# The Spacetime Penrose Inequality for Cohomogeneity One Initial Data

Marcus Khuri  Department of Mathematics
Stony Brook University
Stony Brook, NY 11794, USA<EMAIL_ADDRESS>and  Hari Kunduri
Department of Mathematics & Statistics
and Department of Physics & Astronomy
McMaster University
Hamilton, ON L8S 4M1, Canada<EMAIL_ADDRESS>
###### Abstract.

We prove the spacetime Penrose inequality for asymptotically flat
$2(n+1)$-dimensional initial data sets for the Einstein equations, which are
invariant under a cohomogeneity one action of $\mathrm{SU}(n+1)$. Analogous
results are obtained for asymptotically hyperbolic initial data that arise as
spatial hypersurfaces in asymptotically Anti de-Sitter spacetimes. More
precisely, it is shown that with the dominant energy condition, the total mass
is bounded below by an explicit function of the outermost apparent horizon
area. Furthermore, the inequality is saturated if and only if the initial data
isometrically embed into a Schwarzschild(-AdS) spacetime. This generalizes the
only previously known case of the conjectured spacetime Penrose inequality,
established under the assumption of spherical symmetry. Additionally, in the
time symmetric case, we observe that the inequality holds for
$4(n+1)$-dimensional and 16-dimensional initial data invariant under
cohomogeneity one actions of $\mathrm{Sp}(n+1)$ and $\mathrm{Spin}(9)$,
respectively, thus treating the inequality for all cohomogeneity one actions
in this regime.

M. Khuri acknowledges the support of NSF Grant DMS-2104229. H. Kunduri
acknowledge the support of NSERC Grant RGPIN-2018-04887.

##  1\. Introduction

In an effort to find a counterexample to the weak cosmic censorship conjecture
[36], Penrose put forward a precise inequality [37] relating the ADM mass $m$
of an asymptotically flat 4-dimensional spacetime to any cross-sectional area
$\mathcal{A}_{e}$ of the event horizons it contains, in the form

(1.1) $m\geq\sqrt{\frac{\mathcal{A}_{e}}{16\pi}}.$

It is typical to reformulate this inequality in the setting of initial data
sets. Consider a triple $(M^{d},g,k)$ consisting of a $d$-dimensional
connected and orientable manifold $M^{d}$ with boundary, a complete Riemannian
metric $g$, and a symmetric 2-tensor $k$ denoting the extrinsic curvature of
an embedding into spacetime, with all objects being smooth. These quantities
must satisfy the constraint equations

(1.2) $16\pi\mu=R+(\text{Tr}_{g}k)^{2}-|k|_{g}^{2},\qquad\text{ }8\pi
J=\text{div}_{g}\left(k-(\text{Tr}_{g}k)g\right),$

where $R$ is scalar curvature, and $\mu$, $J$ represent the energy-momentum
density of matter fields. We will say that the dominant energy condition is
satisfied if $\mu\geq|J|_{g}$. Moreover, the data will be referred to as
asymptotically flat if outside a compact set $\mathcal{C}$ there is a
diffeomorphism
$\varphi:M^{d}\setminus\mathcal{C}\rightarrow\mathbb{R}^{d}\setminus B_{1}$,
such that in the Caretesian coordinates $x$ provided by this map

(1.3)
$\varphi_{*}g-\delta=O_{2}(|x|^{-\tau}),\quad\varphi_{*}k=O_{1}(|x|^{-\tau-1}),\quad\mu,J=O(|x|^{-2\tau-2}),\quad\mathrm{Tr}_{g}k=O_{1}(|x|^{-2\tau-1}),$

for some $\tau>\tfrac{d-2}{2}$. The additional decay on the trace of $k$ is
usually not included in the definition of asymptotically flatness, but will be
useful when working with the generalized Jang equation below. With these
asymptotics the ADM energy and linear momentum are well-defined [5, 16] and
given by

(1.4)
$E\\!=\\!\frac{1}{2(d-1)\omega_{d-1}}\lim_{r\rightarrow\infty}\int_{S_{r}}(g_{ij,i}-g_{ii,j})\nu^{j}dV,\quad
P_{i}\\!=\\!\frac{1}{(d-1)\omega_{d-1}}\lim_{r\rightarrow\infty}\int_{S_{r}}(k_{ij}-(\text{Tr}_{g}k)g_{ij})\nu^{j}dV,$

where $S_{r}$ are coordinate spheres with unit outer normal $\nu$ and
$\omega_{d-1}$ is the volume of the unit $(d-1)$-sphere. The ADM mass is then
the Lorentz length of the energy-momentum vector, $m=\sqrt{E^{2}-|P|^{2}}$.

The role of the event horizon is replaced by that of an apparent horizon,
which may be computed directly from the initial data. Recall that the
gravitational field’s strength near a hypersurface $\Sigma\subset M^{d}$ may
be probed by the null expansions (null mean curvatures)

(1.5) $\theta_{\pm}=H_{\Sigma}\pm\text{Tr}_{\Sigma}k,$

where $H_{\Sigma}$ denotes the mean curvature with respect to the normal
pointing towards infinity. These give the rate of change for area of a shell
of light emanating from the surface in the outward future/past direction
($+$/$-$). Future or past trapped surfaces are defined by the inequalities
$\theta_{+}<0$ or $\theta_{-}<0$, respectively, and may be interpreted as
lying within a region of strong gravity. When $\theta_{+}=0$ or $\theta_{-}=0$
the surface is called a future or past apparent horizon; these naturally arise
as boundaries of future or past trapped regions [4]. Furthermore, such a
surface will be called an outermost apparent horizon if it is not enclosed by
any other apparent horizon. The conjectured Penrose inequality for general
dimensions may then be recast as

(1.6)
$m\geq\frac{1}{2}\left(\frac{\mathcal{A}_{h}}{\omega_{d-1}}\right)^{\frac{d-2}{d-1}}$

whenever the dominant energy condition holds, where $\mathcal{A}_{h}$ is the
smallest area required to enclose the outermost apparent horizon. Equality
should be achieved only for slices of the Schwarzschild spacetime.

In the (Riemannian) time symmetric case when $k=0$, the 3-dimensional Penrose
inequality has been confirmed by Huisken-Ilmanen [27] and Agostiniani-
Mantegazza-Mazzieri-Oronzio [1] for a single black hole via inverse mean
curvature flow and $p$-harmonic functions repsectively, and by Bray [7] for
multiple black holes using a conformal flow. The latter approach has been
generalized by Bray-Lee [11] up to dimension 7. Within the context of the
general spacetime setting, there are very few results. In fact, for this
regime the conjectured inequality has only been verified in the spherically
symmetric case [25, 32] with the rigidity statement also obtained in [9, 12],
[30, Theorem 7.46]; these results hold in all dimensions.

In the present note we consider cohomogeneity one initial data sets. Recall
that a Riemannian manifold $(M^{d},g)$ is said to have cohomogeneity one if a
(compact connected) Lie group $\mathcal{G}$ acts by isometries on $M^{d}$ such
that the principal orbits $\mathcal{G}/\mathcal{H}$ are of codimension one,
where $\mathcal{H}$ is a principal isotropy subgroup (for general initial data
sets we also assume that $k$ is invariant under the action of $\mathcal{G}$).
Within the setting of interest, the manifold will be asymptotically flat or
asymptotically hyperbolic with an outermost apparent horizon boundary; note
that smooth outermost apparent horizons inherit the symmetries of the initial
data from which they arise [12, Lemma 3.1] ([3, Theorem 8.1], [35]).
Therefore, the principal orbit theorem implies that the orbit space
$M^{d}/\mathcal{G}$ is diffeomorphic to a half line $[0,\infty)$ with the
origin corresponding to the apparent horizon, and
$M^{d}\cong[0,\infty)\times\mathcal{G}/\mathcal{H}$. This is discussed further
in Section 2, where it is also shown that the structure at infinity ensures
that the surfaces of homogeneity will be spheres. A classification of the
possible homogeneous metrics on spheres has been obtained by Ziller [41]. In
addition to the standard round metric, there are odd-dimensional cases
corresponding to $S^{2n+1}=\mathrm{SU}(n+1)/\mathrm{SU}(n)$,
$S^{4n+3}=\mathrm{Sp}(n+1)/\mathrm{Sp}(n)$, and a homogeneous metric on
$S^{15}=\mathrm{Spin}(9)/\mathrm{Spin}(7)$. We will establish the spacetime
Penrose inequality for the first of these cases in the asymptotically flat and
asymptotically hyperbolic contexts, namely for initial data of dimension
$d=2(n+1)$, $n\geq 1$ which are invariant under the action of
$\mathrm{SU}(n+1)$. Note that in contrast to the spherically symmetric case,
this class of initial data includes those with non-vanishing angular momentum,
and there are explicit rotating black hole solutions arising from data in this
class, see Appendix B. To accomplish this goal we will exploit a method
proposed by Bray and the first author [10, 9] which involves coupling inverse
mean curvature flow to the so called generalized Jang equation. In these
higher dimensions, lack of the Gauss-Bonnet theorem presents difficulties for
monotonicity of Hawking mass, however in the current setting a fortuitous
cancellation occurs which allows the procedure to go through.

######  Theorem 1.1.

Let $(M^{2(n+1)},g,k)$, $n\geq 1$ be an asymptotically flat
$\mathrm{SU}(n+1)$-invariant initial data set, with outermost apparent horizon
boundary of area $\mathcal{A}$. If the dominant energy condition is satisfied
then

(1.7)
$m\geq\frac{1}{2}\left(\frac{\mathcal{A}}{\omega_{2n+1}}\right)^{\frac{2n}{2n+1}},$

and equality occurs if and only if the initial data arise from an isometric
embedding into a Schwarzschild spacetime.

Consider now asymptotically hyperbolic initial data relevant for
asymptotically Anti-de Sitter (AdS) spacetimes. Let $(\mathbb{H}^{d},b)$
denote the $d$-dimensional hyperbolic space with metric expressed in geodesic
polar coordinates as $b=dr^{2}+(\sinh^{2}r)g_{S^{d-1}}$. Recall that
hyperbolic space arises as a totally geodesic spacelike slice of the Anti-de
Sitter (AdS) spacetime,
$(\mathbb{R}\times\mathbb{H}^{d},-(\cosh^{2}r)dt^{2}+b)$. An initial data set
$(M^{d},g,k)$ satisfying the modified constraints

(1.8) $16\pi\mu=R+(\text{Tr}_{g}k)^{2}-|k|_{g}^{2}+d(d-1),\qquad 8\pi
J=\text{div}_{g}\left(k-(\text{Tr}_{g}k)g\right),$

will be referred to as asymptotically hyperbolic if outside a compact set
$\mathcal{C}$ there is a diffeomorphism
$\varphi:M^{d}\setminus\mathcal{C}\rightarrow\mathbb{H}^{d}\setminus B_{1}$
such that

(1.9)
$\mathfrak{g}:=\varphi_{*}g-b=O_{2}(e^{-qr}),\qquad\varphi_{*}k=O_{1}(e^{-qr}),\qquad\mu,J=O(e^{-2qr}),$

for $q>d/2$. The dominant energy condition in this setting is again expressed
as $\mu\geq|J|_{g}$. Initial data satisfying (1.8) and (1.9) arise as
spacelike hypesurfaces in asymptotically AdS spacetimes with (negative)
cosmological constant $\Lambda=-\frac{d(d-1)}{2}$. Well-defined global
Hamiltonian charges, interpreted as the total energy-momentum for such data,
were obtained by Chruściel-Nagy [17] (see also [34] and the exposition in
[13]). In particular, if $U=\cosh r$ denotes the time translation lapse
function then the total energy is given by

(1.10)
$E_{\mathrm{hyp}}=\frac{1}{2(d-1)\omega_{d-1}}\lim_{r\to\infty}\int_{S_{r}}\left[U(\text{div}_{b}\mathfrak{g})-U(d\mathrm{Tr}_{b}\mathfrak{g})+(\mathrm{Tr}_{b}\mathfrak{g})dU-\mathfrak{g}(\nabla_{b}U)\right](\nu_{b})dV,$

where $\nu_{b}$ is the unit outward normal (measured in $b$) to the coordinate
spheres $S_{r}$. An analogous expression gives rise to the total linear
momentum by replacing $U$ with the lapse functions $U_{i}=x^{i}\sinh r$,
$i=1,\ldots d$ defined on $\mathbb{H}^{d}$, where $x^{i}$ are Cartesian
coordinates restricted to the unit sphere $S^{d-1}$. A version of the
spacetime Penrose inequality is conjectured to hold in this asymptotically
hyperbolic setting [8, 14, 22, 33], and is relevant in the context of the
gauge theory/gravity correspondence [20]. The inequality has been established
in spherical symmetry by Engelhardt-Folkestad [19, Theorem 6], Folkestad [21,
Theorem 1], and Husain-Singh [28], with various hypotheses. Moreover, it has
been proved when $k=0$ for small perturbations of the Schwarzschild-AdS
manifold by Ambrozio [2] (see also [29]), and for graphs by Dahl-Gicquaud-
Sakovich [18] and Girão-de Lima [24]. We prove the following result.

