# On Weak Inverse Mean Curvature Flow and Minkowski-type Inequalities in
Hyperbolic Space

Brian Harvie  National Center for Theoretical Sciences, Mathematics Division
<EMAIL_ADDRESS>
###### Abstract.

We prove that a proper weak solution $\\{\Omega_{t}\\}_{0\leq t<\infty}$ to
inverse mean curvature flow in $\mathbb{H}^{n}$, $3\leq n\leq 7$, is smooth
and star-shaped by the time

$T=(n-1)\log\left(\frac{\text{sinh}\left(r_{+}\right)}{\text{sinh}\left(r_{-}\right)}\right),$

where $r_{+}$ and $r_{-}$ are the geodesic out-radius and in-radius of the
initial domain $\Omega_{0}$. The argument is inspired by the Alexandrov
reflection method for extrinsic curvature flows in $\mathbb{R}^{n}$ due to
Chow-Gulliver from [8] and uses a result of Li-Wei from [20].

As applications, we extend the Minkowski inequalities of Brendle-Hung-Wang [4]
and De Lima-Girao [10] to outer-minimizing domains
$\Omega_{0}\subset\mathbb{H}^{n}$ in these dimensions. From this, we also
extend the asymptotically hyperbolic Riemannian Penrose inequality to balanced
asymptotically hyperbolic graphs over the exteriors of outer-minimizing
domains of $\mathbb{H}^{n}$, $3\leq n\leq 7$.

##  1\. Introduction

Hyperbolic space is the Riemannian manifold of constant negative curvature
given by

$(\mathbb{H}^{n},g_{\mathbb{H}^{n}})=(\mathbb{S}^{n-1}\times(0,\infty),dr\otimes
dr+\sinh^{2}(r)d\Omega^{2}),$ (1.1)

where $d\Omega^{2}$ is the unit sphere metric. This paper considers the
inverse mean curvature flow (IMCF) of hypersurfaces in hyperbolic space. Given
a closed smooth manifold $\Sigma^{n-1}$, a one-parameter family of
$C^{\infty}$ embeddings $X:\Sigma^{n-1}\times[0,T)\rightarrow\mathbb{H}^{n}$
solves IMCF if

$\frac{\partial}{\partial t}X(x,t)=\frac{1}{H}\nu(x,t),\hskip
56.9055pt(x,t)\in\Sigma^{n-1}\times[0,T),$ (1.2)

where $\nu$ is the outward unit normal and $H>0$ the mean curvature of the
hypersurface $\Sigma_{t}=X_{t}(\Sigma^{n-1})\subset\mathbb{H}^{n}$. IMCF
enjoys a wide variety of geometric applications, especially to geometric
inequalities for hypersurfaces. However, a key obstacle to many of these
applications is the formation of singularities that terminate the flow in
finite time. One way of overcoming this obstacle is to study weak solutions of
IMCF, which were first introduced by Huisken-Ilmanen in [17] to prove the
Riemannian Penrose inequality. Given a bounded domain
$\Omega_{0}\subset\mathbb{H}^{n}$ with connected $C^{\infty}$ boundary
$\partial\Omega_{0}=\Sigma_{0}$, a proper weak solution to IMCF with initial
condition $\Omega_{0}$ is a proper locally Lipschitz function
$u:\mathbb{H}^{n}\rightarrow\mathbb{R}$ which solves the degenerate-elliptic
exterior Dirichlet problem

$\displaystyle\text{div}\left(\frac{\nabla u}{|\nabla u}\right)$
$\displaystyle=$ $\displaystyle|\nabla u|,$ (1.3) $\displaystyle
u|_{\Sigma_{0}}$ $\displaystyle=$ $\displaystyle 0,$

in a variational sense, and the “weak flow surfaces” are defined as

$\Sigma_{t}=\partial\Omega_{t},\hskip
56.9055pt\Omega_{t}=\\{u<t\\}\subset\mathbb{H}^{n}.$ (1.4)

For a smooth, non-degenerate solution $u$ to (1.3), the $\Sigma_{t}$ in (1.4)
are smooth and strictly mean-convex hypersurfaces and form a solution to
(1.2), but in general these $\Sigma_{t}$ are only $C^{1,\alpha}$ hypersurfaces
with a possible a singular set $Z$ in dimensions $n>7$. Furthermore, a
phenemenon may occur where $\Sigma_{t}$ “jumps” outward at certain times. We
will discuss this more thoroughly in Section 2.

Understanding the long-time behavior of proper weak IMCF is crucial to its
geometric applications. For weak IMCF in $\mathbb{R}^{n}$, a blow-down lemma
due to Huisken and Ilmanen (Lemma 7.1 in [17]) ensures that the weak flow
surfaces $\Sigma_{t}$ converge to a round sphere in a $C^{1}$ sense after
scaling as $t\rightarrow\infty$. By constrant, Hung and Wang show in [18] that
even smooth IMCFs in $\mathbb{H}^{n}$ may not become $C^{1}$ close to a round
sphere as $t\rightarrow\infty$, see also [24]. For weak IMCFs in
$\mathbb{H}^{n}$, the asymptotic behavior is completely unclear a priori.

