and its relations with the
p-Hawking mass (2.12).
Lemma 3.1. Let (M, g) be a strongly p-nonparabolic Riemannian 3-manifold, 1 < p < 3, with
nonnegative scalar curvature and with connected, compact, possibly empty boundary. Assume
also that (M, g) satisfies (†) and H2(M,∂M ;Z) = {0}. Let Ω ⊆ M be a closed bounded subset
with connected C 1-boundary homologous to ∂M and with second fundamental form h ∈ L2(∂Ω),
wp the solution to (1.3) starting at Ω and Ωt = {wp ≤ t}. Then the function t 7→ m̃(p)
H (∂Ωt)
belongs to W 1,1
loc (0,+∞) is monotone nondecreasing. Moreover, we have
m(p)
H (∂Ωt) ≤ m̃(p)
H (∂Ωt) (3.2)
for every t ∈ [0,+∞), and
lim
t→+∞
m(p)
H (∂Ωt) = lim
t→+∞
m̃(p)
H (∂Ωt). (3.3)
Proof. Denote
N(t) = cp(∂Ωt)
− 1
p−1
(
4π −
�
∂Ωt
|Dwp|2
(3− p)2
dσ
)
,
D(t) = cp(∂Ωt)
− 2
(3−p)(p−1) ,
we have that
m̃(p)
H (∂Ωt) = (cp(∂Ωt)
1
3−p
(
4π −
�
∂Ωt
|Dwp|2
(3− p)2
dσ
)
=
N(t)
D(t)
.
Observe that, N(t) is the quantity studied in [14, 38], while 1/D(t) is an exponentially grow-
ing term we multiplied it by. The function N(t), and thus m̃(p)
H (∂Ωt) = N(t)/D(t), belongs
to W 1,1
loc (0,+∞) by Theorem A.2. Moreover, (A.4) in Theorem A.2 gives that N ′(t)/D′(t) =
4π(3 − p)m(p)
H (∂Ωt) which is nondecreasing by Theorem 2.11. Finally, D(t) → 0 as t → +∞,
while N(t) → 0 as t → +∞ by the assumption (†). It is now a general fact that given two
functions f, g ∈ W 1,1
loc (0,+∞), with g(t) ̸= 0 and g′(t) ̸= 0, such that f(t), g(t) → 0 as t → +∞
and with f ′(t)/g′(t) monotone nondecreasing, the function f(t)/g(t) is monotone nondecreasing
as well (see, e.g., [49, Lemma 3.2]). Applying it with f(t) = N(t) and g(t) = D(t), we have that
the function in t 7→ m̃(p)
H (∂Ωt) is nondecreasing. To prove (3.2), it is enough to observe that
0 ≤ d
dt
(
N(t)
D(t)
)
=
N ′(t)D(t)−N(t)D′(t)
D(t)2
=
8π
(p− 1)
(
−m(p)
H (∂Ωt) + m̃(p)
H (∂Ωt)
)
for almost any t. It then remains to prove (3.3). On the other hand, by de L’Hôpital rule
(see [11, Theorem A.1])
lim
t→+∞
4π(3− p)m̃(p)
H (∂Ωt) = lim
t→+∞
N(t)
D(t)
≤ lim
t→+∞
N ′(t)
D′(t)
= lim
t→+∞
4π(3− p)m(p)
H (∂Ωt).
The reverse inequality easily follows by (3.2). ■
14 L. Benatti, M. Fogagnolo and L. Mazzieri
The following result gives the p-capacitary counterpart of [11, Lemma 2.7], that asymptoti-
cally controls the Hawking mass with the quasi-local isoperimetric mass of the evolving sets.
Lemma 3.2 (asymptotic comparison lemma). Let (M, g) be a strongly p-nonparabolic Rieman-
nian 3-manifold, 1 < p < 3, with nonnegative scalar curvature and with connected, compact,
possibly empty boundary. Assume that H2(M,∂M ;Z) = {0} and (M, g) satisfies (†). Let
Ω ⊆ M be a closed bounded subset with connected C 1-boundary homologous to ∂M and with
second fundamental form h ∈ L2(∂Ω), wp the solution to (1.3) starting at Ω and Ωt = {wp ≤ t}.
Then
lim
t→+∞
m(p)
H (∂Ωt) = lim
t→+∞
m̃(p)
H (∂Ωt) ≤ lim inf
t→+∞
m(p)
iso(Ωt), (3.4)
where Ωt = {wp ≤ t}.
Proof. Assume that the right-hand side of (3.4) is finite, otherwise there is nothing to prove.
The function t 7→ |Ωt ∖ Critwp| is monotone continuous in [0,+∞), hence it is absolutely
continuous. The generalised de L’Hôpital rule gives
lim inf
t→+∞
m(p)
iso(Ωt) ≥ lim inf
t→+∞
1
2pπcp(∂Ωt)
2
3−p
(
|Ωt ∖ Critwp| −
4π
3
cp(∂Ωt)
3
3−p
)
≥ lim inf
t→+∞
(3− p)
4pπcp(∂Ωt)
2
3−p
(�
∂Ωt
1
|Dwp|
dσ − 4π
3− p
cp(∂Ωt)
3
3−p
)
. (3.5)
By Hölder inequality, we have that
�
∂Ωt
1
|Dwp|
dσ ≥
( �
∂Ωt
|Dwp|2 dσ
)− p
3−p (
4π(3− p)p−1cp(∂Ωt)
) 3
3−p .
Plugging it in (3.5), we obtain
lim inf
t→+∞
m(p)
iso(Ωt) ≥ lim inf
t→+∞
cp(∂Ωt)
1
3−p
p
( �
∂Ωt
|Dwp|2
(3−p)2
dσ
) p
3−p
(4π) p
3−p −
(�
∂Ωt
|Dwp|2
(3− p)2
dσ
) p
3−p
.
To simplify the notation, denote f(z) = zp/(3−p) and z(t) =
�
∂Ωt
|Dwp|2/(3 − p)2 dσ. Since
z(t) ≤ 4π in virtue of our assumptions, by Lemma 3.1
lim inf
t→+∞
m(p)
iso(Ωt) ≥ lim inf
t→+∞
1
pf(z(t))
f(4π)− f(z(t))
4π − z(t)
cp(∂Ωt)
1
3−p (4π − z(t))
= lim inf
t→+∞
4π(3− p)
pf(z(t))
f(4π)− f(z(t))
4π − z(t)
m̃(p)
H (∂Ωt)
≥ lim inf
t→+∞
4π(3− p)
pf(z(t))
f(4π)− f(z(t))
4π − z(t)
m(p)
H (∂Ωt). (3.6)
The theorem follows once we prove the following claim.
Claim 3.3. There exists a divergent increasing sequence (tn)n∈N realising the rightmost limit
inferior of (3.6) and such that z(tn) → 4π as n→ +∞.
Indeed, we would have
lim
n→+∞
4π
f(z(tn))
= (4π)
3−2p
3−p , lim
n→+∞
f(4π)− f(z(tn))
4π − z(tn)
= f ′(4π) =
p
3− p
(4π)
2p−3
3−p ,
that plugged into (3.6), gives (3.4) in virtue of Theorem 2.11 and Lemma 3.1.