######  Theorem 1.2.

Let $(M^{2(n+1)},g,k)$, $n\geq 1$ be an asymptotically hyperbolic
$\mathrm{SU}(n+1)$-invariant initial data set, with outermost apparent horizon
boundary of area $\mathcal{A}$. If the dominant energy condition is satisfied
then

(1.11)
$E_{\mathrm{hyp}}\geq\frac{1}{2}\left(\frac{\mathcal{A}}{\omega_{2n+1}}\right)^{\frac{2n}{2n+1}}+\frac{1}{2}\left(\frac{\mathcal{A}}{\omega_{2n+1}}\right)^{\frac{2n+2}{2n+1}},$

and equality occurs if and only if the initial data arise from an isometric
embedding into a Schwarzschild-AdS spacetime.

A different type of asymptotically hyperbolic initial data, with second
fundamental form $k$ converging to $g$ rather than vanishing at infinity,
appears naturally as asymptotically umbilic slices in asymptotically flat
spacetimes and are referred to as asymptotically hyperboloidal. In this
setting, the Penrose inequality [40] takes the form (1.7) instead of (1.11),
and the case of equality should give rise to an embedding into the
Schwarzschild spacetime. This has also been established in spherical symmetry
by Hou [26], in the time symmetric graphical context by Girão-de Lima [23],
and we expect that the strategy outlined in [15] together with arguments of
the current paper will yield the $\mathrm{SU}(n+1)$-invariant case as well.

Lastly, in the Riemannian asymptotically flat context, we are able to treat
the Penrose inequality for the full range of cohomogeneity one manifolds. In
addition to the $\mathrm{SU}(n+1)$-invariant case, this also includes the
$\mathrm{Sp}(n+1)$ and $\mathrm{Spin}(9)$-invariant cases. The proof will rely
on a combination of Bray’s conformal flow with inverse mean curvature flow.

######  Theorem 1.3.

Let $(M^{d},g)$, $d\geq 3$ be a cohomogeneity one Riemannian manifold which is
asymptotically flat, with outermost minimal surface boundary. If the scalar
curvature is nonnegative $R\geq 0$, then inequality (1.6) holds. Moreover,
equality is achieved if and only if the manifold is isometric to the canonical
slice of a Schwarzschild spacetime.

This article is organized as follows. In Section 2 we describe cohomogeneity
one initial data sets in detail, while in Section 3 existence is established
for the coupled Jang-IMCF system of equations in this setting. The proofs of
Theorems 1.1, 1.2, and 1.3 are then given in Sections 4, 5, and 6. Finally,
two appendices are provided with the first showing that linear momentum
vanishes in the asymptotically flat case, and the second exhibiting examples
of relevant initial data.

### Acknowledgements

The authors would like to thank McKenzie Wang for helpful comments.

##  2\. Cohomogeneity One Initial Data Sets

Consider a cohomogeneity one Riemannian manifold $(M^{d},g)$ which is
asymptotically flat or asymptotically hyperbolic, and possesses a smooth
outermost apparent horizon boundary. If $\mathcal{G}$ is the compact connected
Lie group of isometries giving rise to the cohomogeneity one structure, then
as discussed in the introduction the manifold is diffeomorphic to a product
$M^{d}\cong[0,\infty)\times\mathcal{G}/\mathcal{H}$, where $\mathcal{H}$ is a
principal isotropy subgroup. To see this, first note that smooth outermost
apparent horizons inherit the symmetries of the initial data from which they
arise [12, Lemma 3.1] ([3, Theorem 8.1], [35]), more precisely $\partial
M^{d}$ is left invariant by $\mathcal{G}$. Therefore, we have a well-defined
$\mathcal{G}$-action on $M^{d}\setminus\partial M^{d}$, and may then apply the
principal orbit theorem to conclude that the orbit space
$(M^{d}\setminus\partial M^{d})/\mathcal{G}$ is diffeomorphic to a open half
line $(0,\infty)$, with the origin corresponding to the apparent horizon, and
thus $M^{d}\setminus\partial
M^{d}\cong(0,\infty)\times\mathcal{G}/\mathcal{H}$. The desired conclusion now
follows.

We next claim that in light of the asymptotics, the orbits
$\mathcal{G}/\mathcal{H}$ must topologically be the sphere $S^{d-1}$. Note
that if $s:M^{d}\rightarrow\mathbb{R}_{+}$ is the distance function to the
boundary $\partial M^{d-1}$, then $g=ds^{2}+g_{s}$ where $g_{s}$ is a
1-parameter family of $\mathcal{G}$-invariant metrics on
$\mathcal{G}/\mathcal{H}$. Let $\Sigma_{s}$ denote the $s$-level set, then
there exists a sufficiently large $s_{0}$ such that $\Sigma_{s_{0}}$ lies
completely within the asymptotic end $M^{d}\setminus\mathcal{C}$. If not, then
there exists a sequence of distances $s_{i}\rightarrow\infty$ with
$|s_{i}-s_{j}|>1$ for $i\neq j$ such that
$\Sigma_{s_{i}}\cap\mathcal{C}\neq\emptyset$. Since $\mathcal{C}$ is compact,
a sequence of points $p_{i}\in\Sigma_{s_{i}}\cap\mathcal{C}$ must have a
convergent subsequence. However this contradicts $|s_{i}-s_{j}|>1$, and we
conclude that there is a $\Sigma_{s_{0}}\subset M^{d}\setminus\mathcal{C}$.
Because $\Sigma_{s_{0}}$ is compact, there exists a large coordinate sphere
$S_{r_{0}}\subset\mathbb{R}^{d}$ whose preimage under the asymptotic
diffeomorphism $\varphi$ separates $\Sigma_{s_{0}}$ from infinity. Let
$\tilde{\Omega}_{s_{0}r_{0}}$ be the image of $\Omega_{s_{0}r_{0}}$ inside
$\mathbb{R}^{d}\setminus B_{1}$, where $\Omega_{s_{0}r_{0}}$ is the region
bounded between $\Sigma_{s_{0}}$ and $\varphi^{-1}(S_{r_{0}})$. By flowing
along radial lines in $\mathbb{R}^{d}$, we may continuously deform
$\tilde{\Omega}_{s_{0}r_{0}}$ into $S_{r_{0}}$. This induces a deformation
retract of $M_{s_{0}}^{d}$ onto $M_{r_{0}}^{d}$, where these are the closure
of the noncompact component of $M^{d}\setminus\Sigma_{s_{0}}$ and
$M^{d}\setminus\varphi^{-1}(S_{r_{0}})$, respectively. Since $M_{s_{0}}^{d}$
is homotopy equivalent to $\Sigma_{s_{0}}$ and $M_{r_{0}}^{d}$ is homotopy
equivalent to $\varphi^{-1}(S_{r_{0}})$, it follows that $\Sigma_{s_{0}}$ is a
homotopy sphere. Therefore, by the resolution of the generalized Poincaré
conjecture we have that $\mathcal{G}/\mathcal{H}=\Sigma_{s_{0}}$ is
homeomorphic to the standard sphere $S^{d-1}$. Moreover, since exotic spheres
do not admit smooth homogeneous space structures [39, Theorem 1.1], it follows
that $\mathcal{G}/\mathcal{H}$ is diffeomorphic $S^{d-1}$.

The classification of homogeneous metrics on spheres has been given by Ziller
[41, page 352]. In all dimensions there is a unique $\mathrm{SO}(d)$-invariant
metric (up to homothety) on $S^{d-1}=\mathrm{SO}(d)/\mathrm{SO}(d-1)$, of
constant curvature. This leads to spherically symmetric initial data, a
setting in which the spacetime Penrose inequality has already been
established. In odd dimensions the Hopf fibrations $S^{1}\hookrightarrow
S^{2n+1}\to\mathbb{CP}^{n}$ give rise to a 1-parameter family or 2-parameter
family of $\mathrm{SU}(n+1)$-invariant metrics on
$S^{2n+1}=\mathrm{SU}(n+1)/\mathrm{SU}(n)$ depending on whether $n>1$ or
$n=1$, respectively. For $n>1$, these metrics have an extra $\mathrm{U}(1)$
symmetry, and will be discussed in detail below. The remaining cases include a
3-parameter family of metrics on $S^{4n+3}=\mathrm{Sp}(n+1)/\mathrm{Sp}(n)$
and a 1-parameter family of metrics on
$S^{15}=\mathrm{Spin}(9)/\mathrm{Spin}(7)$ up to homothety, which will be
examined further in Section 6.

In what follows, an initial data set $(M^{d},g,k)$ will be referred to as
$\mathcal{G}$-invariant if the Riemannian manifold $(M^{d},g)$ is
cohomogeneity one with respect to $\mathcal{G}$, and the Lie derivative
vanishes $L_{\eta}k=0$ for any Killing field $\eta$ associated with the
isometries of $g$.

###  2.1. $\mathrm{SU}(n+1)$-invariant initial data, $n>1$

The 1-parameter family of Berger metrics on the ‘squashed’ sphere $S^{2n+1}$
are given by

(2.1) $g_{\lambda}=\lambda(d\psi+A)^{2}+g_{FS},$

where $\psi$ is a $2\pi$-periodic coordinate on the circles of the Hopf
fibration, $g_{FS}$ is the Fubini-Study metric on $\mathbb{CP}^{n}$ scaled so
that $\text{Ric}(g_{FS})=2(n+1)g_{FS}$, and $A$ is a connection 1-form such
that $\omega=dA/2$ is the associated Kähler form. The normalization is chosen
so that $g_{1}$ is the unit round metric on $S^{2n+1}$. Note that the
squashing parameter $\lambda$ controls the size of the $S^{1}$ fibers in the
fibration, and the metric (2.1) is invariant under the $\mathrm{U}(1)$
isometry generated by the Killing field $\partial_{\psi}$. Moreover, since the
round metric inherits the subgroup of isometries $\mathrm{SU}(n+1)$ from
$\mathbb{C}^{n+1}$ and the action descends by isometries to the complex
projective space quotient, we find that $L_{\eta}(d\psi+A)=0$ for any Killing
field $\eta$ associated with the $\mathrm{SU}(n+1)$ symmetry. It follows that
the Berger metrics inherit this symmetry, and they are the building blocks of
$\mathrm{SU}(n+1)$-invariant initial data.

######  Proposition 2.1.

Let $n>1$. Consider an asymptotically flat or asymptotically hyperbolic
$\mathrm{SU}(n+1)$-invariant initial data set $(M^{2(n+1)},g,k)$, with
outermost apparent horizon boundary. Then $M^{2(n+1)}\cong[0,\infty)\times
S^{2n+1}$ and the metric and extrinsic curvature take the form

(2.2)
$\displaystyle\begin{split}g=&ds^{2}+\rho(s)^{2}\left[e^{-4nB(s)}(d\psi+A)^{2}+e^{2B(s)}g_{FS}\right],\\\
k=&k_{a}ds^{2}+2k_{s}\rho
e^{-2nB}ds(d\psi+A)+\rho^{2}\left[k_{b}e^{-4nB}(d\psi+A)^{2}+k_{c}e^{2B}g_{FS}\right],\end{split}$

for some smooth functions $B$, $k_{a}$, $k_{b}$, $k_{c}$, $k_{s}$, and
$\rho>0$ of $s$ alone. In an asymptotically flat end

(2.3) $\rho(s)=s+O_{2}(s^{1-\tau}),\quad B=O_{2}(s^{-\tau}),\quad
k_{a},k_{b},k_{c},k_{s}=O_{1}(s^{-\tau-1}),\quad\mathrm{Tr}_{g}k=O_{1}(s^{-2\tau-1}),$

while in an asymptotically hyperbolic end

(2.4) $\rho(s)=\sinh s+O_{2}(e^{(1-q)s}),\quad B=O_{2}(e^{-qs}),\quad
k_{a},k_{b},k_{c},k_{s}=O_{1}(e^{-qs}).$

###### Proof.

According to the discussion at the beginning of this section, and the
classification of [41], the manifold is topologically
$M^{2(n+1)}=[0,\infty)\times S^{2n+1}$ and the metric is given by

(2.5) $g=ds^{2}+P(s)^{2}(d\psi+A)^{2}+Q(s)^{2}g_{FS}$

for some smooth positive functions $P(s)$ and $Q(s)$. We may then set

(2.6) $\rho(s):=\left[P(s)Q(s)^{2n}\right]^{\frac{1}{2n+1}},\qquad
e^{B(s)}:=\left(\frac{Q(s)}{P(s)}\right)^{\frac{1}{2n+1}},$

so that

(2.7) $Q(s)=e^{B(s)}\rho(s),\qquad P(s)=e^{-2nB(s)}\rho(s).$

The desired expression for the metric now follows.