In studying the asymptotic behavior of weak solutions, one particularly
important quantity to understand is the support function
$\phi_{t}:\Sigma_{t}\setminus Z\rightarrow\mathbb{R}$,

$\displaystyle\phi_{t}(x)$ $\displaystyle=$ $\displaystyle\langle\nabla
f,\nu(x)\rangle,$ (1.5)

which is taken with respect to the potential function
$f:\mathbb{H}^{n}\rightarrow\mathbb{R}^{+}$ defined by

$f(r)=\cosh(r).$ (1.6)

We call $\Sigma_{t}$ star-shaped if $\Sigma_{t}$ is $C^{1}$ and $\phi_{t}>0$
on $\Sigma_{t}$. Gerhardt [13] and Ding [11] originally proved long-time
existence for the smooth IMCF of a $C^{\infty}$ $H>0$ star-shaped initial
hypersurface $\Sigma_{0}^{n-1}\subset\mathbb{H}^{n}$, c.f. also [22], [30],
and [25]. In [20] and following work of Huisken and Ilmanen in [16], Li and
Wei proved that if a $C^{1}$ hypersurface
$\Sigma_{0}^{n-1}\subset\mathbb{H}^{n}$ is star-shaped with bounded weak mean
curvature $H\geq 0$, then $\mathbb{H}^{n}\setminus\Omega_{0}$ is foliated by a
smooth IMCF $\\{\Sigma_{t}\\}_{0<t<\infty}$. In [16], the analogous result in
Euclidean space was applied to show that weak IMCF in $\mathbb{R}^{n}$ is
eventually smooth. However, there is no immediate way to apply Li-Wei’s result
to weak IMCF in $\mathbb{H}^{n}$ due to the lack of a blow down lemma. In
particular, the weak $\Sigma_{t}\subset\mathbb{H}^{n}$ may never become star-
shaped.

The main result of this paper states that the weak IMCF of any initial domain
$\Omega_{0}\subset\mathbb{H}^{n}$ is indeed star-shaped after an explicit
“waiting time” that depends only on the in-radius and out-radius of
$\Omega_{0}$ in dimension $n<8$, with $\phi_{t}(x)$ obeying a lower bound in
terms of geodesic distance. From this, we conclude smoothing for weak IMCF in
$\mathbb{H}^{n}$. This follows an earlier result by Shi-Zhu [26], who showed
smoothing in the asymptotic regime for asymptotically hyperbolic manifolds of
dimension $n=3$.

######  Theorem 1.1 (Star-shape of Weak IMCF in Hyperbolic Space).

Let $u:\mathbb{H}^{n}\rightarrow\mathbb{R}$, $3\leq n\leq 7$ , be a proper
weak solution to IMCF with initial condition $\Omega_{0}$. Assume WLOG that
$0\in\Omega_{0}$ and define

$T=(n-1)\log\left(\frac{\sinh(r_{+})}{\sinh(r_{-})}\right),$ (1.7)

where $r_{+}=\max_{\Sigma_{0}}r$ and $r_{-}=\min_{\Sigma_{0}}r$. Then
$\Sigma_{t}=\partial\\{u<t\\}$ are a smooth solution to (1.2) for all
$t\in(T,\infty)$. Furthermore, $\Sigma_{t}$ is star-shaped for each
$t\in(T,\infty)$, and for each $x\in\Sigma_{t}$ we have the following lower
bound on the support function:

$\phi_{t}(x)\geq\operatorname{sech}\left(\frac{r_{+}}{2}\right)\sinh\left(r(x)\right).$
(1.8)

A key application of IMCF is to Minkowski-type inequalities, which are
fundamental geometric inequalities in Riemannian geometry. Recall that the
classical Minkowski inequality in Euclidean space for a convex body
$\Omega_{0}^{n}\subset\mathbb{R}^{n}$ with boundary
$\Sigma_{0}=\partial\Omega_{0}$ states

$\frac{1}{(n-1)w_{n-1}}\int_{\Sigma_{0}}Hd\sigma\geq\left(\frac{|\Sigma_{0}|}{w_{n-1}}\right)^{\frac{n-2}{n-1}}.$

In their breakthrough paper [4], Brendle, Hung, and Wang proved a weighted
Minkowski inequality for star-shaped $H>0$ hypersurfaces
$\Sigma_{0}^{n-1}\subset\mathbb{H}^{n}$. In [10], De Lima and Girao proved a
related Minkowski inequality for these hypersurfaces with lower bound
depending on the area of $\Sigma_{0}$. Their respective inequalities read