Let tn be a divergent increasing sequence (tn)n∈N realising the rightmost limit inferior of (3.6).
By Lemma 3.1, we have two possible cases:
Nonlinear Isocapacitary Concepts of Mass 15
(1) there exists T > 0 such that m̃(p)
H (∂Ωtn) ≥ 0 for all tn ≥ T , or
(2) m̃(p)
H (∂Ωtn) < 0 for all n ∈ N.
Case 1. Since m̃(p)
H (∂Ωtn) ≥ 0, z(tn) ≤ 4π for every tn ≥ T . By contradiction, suppose there
exists ε > 0 such that z(tn) ≤ 4π − ε for every n sufficiently large. Then, by (3.6), there exists
C(p, ε) > 0 such that
+∞ > lim inf
t→+∞
m(p)
iso(Ωt) ≥ lim
n→+∞
C(p, ε)cp(∂Ωtn)
1
3−p ,
which is clearly a contradiction. Hence, up to a not relabeled subsequence, z(tn) → 4π as
n→ +∞. This proves the claim in this case.
Case 2. Since m̃(p)
H (∂Ωtn) < 0, z(tn) ≥ 4π for every n ∈ N. Suppose by contradiction
z(tn) ≥ 4π + ε for some ε > 0. Then, by Theorem 2.11, there exists C(p, ε) > 0 such that
m̃
(p)
H (∂Ω) ≤ lim
t→+∞
m̃(p)
H (∂Ωt) ≤ −C(p, ε) lim
n→+∞
cp(∂Ωtn)
1
3−p = −∞,
which is a contradiction since |Dwp| ∈ C 0(∂Ω), proving the claim also in this case. ■
Differently from [11, Lemma 2.7], here we assumed (†). We already mentioned in the In-
troduction that this condition is very mild. In the following remark, we better specify our
assertion.
Remark 3.4. First of all, observe that
�
∂Ωt
|Dwp|2
(3− p)2
dσ ≤ 4π et cp(∂Ω) sup
∂Ωt
|Dwp|3−p
(3− p)3−p
. (3.7)
If Ric(x) ≥ −2κ2 for some κ ∈ R and every x ∈ M , by [46, Theorem 1.1] we have that
|Dwp| ≤ C1 for some constant depending on p and κ. In particular, for 1 < p < 2, (†) is fulfilled.
The case p = 2 may be treated as in [14, Corollary 1.1]. For the same reason, if (M, g) is
C 1-asymptotically flat Riemannian manifold (†) is implied for every 1 < p ≤ 2 by Lemma 2.5
(see also Remark 2.6).
Alternatively, assuming that (M, g) is C 0-asymptotically flat and the Ricci tensor satisfies
Ric(x) ≥ −2κ2/(1 + d(x, o))2 for some κ ∈ R a fixed o ∈M and for every x ∈M , one can cover
the whole range 1 < p < 3. Indeed, by [46, Theorem 1.1] and [9, Theorem 1.1], |Dwp| ≤ C3 e
−t
for some positive constat C3 depending only on p, κ and Ω. Plugging it into (3.7), we infer that�
∂Ωt
|Dwp|2 dσ ≤ C4 for a positive constant C4.
We establish a nonsharp Penrose inequality for the p-isocapacitary mass in the generality of
Theorem 1.2.
Proof of Theorem 1.2. Assume first that ∂M = ∅. Then, we let wp = −(p−1)Gp, where Gp
is the p-Green function issuing from some point o ∈M . Then, all of results stated above for wp
starting from a given set can be obtained with no modifications in the proofs for such limit
case. Moreover, by the asymptotic development of the p-Green function at the pole (see [35,
Theorem 2.4]), we have
lim
t→−∞
m(p)
H (∂Ωt) = 0.
Applying Lemma 3.2 and Theorem 2.11, we deduce that m(p)
iso ≥ 0, as claimed.
16 L. Benatti, M. Fogagnolo and L. Mazzieri
We do now treat the case ∂M ̸= ∅. Let wp the solution to (1.3) and define Ωt = {wp ≤ t}.
Observe that we can write
2cp(∂M)
1
(3−p) = 2m(p)
H (∂M) + (3− p)m̃(p)
H (∂M).
Then, combining the monotonicity of m(p)
H , that of m̃(p)
H following from (A.4), and the asymptotic
comparison Lemma 3.2, we obtain
2cp(∂M)
1
(3−p) ≤ lim
t→+∞
2m(p)
H (∂Ωt) + (3− p)m̃(p)
H (∂Ωt)
≤ lim inf
t→+∞
(5− p)m(p)
iso(Ωt) ≤ (5− p)m(p)
iso. (3.8)
Finally, we just have to discuss the equality case in the positive mass theorem. By the just
prove Penrose-type inequality, ∂M must be empty. Let then again, as above wp = −(p−1) logGp.
We deduce from the argument that yielded the positivity of the mass that m(p)
H must actually
be constant. In particular, the right-hand side of (2.13) constantly vanishes along the flow. The
isometry with flat Rn then follows through very classical computations, that can be performed
following the lines of [27, proof of Main Theorem 2]. ■
Remark 3.5. We could have used the monotonicity of m(p)
H alone in (3.8), instead of combining
with the monotonicity of m̃(p)
H . However, this would have led to the worse constant 2 in the
right-hand side of (1.5).
4 Proof of Theorem 1.4
The proof of Theorem 1.4 follows from an asymptotic equivalence of p-Hawking masses. As one
can expect, the p-Hawking mass has a better behaviour along the level set flow of wp, which is
the solution to (1.3). But interestingly, under the right assumption on the asymptotic flatness,
it tends to coincide with (the superior limit of) the Hawking mass on large sets. Moreover, it
is asymptotically controlled by the other q-Hawking mass for 1 < q < 3. Here we employ both
the monotonicity of the mass m(p)
H and the better asymptotic behaviour of m̃(p)
H defined in (3.1).
Indeed, we will use the latter one to ensure that e−t/(3−p)m(p)
H (∂{wp ≤ t}) = o(1) as t → +∞,
which permits to trigger the computations in [3, formula (2.12)].
Proposition 4.1. Let (M, g) be a C 1-asymptotically flat Riemannian 3-manifold with nonneg-
ative scalar curvature and with connected, compact, possibly empty boundary. Assume also that
H2(M,∂M ;Z) = {0}. Fix 1 < p < 3. Let Ω ⊆ M be homologous to ∂M with connected
C 1-boundary and h ∈ L2(∂Ω), wp the solution to (1.3) starting at Ω and Ωt = {wp ≤ t}. Then
lim
t→+∞
m(p)
H (∂Ωt) = lim sup
t→+∞
mH(∂Ωt) ≤ lim sup
t→+∞
m(q)
H (∂Ωt) (4.1)
for every 1 < q < 3.