Consider now the structure of $k$. It is useful to introduce the orthonormal
coframe

(2.8)
$\mathbf{e}^{1}=ds,\qquad\mathbf{e}^{2}=\rho(s)e^{-2nB(s)}(d\psi+A),\qquad\mathbf{e}^{i}=\rho(s)e^{B(s)}\hat{e}^{i},\quad
i=3,\ldots,2(n+1),$

where $\hat{e}^{i}$ are members of an orthornoaml coframe for the Fubini-Study
metric. The dual basis vectors are given by

(2.9)
$\mathbf{e}_{1}=\partial_{s},\qquad\mathbf{e}_{2}=\rho(s)^{-1}e^{2nB(s)}\partial_{\psi},\qquad\mathbf{e}_{i}=\rho(s)^{-1}e^{-B(s)}\left(\hat{e}_{i}-A(\hat{e}_{i})\partial_{\psi}\right).$

We may then expand the extrinsic curvature as
$k=k_{\mathrm{ij}}\mathbf{e}^{\mathrm{i}}\otimes\mathbf{e}^{\mathrm{j}}$ for
some symmetric matrix of functions $k_{\mathrm{ij}}$, with
$\mathrm{i},\mathrm{j}=1,\ldots,2(n+1)$. First observe that $k_{\mathrm{ij}}$
has no dependence on $\psi$, since the Killing field $\partial_{\psi}$
commutes with the frame (2.9). Next, note that $g_{FS}$ is the unique
$\mathrm{SU}(n+1)$-invariant metric (up to a scaling) on $\mathbb{CP}^{n}$.
This implies that there can be no nontrivial $\mathrm{SU}(n+1)$-invariant
1-forms $\sigma$ on $\mathbb{CP}^{n}$, for otherwise
$g_{FS}+\lambda\sigma^{2}$ would produce another invariant family of metrics.
It follows that an invariant $k$ cannot contain terms of the form
$\mathbf{e}^{1}\otimes\mathbf{e}^{i}$ or $\mathbf{e}^{2}\otimes\mathbf{e}^{i}$
for any $i>2$. To see this, write $k_{\ell
i}\mathbf{e}^{1}\otimes\mathbf{e}^{i}=\mathbf{e}^{\ell}\otimes K_{\ell}$ where
$K_{\ell}=k_{\ell i}\mathbf{e}^{i}$ and $\ell=1,2$. Let $\eta$ be a Killing
field associated with the $\mathrm{SU}(n+1)$ symmetry, then
$L_{\eta}(\mathbf{e}^{\ell}\otimes K_{\ell})=\mathbf{e}^{\ell}\otimes
L_{\eta}K_{\ell}$. However, since $L_{\eta}K_{\ell}$ is a linear combination
of $\mathbf{e}^{i}$ for $i>2$, there is no possibility of canceling this term
with any other expression in $L_{\eta}k$. We then have that
$L_{\eta}K_{\ell}=0$. Thus, $K_{\ell}$ is a $\mathrm{SU}(n+1)$-invariant
1-form as $\eta$ was arbitrary, so that $K_{\ell}=0$. From this, one may
further conclude that $k_{11}$, $k_{12}$, and $k_{22}$ depend only on $s$, and
that $\alpha=\sum_{i,j>2}k_{ij}e^{i}\otimes e^{j}$ is
$\mathrm{SU}(n+1)$-invariant. The latter statement implies that $\alpha$ must
be a multiple of $g_{FS}$, otherwise for small $\varepsilon$ the expression
$g_{FS}+\varepsilon\alpha$ would furnish an invariant metric on
$\mathbb{CP}^{n}$ in violation of the uniqueness property of the Fubini-Study
metric. Putting this all together yields the desired structure for $k$.

Lastly, to obtain the fall-off in (2.3), write the Euclidean metric of (1.3)
in polar form with the coordinate spheres expressed with respect to the Hopf
fibration, and compare with the coefficients of $\varphi_{*}g$. This gives
relations between the two sets of coordinates and the quantities $\rho$, $B$,
from which the desired decay may be derived. An analogous procedure applies to
the extrinsic curvature. Moreover, the asymptotically hyperbolic case is
treated similarly. ∎

It will be useful to record some facts about the class of metrics (2.2). First
note that the $s$-level set spheres $\Sigma_{s}$ have volume form

(2.10) $dV_{\Sigma}=\rho^{2n+1}d\psi\wedge
dV_{\mathbb{CP}^{n}}=\rho^{2n+1}dV_{S^{2n+1}},$

and scalar curvature

(2.11)
$R_{\Sigma}=-\frac{2nP^{2}}{Q^{4}}+\frac{4n(n+1)}{Q^{2}}=-\frac{2n}{\rho^{2}}\left(e^{-4(n+1)B}-2(n+1)e^{-2B}\right),$

while the scalar curvature of $g$ as expressed in (2.5) is given by

(2.12)
$R=-\frac{4nQ^{\prime\prime}}{Q}-\frac{2P^{\prime\prime}}{P}-\frac{2nP^{2}}{Q^{4}}-\frac{4nP^{\prime}Q^{\prime}}{PQ}+\frac{4n(n+1)}{Q^{2}}-\frac{2n(2n-1)Q^{\prime
2}}{Q^{2}}$

where the prime notation indicates differentiation with respect to $s$.
Furthermore

(2.13)
$\mathrm{Tr}_{g}k=k_{a}+k_{b}+2nk_{c},\qquad\mathrm{Tr}_{\Sigma}k=k_{b}+2nk_{c},$

and the mean curvature of the level sets is

(2.14) $H_{\Sigma}=(2n+1)\frac{\rho^{\prime}}{\rho}.$

###  2.2. $\mathrm{SU}(2)$-invariant initial data

For $n=1$, $S^{3}$ is diffeomorphic to the Lie group $\mathrm{SU}(2)$ and
hence there exist homogeneous metrics invariant under the left or right action
of $\mathrm{SU}(2)$ that do not possess the enhanced $\mathrm{U}(1)$ isometry.
Consider the following right-invariant 1-forms [31, Appendix A]

(2.15) $\sigma^{1}=\sin\psi d\theta-\cos\psi\sin\theta
d\phi,\quad\sigma^{2}=\cos\psi d\theta+\sin\psi\sin\theta
d\phi,\quad\sigma^{3}=d\psi+\cos\theta d\phi,$

where $\theta\in(0,\pi),\phi\in(0,2\pi)$, and $\psi\in(0,4\pi)$. These satisfy
$d\sigma^{i}=-\tfrac{1}{2}\epsilon_{ijk}\sigma^{j}\wedge\sigma^{k}$, where
$\epsilon_{ijk}$ is totally antisymmetric and $\epsilon_{123}=1$. We then have
the following $\mathrm{SU}(2)$-invariant metrics on the 3-sphere
$g_{SU}=h_{ij}\sigma^{i}\sigma^{j}$, in which $h$ is a constant positive
definite matrix. Note that $h_{ij}=\tfrac{1}{4}\delta_{ij}$ gives rise to the
unit round metric. Furthermore, an orthogonal frame may be obtained with a
diagonalizing $O\in\mathrm{SO}(3)$ by setting $\hat{\sigma}^{i}\to
O^{i}_{j}\sigma^{j}$. It follows that

(2.16)
$g_{SU}=\frac{1}{4}\left(c_{1}^{2}(\hat{\sigma}^{1})^{2}+c_{2}^{2}(\hat{\sigma}^{2})^{2}+c_{3}^{2}(\hat{\sigma}^{3})^{2}\right),$

where $c_{i}>0$ are positive constants. The Berger class of
$\mathrm{SU}(2)\times\mathrm{U}(1)$-invariant metrics is obtained by setting
$c_{1}=c_{2}$; in this case the additional Killing field is given by
$\partial_{\psi}$. A computation shows that the volume form is
$dV_{g_{SU}}=c_{1}c_{2}c_{3}dV_{g_{S^{3}}}$, and the scalar curvature is given
by

(2.17)
$R_{SU}=\frac{2}{c_{1}^{2}c_{2}^{2}c_{3}^{2}}\left(2c_{1}^{2}(c_{2}^{2}+c_{3}^{2})-c_{1}^{4}-(c_{2}^{2}-c_{3}^{2})^{2}\right).$

Consider now $\mathrm{SU}(2)$-invariant initial data $(M^{4},g,k)$ satisfying
the hypotheses of Proposition 2.1. Then $M^{4}\cong[0,\infty)\times S^{3}$,
the metric takes the form

(2.18)
$g=ds^{2}+\frac{1}{4}\left(c_{1}^{2}(s)(\hat{\sigma}^{1})^{2}+c_{2}^{2}(s)(\hat{\sigma}^{2})^{2}+c_{3}^{2}(s)(\hat{\sigma}^{3})^{2}\right)$

where $\hat{\sigma}_{i}$ also depends on $s$ but only through the orthogonal
transformation $O$, and the extrinsic curvature may be expressed as

(2.19) $k=k_{a}(s)ds^{2}+2\hat{k}_{i}(s)e^{i}ds+\hat{k}_{ij}(s)e^{i}e^{j}$

for some smooth functions $k_{a}$, $\hat{k}_{i}$, and $\hat{k}_{ij}$ of $s$
alone, where $e^{i}(s)=(c_{i}(s)/2)\hat{\sigma}^{i}$ is an orthonormal frame
for each coordinate sphere. Moreover, as in the proof of Proposition 2.1, it
may be shown that in an asymptotically flat end

(2.20) $c_{i}(s)=s+O_{2}(s^{1-\tau}),\quad\quad\quad
k_{a}(s),\hat{k}_{i}(s),\hat{k}_{ij}(s)=O_{1}(s^{-\tau-1}),$

while in an asymptotically hyperbolic end

(2.21) $c_{i}(s)=\sinh s+O_{2}(e^{(1-q)s}),\quad\quad\quad
k_{a}(s),\hat{k}_{i}(s),\hat{k}_{ij}(s)=O_{1}(e^{-qs}).$

##  3\. The Cohomogeneity One Generalized Jang Equation

The goal of this section is to obtain existence and asymptotics for solutions
of a coupled system involving inverse mean curvature flow and the generalized
Jang equation, in the setting of $\mathrm{SU}(n+1)$-invariant initial data
sets $(M^{2(n+1)},g,k)$. Recall that the generalized Jang equation [10, 9]
associated with the data set is given by

(3.1) $\left(g^{ij}-\frac{\phi^{2}f^{i}f^{j}}{1+\phi^{2}|\nabla
f|^{2}}\right)\left(\frac{\phi\nabla_{ij}f+\phi_{i}f_{j}+\phi_{j}f_{i}}{\sqrt{1+\phi^{2}|\nabla
f|^{2}}}-k_{ij}\right)=0,$

where $f^{i}=g^{ij}f_{j}$, $f_{j}=\partial_{j}f$, and $\nabla$ is the Levi-
Civita connection. Utilizing the structure of the metric and extrinsic
curvature from Proposition 2.1, and assuming that $\phi$, $f$ are functions of
$s$ alone, we find that this equation is equivalent to

(3.2)
$v^{\prime}+\frac{(2n+1)\rho^{\prime}}{\rho}v+\left(\frac{\phi^{\prime}}{\phi}v-k_{a}\right)(1-v^{2})-k_{b}-2nk_{c}=0,$

where prime indicates differentiation with respect to $s$ and

(3.3) $v=\frac{\phi f^{\prime}}{\sqrt{1+\phi^{2}f^{\prime 2}}}.$

Since inverse mean curvature flow emanating from a
$\mathrm{SU}(n+1)$-invariant apparent horizon remains $SU(n+1)$-invariant,
following the proposal in [10, 9] to couple the generalized Jang equation with
inverse mean curvature flow leads to

(3.4)
$\phi=\frac{d\rho}{d\bar{s}}=\sqrt{1-v^{2}}\rho^{\prime}=\left(\frac{|\Sigma_{\bar{s}}|}{\omega_{2n+1}}\right)^{\frac{1}{2n+1}}\frac{\bar{H}}{2n+1},$

where $\Sigma_{\bar{s}}$ is the $\bar{s}$-level set surface with mean
curvature $\bar{H}$ with respect to the Jang metric
$\bar{g}=g+\phi^{2}df^{2}$, $|\Sigma_{\bar{s}}|$ is the area, and $\bar{s}$ is
the corresponding radial arclength parameter

(3.5) $\bar{s}=\int_{0}^{s}\sqrt{1+\phi^{2}f^{\prime
2}}=\int_{0}^{s}\frac{1}{\sqrt{1-v^{2}}}.$

This choice of $\phi$ ensures monotonicity of the Hawking mass along inverse
mean curvature flow in the Jang manifold $(M^{2(n+1)},\bar{g})$.