$\displaystyle\frac{1}{(n-1)w_{n-1}}\int_{\Sigma_{0}}fHd\sigma$
$\displaystyle\geq$
$\displaystyle\left(\frac{|\Sigma_{0}|}{w_{n-1}}\right)^{\frac{n-2}{n-1}}+\frac{n}{w_{n-1}}\int_{\Omega_{0}}fd\Omega,$
(1.9) $\displaystyle\frac{1}{(n-1)w_{n-1}}\int_{\Sigma_{0}}fHd\sigma$
$\displaystyle\geq$
$\displaystyle\left(\frac{|\Sigma_{0}|}{w_{n-1}}\right)^{\frac{n-2}{n-1}}+\left(\frac{|\Sigma_{0}|}{w_{n-1}}\right)^{\frac{n}{n-1}}.$
(1.10)

The proofs of these inequalities use quantities which are monotone under
smooth IMCF, and Theorem 1.1 allows for proofs involving weak IMCF. Before
stating our extension of these inequalities, we remind the reader that a
domain $\Omega_{0}\subset\mathbb{H}^{n}$ is outer-minimizing if $|\partial
F|\geq|\partial\Omega|$ for every finite-perimeter domain $F$ containing
$\Omega_{0}$. Every $H>0$ star-shaped domain $\Omega_{0}\subset\mathbb{H}^{n}$
is outer-minimizing. More generally, $\Omega_{0}$ is outer-minimizing if, for
example, $\mathbb{H}^{n}\setminus\Omega_{0}$ is foliated by hypersurfaces with
non-negative mean curvature.

######  Theorem 1.2 (Minkowski Inequalities for Outer-Minimizing Domains in
Hyperbolic Space).

Let $3\leq n\leq 7$. Then the inequalities (1.9) and (1.10) hold for outer-
minimizing domains $\Omega_{0}\subset\mathbb{H}^{n}$. In both cases, equality
holds if and only if $\Sigma_{0}=\mathbb{S}^{n-1}\times\\{r_{0}\\}$ is a slice
of (1.1).

One immediate application of the extended Minkowski inequalities is to the
conjectured Riemannian Penrose inequality for asymptotically hyperbolic
manifolds. Consider the following model for $(n+1)$-dimensional hyperbolic
space:

$(\mathbb{H}^{n+1},\overline{g})=(\mathbb{H}^{n}\times\mathbb{R},f^{2}(r)d\tau^{2}+g_{\mathbb{H}^{n}}).$
(1.11)

In [10] and following earlier work by Dahl, Gicquaud, and Sakovich in [9], de
Lima and Girao considered non-compact hypersurfaces
$M^{n}\subset\mathbb{H}^{n+1}$ satisfying the following:

  1. (A)

$M^{n}\subset(\mathbb{H}^{n+1},\overline{g})$ is asymptotically hyperbolic and
balanced in the sense of [10].

  2. (B)

$M^{n}=\text{graph}(h)$ for a function $h\in
C^{\infty}(\mathbb{H}^{n}\setminus\overline{\Omega_{0}})$ defined over the
exterior of a smooth domain $\Omega_{0}\subset\mathbb{H}^{n}$ containing $0$.

  3. (C)

$\tau\equiv\tau_{0}$ on $\partial M^{n}\subset\mathbb{H}^{n+1}$, and $M^{n}$
intersects $\mathbb{H}^{n}\times\\{\tau_{0}\\}$ orthogonally along $\partial
M$.

Note that the last item implies that $\partial M$ considered as a hypersurface
in $(M^{n},\overline{g}|_{M^{n}})$ is totally geodesic. Balanced
asymptotically hyperbolic manifolds $(M^{n},g)$ carry a natural mass-like
invariant $m$, which is studied in [9]. When a balanced asympotically
hyperbolic $(M^{n},g)$ satisfies (B) and (C), it is also shown in [9] that
this hyperbolic mass $m$ is bounded below by the integral quantity

$\frac{1}{2(n-1)w_{n-1}}\int_{\Sigma_{0}}fHd\sigma,$

where $\Sigma_{0}=\partial\Omega_{0}$ for the domain $\Omega_{0}$ in item (B)
is considered as a hypersurface in $\mathbb{H}^{n}$. This is the left-hand
side of (1.10), and so de Lima and Girao concluded from this inequality that

$m\geq\frac{1}{2}\left(\left(\frac{|\partial
M|}{w_{n-1}}\right)^{\frac{n-2}{n-1}}+\left(\frac{|\partial
M|}{w_{n-1}}\right)^{\frac{n}{n-1}}\right)$ (1.12)

whenever $\Sigma_{0}^{n-1}\subset\mathbb{H}^{n}$ is star-shaped with $H>0$.
Inequality (1.12) is known as the Riemannian Penrose inequality in general
relativity, and it is conjectured to hold for all asymptotically hyperbolic
manifolds $(M^{n},g)$ with horizon boundary $\partial M$. The idea of
considering the Penrose inequality for graphical submanifolds and applying the
Minkowski inequality goes back to the work of Lam [19].