Proof. The inequality appearing in (4.1) is obtained arguing as in Lemma 2.12. Indeed, we get
lim sup
t→+∞
mH(∂Ωt) ≤ lim sup
t→+∞
cq(∂Ωt)
− 1
3−q
√
|∂Ωt|
4π
m(q)
H (∂Ωt) = lim sup
p→+∞
m(q)
H (∂Ωt), (4.2)
where the last identity follows by Lemma 2.9 (3) (4). In order to show the identity appearing
in (4.1), we are thus left to show the inequality
lim
t→+∞
m(p)
H (∂Ωt) ≤ lim sup
t→+∞
mH(∂Ωt), (4.3)
Nonlinear Isocapacitary Concepts of Mass 17
the reverse one consisting in (4.2) with p = q. To do so, we claim that[
4π +
�
∂Ω
|Dwp|2
(3− p)2
dσ −
�
∂Ω
|Dwp|
(3− p)
H dσ
]
= o(1) (4.4)
as t → +∞. Indeed, if this happens, we can follow the chain of inequalities in [3, (2.12)] (see
also [11, Theorem 4.11]) and obtain
lim
t→+∞
m(p)
H (∂Ωt) ≤ lim sup
t→+∞
cp(∂Ωt)
1
3−p
√
4π
|∂Ωt|
mH(∂Ωt) = lim sup
t→+∞
mH(∂Ωt),
where again we applied Lemma 2.9 (3) (4), proving (4.3).
We then proceed to prove (4.4). If m(p)
H (∂Ωt) < 0 for every t ∈ [0,+∞), arguing as in Case 2
of the proof of Lemma 3.2, we deduce that (4.4) must hold. Otherwise, we would contradict
the monotonicity formulas in Theorem 2.11. Conversely, appealing again to the monotonicity
formulas in Theorem 2.11, t 7→ m(p)
H (∂Ωt) must be definitely nonnegative. Observe that by
Lemma 2.9 (1) (2), we have
4π −
�
∂Ωt
|Dwp|2
(3− p)2
dσ = o(1)
as t→ +∞. Hence, Lemma 3.1 implies
0 ≤ m(p)
H (∂Ωt) ≤ m̃(p)
H (∂Ωt) = o(et)
as t→ +∞. Dividing both sides by cp(∂Ωt)
1/(3−p) we get (4.4). ■
Conclusion of the proof of Theorem 1.4. Differently from the case p = 1, corresponding
to the classical Hawking mass, here we assume connectedness of the boundary of the manifold.
In fact, it is not clear to us how to adapt the argument employed in [27, Section 6], where
the authors took advantage of the horizons being minimal and outward minimizing in order
to prescribe a jump that maintains the monotonicity of the Hawking mass. The difficulties
when dealing with the p-Hawking mass arise in connection with the gradient of wp appearing
in its expression. Assuming ∂M to be connected, we can consider the solution wp to (1.3)
starting at Ω = ∂M and Ωt = {wp ≤ t}. The boundary of M being outermost implies that
H2(M,∂M ;Z) = {0} (see [27, Lemma 4.1], or the alternative argument in the proof of [11,
Lemma 2.8]). Applying Proposition 4.1 for q = 2, we have
cp(∂M)
1
3−p
2
≤ lim
t→+∞
m(p)
H (∂Ωt) ≤ lim sup
t→+∞
m(2)
H (∂Ωt).
Since by Lemma 2.9 (2), ∂Ωt is regular for any t large enough, we can use [11, Theorem 4.11]
to control the right-hand side with mADM, concluding the proof of (1.6). Observe now that an
outermost minimal boundary is outward minimising. If this were not the case, the outward min-
imising hull [20, 27] would be a closed minimal surface homologous to it and, by the Maximum
Principle, disjoint from ∂M . Then, letting p→ 1+ and appealing to [20, Theorem 1.2] recovers
the sharp Penrose inequality (1.7). ■
5 Relation between the isoperimetric mass and the
p-isocapacitary mass
We now employ the explicit control of the Hawking mass in terms of the p-Hawking mass
Lemma 2.12 to produce an upper bound on the isoperimetric mass in terms of p-isocapacitary
mass. This bound is not sharp but sharpens as p→ 1+.
18 L. Benatti, M. Fogagnolo and L. Mazzieri
Lemma 5.1. Let (M, g) be a C 0-asymptotically flat Riemannian 3-manifold with nonnega-
tive scalar curvature and with smooth, compact, minimal, possibly empty boundary. Assume
that (M, g) satisfies (†) for some 1 < p < 3. Then
miso ≤
(
22p−1π
p−1
2 ppC
3
2
(p−1)
S
(3− p)(p− 1)p−1
) 1
3−p
m(p)
iso,
where CS is the global Sobolev constant of (M, g).
Proof. By the topological description of manifolds like these, reworked in [11, Lemma 2.8], we
can assume that our Riemannian manifold has a (possibly empty) minimal, outermost boundary
such that H2(M,∂M ;Z) = {0}. Let E ⊂M be a closed bounded subset with smooth boundary
such that any connected component of ∂M is either contained in E or disjoint from E. Using [27,
Theorem 6.1] (see also [11, Proposition 2.5]), we can find a subset Ω closed bounded with C 1-
boundary homologous to ∂M and with h ∈ L2(∂Ω) such that mH(∂E) ≤ mH(∂Ω). Let wp the
solution to (1.3) starting at Ω and Ωt = {wp ≤ t} . By Lemmas 2.12 and 3.2, we now have
mH(∂E) ≤
(
22p−1π
p−1
2 ppC
3
2
(p−1)
S
(3− p)(p− 1)p−1
) 1
3−p
lim sup
t→+∞
m(p)
H (∂Ωt) ≤
(
22p−1π
p−1
2 ppC
3
2
(p−1)
S
(3− p)(p− 1)p−1
) 1
3−p
m(p)
iso.
Since we have a control on the Hawking mass of every E, we can apply [29] (see [11, Theorem 2.6]
for the precise statement and remarks) to control the isoperimetric mass with the same quan-
tity. ■
We prove a family of equivalent formulations for the p-isocapacitary masses, as well as for the
isoperimetric one. The proofs will follow the one given in [28, Lemma 10] for the 2-isocapacitary
mass.
Proposition 5.2. Let (M, g) be a C 0-asymptotically flat Riemannian 3-manifold with compact,
possibly empty boundary. Then, for 1 < p < 3, we have
m(p)
iso = sup
(Ωj)j∈N
lim sup
j→+∞
2cp(∂Ω)
1−3α
3−p
3pα
((
3|Ωj |
4π
)α
− cp(∂Ωj)
3α
3−p
)
(5.1)
for every α ≥ 1/3.
The main computation performed in order to prove the result above is the following one, that
we isolate for future reference.
Lemma 5.3. Let (M, g) be a C 0-asymptotically flat Riemannian 3-manifold with compact pos-
sibly empty boundary, and 1 < p < 3. Let (Ωj)j∈N be an exhaustion of M such that
lim
j→+∞
|Ωj |
cp(∂Ωj)
3
3−p
=
4π
3
. (5.2)
Then, we have
lim sup
j→+∞
m(p)
iso(Ωj) = lim sup
j→+∞
2cp(∂Ωj)
1−3α
3−p
3pα
((
3|Ωj |
4π
)α
− cp(∂Ωj)
3α
3−p
)
for any α ∈ R \ {0}.