The boundary of the initial data is assumed to be an outermost apparent
horizon, and therefore we must have

(3.6)
$0<\theta_{\pm}=H\pm\text{Tr}_{\Sigma_{s}}k=(2n+1)\frac{\rho^{\prime}}{\rho}\pm\left(k_{b}+2nk_{c}\right)$

for $s>0$, and $\theta_{+}(0)=0$ and/or $\theta_{-}(0)=0$ depending on whether
$\Sigma_{0}=\partial M^{2(n+1)}$ is a future and/or past horizon. It follows
that the $g$-mean curvature of radial surfaces is positive
$H=\tfrac{1}{2}(\theta_{+}+\theta_{-})>0$ away from the boundary, and hence on
this region $\phi>0$ as long as $f^{\prime}$ is bounded. Inserting the
expression (3.4) for $\phi$ into the generalized Jang equation (3.2) and
rearranging terms produces

(3.7) $(1-v^{2})v^{\prime}+(1-v^{2})F_{\mp}(s,v)\pm\theta_{\mp}=0,$

where

(3.8) $F_{\mp}(s,v):=\mp\frac{(2n+1)}{1\pm
v}\frac{\rho^{\prime}}{\rho}+\frac{v\rho^{\prime\prime}}{\rho^{\prime}}-k_{a}.$

######  Remark 3.1.

The above computations have been performed for $\mathrm{SU}(n+1)$-invariant
initial data with $n>1$. In particular, Proposition 2.1 has been utilized. For
the special case when $n=1$, we may appeal to Section 2.2 to obtain an
analogous equation (3.7). In what follows we will treat the two cases
simultaneously without noting differences when $n=1$, as the modifications
needed for this dimension are straightforward.

######  Theorem 3.2.

Let $(M^{2(n+1)},g,k)$, $n\geq 1$ be an asymptotically flat or asymptotically
hyperbolic $\mathrm{SU}(n+1)$-invariant initial data set, with outermost
apparent horizon boundary. If the boundary mean curvature satisfies $H(0)\neq
0$, then given $\alpha\in(-1,1)$ there exists a unique solution $v\in
C^{1}([0,\infty))\cap C^{\infty}((0,\infty))$ of (3.7) such that $-1<v(s)<1$
for $s>0$ and $v(0)=\alpha$. This remains true when $\alpha=\pm 1$ if
$\partial M^{2(n+1)}$ is a past (future) apparent horizon, respectively. If
$H(0)=0$, then the same existence statement holds with $v(0)=0$ and $v\in
C^{0}([0,\infty))\cap C^{\infty}((0,\infty))$. Furthermore, in all cases the
solution admits the following decay

(3.9) $|v(s)|+s|v^{\prime}(s)|\leq Cs^{-2\tau_{0}}$

in the asymptotically flat setting for any
$\tau_{0}<\min\\{\tau,n+\tfrac{1}{2}\\}$, and the decay

(3.10) $|v(s)|+|v^{\prime}(s)|\leq Ce^{-q_{0}s}$

in the asymptotically hyperbolic setting for any $q_{0}<\min\\{q,2n+2\\}$,
where $C$ is a constant depending on $g$ (up to second derivatives) and $k$ as
well as on $\tau_{0}$ or $q_{0}$, respectively.

###### Proof.

Assume first that the initial data are asymptotically flat. Consider the case
when $v(0)=\alpha$ with $|\alpha|<1$ and $H(0)\neq 0$. By standard methods
there exists a $C^{1}$ solution on some maximal interval $[0,\bar{s})$ of
nonzero length. We claim that the solution must satisfy the basic uniform
pointwise bound

(3.11) $|v(s)|<1\quad\quad\text{ for all }s.$

To see this, proceed by contradiction and assume that the estimate fails, then
there exists a first $s_{0}>0$ such that $v(s_{0})=\pm 1$. Without loss of
generality we may take $v(s_{0})=1$, then there is an increasing sequence
$s_{i}\rightarrow s_{0}$ with $v^{\prime}(s_{i})\geq 0$ and $|v(s_{i})|<1$. It
follows that

(3.12)
$0\leq\lim_{i\rightarrow\infty}(1-v^{2})v^{\prime}(s_{i})=\lim_{i\rightarrow\infty}\left[-\theta_{-}(s_{i})-(1-v^{2})F_{-}(s_{i},v(s_{i}))\right]=-\theta_{-}(s_{0})<0,$

which yields a contradiction. We conclude that (3.11) is valid. In fact, this
estimate may be improved. Observe that since $H(0)>0$, with the help of
Proposition 2.1 and the discussion preceding this theorem, we have that
$\rho^{\prime}(s)\geq c$ globally for some constant $c>0$. Thus, there exists
a positive radial function $\bar{F}_{-}$ satisfying

(3.13) $|F_{-}(v,s)|\leq\bar{F}_{-}(s)\leq\frac{c_{1}}{1+s}\quad\text{ for all
}s\in[0,\infty)\text{ whenever }v(s)\geq 0,$

for some constant $c_{1}$. If $\partial M^{2(n+1)}$ is not a past apparent
horizon, that is $\theta_{-}(0)>0$, then there exists a constant $\delta>0$
such that $\bar{F}_{-}(s)^{-1}\theta_{-}(s)>\delta$; the constant may also be
chosen small enough to satisfy $v(0)=\alpha<\sqrt{1-\delta}$. If $s_{1}$ is
the first point at which $v(s_{1})=\sqrt{1-\delta}$ then
$v^{\prime}(s_{1})\geq 0$ and consequently

(3.14) $0\leq-\theta_{-}(s_{1})+(1-v^{2}(s_{1}))\bar{F}_{-}(s_{1}),$

which implies that $1-v^{2}(s_{1})>\delta$ yielding a contradiction. We
conclude that $v(s)<\sqrt{1-\delta}$ at all points, and a similar argument
with $F_{-}$ replaced by $F_{+}$ provides a lower bound when $\partial
M^{2(n+1)}$ is not a future apparent horizon. If $\partial M^{2(n+1)}$ is a
past apparent horizon, that is $\theta_{-}(0)=0$, then since $\theta_{-}\geq
0$ we have

(3.15)
$v^{\prime}(s)=-\theta_{-}(s)(1-v^{2}(s))^{-1}-F_{-}(s,v(s))\leq\bar{F}_{-}(s)\leq
c_{1}$

whenever $v$ is nonnegative. It follows that $v(s)\leq c_{1}s+\alpha$ for
$s\in[0,\bar{s}_{1})$ where
$\bar{s}_{1}=\min\\{\bar{s},c_{1}^{-1}(1-\alpha)\\}$. Since $\theta_{-}$ is
strictly positive at $\bar{s}_{1}/2$, by applying the same arguments in the
non-past apparent horizon case to the region $s\geq\bar{s}_{1}/2$, we find
that globally $v(s)<\sqrt{1-\delta}$ for some $\delta>0$ depending on
$\alpha$. As before, analogous arguments with $F_{-}$ replaced by $F_{+}$
yield a lower bound when $\partial M^{2(n+1)}$ is a future apparent horizon.
Therefore, in all cases

(3.16) $|v(s)|<\sqrt{1-\delta}$

on the domain of existence. These pointwise estimates imply uniform control on
first derivatives immediately from equation (3.7). The function $v$ is then
uniformly continuous, and thus has a unique and continuous extension to
$[0,\bar{s}]$. By differentiating the equation, the same holds true for all
derivatives of $v$. Hence the solution has an unrestricted domain, that is
$v\in C^{\infty}([0,\infty))$.

Consider now the case when $\alpha=1$, $\partial M^{2(n+1)}$ is a past
apparent horizon, and $H(0)\neq 0$. Observe that for $-\tfrac{3}{2}\leq
w\leq\tfrac{1}{2}$ or equivalently $|v|\leq 1$ equation (3.7) may be expressed
as

(3.17)
$(1+v(w))w^{\prime}=-\theta_{-}+F_{-}(s,v(w))\left(1-2w-2\sqrt{1-2w}\right),$

where $w=v-\frac{v^{2}}{2}$ and $v(w)=1-\sqrt{1-2w}$. Notice that (3.17) is no
longer degenerate in that the principal symbol does not vanish. To produce a
solution in the required range, we may take a limit of solutions
$w_{\varepsilon}$ with initial condition
$w_{\varepsilon}(0)=\tfrac{1}{2}-\varepsilon$. Due to the nondegeneracy of
this equation and the estimate (3.16), the solution
$w:=\lim_{\varepsilon\rightarrow 0}w_{\varepsilon}$ is $C^{1}$ up to the
boundary, and satisfies $-\tfrac{3}{2}\leq w\leq\tfrac{1}{2}$. In fact, since
$\theta_{-}(s)>0$ for $s>0$ we must have that $w(s)<\tfrac{1}{2}$ for $s>0$;
similarly, replacing the $F_{-}$ with $F_{+}$ as before shows that also
$w(s)>-\tfrac{3}{2}$. It may further be shown that the corresponding $v$ has
continuous first derivatives up to the boundary. To see this rewrite equation
(3.7) once more as

(3.18) $\left(u+\mathcal{F}_{-}\right)u^{\prime}=2v\theta_{-},$

where $u=1-v^{2}-\mathcal{F}_{-}$ and $\mathcal{F}_{-}=\int_{0}^{s}2vF_{-}$.
Integrating (3.18) and then dividing by $s^{2}$ yields a quadratic equation
for $u/s$, with coefficients that have finite limits as $s\rightarrow 0$. This
implies existence of the first derivative for $v$ at $s=0$, which may be found
as the nonpositive solution of

(3.19) $2v^{\prime}(0)^{2}+2F_{-}(0)v^{\prime}(0)-\theta^{\prime}_{-}(0)=0.$

It follows that $v\in C^{1}([0,\infty))\cap C^{\infty}((0,\infty))$ satisfies
(3.7) with $v(0)=1$. An analogous argument shows that one may solve the
equation with $v(0)=-1$ when $\partial M^{2(n+1)}$ is a future apparent
horizon.

Lastly, consider the case when $H(0)=0$. The primary difference in this
situation concerns the functions $F_{\mp}$, namely they must blow-up at the
boundary due to the term $v\tfrac{\rho^{\prime\prime}}{\rho^{\prime}}$, unless
$v(0)=0$. Thus, in this case we will only consider the initial condition
$v(0)=0$. Note that with this condition, the principal part of the equation is
nondegenerate near $s=0$, however there is a singular behavior in the
coefficients:

(3.20)
$v^{\prime}+\frac{\rho^{\prime\prime}}{\rho^{\prime}}v=k_{a}+\frac{(2n+1)\rho^{\prime}}{(1+v)\rho}-\frac{\theta_{-}}{1-v^{2}}.$

Approximate solutions $v_{\varepsilon}\in C^{\infty}([\varepsilon,1))$, with
$v_{\varepsilon}(\varepsilon)=0$ and $|v_{\varepsilon}|<1$, may be obtained
with uniform estimates on compact subsets of $(0,1)$ as
$\varepsilon\rightarrow 0$. Thus, a diagonal argument ensures subconvergence
to a solution $v\in C^{\infty}((0,1))$. Using the inverse of the linear
operator on the left-hand side of (3.20), we find that the limit function may
be represented by $v(s)=\rho^{\prime}(s)^{-1}\int_{0}^{s}\rho^{\prime}u$ where
$u$ is uniformly bounded. It follows that the solution is continuous up to the
boundary with $v(0)=0$, and this may then be extended globally using the
methods above, so that $v\in C^{0}([0,\infty))\cap C^{\infty}((0,\infty))$.

Finally, it will be shown that $v$ has the correct asymptotics at infinity.
Recall that we have already assumed the asymptotically flat hypothesis. By the
fall-off conditions (2.3) and the estimate (3.16), we may write equation (3.7)
as

(3.21)
$v^{\prime}+\frac{(2n+1)s^{-1}}{1-v^{2}}v=O(s^{-2\tau-1}+s^{-\tau-1}|v|),\quad\quad
s\geq s_{0}>0.$

The solution on this exterior region can then be represented by

(3.22)
$v(s)\\!=\\!\exp\left(\\!\\!-\\!\\!\int_{s_{0}}^{s}\\!\\!\frac{(2n+1)r^{-1}}{1-v^{2}}dr\\!\right)\\!\\!\left[\int_{s_{0}}^{s}\\!\\!\\!O(r^{-2\tau-1}\\!\\!+\\!r^{-\tau-1}|v|)\exp\left(\int_{s_{0}}^{r}\\!\frac{(2n+1)t^{-1}}{1-v^{2}}dt\right)dr\\!+\\!v(s_{0})\right].$

From this it follows that $v$ decays to zero polynomially. To estimate the
rate, choose any positive $\tau_{0}<\min\\{\tau,n+\tfrac{1}{2}\\}$ and denote
the exponential function inside the integral as $\mathcal{E}(r)$. Note that
$r^{-2\tau_{0}}\mathcal{E}(r)$ is increasing for $r\geq s_{0}$ if $s_{0}$ is
large enough. Then extracting this function from the integral at its maximum
value shows that the term involving $O(r^{-2\tau-1})$ decays on the order of
$s^{-2\tau_{0}}$; clearly also the last term involving $v(s_{0})$ decays at
this rate as well. For the term involving $|v|$, extracting
$r^{-\tau_{0}}\mathcal{E}(r)$ in the same way yields decay for $v$ on the
order of $s^{-\tau_{0}}$. Inserting this estimate back into the same term and
extracting $r^{-2\tau_{0}}\mathcal{E}(r)$ produces the desired decay, namely,
with the help of (3.21) for the derivative fall-off we find that

(3.23) $|v(s)|+s|v^{\prime}(s)|\leq Cs^{-2\tau_{0}}$

for some constant $C$.