By extending (1.10), we can remove the additional assumptions from [10] on the
underlying $\Omega_{0}$. Thus, we obtain the Penrose inequality for balanced
asymptotically hyperbolic graphs under the geometrically natural assumption
that $\Omega_{0}$ is outer-minimizing.

######  Corollary 1.3 (Riemannian Penrose Inequality for Balanced
Asymptotically Hyperbolic Graphs).

Let $M^{n}\subset(\mathbb{H}^{n+1},\overline{g})$, $3\leq n\leq 7$, satisfy
(A)-(C). Suppose that the domain $\Omega_{0}$ in item (B) is outer-minimizing.
Then the Riemannian Penrose inequality (1.12) holds for
$(M^{n},\overline{g}|_{M^{n}})$. Furthermore, equality holds if and only if
$\Sigma_{0}=\mathbb{S}^{n-1}\times\\{r_{0}\\}$ and
$(M^{n},\overline{g}|_{M^{n}})$ is isometric to anti de Sitter Schwarzschild
space of mass $m$.

We now give an outline of the paper. In Section 2, we review the variational
formulation of (1.3). The aspects of weak solution theory which are especially
important to our approach are the comparison principle for weak solutions and
the approximation theorems for the level sets.

In Section 3, we prove Theorem 1.1. The approach is similar to an Alexandrov
reflection method for viscosity solutions of extrinsic curvature flows in
$\mathbb{R}^{n}$ that was developed by Chow-Gulliver [8]. In [14], I applied a
version of their reflection method to proper weak IMCF in $\mathbb{R}^{n}$.
Here, reflections about a plane in Euclidean space are replaced with
inversions about a sphere in the Poincare ball. We will derive a comparison
principle for weak solutions under these inversions which may be used to show
in arbitrary dimensions that $\Sigma_{t}$ are each radial Lipschitz graphs
once they have escaped $B_{r_{+}}(0)$, with a Lipschitz constant depending
only on $r_{+}$. For $n>8$, this then implies that $\Sigma_{t}$ satisfy (1.8)
for $t\in(T,\infty)$. I highly suspect that this Alexandrov reflection method
in $\mathbb{H}^{n}$ also works the class of viscosity solutions of flows
considered by Chow-Gulliver. Finally, the smoothing comes from Theorem 1.2 in
[20], which states that the region outside of a $C^{1}$ star-shaped domain
$\Omega_{0}\subset\mathbb{H}^{n}$ with bounded non-negative weak mean
curvature is foliated by a star-shaped solution to (1.2).

In Section 4, we consider the inequality of Brendle from [3], which states
that

$(n-1)\int_{\Sigma_{0}}\frac{f}{H}d\sigma\geq n\int_{\Omega_{0}}fd\Omega$

for a domain $\Omega_{0}\subset\mathbb{H}^{n}$ with $C^{\infty}$ $H>0$
boundary. This inequality is key to showing monotonicity of the Minkowski
functional under smooth IMCF, and so we must also apply it to obtain
monotonicity under weak IMCF. When $n<8$, the $\Sigma_{t}$ are $C^{1}$
hypersurfaces with bounded non-negative weak mean curvature $H$. Our approach
is to approximate $\Sigma_{t}$ by mean curvature flow (MCF), an approach that
was utilized in [16] and subsequently in [20] and [26]. Brendle’s inequality
holds on the approximating surfaces $\Sigma_{\epsilon}$ because they are
smooth with $H_{\epsilon}>0$, and so an analysis of $H^{-1}_{\epsilon}$ as
$\epsilon\rightarrow 0$ allows us to pass Brendle’s inequality to
$\Sigma_{t}$. An approximation procedure would likely be more complicated for
$n\geq 8$, and so we ignore these dimensions here.

In Section 5, we extend inequality (1.9) to smooth, outer-minimizing
$\Omega_{0}\subset\mathbb{H}^{n}$ for $3\leq n\leq 7$. To accomplish this, we
show monotonicity of the functional

$Q(t)=|\Sigma_{t}|^{\frac{2-n}{n-1}}\left(\int_{\Sigma_{t}}fHd\sigma-n(n-1)\int_{\Omega_{t}}fd\Omega\right)$

under proper weak IMCF in these dimensions. We estimate the growth of the bulk
term and the surface integral term separately. First for the bulk term, we use
Brendle’s inequality for the weak $\Sigma_{t}$ and the co-area formula for
Lipschitz functions in order to derive a lower bound on
$\int_{\Omega_{t}}fd\Omega$. For the growth of the surface integral term, we
employ the method of Wei in [29] of approximation by elliptically-regularized
solutions $u^{\epsilon}$. In particular, we prove an inequality over the level
sets of $u^{\epsilon}$ involving the potential $f$ which can then by used to
estimate $\int_{\Sigma_{t}}fHd\sigma$. Altogether, we get $Q(0)\geq Q(t)$ for
any $t$, and Theorem 1.1 then completely solves the convergence issue since
$Q(t)\geq(n-1)w_{n-1}^{\frac{1}{n-1}}$ for $t>T$.