Nonlinear Isocapacitary Concepts of Mass 19
Proof. Let (Ωj)j∈N be a sequence such that (5.2) holds. Up to considering a subsequence, we
can assume that (Ωj)j∈N realises the superior limit. Denote f(z) = zα, zj = |Ωj |/cp(∂Ωj)
3/(3−p),
we have that
lim sup
j→+∞
m(p)
iso(Ωj) = lim
j→+∞
cp(∂Ωj)
1
3−p
2pπ
zj − 4π/3
f(zj)− f(4π/3)
(f(zj)− f(4π/3)). (5.3)
Since f is differentiable at 4π/3 and zj → 4π/3 ̸= 0 as j → +∞ by (5.2), we have
lim
j→+∞
zj − 4π/3
f(zj)− f(4π/3)
=
1
f ′(4π/3)
=
3α−1
α(4π)α−1
.
Plugging this into (5.3) we conclude. ■
Proof of Proposition 5.2. We claim that it is enough to prove the equivalence on sequences
such that (5.2) holds, so that Proposition 5.2 follows from Lemma 5.3. Let then (Ωj)j∈N be an
exhaustion. By the p-isocapacitary inequality, we have that
lim sup
j→+∞
|Ωj |
cp(∂Ωj)
3
3−p
≤ 4π
3
.
Indeed, the metric g becomes uniformly equivalent to the flat Euclidean metric on M ∖ Ωj as
j → +∞. Moreover, for sufficiently large j there exists a unique rj > 0 such that the coordinate
ball Brj has the same volume of Ωj . Define
Ω′
j =
{
Ωj if Capp(Ωj) ≤ Capp(Brj ),
Brj if Capp(Ωj) > Capp(Brj ).
The sequence
(
Ω′
j
)
j∈N is an exhaustion of M and
lim inf
j→+∞
∣∣Ω′
j
∣∣
cp(∂Ω′
j)
3
3−p
≥ lim inf
j→+∞
∣∣Brj
∣∣
cp
(
∂Brj
) 3
3−p
=
4π
3
,
where the right-hand side is computed using the asymptotic flatness. In particular, the se-
quence (Ω′
j)j∈N fulfils (5.2), |Ω′
j | = |Ωj | and cp(∂Ω
′
j) ≤ cp(∂Ωj). Then, when α ≥ 1/3, (Ω′
j)j∈N
is a better competitor both for mp
iso as in the definition of p-isocapacitary mass (1.4) and for the
right-hand side of (5.1). This completes the proof. ■
Completely analogous results hold for the perimeter and the isoperimetric mass. We gather
them in the following statement.
Proposition 5.4. Let (M, g) be a C 0-asymptotically flat Riemannian 3-manifold with compact
possibly empty boundary. Let (Ωj)j be an exhaustion of M such that
lim
j→+∞
|Ωj |
|∂Ωj |
3
2
=
1
6
√
π
.
Then
lim sup
j→+∞
2
|∂Ωj |
(
|Ωj | −
|∂Ωj |
3
2
6
√
π
)
= lim sup
j→+∞
|∂Ωj |
1−3α
2
3α
√
π
(
(6
√
π|Ωj |)α − |∂Ωj |
3α
2
)
(5.4)
holds for any α ∈ R∖ {0}. As a consequence, we have
miso = sup
(Ωj)j∈N
lim sup
j→+∞
|∂Ωj |
1−3α
2
3α
√
π
(
(6
√
π|Ωj |)α − |∂Ωj |
3α
2
)
for every α ≥ 1/3.
20 L. Benatti, M. Fogagnolo and L. Mazzieri
The inequality m(p)
iso ≤ miso will substantially be a consequence of the following p-isocapacitary
inequality for sets with volume going to infinity. Its isoperimetric version was pointed out in [15,
Corollary C.3].
Theorem 5.5 (sharp asymptotic p-isocapacitary inequality). Let (M, g) be a C 0-asymptotically
flat Riemannian 3-manifold with compact possibly empty boundary ∂M . Then, for every 1 <
p < 3, we have that
|Ω|
3−p
3 ≤
(
4π
3
) 3−p
3
cp(∂Ω) +
p(3− p)
2
m
(
4π
3
) 3−p
3
cp(∂Ω)
2−p
3−p (1 + o(1)) (5.5)
as |Ω| → +∞, where Ω closed and bounded with C 1,α-boundary containing ∂M and m > −∞ is
such that m ≥ miso.
Proof. Assume that m < +∞, otherwise there is nothing to prove. We claim that for ev-
ery ε > 0 small enough, there exists Vε > ε−3 such that(
6
√
π|Ω|
) 2p
3 ≤ |∂Ω|p + 2p
√
π(m+ ε)|∂Ω|
2p−1
2 (5.6)
for every Ω ⊆ M such that |Ω| ≥ Vε. Indeed, if this were not the case, we would find a se-
quence (Ωj)j∈N with |Ωj | → +∞ such that the right-hand side with α = 2p/3 ≥ 1/3, and thus
the left-hand side, of (5.4), is strictly bigger than m ≥ miso. Since, by the isoperimetric inequal-
ity, the perimeters of the Ωj ’s diverge at infinity too, this would contradict [29, Proposition 37],
stating that one can relax the competitors in the definition of the isoperimetric mass in order
to include any sequence of bounded sets containing ∂M with diverging perimeters.