Assume now that the initial data are asymptotically hyperbolic. Only a slight
modification of the above arguments is needed to obtain global existence of a
solution satisfying the required initial conditions. More precisely, in this
case $F_{-}=-(2n+1)+O(e^{-2s})$ and $\theta_{-}=2n+1+O(e^{-2s})$ in the
asymptotic end, and therefore the function $\bar{F}_{-}(s)$ used in
(3.13)-(3.15) should be taken instead to be a large constant. Consider now the
asymptotics of the solution, and with the help of (2.4) rewrite equation (3.7)
as

(3.24)
$v^{\prime}+\left(\frac{(2n+1)\rho^{\prime}}{(1-v^{2})\rho}+\frac{\rho^{\prime\prime}}{\rho^{\prime}}\right)v=O(e^{-qs}+e^{-qs}|v|),\quad\quad
s\geq s_{0}>0.$

The solution on this exterior region can then be represented by

(3.25)
$v(s)=e^{-\int_{s_{0}}^{s}\mathbf{A}(r)dr}\left(\int_{s_{0}}^{s}O(e^{-qr}+e^{-qr}|v|)e^{\int_{s_{0}}^{r}\mathbf{A}(t)dt}dr+v(s_{0})\right),\quad\text{
}\mathbf{A}(s)=\frac{(2n+1)\rho^{\prime}}{(1-v^{2})\rho}+\frac{\rho^{\prime\prime}}{\rho^{\prime}}.$

From this it follows that $v$ decays to zero exponentially. Since
$\mathbf{A}(s)=2n+2+o(1)$, an argument analogous to that in the asymptotically
flat case may be used to obtain

(3.26) $|v(s)|+|v^{\prime}(s)|\leq Ce^{-q_{0}s},$

for any $q_{0}<\min\\{q,2n+2\\}$. ∎

##  4\. Proof of Theorem 1.1: Asymptotically Flat Penrose Inequality

Here we will establish the spacetime Penrose inequality for asymptotically
flat $\mathrm{SU}(n+1)$-invariant initial data sets. This will be accomplished
with an averaged monotonicity for the Hawking mass along inverse mean
curvature flow within the Jang deformed manifold. A crucial observation in
this setting, which replaces the use of Gauss-Bonnet, is that the scalar
curvature of the leaves cannot be ‘too large’. Let $\Sigma\subset(M^{d},g)$ be
a smooth hypersurface in a $d$-dimensional Riemannian manifold, then the
Hawking mass of $\Sigma$ is given by

(4.1)
$\mathbf{m}_{H}(\Sigma):=\frac{1}{2}\left(\frac{|\Sigma|}{\omega_{d-1}}\right)^{\frac{d-2}{d-1}}\left[1-\frac{1}{(d-1)^{2}\omega_{d-1}^{\frac{2}{d-1}}|\Sigma|^{\frac{d-3}{d-1}}}\int_{\Sigma}H^{2}dV\right],$

where $H$ is the mean curvature of $\Sigma$ and $|\Sigma|$ denotes its area.
If $\\{\Sigma_{t}\\}_{t=0}^{\infty}$ denotes a smooth inverse mean curvature
flow, then a direct computation (see e.g. [30, Theorem 4.27] for the case
$d=3$) shows that

(4.2)
$\displaystyle\begin{split}\frac{d}{dt}\mathbf{m}_{H}(\Sigma_{t})=&\frac{1}{2}\left(\frac{|\Sigma_{t}|}{\omega_{d-1}}\right)^{\frac{d-2}{d-1}}\left(\frac{d-2}{d-1}-\frac{1}{(d-1)^{2}\omega_{d-1}^{\frac{2}{d-1}}|\Sigma_{t}|^{\frac{d-3}{d-1}}}\int_{\Sigma_{t}}R_{\Sigma}dV\right)\\\
+&\frac{1}{2}\left(\frac{|\Sigma_{t}|}{\omega_{d-1}}\right)^{\frac{d-2}{d-1}}\\!\frac{1}{(d-1)^{2}\omega_{d-1}^{\frac{2}{d-1}}|\Sigma_{t}|^{\frac{d-3}{d-1}}}\int_{\Sigma_{t}}\left(\frac{|\nabla_{\Sigma}H|^{2}}{H^{2}}+|II|^{2}-\frac{H^{2}}{d-1}+R\right)dV,\end{split}$

where $R_{\Sigma}$ is the scalar curvature and $II$ is the second fundamental
form of $\Sigma_{t}$.

######  Proposition 4.1.

Let $(M^{2(n+1)},g)$ be an asymptotically flat or asymptotically hyperbolic
$\mathrm{SU}(n+1)$-invariant Riemannian manifold with outermost minimal
surface boundary, for $n\geq 1$. If $\Sigma$ is a level set of the distance
function to the boundary then

(4.3) $\int_{\Sigma}R_{\Sigma}dV\leq c_{n}|\Sigma|^{\frac{2n-1}{2n+1}},$

where $c_{n}=2n(2n+1)\omega_{2n+1}^{\frac{2}{2n+1}}$. Moreover, equality is
achieved if and only if $\Sigma$ is a round sphere.

###### Proof.

Consider first the case $n>1$. According to Proposition 2.1, (2.10), and
(2.11) we have

(4.4)
$2n+1-\frac{1}{2n\omega_{2n+1}^{\frac{2}{2n+1}}|\Sigma|^{\frac{2n-1}{2n+1}}}\int_{\Sigma}R_{\Sigma}dV=2n+1+e^{-4(n+1)B(s)}-2(n+1)e^{-2B(s)}.$

Setting $\varrho=e^{-2B}$ yields the polynomial

(4.5) $I_{n}(\varrho)=2n+1+\varrho^{2(n+1)}-2(n+1)\varrho.$

This function satisfies $I_{n}(0)=2n+1$, it monotonically decreases on $(0,1)$
to a minimum $I_{n}(1)=0$, and then increases monotonically for $\varrho\geq
1$. It follows that (4.4) is always nonnegative, and vanishes only when $B=0$
which coincides with a round $\Sigma$.

Consider now the case $n=1$. According to Section 2.2 we have

(4.6)
$3-\frac{1}{2\omega_{3}^{2/3}|\Sigma|^{1/3}}\int_{\Sigma}R_{\Sigma}dV=3-\frac{1}{(c_{1}c_{2}c_{3})^{4/3}}\left[2c_{1}^{2}(c_{2}^{2}+c_{3}^{2})-c_{1}^{4}-(c_{2}^{2}-c_{3}^{2})^{2}\right].$

Since this quantity is invariant under rescalings of the metric, we may set
$c_{3}=1$ and rearrange terms to find that it becomes

(4.7)
$I_{1}(c_{1},c_{2})=3+\frac{(1-c_{1}^{2}-c_{2}^{2})^{2}}{(c_{1}c_{2})^{4/3}}-4(c_{1}c_{2})^{2/3}.$

In the first quadrant $c_{1},c_{2}>0$ this function has a local minimum
$I_{1}(1,1)=0$, which corresponds to the round metric, and there are no other
critical points on this domain. Limits to the $c_{1}$ and $c_{2}$-axes are
infinitely positive, except at the two points $(c_{1},c_{2})=(1,0)$ or $(0,1)$
where the limit is 3. Moreover, limits to infinity are also infinitely
positive, except in one direction when $c_{1}=c_{2}$ where the limit is 3. It
follows again that (4.4) is nonnegative, and vanishes only for a round
$\Sigma$. ∎

This proposition shows that the first line of (4.2) is nonnegative along a
$\mathrm{SU}(n+1)$-invariant inverse mean curvature flow. When combined with
weak nonnegativity of the scalar curvature for the Jang deformation, it will
lead to an averaged monotonicity for Hawking mass in the Jang setting. The
next result gives the expected upper bound for the limiting Hawking mass in
terms of the ADM energy $E$. In dimension three this has been established [30,
Proposition 4.52], [27, Lemma 7.4] using the Gauss-Bonnet Theorem. Here we
avoid the need for Gauss-Bonnet with help from the proof of Proposition 4.1.

######  Proposition 4.2.

Let $(M^{2(n+1)},g)$ be an asymptotically flat $\mathrm{SU}(n+1)$-invariant
Riemannian manifold with outermost minimal surface boundary, for $n\geq 1$. If
$\Sigma_{s}$ denotes the surface of distance $s$ from the boundary, then

(4.8) $\lim_{s\rightarrow\infty}\mathbf{m}_{H}(\Sigma_{s})\leq E.$

###### Proof.

We will closely follow the arguments of [30, Proposition 4.52]. Taking two
traces of the Gauss equations along $\Sigma_{s}$ yields

(4.9) $H^{2}=2\text{Ric}(\nu,\nu)-R+R_{\Sigma}+|II|^{2},$

where $\nu$ is the unit normal pointing towards infinity. Let $d=2(n+1)$ and
note that by Cauchy-Schwarz $(d-1)|II|^{2}\geq H^{2}$, therefore

(4.10) $-(d-2)H^{2}\leq-(d-1)R_{\Sigma}-2(d-1)\text{Ric}(\nu,\nu)+(d-1)R.$

We then have

(4.11)
$\displaystyle\begin{split}\mathbf{m}_{H}(\Sigma_{s})\leq&\frac{(d-1)}{2(d-2)}\left(\frac{|\Sigma_{s}|}{\omega_{d-1}}\right)^{\frac{d-2}{d-1}}\left[\frac{d-2}{d-1}-\frac{1}{(d-1)^{2}\omega_{d-1}^{\frac{2}{d-1}}|\Sigma_{s}|^{\frac{d-3}{d-1}}}\int_{\Sigma_{s}}R_{\Sigma}dV\right]\\\
&-\frac{1}{(d-2)(d-1)\omega_{d-1}}\left(\frac{|\Sigma_{s}|}{\omega_{d-1}}\right)^{\frac{1}{d-1}}\int_{\Sigma_{s}}G(\nu,\nu)dV,\end{split}$

where $G=\mathrm{Ric}-\tfrac{1}{2}Rg$ is the Einstein tensor. According to the
discussion in Section 2 it holds that

(4.12)
$\left(\frac{|\Sigma_{s}|}{\omega_{d-1}}\right)^{\frac{1}{d-1}}=s+O(s^{1-\tau}),$

and hence [30, Theorem 3.14] implies the last term converges to the ADM energy
to produce

(4.13)
$\lim_{s\to\infty}\mathbf{m}_{H}(\Sigma_{s})\leq\lim_{s\to\infty}\frac{1}{2(d-1)}\left(\frac{|\Sigma_{s}|}{\omega_{d-1}}\right)^{\frac{d-2}{d-1}}I_{n}(s)+E,$

where $I_{n}(s)$ is the function from the proof of Proposition 4.1. Since
$B=O(s^{-\tau})$ for $n>1$ and $c_{i}=1+O(s^{-\tau})$ for $n=1$, a calculation
shows that $I_{n}(s)=O(s^{-2\tau})$. Hence

(4.14) $\lim_{s\to\infty}\mathbf{m}_{H}(\Sigma_{s})\leq
E+\lim_{s\to\infty}O(s^{d-2-2\tau})=E,$

as $\tau>\frac{d-2}{2}$. ∎

We are now in a position to complete the proof of Theorem 1.1. Let
$(M^{2(n+1)},\bar{g})$ be the Jang manifold associated with the given initial
data set, which is constructed from the solution given in Theorem 3.2 with
$v(0)=\pm 1$ if $\partial M^{2(n+1)}$ is a past (future) apparent horizon
having $H(0)\neq 0$, or with $v(0)=0$ if $H(0)=0$; in this last case the
boundary is both a past and future apparent horizon. Note that this manifold
is asymptotically flat, $\mathrm{SU}(n+1)$-invariant, and possesses an
outermost minimal surface boundary. This last assertion follows from the fact
that the $\bar{g}$-mean curvature of surfaces having constant distance to the
boundary is $\bar{H}=\sqrt{1-v^{2}}H$, so that $\bar{H}(0)=0$ and
$\bar{H}(\bar{s})>0$ for $\bar{s}>0$. Moreover, these surfaces provide a
smooth inverse mean curvature $\\{\Sigma_{\bar{t}}\\}_{\bar{t}=0}^{\infty}$ in
the Jang manifold. Next, recall that the scalar curvature of the Jang metric
[10, 9] takes the form