Finally in Section 6, we extend the pure area Minkowski inequality (1.10).
Like in [10], it is sufficient to prove monotonicity only for times $t$
satisfying

$\frac{n}{w_{n-1}}\int_{\Omega_{t}}fd\sigma<\left(\frac{|\Sigma_{t}|}{w_{n-1}}\right)^{\frac{n}{n-1}},$

and this follows immediately from the estimates in Section 5. We will not
discuss the RPI for asymptotically hyperbolic graphs in this section– see [9]
for the derivation of the inequality
$m\geq\frac{1}{2(n-1)w_{n-1}}\int_{\Sigma_{0}}fHd\sigma$.

### Acknowledgements

I would like to thank Professor Yong Wei for explaining his work on IMCF and
Minkowski inequalities and Professor Martin Li for suggestions about the $n>7$
case of Theorem 1.1. I would also like to thank the National Center for
Theoretical Sciences mathematics division for continued financial and career
support.

##  2\. Preliminaries on Weak IMCF

To introduce weak solutions of IMCF, let us consider the functional

$J_{u}(F)=|\partial F|-\int_{F}|\nabla u|d\Omega,$ (2.1)

over finite-perimeter domains $F\subset\mathbb{H}^{n}$, where $u\in
C^{0,1}_{loc}(\mathbb{H}^{n})$ is a fixed locally Lipschitz function. Given a
proper smooth $u:\mathbb{H}^{n}\rightarrow\mathbb{R}$ with $du\neq 0$, one may
check that the domains $\Omega_{t}=\\{u<t\\}$ minimize $J_{u}$ over
$\mathbb{H}^{n}\setminus\Omega_{0}$ for each $t>0$ if and only if $u$ solves
the Dirichlet problem (1.3) on $\mathbb{H}^{n}\setminus\Omega_{0}$. In turn,
the level sets $\Sigma_{t}=\\{u=t\\}$ are a solution to classical IMCF (1.2),
and this leads to the following definition.

######  Definition 2.1.

Let $\Omega_{0}\subset\mathbb{H}^{n}$ be a bounded domain with connected
$C^{\infty}$ boundary $\Sigma_{0}$. A weak solution of IMCF with initial
condition $\Omega_{0}$ is a function $u\in
C^{0,1}_{\text{loc}}(\mathbb{H}^{n})$ with $u|_{\Sigma_{0}}=0$ and
$u|_{\mathbb{H}^{n}\setminus\overline{\Omega_{0}}}\geq 0$ such that the sub-
level sets $\Omega_{t}=\\{u<t\\}$ satisfy

$J_{u}(\Omega_{t})\leq J_{u}(F)\hskip 28.45274pt\forall F\supset\Omega_{0}$
(2.2)

for every $t>0$.

This formulation is inspired by the level-set formulation for mean curvature
flow due to Evans-Spruck [12] and Chen-Goto-Giga [6]. Because $\Omega_{t}$
solve a minimization problem, the weak flow surfaces
$\Sigma_{t}=\partial\Omega_{t}$ enjoy several of the same regularity
properties as area-minimizers in Riemannian manifolds. Before stating these,
we remind the reader that the weak mean curvature of $\Sigma_{0}$ is the
unique measurable function $H$ on a $C^{1}$ hypersurface
$\Sigma_{0}=X_{0}(\Sigma^{n-1})\subset\mathbb{H}^{n}$ satisfying

$\frac{d}{d\epsilon}\bigg{\rvert}_{\epsilon=0}|\Sigma_{\epsilon}|=\int_{\Sigma_{0}}HV^{\perp}d\sigma$
(2.3)

for the flow
$X:\Sigma^{n-1}\times(-\epsilon_{0},\epsilon_{0})\rightarrow\mathbb{H}^{n}$ of
any variation variation field $V\in C^{0}_{c}(\Gamma\mathbb{H}^{n})$.

######  Theorem 2.2 (Elementary Properties of Proper Weak IMCF).

Let $\Omega_{0}\subset\mathbb{H}^{n}$ be a bounded domain with connected
$C^{\infty}$ boundary $\partial\Omega_{0}=\Sigma_{0}$, and let
$u:\mathbb{H}^{n}\rightarrow\mathbb{R}$ be a proper weak IMCF with initial
condition $\Omega_{0}$. Then the following properties hold with references
from [17] provided:

  1. i

(Theorem 1.3) If $n<8$, then $\Sigma_{t}=\partial\\{u<t\\}$ and
$\Sigma^{+}_{t}=\partial\\{u>t\\}$ are $C^{1,\alpha}$ hypersurfaces for every
$t>0$.