We can now assume that
|Ω|
3−p
2p ≥
(
4π
3
) 3−p
2p
cp(∂Ω)
3
2p (1− ε)
3−p
2p , (5.7)
otherwise (5.5) is trivially satisfied. Let wp : M ∖ Ω → R be the solution to (1.3) starting at Ω,
wp = −(p − 1) log up and let Ωt = {up ≥ t} ∪ Ω and V (t) = |Ωt| ≥ Vε for every t ∈ (0, 1). The
Hölder’s inequality with exponents a = p and b = p/(p− 1) gives
|∂Ωt|p ≤
( �
∂Ωt
|Dup|p−1 dσ
)( �
∂Ωt
1
|Dup|
dσ
)p−1
= 4πcp
(
3− p
p− 1
)p−1
[−V ′(t)]p−1 (5.8)
for almost every t ∈ (0, 1], where cp = cp(∂Ω). Plugging it into (5.6) and integrating on (0, 1)
we obtain
� 1
0
[
6
√
πV (t)
] 2p
3
(−V ′(t))p−1
dt
≤ 4πcp
(
3− p
p− 1
)p−1
+
[
4πcp
(
3− p
p− 1
)p−1
] 2p−1
2p � 1
0
2p
√
π(m+ ε)
(−V ′(t))
p−1
2p
dt. (5.9)
Applying (5.6) and the isoperimetric inequality, we get
(6
√
πV (t))
2p
3 ≤ |∂Ωt|p
(
1 +
2p
√
π|m+ ε|√
|∂Ωt|
)
≤ |∂Ωt|p
(
1 +
C
V (t)
1
3
)
,
where C depends only on m, p and the isoperimetric constant. Plugging it into (5.8), we have
that
[−V ′(t)]p−1 ≥
(
p− 1
3− p
)p−1 3
2p
3 V (t)
2p
3
(4π)
3−p
p cp
(
1 +
C
V (t)
1
3
)−1
. (5.10)
Nonlinear Isocapacitary Concepts of Mass 21
Hence, using (5.10), the assumption V (t) ≥ |Ω| ≥ ε−3, a change of variable in the integral
and (5.7) yield
� 1
0
[
−V ′(t)
]− p−1
2p dt = −
� 1
0
[
−V ′(t)
]− 3p−1
2p V ′(t) dt
≤ −
� 1
0
[(
3− p
p− 1
)p−1 (4π)
3−p
p cp
3
2p
3 V (t)
2p
3
(
1 +
C
V (t)
1
3
)] 3p−1
2p(p−1)
V ′(t) dt
≤
[(
3− p
p− 1
)p−1 (4π)
3−p
3
3
2p
3
cp(1 + Cε)
] 3p−1
2p(p−1) � +∞
|Ω|
V
− 3p−1
3(p−1) dV
≤ 3− p
2
(3− p
p− 1
) (p−1)2
3p−1 (4π)
3−p
3
3
4p
3(3p−1)
cp(1 + Cε)
3p−1
2p(p−1)
|Ω|−
2
3(p−1)
≤ 3− p
2
(4π)
− p−1
2p c
− 3(p−1)
2p(3−p)
p
(
3− p
p− 1
) p−1
2p (1 + Cε)
3p−1
2p(p−1)
(1− ε)
2
3(p−1)
. (5.11)
On the other hand, let v : {|x| ≥ R(1)} ⊂ Rn → (0, 1] be the function such that {v = t} =
{|x| = R(t)} and |Ωt| = 4πR(t)3/3. Since by construction |Dv| = −4πR(t)2/V ′(t), the function v
is locally Lipschitz. By coarea formula, we have
� 1
0
V (t)
2p
3
(−V ′(t))p−1
dt =
1
(36π)
p
3
� 1
0
�
{v=t}
|Dv|p−1 dσ dt =
1
(36π)
p
3
�
{|x|≥R(1)}
|Dv|p dx
≥ (4π)
3−p
3
3
2p
3
(
3− p
p− 1
)p−1
cp({|x| = R(1)})
=
1
3p−1
(
3− p
p− 1
)p−1
|Ω|
3−p
3 . (5.12)
Plugging (5.12) and (5.11) into (5.9), we conclude the proof by arbitrariness of ε. ■
We are ready to prove the claimed upper bound of the p-isocapacitary mass in terms of the
isoperimetric mass.
Theorem 5.6. Let (M, g) a C 0-asymptotically flat Riemannian 3-manifold with possibly empty
compact boundary. Then, for every 1 < p ≤ 2, we have that
m(p)
iso ≤ miso.
Proof. Let (Ωj)j∈N be an exhaustion of (M, g), then |Ωj | → +∞ as j → +∞. In particular,
by Theorem 5.5, we have that
1
cp(∂Ωj)
2−p
3−p
[(
3|Ωj |
4π
) 3−p
3
− cp(∂Ωj)
]
≤ p(3− p)
2
m(1 + o(1))
as j → +∞, where m ∈ R ∪ {+∞} is such that m ≥ miso. Hence
lim sup
j→+∞
1
cp(∂Ωj)
2−p
3−p
[(
3|Ωj |
4π
) 3−p
3
− cp(∂Ωj)
]
≤ p(3− p)
2
m.
Taking the supremum among all exhaustions (Ωj)j∈N, we conclude employing Proposition 5.2
for α = (3− p)/3 and sending m → −∞ if miso = −∞. ■
22 L. Benatti, M. Fogagnolo and L. Mazzieri
Combining Lemma 5.1 and Theorem 5.6, we directly get the convergence of the p-isocapaci-
tary masses to the isoperimetric mass as p→ 1+.
Corollary 5.7. Let (M, g) be a C 0-asymptotically flat Riemannian 3-manifold with nonnegative
scalar curvature and smooth, compact, minimal, possibly empty boundary. Assume that (M, g)
satisfies (†) for any 1 < p < 1 + δ, for a fixed 0 < δ < 2. Then
lim
p→1+
m(p)
iso = miso.
We are ready to prove, in the stronger C 1
τ -asymptotically flat assumptions, τ > 1/2, that the
p-isocapacitary masses do actually coincide with each other.
Proof of Theorem 1.3. Assume that mADM is finite, otherwise [11, Theorem 4.13] yields
miso = +∞ and Lemma 5.1 implies m(p)
iso = +∞. The first inequality, under these assump-
tions, is the content of [11, Theorem 4.13]. Following the same lines of [28, Proposition 14]
(based on computations contained in [19], which in fact only relies on the C 1-character of the
metric), we have
1
16π
�
∂Br
H2 dσ = 1− 2mADM
r
+ o
(
r−1
)
,
|∂Br| = 4πr2 + 4πη(r) + o(r),
3|Br|
4π
= r3 +
3mADM
2
r2 +
3
2
η(r)r + o
(
r2
)
,
as r → +∞, where Br = {|x| ≤ r} and |η(r)| ≤ Cr2−τ . Employing Proposition 2.13 and using
Taylor’s expansion of 2F1 around 0 (see (2.14)), we have
cp(∂Br) ≤
(
r2 + η(r) + o(r)
) 3−p
2
2F1
(
1
2
,
3− p
p− 1
,
2
p− 1
;
2mADM
r
+ o
(
r−1
))−(p−1)
≤
(
r2 + η(r) + o(r)
) 3−p
2
(
1 +
3− p
2r
mADM + o
(
r−1
))−(p−1)
= r3−p
(
1 +
3− p
2r2
η(r) + o
(
r−1
))(
1− (3− p)(p− 1)
2r
η(r) + o
(
r−1
))
= r3−p +
3− p
2
η(r)r1−p − (3− p)(p− 1)
2
r2−pmADM + o
(
r2−p
)
.