(4.15)
$\bar{R}=16\pi(\mu-J(w))+|h-k|_{\bar{g}}^{2}+2|X|_{\bar{g}}^{2}-2\phi^{-1}\mathrm{div}_{\bar{g}}(\phi
X),$

where

(4.16) $w_{i}=\frac{\phi f_{i}}{\sqrt{1+\phi^{2}|\nabla
f|_{g}^{2}}},\quad\quad
h_{ij}=\frac{\phi\nabla_{ij}f+\phi_{i}f_{j}+\phi_{j}f_{i}}{\sqrt{1+\phi^{2}|\nabla
f|_{g}^{2}}},\quad\quad X_{i}=\frac{\phi f^{j}}{\sqrt{1+\phi^{2}|\nabla
f|_{g}^{2}}}(h_{ij}-k_{ij}).$

Here $h$ is the second fundamental form of the Jang graph in a dual Lorentzian
setting, and $|w|_{g}\leq 1$. Although the computation appearing in [10, 9]
was set in dimension three, it extends without modification to all higher
dimensions. We may now integrate (4.2) from $\bar{t}=0$ to $\infty$, apply the
coarea formula as well as Propositions 4.1 and 4.2, to find

(4.17)
$\displaystyle\begin{split}\bar{E}-\frac{1}{2}\left(\frac{\mathcal{A}}{\omega_{2n+1}}\right)^{\frac{2n}{2n+1}}\geq&\frac{1}{2(2n+1)^{2}\omega_{2n+1}}\int_{M^{2(n+1)}}\left(\frac{|\Sigma_{\bar{t}}|}{\omega_{2n+1}}\right)^{\frac{1}{2n+1}}\bar{H}\bar{R}dV_{\bar{g}}\\\
\geq&\frac{1}{(2n+1)\omega_{2n+1}}\left(\int_{\partial M^{2(n+1)}}\phi
X(\bar{\nu})dV-\int_{\Sigma_{\infty}}\phi X(\bar{\nu})dV\right)\end{split}$

where $\bar{E}$ is the ADM energy with respect to the Jang metric and
$\mathcal{A}=|\partial M^{2n+1}|$. In the second inequality the dominant
energy condition used, as well as the expression (3.4) for $\phi$ together
with the divergence theorem for the last term in (4.15). Here
$\bar{\nu}=\partial_{\bar{s}}=\sqrt{1-v^{2}}\partial_{s}$, and
$\Sigma_{\infty}$ indicates a limit to the asymptotic end along $s$-level
sets.

Consider now the boundary terms of (4.17). According to [10, pg. 582], the
Jang equation implies that

(4.18) $(h-k)(\nu,\nu)=(1+\phi^{2}|\nabla
f|_{g}^{2})\left(-\frac{\phi\nu(f)}{\sqrt{1+\phi^{2}|\nabla
f|_{g}^{2}}}H+\mathrm{Tr}_{\Sigma}k\right)=(1-v^{2})^{-1}(-vH+\mathrm{Tr}_{\Sigma}k)$

for any level set $\Sigma$ of $f$ with unit normal $\nu$. Therefore, on each
$\Sigma_{s}$ it holds that

(4.19) $\phi X(\bar{\nu})=\phi\sqrt{1-v^{2}}\left(\frac{\phi
f^{\prime}}{\sqrt{1+\phi^{2}f^{\prime
2}}}\right)(h_{ss}-k_{ss})=v\rho^{\prime}\left((\pm
1-v)H\mp\theta_{\mp}\right).$

Since $|v(s)|=O(s^{-2\tau_{0}})$ for $2\tau_{0}>d-2=2n$ and
$H,\theta_{\pm}=O(s^{-1})$, the boundary integral at infinity converges to
zero. Moreover, if the boundary is a past (future) apparent horizon with
$H(0)\neq 0$ then $v(0)=\pm 1$, and if the boundary is an apparent horizon
with $H(0)=0$ then $v(0)=0$. It follows that the inner boundary integral
vanishes in all cases. The desired Penrose inequality (1.7) involving ADM mass
now follows from the agreement between the ADM energies $\bar{E}=E$ in light
of the decay of $v$, and the vanishing ADM linear momentum as shown in
Appendix A.

It remains to establish the rigidity statement. If equality holds in (1.7),
then equality must hold in Proposition 4.1 for each surface $\Sigma_{s}$. This
shows that each such surface is round, implying that the original initial data
set $(M^{2(n+1)},g,k)$ is spherically symmetric. In this case it is known that
the Jang graph yields the desired embedding into the Schwarzschild spacetime
[12, Theorem 3.4].

##  5\. Proof of Theorem 1.2: Asymptotically Hyperbolic Penrose Inequality

In this section we will show how the arguments presented for the
asymptotically flat case can be modified to establish the spacetime Penrose
inequality in the asymptotically hyperbolic setting. A 3-dimensional
‘hyperbolic Hawking mass’ was defined by Bray-Chuściel in [8, Section 4.1],
and its generalization to $d$-dimensions may be expressed as

(5.1)
$\mathbf{m}^{\mathrm{hyp}}_{H}(\Sigma):=\frac{1}{2}\left(\frac{|\Sigma|}{\omega_{d-1}}\right)^{\frac{d-2}{d-1}}\left[1+\left(\frac{|\Sigma|}{\omega_{d-1}}\right)^{\frac{2}{d-1}}-\frac{1}{(d-1)^{2}\omega_{d-1}^{\frac{2}{d-1}}|\Sigma|^{\frac{d-3}{d-1}}}\int_{\Sigma}H^{2}dV\right].$

Under a smooth inverse mean curvature flow its first variation is given by

(5.2) $\displaystyle\frac{d}{dt}\mathbf{m}^{\mathrm{hyp}}_{H}$
$\displaystyle(\Sigma_{t})=\frac{1}{2}\left(\frac{|\Sigma_{t}|}{\omega_{d-1}}\right)^{\frac{d-2}{d-1}}\left(\frac{d-2}{d-1}-\frac{1}{(d-1)^{2}\omega_{d-1}^{\frac{2}{d-1}}|\Sigma_{t}|^{\frac{d-3}{d-1}}}\int_{\Sigma_{t}}R_{\Sigma}dV\right)$
$\displaystyle+\frac{1}{2}\left(\frac{|\Sigma_{t}|}{\omega_{d-1}}\right)^{\frac{d-2}{d-1}}\\!\\!\\!\frac{1}{(d-1)^{2}\omega_{d-1}^{\frac{2}{d-1}}|\Sigma_{t}|^{\frac{d-3}{d-1}}}\int_{\Sigma_{t}}\\!\left(\frac{|\nabla_{\Sigma}H|^{2}}{H^{2}}\\!+\\!|II|^{2}\\!-\\!\frac{H^{2}}{d-1}+R+d(d-1)\\!\right)\\!dV.$

Note that this differs from the asymptotically flat formula only by the
addition of the last term involving $d(d-1)$, which is relevant for the
hyperbolic dominant energy condition. In the $\mathrm{SU}(n+1)$-invariant
case, Proposition 4.1 implies that the first line is again nonnegative,
moreover the following analogue of Proposition 4.2 yields the appropriate
asymptotics for this quasi-local mass.

######  Proposition 5.1.

Let $(M^{2(n+1)},g)$ be an asymptotically hyperbolic
$\mathrm{SU}(n+1)$-invariant Riemannian manifold with outermost minimal
surface boundary, for $n\geq 1$. If $\Sigma_{s}$ denotes the surface of
distance $s$ from the boundary, then

(5.3)
$\lim_{s\rightarrow\infty}\mathbf{m}^{\mathrm{hyp}}_{H}(\Sigma_{s})=E_{\mathrm{hyp}}.$

###### Proof.

We will assume that $n>1$, as the case $n=1$ may be treated similarly. The
hyperbolic defect tensor from the definition of hyperbolic mass (1.10), when
expressed in the coordinates provided by Proposition 2.1, is given by

(5.4)
$\mathfrak{g}=g-b=\left(\rho^{2}e^{-4nB}-\sinh^{2}s\right)(d\psi+A)^{2}+\left(\rho^{2}e^{2B}-\sinh^{2}s\right)g_{FS}.$

We proceed to compute the relevant terms of (1.10). Note that
$\mathfrak{g}(\nu_{b},\cdot)=0$, and therefore

(5.5)
$(\mathrm{div}_{b}\mathfrak{g})(\nu_{b})=\nu_{b}^{j}\boldsymbol{\nabla}^{i}\mathfrak{g}_{ij}=-(\boldsymbol{\nabla}^{i}\nu_{b}^{j})\mathfrak{g}_{ij}=-(\coth
s)\mathrm{Tr}_{b}\mathfrak{g},$

where $\boldsymbol{\nabla}$ denotes covariant differentiation with respect to
$b$. Moreover

(5.6) $\mathrm{Tr}_{b}\mathfrak{g}=-(2n+1)+\left(\frac{\rho}{\sinh
s}\right)^{2}\left(e^{-4nB}+2ne^{2B}\right),\quad\quad\quad\mathfrak{g}(\nabla_{b}U,\nu_{b})=0,$

in which $U=\cosh s$ is the lapse function. A further calculation and
rearrangement shows that the energy becomes

(5.7)
$\displaystyle\begin{split}E_{\mathrm{hyp}}=&\frac{1}{2(d-1)\omega_{d-1}}\lim_{s\rightarrow\infty}\int_{\Sigma_{s}}\left[U(\text{div}_{b}\mathfrak{g})-U(d\mathrm{Tr}_{b}\mathfrak{g})+(\mathrm{Tr}_{b}\mathfrak{g})dU-\mathfrak{g}(\nabla_{b}U)\right](\nu_{b})dV\\\
=&\lim_{s\rightarrow\infty}\frac{(\sinh
s)^{2n}}{2}\left[1-\gamma\left(\frac{\rho^{2}}{\sinh^{2}s}-2\frac{\rho^{2}\cosh^{2}s}{\sinh^{2}s}+2\frac{\rho\rho^{\prime}\cosh
s}{\sinh s}\right)\right]\\\ &+\lim_{s\rightarrow\infty}\frac{2n\rho^{2}\cosh
s(\sinh s)^{2n-1}}{(2n+1)}B^{\prime}(e^{-4nB}-e^{2B}),\end{split}$

with

(5.8) $\gamma=\frac{e^{-4nB}+2ne^{2B}}{2n+1}=1+O(e^{-2qs}).$

Observe that the last term of (5.7) decays on the order of
$O(e^{(2n+2-2q)s})$, and hence

(5.9) $E_{\mathrm{hyp}}\\!=\\!\lim_{s\rightarrow\infty}\\!\left[\frac{(\sinh
s)^{2n}}{2}\\!\left(\\!1+\rho^{2}-\rho^{\prime
2}+\left(\\!\rho^{\prime}\\!-\frac{\rho\cosh s}{\sinh
s}\right)^{2}\right)\\!+O\\!\left(e^{(2n+2-2q)s}\right)\right]\\!=\\!\lim_{s\rightarrow\infty}m_{H}^{\mathrm{hyp}}(\Sigma_{s}),$

since

(5.10)
$\mathbf{m}_{H}^{\mathrm{hyp}}(\Sigma_{s})=\frac{\rho^{2n}}{2}(1+\rho^{2}-\rho^{\prime
2})$

and $q>n+1$. ∎

We are now in a position to complete the proof of Theorem 1.2. As in the
asymptotically flat case, let $(M^{2(n+1)},\bar{g})$ be the Jang manifold
associated with the given initial data set, which is constructed from the
solution given in Theorem 3.2 with $v(0)=\pm 1$ if $\partial M^{2(n+1)}$ is a
past (future) apparent horizon having $H(0)\neq 0$, or with $v(0)=0$ if
$H(0)=0$. The asymptotics of $v$ imply that $\bar{g}=g+O_{2}(e^{-2q_{0}s})$
where $q_{0}>n+1$, and hence the Jang manifold is asymptotically hyperbolic.
It is also $\mathrm{SU}(n+1)$-invariant, has an outermost minimal surface
boundary, and the surfaces $\\{\Sigma_{\bar{t}}\\}_{\bar{t}=0}^{\infty}$ of
constant $\bar{g}$-distance from the boundary give a smooth inverse mean
curvature flow. Utilizing (5.2), the hyperbolic dominant energy condition, and
the weak nonnegativity of the Jang scalar curvature (4.15), together with the
arguments leading to (4.17) produces

(5.11)
$\bar{E}_{\mathrm{hyp}}-\frac{1}{2}\\!\left(\frac{\mathcal{A}}{\omega_{2n+1}}\right)^{\frac{2n}{2n+1}}\\!\\!\\!\\!-\frac{1}{2}\\!\left(\frac{\mathcal{A}}{\omega_{2n+1}}\right)^{\frac{2n+2}{2n+1}}\\!\\!\geq\\!\frac{1}{(2n+1)\omega_{2n+1}^{\frac{2}{2n+1}}}\left(\int_{\partial
M^{2(n+1)}}\\!\\!\\!\\!\\!\\!\phi
X(\bar{\nu})dV-\\!\int_{\Sigma_{\infty}}\\!\\!\phi X(\bar{\nu})dV\\!\right),$

where $\bar{E}_{\mathrm{hyp}}$ is the total energy with respect to the Jang
metric and Propition 5.1 has been employed. The boundary integral vanishes for
the same reasons as presented in the proof of Theorem 1.1. Furthermore the
decay recorded in Section 2 and (3.10) imply

(5.12) $\phi
X(\bar{\nu})=v\rho^{\prime}(-vH+\mathrm{Tr}_{\Sigma}k)=O\left(e^{(1-2q_{0})s}\right).$

It follows that the last term in (5.11) is also zero, since the integrals of
$\Sigma_{s}$ fall-off on the order of $O(e^{(2n+2-2q_{0})s})$. Therefore, if
the original energy $E_{\mathrm{hyp}}$ agrees with that of the Jang manifold
$\bar{E}_{\mathrm{hyp}}$, then (5.9) confirms the desired Penrose inequality
(1.11). To see that this is indeed valid, first observe that

(5.13)
$\left(\frac{d\rho}{d\bar{s}}\right)^{2}=(1-v^{2})\left(\frac{d\rho}{ds}\right)^{2}=\left(\frac{d\rho}{ds}\right)^{2}+O\left(e^{(2-2q_{0})s}\right),$

and then apply (5.10) as well as Proposition 5.1 to find

(5.14)
$\bar{E}_{\mathrm{hyp}}=\lim_{\bar{s}\rightarrow\infty}\bar{\mathbf{m}}_{H}^{\mathrm{hyp}}(\Sigma_{\bar{s}})=\lim_{s\rightarrow\infty}\left(\mathbf{m}_{H}(\Sigma_{s})+O\left(e^{(2n+2-2q_{0})s}\right)\right)=E_{\mathrm{hyp}},$

where $\bar{\mathbf{m}}_{H}^{\mathrm{hyp}}$ denotes Hawking mass with respect
to the Jang metric.