  2. ii

((1.10)) If $n<8$, we have the convergence

$\Sigma_{s}\rightarrow\Sigma_{t}\hskip 28.45274pt\text{ as }\hskip
14.22636pts\nearrow t,\hskip
28.45274pt\Sigma_{s}\rightarrow\Sigma^{+}_{t}\hskip 28.45274pt\text{ as
}s\searrow t,$ (2.4)

locally in $C^{1,\alpha}$ for every $t>0$.

  3. iii

((1.12) and Lemma 5.1) The weak mean curvature $H$ of $\Sigma_{t}$ is
essentially bounded on $\Sigma_{t}$. Furthermore, for a.e. $t\in(0,\infty)$
and $\mathcal{H}^{n-1}$ a.e. $x\in\Sigma_{t}$ we have $H(x)=|\nabla u(x)|>0$.

  4. iv

(Lemma 5.6) If $\Sigma_{0}$ is outer-minimizing, then the area of $\Sigma_{t}$
is given by $|\Sigma_{t}|=e^{t}|\Sigma_{0}|$.

When $n\geq 8$, the first two items are true away from a singular set $Z$ with
Hausdorff dimension at most $n-8$ and disjoint from $\Omega_{0}$.

Given the functional $J_{u}$ in (2.1) and the definition of a weak solution,
the domains $\Omega_{t}=\\{u<t\\}$ and $\Omega^{+}_{t}=\\{u\leq t\\}$ are
respectively outer-minimizing and strictly outer-minimizing domains of
$\mathbb{H}^{n}$ for each $t>0$. If $\Omega_{t}\neq\Omega^{+}_{t}$ for some
$t$, then the weak flow $\Sigma_{t}$ jumps across the region $\\{u=t\\}\subset
M^{n}$ and continues from $\Sigma^{+}_{t}$ according to property (ii). It is
important to note, however, that these jumps can only happen at a measure-zero
subset of times, i.e. $\Sigma_{t}=\Sigma^{+}_{t}$ for a.e. $t\in(0,\infty)$,
by property (iii).

In general, weak solutions of IMCF are not unique: for example, if $u$
satisfies (2.2), then so does $u^{t}=\min\\{u,t\\}$ for any choice of $t$, a
fact which we will also use later. However, a comparison principle from [17]
guarantees that a proper weak solution, i.e. one with compact level sets, is
uniquely determined by the initial condition $\Omega_{0}$. We will need their
entire comparison theorem in order to prove Theorem 1.1, so we will re-state
it here.

######  Theorem 2.3 ([17], Theorem 2.2).

Let $u$ and $v$ be weak solutions of IMCF on $\mathbb{H}^{n}$ with initial
conditions $\Omega_{0}$ and $\widetilde{\Omega}_{0}$, respectively.

  1. i

If $\\{v>u\\}\subset\subset W$ on an open set
$W\subset\mathbb{H}^{n}\setminus\left(\Omega_{0}\cup\widetilde{\Omega_{0}}\right)$,
then $v\leq u$ on $W$.

  2. ii

If $\Omega_{0}\subset\widetilde{\Omega}_{0}$ and $u$ is proper, then $v\leq u$
on $\mathbb{H}^{n}\setminus\Omega_{0}$, or equivalently
$\Omega_{t}\subset\widetilde{\Omega}_{t}$ for every $t>0$.

  3. iii

Given any $C^{\infty}$ domain $\Omega_{0}\subset\mathbb{H}^{n}$, the proper
weak solution $u:\mathbb{H}^{n}\rightarrow\mathbb{R}$ to IMCF with initial
condition $\Omega_{0}$ is unique.

To conclude, we adress the approximating solutions of a proper weak IMCF. In
order to prove existence of proper weak solutions to (2.2), Huisken and
Ilmanen in [17] considered the elliptic Dirichlet problem

$\displaystyle\text{div}\left(\frac{\nabla u^{\epsilon}}{\sqrt{|\nabla
u^{\epsilon}|^{2}+\epsilon^{2}}}\right)$ $\displaystyle=$
$\displaystyle\sqrt{|\nabla u^{\epsilon}|^{2}+\epsilon^{2}}\hskip
28.45274pt\text{on}\hskip 28.45274ptF_{L},$ $\displaystyle u^{\epsilon}$
$\displaystyle\equiv$ $\displaystyle 0\hskip 92.47145pt\text{on}\hskip
28.45274pt\Sigma_{0}=\partial\Omega_{0},$ (2.5) $\displaystyle
u^{\epsilon}_{\partial F_{L}}$ $\displaystyle=$ $\displaystyle L\hskip
92.47145pt\text{on}\hskip 28.45274pt\partial F_{L}\setminus\Sigma_{0}.$

Here, $F_{L}=\\{v<L\\}$ is defined using a subsolution at infinity of (2.2),
but in $\mathbb{H}^{n}$ we can simply pick $v$ to be the expanding sphere
solution. We will not summarize the existence argument here (See Section 2 of
[29] for a summary), but these $u^{\epsilon}$ are useful to consider for
geometric applications because they are smooth and proper. As in [29], they
will be a key ingredient in the monotonicity proof, and so we mention the
important convergence properties of these.