Proposition 5.2 for α = (3− p)/3 gives
m(p)
iso ≥ lim sup
r→+∞
2cp(∂Br)
p−2
3−p
p(3− p)
((
3|Br|
4π
) 3−p
3
− cp(∂Br)
)
≥ lim sup
r→+∞
2cp(∂Br)
p−2
3−p
p(3− p)
(
p(3− p)
2
r2−pmADM + o(r2−p)
)
= mADM,
where the last identity is given by [9, Lemma 2.21]. The conclusion follows by Theorem 5.6,
since miso = mADM [11, Theorem 4.13]. ■
A Monotonicities along the p-capacitary potential
Here we slightly improve the monotonicity results in [3, 14]. Inspired by these two works, we
are approximating the p-capacitary potential with a family of smooth functions. To enter more
Nonlinear Isocapacitary Concepts of Mass 23
in detail, let (M, g) be a strongly p-nonparabolic Riemannian manifold with (possibly empty)
boundary. Let Ω ⊂ M be homologous to ∂M and up is the solution to (1.3) starting at Ω. For
every T > 1 let ΩT be strictly homologous to ∂M with connected boundary and containing
{up > αp(T )}, where αp(T ) = T−(3−p)/(p−1). Then, we define uεp as the solution to the following
boundary value problem:
∆ε
pu
ε
p = 0 on IntΩT ∖ Ω,
uεp = 1 on ∂Ω,
uεp = up on ∂ΩT ,
(A.1)
where
∆ε
pf = div
(
|Df |p−2
ε Df
)
and | · |ε =
√
| · |2 + ε2.
The function uεp is smooth away from the exterior boundary and converges in C 1,β
loc to the p-
capacitary potential up as ε→ 0+. Indeed, this family was used in [17] to prove C 1,β
loc -regularity
of p-harmonic functions. Moreover, looking more carefully at the proof of [34, Lemma 2.1],
|Duεp|
p−1 is uniformly bounded inW 1,2
loc . Hence, up to a not relabeled subsequence, we can always
assume that |Duεp|
p−1 weakly converges in W 1,2
loc . Moreover, since |Duεp| converges uniformly
to |Dup|, the weak limit of D|Duεp|
p−1 must be D|Dup|p−1.
We are now ready to prove Theorem 2.11.
Proof of Theorem 2.11. For ease of computations, we rewrite the function t 7→ m(p)
H (∂Ωt) in
terms of the p-capacitary potential, that is,
Up(t) = 4πt+
(p− 1)2
(3− p)2
t
5−p
p−1
�
{up=αp(t)}
|Dup|2 dσ − (p− 1)
(3− p)
t
2
p−1
�
{up=αp(t)}
|Dup|H dσ.
Observe that
cp(∂Ω)
1
3−p
8π
Up
(
e
t
3−p
)
= m(p)
H (∂Ωt) =
1
8π
Fp
(
cp(∂Ω)
1
3−p e
t
3−p
)
,
where Fp is the function defined in [3]. Equation (2.13) now follows from computations in [3,
Section 1.2]. The function Up is well defined on [0,+∞). Indeed, one can observe that
|H||Dup|2 = |Dup|3−p|D|Dup|p−1| ∈ L2
loc(M ∖ Ω),
since 1 < p < 3 and |Dup|p−1 ∈ L∞
loc ∩ W 1,2
loc (M ∖ Ω) by [34, Lemma 2.1]. Then, by coarea
formula, the function
t 7→
�
{up=αp(t)}
|Dup|H dσ ∈ L1
loc(0,+∞)
and its equivalence class does not depend on the representative of |Dup|H. In particular, the
function t 7→ m(p)
H (∂Ωt) ∈ L1
loc(0,+∞).
It only remains to prove that it has nonnegative first derivative in the sense of distribu-
tions, which both gives that t 7→ m(p)
H (∂Ωt) ∈ BVloc(0,+∞) and that admits a nondecreasing
representative. Fix T > 0 and uεp be the solution to (A.1). One can now define the function
F ε
p (t) = 4πt+
(p− 1)2
(3− p)2
t
5−p
p−1
�
{uε
p=αp(t)}
∣∣Duεp∣∣2 dσ − (p− 1)
(3− p)
t
2
p−1
�
{uε
p=αp(t)}
∣∣Duεp∣∣Hε dσ
24 L. Benatti, M. Fogagnolo and L. Mazzieri
for every t ∈ (1, T ), where Hε is the mean curvature of the level
{
uεp = αp(t)
}
. By [3, Lemma 1.2],
we have that the function F ε
p is almost monotone, in the sense that
F ε
p (t)− F ε
p (s) ≥ −ε(p+ 1)2(p− 1)
(3− p)3
�
{αp(t)<uε
p<αp(s)}
∣∣Duεp∣∣2(uεp)− p−1
3−p
−3
dµ
holds for almost every 0 ≤ s ≤ t ≤ T . In particular, by coarea formula and Lebesgue differenti-
ation theorem,
(F ε
p )
′(t) ≥ −ε(p+ 1)2
(3− p)2
t
2(3−p)
p−1
�
{uε
p=αp(t)}
∣∣Duεp∣∣ dσ
holds for almost every t ∈ (1, T ).
The monotonicity follows if one can prove the following claim.
Claim A.1. F ε
p converges to Fp in the sense of distributions as ε→ 0+.
Indeed, if that is the case, for every nonnegative test function φ ∈ C∞
c (1, T ), we obtain that
−
� T
1
φ′(t)Fp(t) dt = − lim
ε→0
� T
1
φ′(t)F p
ε (t) dt
≥ −(p+ 1)2
(3− p)2
lim
ε→0
ε
� T
1
φ(t)t
2(3−p)
p−1
�
{uε
p=αp(t)}
∣∣Duεp∣∣dσ dt
= −(p+ 1)2
(3− p)2
lim
ε→0
ε
�
ΩT∖Ω
φ
(
α−1
p
(
uεp
)) ∣∣Duεp∣∣2(
uεp
) 3−p
p−1
+3
dµ = 0,
since uεp converges to up in C 1,β
loc . This shows that Fp has nonnegative first derivative in the sense
of distributions, proving its monotonicity.
We now turn to prove the claim. Consider any φ ∈ C∞
c (0,+∞). The first term is independent
of ε. As far as the second term is concerned, by coarea formula, we have that
� T
1
φ(t)t
5−p
p−1
�
{uε
p=αp(t)}
∣∣Duεp∣∣2 dσ dt = (p− 1)
(3− p)
�
ΩT∖Ω
(
uεp
)− 7−p
3−pφ
(
α−1
p
(
uεp
))∣∣Duεp∣∣3 dµ.
Since the function φ is smooth with compact support in (1, T ) and uεp converges to up in C 1,β
loc ,
the right-hand side converges to
(p− 1)
(3− p)
�
ΩT∖Ω
u
− 7−p
3−p
p φ
(
α−1
p
(
uεp
))
|Dup|3 dµ =
� T
1
φ(t)t
5−p
p−1
�
{uε
p=αp(t)}
|Dup|2 dσ dt,
where the identity follows by coarea formula. The last term is a little trickier since it involves
second derivatives of the function uεp that are not converging uniformly as ε → 0 to the corre-
sponding ones for up. Employing again the coarea formula and by straightforward computations,
we then have that
� T
1
φ(t)t
2
p−1
�
{uε
p=αp(t)}
∣∣Duεp∣∣Hε dσ dt
=
� T
1
φ(t)t
2
p−1
�
{uε
p=αp(t)}
〈
D
∣∣Duεp∣∣p−1∣∣Duεp〉∣∣Duεp∣∣p−1
(
1− (p− 2)
(p− 1)
ε2∣∣Duεp∣∣2ε
)
dσ dt
=
(p− 1)
(3− p)
�
ΩT∖Ω
(
uεp
)− 4
3−pφ
(
α−1
p
(
uεp
))〈D∣∣Duεp∣∣p−1∣∣Duεp〉∣∣Duεp∣∣p−2
(
1− (p− 2)
(p− 1)
ε2∣∣Duεp∣∣2ε
)
dµ.