It remains to establish the rigidity statement. If equality holds in (1.11),
then equality must hold in Proposition 4.1 for each surface
$\Sigma_{\bar{s}}$. This shows that each such surface is round, implying that
the Jang manifold is spherically symmetric. Furthermore, from (5.2) it follows
that $\bar{R}=-(2n+1)(2n+2)$ is constant, and we conclude that
$(M^{2(n+1)},\bar{g})$ is isometric to a constant time slice of the
Schwarzschild-AdS spacetime. Therefore, (3.4) implies the following expression
for the change of raidal coordinates

(5.15)
$\bar{g}=d\bar{s}^{2}+\rho^{2}(\bar{s})g_{S^{2n+1}}=\frac{d\rho^{2}}{\phi^{2}(\rho)}+\rho^{2}g_{S^{2n+1}}$

where $g_{S^{2n+1}}$ is the unit sphere metric, with

(5.16) $\phi^{2}(\rho)=1-\frac{2E_{\mathrm{hyp}}}{\rho^{2n}}+\rho^{2}.$

Note that this formula $\phi$ may also be found from (5.10), together with the
fact that the Hawking mass is constant
$\bar{\mathbf{m}}_{H}^{\mathrm{hyp}}(\Sigma_{\bar{s}})=E_{\mathrm{hyp}}$. The
original metric $g=\bar{g}-\phi^{2}df^{2}$ is then induced from the graph
(over a constant time slice) given by $f$ inside Schwarzschild-AdS. Lastly,
the second fundamental form of this isometric embedding agrees with $k$ due to
the fact that the second term (in fact each term except for $\mu$) of (4.15)
vanishes.

##  6\. Proof of Theorem 1.3: All Cohomogeneity One Cases

In this section we will establish the Riemannian Penrose inequality for all
cohomogeneity one manifolds. According to the discussion at the beginning of
Section 2, in addition to the $\mathrm{SU}(n+1)$-invariant setting considered
above there are two other cases to consider, namely the $\mathrm{Sp}(n+1)$ and
$\mathrm{Spin}(9)$-invariant cases. Let $(M^{d},g)$, $d\geq 3$ be an
asymptotically flat cohomogeneity one Riemannian manifold with outermost
minimal surface boundary and nonnegative scalar curvature. As previously
mentioned, $M^{d}\cong[0,\infty)\times S^{d-1}$ and $g=ds^{2}+g_{s}$ where $s$
is the distance function to the boundary, and $g_{s}$ is a 1-parameter family
of $\mathcal{G}$-invariant metrics on $S^{d-1}$. Interestingly, when
$\mathcal{G}$ is either $\mathrm{Sp}(n+1)$ or $\mathrm{Spin}(9)$, the analogue
of Proposition 4.1 does not hold in general, and therefore the inverse mean
curvature flow approach breaks down. To deal with this issue, we utilize the
conformal flow method of Bray [7] until the outermost minimal surface reaches
an appropriate location in the asymptotic end. From there, a modified version
of this proposition may be implemented to complete the argument with inverse
mean curvature flow. This bypasses the more involved problem of showing that
the conformal flow converges to Schwarzschild in the asymptotic end.

###  6.1. $\text{Sp}(n+1)$-invariant metrics

Let $d=4n+4$ for $n\geq 0$. The group $\mathcal{G}=\mathrm{Sp}(n+1)$ acts
transitively on the sphere $S^{4n+3}$ with isotropy subgroup
$\mathcal{H}=\mathrm{Sp}(n)$. Any $\mathcal{G}$-invariant metric
$g_{\mathbf{c}}$ on this sphere depends (up to an overall scaling) on three
positive parameters $\mathbf{c}=(c_{1},c_{2},c_{3})$, and arises as a
Riemannian submersion for the Hopf fibration $S^{3}\hookrightarrow
S^{4n+3}\to\mathbb{HP}^{n}$ in which the parameters scale the fiber directions
and the base is equipped with the canonical Einstein metric on quaternionic
projective space. In this notation, the round metric of unit curvature is then
described by $\mathbf{c}=(1,1,1)$. Moreover, the $s$-level set metrics from
the discussion above are then given by $g_{s}=\rho^{2}(s)g_{\mathbf{c}(s)}$
for some positive function $\rho(s)$. According to [6, Section 8] the scalar
curvature of $g_{\mathbf{c}}$ takes the form

(6.1)
$\displaystyle\begin{split}R_{\mathbf{c}}=&\frac{2}{c_{1}c_{2}c_{3}}\left(c_{1}^{2}+c_{2}^{2}+c_{3}^{2}-(c_{2}-c_{3})^{2}-(c_{3}-c_{1})^{2}-(c_{1}-c_{2})^{2}\right)\\\
&-4n(c_{1}+c_{2}+c_{3})+16n^{2}+32n.\end{split}$

Notice that setting $c_{i}=c^{2}$ yields

(6.2) $R_{\mathbf{c}}=\frac{6}{c^{2}}-12nc^{2}+16n^{2}+32n,$

showing that a collapse of the Hopf fibers results in blow-up of curvature,
which is in contrast to the $\mathrm{SU}(n+1)$-invariant setting. Furthermore
due to this behavior, Proposition 4.1 does not hold here without additional
qualification. The next result provides a regime for which the result remains
valid.

######  Proposition 6.1.

Let $(M^{4(n+1)},g)$ be an asymptotically flat $\mathrm{Sp}(n+1)$-invariant
Riemannian manifold with outermost minimal surface boundary, for $n\geq 0$.
There exists a distance $s_{0}>0$ such that if $\Sigma_{s}$ denotes the
surface of distance $s$ to the boundary and $s\geq s_{0}$, then

(6.3) $\int_{\Sigma_{s}}R_{\Sigma}dV\leq
C_{n}|\Sigma_{s}|^{\frac{4n+1}{4n+3}}$

where $C_{n}=(4n+2)(4n+3)\omega_{4n+3}^{\frac{2}{4n+3}}$, with equality
achieved in this regime if and only if $\Sigma_{s}$ is a round sphere.

###### Proof.

Since the volume form for $g_{\mathbf{c}}$ is given by
$dV=(c_{1}c_{2}c_{3})^{1/2}dV_{S^{4n+3}}$, we have

(6.4)
$I(\mathbf{c}):=(4n+2)(4n+3)-\frac{1}{\omega_{4n+3}^{\frac{2}{4n+3}}|\Sigma_{s}|^{\frac{4n+1}{4n+3}}}\int_{\Sigma_{s}}R_{\Sigma}dV=(4n+2)(4n+3)-(c_{1}c_{2}c_{3})^{1/(4n+3)}R_{\mathbf{c}}.$

A direct calculation shows that $\mathbf{c}=(1,1,1)$ is a local isolated
minimum for the function $I$. Thus, since $\Sigma_{s}$ uniformly approaches
the unit round sphere as $s\rightarrow\infty$, there exists $s_{0}$ such that
$s\geq s_{0}$ implies $I(\mathbf{c(s)})\geq I(1,1,1)$ with equality if and
only if $g_{s}$ is a round metric. Moreover, as in the proof of Proposition
2.1, asymptotic flatness produces

(6.5) $\rho(s)=s+O_{2}(s^{1-\tau}),\quad\quad\quad\quad
c_{i}(s)=1+O_{2}(s^{-\tau}),\quad i=1,2,3.$

Hence, since $I(1,1,1)=|\nabla I(1,1,1)|=0$ we have
$I(\mathbf{c(s)})=O(s^{-2\tau})$ as $s\rightarrow\infty$. ∎

###  6.2. $\text{Spin}(9)$-invariant metrics

The group $\mathcal{G}=\mathrm{Spin}(9)$ acts transitively on $S^{15}$ with
isotropy subgroup $\mathcal{H}=\mathrm{Spin}(7)$. Any $\mathcal{G}$-invariant
metric $g_{c}$ on this sphere depends (up to an overall scaling) on one
positive parameters $c$, and arises as a Riemannian submersion for the Hopf
fibration $S^{7}\hookrightarrow S^{15}\to S^{8}$ in which the parameter scales
the fiber and the base is equipped with the unit round metric. In this
notation, the round metric of unit curvature on $S^{15}$ is then described by
$c=1$. Moreover, the $s$-level set metrics from the discussion at the
beginning of this section are then given by $g_{s}=\rho^{2}(s)g_{c(s)}$ for
some positive function $\rho(s)$. According to [6, Section 8] the scalar
curvature of $g_{c}$ takes the form

(6.6) $R_{c}=\frac{42}{c}-56c+224.$

Notice that as in the $\mathrm{Sp}(n+1)$-invariant case, collapse of the Hopf
fibers results in blow-up of curvature, again in contrast to the
$\mathrm{SU}(n+1)$-invariant setting. Thus, we must again replace Proposition
4.1 with an asymptotic version.

######  Proposition 6.2.

Let $(M^{16},g)$ be an asymptotically flat $\mathrm{Spin}(9)$-invariant
Riemannian manifold with outermost minimal surface boundary. There exists a
distance $s_{0}>0$ such that if $\Sigma_{s}$ denotes the surface of distance
$s$ to the boundary and $s\geq s_{0}$, then

(6.7)
$\int_{\Sigma_{s}}R_{\Sigma}dV\leq\mathcal{C}|\Sigma_{s}|^{\frac{13}{15}}$

where $\mathcal{C}=210\omega_{15}^{\frac{2}{15}}$, with equality achieved in
this regime if and only if $\Sigma_{s}$ is a round sphere.

###### Proof.

Since the volume form for $g_{c}$ is given by $dV=c^{7/2}dV_{S^{15}}$, we have

(6.8)
$I(c):=210-\frac{1}{\omega_{15}^{\frac{2}{15}}|\Sigma_{s}|^{\frac{13}{15}}}\int_{\Sigma_{s}}R_{\Sigma}dV=210-c^{7/15}R_{c}.$

A direct calculation shows that $c=1$ is a local isolated minimum for the
function $I$. Thus, since $\Sigma_{s}$ uniformly approaches the unit round
sphere as $s\rightarrow\infty$, there exists $s_{0}$ such that $s\geq s_{0}$
implies $I(c(s))\geq I(1,1,1)$ with equality if and only if $g_{s}$ is a round
metric. Moreover, as in the proof of Proposition 2.1, asymptotic flatness
produces

(6.9) $\rho(s)=s+O_{2}(s^{1-\tau}),\quad\quad\quad\quad
c(s)=1+O_{2}(s^{-\tau}),\quad i=1,2,3.$

Hence, since $I(1)=|I^{\prime}(1)|=0$ we have $I(c(s))=O(s^{-2\tau})$ as
$s\rightarrow\infty$. ∎

###  6.3. Combining the conformal flow with inverse mean curvature flow

Let $(M^{d},g)$ be as described at the start of this section, with
$\mathcal{G}=\mathrm{Sp}(n+1)$ or $\mathrm{Spin}(9)$. Consider the conformal
flow of metrics defined by $g_{t}=u_{t}^{\frac{4}{n-2}}g$ where
$\tfrac{d}{dt}u_{t}=v_{t}$ satisfies

(6.10) $\Delta_{g}v_{t}=0\quad\text{ on }\quad M^{d}_{t},\quad\quad
v_{t}=0\quad\text{ on }\quad\partial M^{d}_{t},\quad\quad
v_{t}(x)\rightarrow-e^{-t}\quad\text{ as }\quad|x|\rightarrow\infty,$

and $v_{t}=0$ on $M^{d}\setminus M^{d}_{t}$. Here $M^{d}_{t}$ denotes the
region outside of the outermost minimal surface (denoted $\partial M^{d}_{t}$)
in $(M^{d},g_{t})$. The conformal flow was initially studied [7] in dimension
3, and was extended to higher dimensions in [11]. In particular, the flow
exists as long as the outermost minimal surfaces involved remain smooth. In
the current cohomogeneity one setting, the relevant minimal surfaces $\partial
M^{d}_{t}$ must be level sets of the distance function to $\partial M^{d}$,
and are therefore smooth. Moreover, the functions $v_{t}$ as well as $u_{t}$
depend only on $s$, showing that the conformal metrics $g_{t}$ are also
$\mathcal{G}$-invariant. According to [11, Lemma 2.3], existence of the flow
guarantees that the areas $|\partial M_{t}^{d}|$ remain constant in $t$.
Furthermore, the mass decrease law, which was partially responsible for the
dimensional restriction in [11] due to its reliance on the positive mass
theorem, is valid here since each $M^{d}_{t}$ is a spin manifold. More
precisely, the positive mass theorem is applied to a doubled manifold with one
of the two ends being compactified, and in this context we may apply the
‘corners’ version of this result obtained with harmonic spinors [38, Theorem
3.1]. It follows that the mass $m(t)$ of $(M^{d}_{t},g_{t})$ is nonincreasing
in $t$.