Before stating these properties, we highlight that because $u^{\epsilon}$ are
proper and smooth, the level set $\Sigma^{\epsilon}_{t}=\\{u^{\epsilon}=t\\}$
is regular for a.e. $t\in(0,\infty)$ by the $C^{1}$ Sard Theorem (see, for
example, Section 2.7 of [27]). Moreover, the Hessian of $u^{\epsilon}$ solving
(2.5) is non-vanishing at critical points, and so a.e.
$x\in\Omega^{\epsilon}_{t}=\\{u^{\epsilon}<t\\}$ belongs to a regular level
set.

######  Theorem 2.4 ([29] Secs 2 and 4, Convergence of Approximate Solutions).

Let $\Omega_{0}\subset\mathbb{H}^{n}$ be a bounded domain with connected
$C^{\infty}$ boundary $\Sigma_{0}$, and let $u^{\epsilon}$ solve (2.5) on
$F_{L}$. Then there is a subsequence $\epsilon_{i}\rightarrow 0$ so that for
a.e. $t\geq 0$, the mean curvature $H_{\Sigma^{i}_{t}}=H^{i}$ of
$\Sigma^{i}_{t}=\partial\Omega^{i}_{t}$,
$\Omega^{i}_{t}=\partial\\{u_{\epsilon_{i}}<t\\}$, converges to the weak mean
curvature $H_{\Sigma_{t}}=H$ of $\Sigma_{t}$ locally in $L^{2}$. Here
$\Sigma_{t}$ correspond the proper weak IMCF with initial condition
$\Omega_{0}$. Furthermore, for any $\phi\in C^{0}_{c}(\Omega_{t})$ we have the
convergences

$\displaystyle\int_{\Omega^{i}_{t}}\phi H_{i}d\Omega$
$\displaystyle\rightarrow$ $\displaystyle\int_{\Omega_{t}}\phi Hd\Omega,$
(2.6) $\displaystyle\int_{\Omega^{i}_{t}}\phi H_{i}^{2}d\Omega$
$\displaystyle\rightarrow$ $\displaystyle\int_{\Omega_{t}}\phi H^{2}d\Omega.$

##  3\. An Alexandrov Reflection Principle in Hyperbolic Space

To prove Theorem 1.1, we will use the Poincare ball model for hyperbolic
space. This consists of the unit ball $B_{1}(0)\subset\mathbb{R}^{n}$ equipped
with the metric

$g_{\mathbb{H}^{n}}=\frac{4}{(1-\rho^{2})^{2}}g_{\mathbb{R}^{n}},$

where $\rho$ is the radial coordinate of $B_{1}(0)$. This model is related to
(1.1) by the change of variable

$\displaystyle r(\rho)$ $\displaystyle=$
$\displaystyle\log\left(\frac{1+\rho}{1-\rho}\right).$ (3.1)

Throughout this section, we will denote the position vector of a point
$x=(x_{1},\dots,x_{n})\in\mathbb{R}^{n}$ by $\mathbf{x}$.

In [14] and following earlier work by Chow-Gulliver in [8], I studied weak
solutions of IMCF in $\mathbb{R}^{n}$ using a parabolic analogue to the moving
plane method of Alexandrov [1] for constant mean curvature hypersurfaces. In
$\mathbb{R}^{n}$, composition of a weak solution of IMCF with reflection about
a fixed hyperplane produces a new weak solution. The key insight in [14] is a
comparison principle within the lower half-plane between the original solution
and the reflected one. For a survey of the Euclidean version of this method,
see [7].

Here we consider sphere inversions, which may be thought of as the reflections
of the Poincare ball. For every constant $\lambda\in(0,1)$ and point
$\theta\in\partial B_{1}(0)$, we consider the ball
$B_{R_{\lambda}}(x_{\lambda,\theta})\subset\mathbb{R}^{n}$ with radius

$R_{\lambda}=\frac{1}{2}(\lambda^{-1}-\lambda)$ (3.2)

and centered at the point

$\mathbf{x}_{\lambda,\theta}=\frac{1}{2}(\lambda^{-1}+\lambda)\boldsymbol{\theta}\in\mathbb{R}^{n}\setminus
B_{1}(0).$ (3.3)

$B_{R_{\lambda}}(x_{\lambda,\theta})$ lies a distance $\lambda$ from $0\in
B_{1}(0)$, is centered on the line containing $0$ and $\theta$, and intersects
$\partial B_{1}(0)$ orthogonally. Given these facts, one may verify that
inversion
$F_{\lambda,\theta}:\mathbb{R}^{n}\setminus\\{x_{\lambda,\theta}\\}\rightarrow\mathbb{R}^{n}\setminus\\{x_{\lambda,\theta}\\}$
about $B_{R_{\lambda,\theta}}(x_{\lambda,\theta})$ defined by