Nonlinear Isocapacitary Concepts of Mass 25
As before we want to prove that the right-hand side converges to the corresponding term for up.
Since uεp → up in C 1,β
loc and D
∣∣Duεp∣∣p−1
⇀ D|Dup|p−1 weakly in L2
loc, we have that
lim
ε→0
(p− 1)
(3− p)
�
ΩT∖Ω
(
uεp
)− 4
3−pφ
(
αp
(
uεp
))〈D∣∣Duεp∣∣p−1∣∣Duεp〉∣∣Duεp∣∣p−2 dµ
=
(p− 1)
(3− p)
�
ΩT∖Ω
(up)
− 4
3−pφ
(
α−1
p
(
uεp
))〈D|Dup|p−1|Dup
〉
|Dup|p−2
dµ
=
� T
1
φ(t)t
2
p−1
�
{up=αp(t)}
|Dup|H dσ dt.
Moreover, the remaining term vanishes. Hölder’s inequality and equi-boundedness in L2
loc(ΩT ∖
Ω) of
∣∣D∣∣Duεp∣∣p−1∣∣ yield
�
K
∣∣Duεp∣∣3−p∣∣D∣∣Duεp∣∣p−1∣∣ ε2∣∣Duεp∣∣2ε dµ ≤ C1
(�
K
ε4∣∣Duεp∣∣4ε
∣∣Duεp∣∣6−2p
dµ
) 1
2
(A.2)
for every K compactly contained in
{
αp(T ) < uεp < 1
}
and for some positive constant C1. Ob-
serve that∣∣Duεp∣∣6−2p ε4∣∣Duεp∣∣4ε ≤
∣∣Duεp∣∣6−2p ≤ C2,
since the function
∣∣Dupε∣∣ converges locally uniformly and 1 < p < 3. The left-hand side converges
almost everywhere to 0. Indeed, if a point belongs to critical set of up,
∣∣Dupε∣∣6−2p → 0 as ε→ 0.
Otherwise,
∣∣Dupε∣∣ is definitely bounded away from 0, then
∣∣Duεp∣∣4ε is not vanishing, thus the
left-hand side is controlled by ε4 up to a constant. By dominated convergence theorem, the
right-hand side in (A.2) approaches 0 as ε → 0, so does the left-hand side, concluding the
step. ■
We use this theorem to study the quantity introduced in [14, Theorem 1.2] along the level
set of the solution wp to (1.3).
Theorem A.2. Let (M, g) be a strongly p-nonparabolic Riemannian 3-manifold with smooth,
compact and connected possibly empty boundary ∂M . Assume that H2(M,∂M ;Z) = {0}. Let
Ω ⊆M be bounded closed with connected C 1-boundary homologous to ∂M and with h ∈ L2(∂Ω).
Let wp be the solution to (1.3) starting at Ω. Then, denoting Ωt = {wp ≤ t}, the function
t 7→ cp(∂Ωt)
− 1
p−1
4π(3− p)
(
4π −
�
∂Ωt
|Dwp|2
(3− p)2
dµ
)
(A.3)
belongs to W 1,1
loc (0,+∞) and
d
dt
[
cp(∂Ωt)
− 1
p−1
(
4π −
�
∂Ωt
|Dwp|2
(3− p)2
dσ
)]
= − 8π
p− 1
cp(∂Ωt)
− 2
(3−p)(p−1)m(p)
H (∂Ωt), (A.4)
for almost every t ∈ [0,+∞).
Remark A.3. Observe that (A.3) is, up to multiplying by a constant and changing variables, the
quantity studied in [14, 38] for p ∈ (1, 2] along the level set of the p-Green function. In particular,
coupled with Theorem 2.11, the above result implies that the monotonicity property of (A.3)
26 L. Benatti, M. Fogagnolo and L. Mazzieri
is preserved if, in place of the p-Green’s function, the p-capacitary potential of a connected ∂Ω
with nonnegative p-Hawking mass is considered. Clearly, the monotonicity results in [14, 38] are
recovered applying Theorem A.2 to the p-Green’s function. Hence, we settle the question raised
in [14] about the monotonicity of their quantities in the range p ∈ (2, 3).
Finally, observe that m̃(p)
H is obtained multiplying the function (A.3) by the p-capacity term
cp(∂Ωt)
2/(3−p)(p−1) which is exponentially growing as t → +∞. This term forces the quan-
tity m̃(p)
H to be monotone nondecreasing under assumption (†) (see Lemma 3.1) even when (A.3)
is monotone nonincreasing.
Proof of Theorem A.2. Fix T > 1 and let uεp the solution to the problem (A.1). Consider
any φ ∈ C∞
c (0, T ). Employing coarea formula and integration by parts, we have that
� T
1
φ′(t)
�
{uε
p=αp(t)}
∣∣Duεp∣∣2 dσ dt = (p− 1)
(3− p)
�
ΩT∖Ω
φ′((uεp)− p−1
3−p
)(
uεp
)− 2
3−p
∣∣Duεp∣∣3 dµ
= −
�
ΩT∖Ω
〈∣∣Duεp∣∣Duεp∣∣D[φ((uεp)− p−1
3−p
)]〉
dµ
=
�
ΩT∖Ω
div
(∣∣Duεp∣∣Duεp)φ((uεp)− p−1
3−p
)
dµ. (A.5)
Clearly, employing C 1,β
loc convergence of uεp → up as ε→ 0, we obtain that
lim
ε→0
� T
1
φ′(t)
�
{uε
p=αp(t)}
∣∣Duεp∣∣2 dσ dt = � T
1
φ′(t)
�
{up=αp(t)}
|Dup|2 dσ dt.
Moreover, a straightforward computation leads to
div
(∣∣Duεp∣∣Duεp) =
∣∣Duεp∣∣2−p
(p− 1)
(
(3− p)
∣∣Duεp∣∣2∣∣Duεp∣∣2ε − ε2∣∣Duεp∣∣2ε
)〈
D
∣∣Duεp∣∣p−1∣∣Duεp〉.
Arguing as in the previous theorem, since
∣∣Duεp∣∣ → |Dup| locally uniformly and D
∣∣Duεp∣∣p−1
⇀
D|Dup|p−1 weakly L2
loc as ε→ 0, one gets
lim
ε→0
�
ΩT∖Ω
∣∣Duεp∣∣2−p
(p− 1)
(
(3− p)
∣∣Duεp∣∣2∣∣Duεp∣∣2ε − ε2∣∣Duεp∣∣2ε
)〈
D
∣∣Duεp∣∣p−1∣∣Duεp〉φ((uεp)− p−1
3−p
)
dµ
=
(3− p)
(p− 1)
�
ΩT∖Ω
〈
D|Dup|p−1|Dup
〉
|Dup|p−2
φ
(
(up)
− p−1
3−p
)
dµ
=
(3− p)2
(p− 1)2
� T
1
φ(t)t
− 2
p−1
�
ΩT∖Ω
H |Du| dσ dt,
where the last equality follows by coarea formula and H =
〈
D|Dup|p−1|Dup
〉
/|Dup|p. Then,
passing to the limit as ε→ 0 in (A.5), we obtain
� T
1
φ′(t)
�
{up=αp(t)}
|Dup|2 dσ dt =
(3− p)2
(p− 1)2
� T
1
φ(t)t
− 2
p−1
�
{up=αp(t)}
H |Du|dσ dt.