We now claim that the flow surfaces $\partial M^{d}_{t}$ reach the asymptotic
end of $M^{d}$ in finite time, and in fact eventually leave every compact set.
This was originally established in dimension 3 [7, Theorem 13], while in [11]
this issue is avoided altogether. Although the original proof relied on the
Gauss-Bonnet theorem which is not available here, a weaker version of this
result [7, Theorem 12] extends to higher dimensions with only minor changes.
An immediate corollary of this theorem shows that $\partial M^{d}_{t}$ cannot
be entirely enclosed by a single coordinate sphere in the asymptotic end, for
all $t$. Because the flow surfaces in the current setting are
$\mathcal{G}$-invariant, if they did not eventually leave every compact set
then they would be entirely enclosed by a coordinate sphere. Thus, we may
apply the higher dimensional analogue of Bray’s observation to obtain the
desired conclusion. Note that the proof of [7, Theorem 12] makes use of
harmonic asymptotics, however this is not necessary as the relevant harmonic
functions may be expanded in spherical harmonics to produce the same outcome.

To complete the proof, run the conformal flow until time $t_{0}$ when
$\partial M^{d}_{t_{0}}$ reaches sufficiently far out in the asymptotic end to
enclose $\Sigma_{s_{0}}$, the designated surface appearing in Propositions 6.1
and 6.2. Properties of the flow discussed above guarantee that

(6.11) $m\geq m(t_{0}),\quad\quad\quad\quad|\partial
M^{d}_{t_{0}}|=\mathcal{A},$

where $m$ is the ADM mass of $(M^{d},g)$ and $\mathcal{A}$ is the area of its
boundary. On the other hand, the aforementioned propositions give rise to
monotonicity for the Hawking mass in $(M^{d}_{t_{0}},g_{t_{0}})$, as in
Section 4. Moreover, since the functions $I$ from the proof of these
propositions vanishes to second order at the round metric, the asymptotic
limit of the Hawking mass is no greater than $m(t_{0})$; this is established
in the same manner as Proposition 4.2. Therefore

(6.12) $m(t_{0})\geq\frac{1}{2}\left(\frac{|\partial
M^{d}_{t_{0}}|}{\omega_{d-1}}\right)^{\frac{d-2}{d-1}},$

and combining (6.11) with (6.12) produces the desired Penrose inequality since
$\mathcal{A}_{h}=\mathcal{A}$. Lastly, consider the case of equality for
(1.6). This forces equality between the masses of (6.11), and thus the
rigidity statement of the positive mass theorem used for the (conformal flow)
mass decrease law, implies that $(M^{d}_{t},g_{t})$ is spherically symmetric
for $t\leq t_{0}$. In particular, $(M^{d},g)$ is spherically symmetric and is
therefore isometric to the canonical slice of a Schwarzschild spacetime [12,
Theorem 3.4].

##  Appendix A Vanishing of Linear Momenta

In this appendix we show that under the hypotheses of Theorem 1.1, the ADM
linear momenta $P_{i}$ vanish so that the ADM mass agrees with the ADM energy
$m=E$. Let $n>1$ and use the expressions for the metric and extrinsic
curvature given in Proposition 2.1, to compute the following two components of
the momentum density

(A.1)
$J(\mathbf{e}_{1})=J(\partial_{s})=-k^{\prime}_{b}-2nk^{\prime}_{c}+\frac{(2n+1)\dot{\rho}}{\rho}k_{a}-(k_{b}+2nk_{c})\frac{\rho^{\prime}}{\rho}+2nB^{\prime}(k_{b}-k_{c}),$
(A.2)
$J(\mathbf{e}_{2})=k^{\prime}_{s}+k_{s}\left((2n+2)\frac{\dot{\rho}}{\rho}-2nB^{\prime}\right),$

where $\mathbf{e}_{1}$ and $\mathbf{e}_{2}$ are a part of the orthonormal
frame from (2.9). Recalling the formula for $\mathrm{Tr}_{g}k$ in (2.13)
motivates a rewriting of (A.1) by

(A.3)
$k^{\prime}_{a}+(2n+2)\frac{\rho^{\prime}}{\rho}k_{a}=J(\mathbf{e}_{1})+\left(\mathrm{Tr}_{g}k\right)^{\prime}+\frac{\rho^{\prime}}{\rho}\mathrm{Tr}_{g}k-2nB^{\prime}(k_{b}-k_{c}).$

Since the asymptotically flat fall-off conditions (1.3) imply that the right-
hand side is $O(s^{-2\tau-2})$, it follows that

(A.4)
$k_{a}=\rho^{-2\tau-2}\left[\int_{s_{0}}^{s}\rho^{2\tau+2}O(t^{2\tau+2})dt+C\right]=O(s^{-2\tau-2})$

for some constant $C$. Similarly, we also find that $k_{s}=O(s^{-2\tau-2})$
from (A.2). Moreover, because the only possible nonzero components of the ADM
linear momentum involve only $k(e_{1},e_{1})=k_{a}$ or $k(e_{1},e_{2})=k_{s}$,
and $\tau>n$, we conclude that the ADM linear momentum vanishes. The case
$n=1$ is treated analogously using Section 2.2. Note that the additional decay
of $\mathrm{Tr}_{g}k$ from (1.3) is only used here, to obtain the vanishing
linear momentum.

##  Appendix B An Example

In this last appendix we will exhibit cohomogeneity one asymptotically flat
and asymptotically hyperbolic initial data, which deviate in a significant
manner from spherical symmetry by allowing for nonzero angular momentum.
Consider the 2-parameter family of asymptotically flat
$\mathrm{SU}(n+1)$-invariant initial data $([r_{+},\infty)\times
S^{2n+1},g,k)$ in which

(B.1)
$g=U(r)^{2}dr^{2}+P(r)^{2}\left(d\psi+A\right)^{2}+r^{2}g_{FS},\quad\quad\quad
k=-r^{-1}U(r)W^{\prime}(r)P(r)^{3}dr(d\psi+A),$

where $\psi$, $A$, and $g_{FS}$ are as in (2.1) and

(B.2)
$U(r)^{2}=\left(1-\frac{2m}{r^{2n}}+\frac{2ma^{2}}{r^{2n+2}}\right)^{-1},\quad\quad
P(r)=r\left(1+\frac{2ma^{2}}{r^{2n+2}}\right)^{1/2},\quad\quad
W(r)=\frac{2ma}{r^{2n}P(r)^{2}},$

with $m$ and $a$ denoting the mass and angular momentum parameters. The value
$r_{+}$ is the largest positive root of $U(r)^{-2}$, and corresponds to an
outermost minimal surface. This initial data set arises from the 2-parameter
family of Myers-Perry stationary asymptotically flat black hole solutions of
the vacuum Einstein equations given by

(B.3)
$\mathbf{g}=-r^{2}U(r)^{-2}P(r)^{-2}dt^{2}+U(r)^{2}dr^{2}+P(r)^{2}\left(d\psi+A-W(r)dt\right)^{2}+r^{2}g_{FS}.$

Note that when $a=0$ the Schwawrzschild solution is recovered. The event
horizon is located at $r=r_{+}$ and has null generator

(B.4)
$\xi=\partial_{t}+\Omega\partial_{\psi},\qquad\quad\Omega=\frac{2ma}{r_{+}^{2(n+1)}+2ma^{2}},$

where $\Omega$ is the angular velocity. Moreover, a calculation shows that for
the black hole to be subextremal the parameters must satisfy
$r_{+}^{2}>n^{-1}(n+1)a^{2}$, with equality corresponding to an extreme Myers-
Perry solution having a degenerate horizon. It is convenient to use $r_{+}$
and $a$ express relevant quantities. In particular, the mass parameter $m$
coincides with the ADM mass and takes the form

(B.5) $m=\frac{r_{+}^{2(n+1)}}{2(r_{+}^{2}-a^{2})},$

while the area of a cross section of the event horizon becomes

(B.6)
$\mathcal{A}=\omega_{2n+1}P(r_{+})r_{+}^{2n}=\frac{\omega_{2n+1}r_{+}^{2(n+1)}}{\sqrt{r_{+}^{2}-a^{2}}},$

and the only nonzero ADM angular momentum occurs in the $\psi$-direction and
is given by $\mathcal{J}_{\psi}=ma$ where by definition

(B.7)
$\mathcal{J}_{\psi}=\frac{1}{2(n+1)\omega_{2n+1}}\int_{S_{\infty}}(k-(\text{Tr}k)g)(\partial_{r},\partial_{\psi})dV.$

We now have the relation

(B.8)
$m=\left(\frac{r_{+}^{2}}{r_{+}^{2}-a^{2}}\right)^{\frac{n+1}{2n+1}}\frac{1}{2}\left(\frac{\mathcal{A}}{\omega_{2n+1}}\right)^{\frac{2n}{2n+1}}.$

Note that this satisfies the Penrose inequality since $a^{2}r_{+}^{-2}<1$, and
equality holds only for the case of Schwarzschild, when $a=0$.

An asymptotically hyperbolic generalization of the above rotating black hole
initial data may be obtained by setting

(B.9)
$U(r)^{2}=\left(1+r^{2}-\frac{2m(1-a^{2})}{r^{2n}}+\frac{2ma^{2}}{r^{2n+2}}\right)^{-1}.$

The resulting data coincides with the canonical slice of a Myers-Perry-AdS
black hole, a solution of the stationary vacuum Einstein equations with
negative cosmological constant, having equal angular momenta. Again letting
$r_{+}$ denote the largest positive root of $U(r)^{-2}$, there will be an
event horizon at $r=r_{+}$ provided $m>0$ and $0\leq a<1$. Notice that when
$m=0$ the spacetime is the $(2n+3)$-dimensional AdS space, whereas when $a=0$
the solution reduces to Schwarzschild-AdS. A computation reveals the total
energy to be

(B.10) $E_{\mathrm{hyp}}=m\left(1+\frac{a^{2}}{2n+1}\right),$

while as before the angular momentum is $\mathcal{J}_{\psi}=ma$. Expressing
the mass parameter in terms of $r_{+}$ and $a$ yields

(B.11) $m=\frac{(1+r_{+}^{2})r_{+}^{2(n+1)}}{2(r_{+}^{2}(1-a^{2})-a^{2})},$

and therefore

(B.12)
$m=\frac{1}{2\left(1-\alpha\right)^{\frac{n+1}{2n+1}}(1-\beta)}\left(\frac{\mathcal{A}}{\omega_{2n+1}}\right)^{\frac{2n}{2n+1}}+\frac{1}{2\left(1-\alpha\right)^{\frac{n}{2n+1}}(1-\beta)}\left(\frac{\mathcal{A}}{\omega_{2n+1}}\right)^{\frac{2n+2}{2n+1}}$

where

(B.13)
$\alpha=\frac{a^{2}}{r_{+}^{2}},\quad\qquad\beta=\frac{a^{2}}{r_{+}^{2}(1-a^{2})}.$

Next observe that a calculation provides the nondegeneracy condition

(B.14) $r_{+}^{2}(n+(n+1)r_{+}^{2})-a^{2}(1+n)(1+r_{+}^{2})^{2}\geq 0.$

Hence, because restrictions on the parameters imply that $\alpha<1$ and
$\beta<1$, and $E_{\mathrm{hyp}}\geq m$ with equality only when $a=0$, it
follows that the hyperbolic Penrose inequality holds with saturation only for
Schwarzschild-AdS.

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