By arbitrariness of T and φ one has that the function
t 7→ Hp(t) =
(
4πt
− 3−p
p−1 − (p− 1)2
(3− p)2
t
3−p
p−1
�
{u=αp(t)}
|Dup|2 dσ
)
Nonlinear Isocapacitary Concepts of Mass 27
belongs to W 1,1
loc (1,+∞) and its derivative is given by
H ′
p(t) = − (3− p)
(p− 1)
t
− p+1
p−1
(
4πt+
(p− 1)2
(3− p)2
t
5−p
p−1
×
�
{u=αp(t)}
|Dup|2 dσ − (p− 1)
(3− p)
t
2
p−1
�
{up=αp(t)}
H |Du| dσ
)
holds for almost every t ∈ (1,+∞). Then
d
dt
[
cp(∂Ωt)
− 1
p−1
(
4π −
�
∂Ωt
∣∣Dwp
∣∣2
(3− p)2
dµ
)]
=
cp(∂Ω)
− 1
p−1
3− p
H ′
p
(
e
t
3−p
)
e
t
3−p
= − 8π
p− 1
cp(∂Ωt)
− 2
(3−p)(p−1)m(p)
H (∂Ωt),
concluding the proof. ■
Acknowledgements
Part of this work has been carried out during the authors’ attendance to the Thematic Program
on Nonsmooth Riemannian and Lorentzian Geometry that took place at the Fields Institute in
Toronto. The authors warmly thank the staff, the organizers and the colleagues for the won-
derful atmosphere and the excellent working conditions set up there. L.B. is supported by the
European Research Council’s (ERC) project n.853404 ERC VaReg – Variational approach to the
regularity of the free boundaries, financed by the program Horizon 2020, by PRA 2022 11 and
by PRA 2022 14. M.F. has been supported by the European Union – NextGenerationEU and
by the University of Padova under the 2021 STARS Grants@Unipd programme “QuASAR”.
The authors are members of Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro
Applicazioni (GNAMPA), which is part of the Istituto Nazionale di Alta Matematica (INdAM),
and are partially funded by the GNAMPA project “Problemi al bordo e applicazioni geomet-
riche”. The authors are grateful to S. Hirsch and F. Oronzio for their interest in the work and for
pleasureful and useful conversations on the subject. The authors warmly thank the anonymous
referees for their thorough reading of the paper, and for the precious suggestions that allowed
to improve the quality of the paper.
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1 Introduction
2 Preliminaries in nonlinear potential theory
2.1 Estimates on asymptotically flat Riemannian manifolds
2.2 Concepts of mass in nonlinear potential theory
3 p-isocapacitary Riemannian Penrose inequality
4 Proof of Theorem 1.4
5 Relation between the isoperimetric mass and the p-isocapacitary mass
A Monotonicities along the p-capacitary potential
References
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| id | nasplib_isofts_kiev_ua-123456789-212040 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-13T05:04:08Z |
| publishDate | 2023 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Benatti, Luca Fogagnolo, Mattia Mazzieri, Lorenzo 2026-01-23T10:10:45Z 2023 Nonlinear Isocapacitary Concepts of Mass in 3-Manifolds with Nonnegative Scalar Curvature. Luca Benatti, Mattia Fogagnolo and Lorenzo Mazzieri. SIGMA 19 (2023), 091, 29 pages 1815-0659 2020 Mathematics Subject Classification: 83C99; 35B40; 35A16; 31C15; 53C21 arXiv:2305.01453 https://nasplib.isofts.kiev.ua/handle/123456789/212040 https://doi.org/10.3842/SIGMA.2023.091 We deal with suitable nonlinear versions of Jauregui's isocapacitary mass in 3-manifolds with nonnegative scalar curvature and compact outermost minimal boundary. These masses, which depend on a parameter 1 < 𝑝 ≤ 2, interpolate between Jauregui's mass 𝑝 = 2 and Huisken's isoperimetric mass, as 𝑝 → 1⁺. We derive positive mass theorems for these masses under mild conditions at infinity, and we show that these masses do coincide with the ADM mass when the latter is defined. We finally work out a nonlinear potential theoretic proof of the Penrose inequality in the optimal asymptotic regime. Part of this work has been carried out during the authors’ attendance at the Thematic Program on Nonsmooth Riemannian and Lorentzian Geometry that took place at the Fields Institutein Toronto. The authors warmly thank the staff, the organizers, and the colleagues for the wonderful atmosphere and the excellent working conditions setupthere. L.B. is supported by the European Research Council’s (ERC) Project n.853404 ERC VaReg–Variational approach to the regularity of the free boundaries, financed by the program Horizon 2020, by PRA_2022_11 and by PRA_2022_14. M.F. has been supported by the European Union – Next Generation EU and by the University of Padova under the 2021 STARS Grants@Unipd programme “QuASAR”. The authors are members of Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA), which is part of the Istituto Nazionale di Alta Matematica (INdAM), and are partially funded by the GNAMPA Project “Problemi al bordo e applicazioni geometriche”. The authors are grateful to S. Hirsch and F. Oronzio for their interest in the work and for pleasant and useful conversations on the subject. The authors warmly thank the anonymous referees for their thorough reading of the paper and for the valuable suggestions that allowed them to improve the quality of the paper. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Nonlinear Isocapacitary Concepts of Mass in 3-Manifolds with Nonnegative Scalar Curvature Article published earlier |
| spellingShingle | Nonlinear Isocapacitary Concepts of Mass in 3-Manifolds with Nonnegative Scalar Curvature Benatti, Luca Fogagnolo, Mattia Mazzieri, Lorenzo |
| title | Nonlinear Isocapacitary Concepts of Mass in 3-Manifolds with Nonnegative Scalar Curvature |
| title_full | Nonlinear Isocapacitary Concepts of Mass in 3-Manifolds with Nonnegative Scalar Curvature |
| title_fullStr | Nonlinear Isocapacitary Concepts of Mass in 3-Manifolds with Nonnegative Scalar Curvature |
| title_full_unstemmed | Nonlinear Isocapacitary Concepts of Mass in 3-Manifolds with Nonnegative Scalar Curvature |
| title_short | Nonlinear Isocapacitary Concepts of Mass in 3-Manifolds with Nonnegative Scalar Curvature |
| title_sort | nonlinear isocapacitary concepts of mass in 3-manifolds with nonnegative scalar curvature |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/212040 |
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