$\mathbf{F}_{\lambda,\theta}(\mathbf{x})=\frac{R_{\lambda}^{2}}{|\mathbf{x}-\mathbf{x_{\lambda,\theta}}|^{2}}\left(\mathbf{x}-\mathbf{x}_{\lambda,\theta}\right)+\mathbf{x}_{\lambda,\theta}$
(3.4)

restricts to an isometry over $(B_{1}(0),g_{\mathbb{H}^{n}})$. For a domain
$\Omega_{0}\subset B_{1}(0)$ we denote

$\Omega^{\lambda,\theta}_{0}=F_{\lambda,\theta}(\Omega_{0}),$ (3.5)

and we consider the “lower half-space” region $H_{\lambda,\theta}\subset
B_{1}(0)$ given by

$H_{\lambda,\theta}=B_{1}(0)\cap B_{R_{\lambda}}(x_{\lambda,\theta}).$ (3.6)

Given a proper weak solution $u:(B_{1}(0),g_{\mathbb{H}^{n}})$ of IMCF, we
define another weak solution
$u_{\lambda,\theta}:(B_{1}(0),g_{\mathbb{H}}^{n+1})\rightarrow\mathbb{R}$ by

$u_{\lambda,\theta}(x)=u\circ F_{\lambda,\theta}(x).$ (3.7)

First, we derive a comparison principle between $u$ and $u_{\lambda,\theta}$.
Geometrically, the following proposition gives that if
$\Omega_{0}^{\lambda,\theta}\setminus\overline{H_{\lambda,\theta}}\subset\Omega_{0}$,
then the inverted image
$\Omega_{t}^{\lambda,\theta}\setminus\overline{H_{\lambda,\theta}}$ gets
trapped in the evolving domain $\Omega_{t}$ for every $t>0$, see Figure 1.

######  Proposition 3.1 (Comparison Principle for Inverted Weak Solutions).

Let $u:(B_{1}(0),g_{\mathbb{H}^{n}})\rightarrow\mathbb{R}$ be a variational
solution to IMCF with initial condition $\Omega_{0}$. Fix $\theta\in\partial
B_{1}(0)$ and $\lambda\in(0,1)$. If
$\Omega^{\lambda,\theta}_{0}\setminus\overline{H_{\lambda,\theta}}\subset\Omega_{0}$,
then
$\Omega_{t}^{\lambda,\theta}\setminus\overline{H_{\lambda,\theta}}\subset\Omega_{t}$
for every $t>0$. Equivalently, $u_{\lambda,\theta}(x)\leq u(x)$ for every
$x\in H_{\lambda,\theta}\setminus\overline{\Omega_{0}}$.

###### Proof.

For any $t>0$, the function $u^{t}(x)=\min\\{t,u(x)\\}$ is a variational
solution to IMCF on $\mathbb{H}^{n}\setminus\Omega_{0}$. Define the set

$W=\Omega^{\lambda,\theta}_{t}\setminus\overline{\left(\Omega_{0}\cup
H_{\lambda,\theta}\right)}.$

For $\delta>0$, we compare $u^{t}$ and $u_{\lambda,\theta}+\delta$ on $W$. By
definition, $u^{t}<u_{\lambda,\theta}+\delta=t+\delta$ on
$\partial\Omega^{\lambda,\theta}_{t}$, and by assumption
$u_{\lambda,\theta}+\delta>u^{t}=0$ on $\partial\Omega_{0}\setminus
H_{\lambda,\theta}$. Finally, since $F_{\lambda,\theta}$ acts as the identity
on $\partial B_{\lambda}(x_{\lambda,\theta})\cap B_{1}(0)$, we have that
$u_{\lambda,\theta}=u$ at these points and so altogether

$u^{t}<u_{\lambda,\theta}+\delta\hskip 56.9055pt\text{ on }\hskip
28.45274pt\partial W\subset\partial
B_{\lambda}(x_{\lambda},\theta)\cup\partial\Omega^{\lambda,\theta}_{t}\cup\left(\partial\Omega_{0}\setminus
H_{\lambda,\theta}\right).$

By Theorem 2.3(i), this implies that $u^{t}<u_{\lambda,\theta}+\delta$ on $W$,
and since $\delta$ is arbitrary we get $u\leq u_{\lambda,\theta}<t$ in $W$.
Hence
$\Omega^{\lambda,\theta}_{t}\setminus\overline{H_{\lambda,\theta}}\subset\Omega_{t}\setminus\overline{H_{\lambda,\theta}}$.
This is equivalent to $u_{\lambda,\theta}(x)\geq u(x)$ for $x\in
B_{1}(0)\setminus\overline{\left(H_{\lambda,\theta}\cup\Omega^{\lambda,\theta}_{0}\right)}$,
and also to $u(x)\leq u_{\lambda,\theta}(x)$ for $x\in
H_{\lambda,\theta}\setminus\Omega_{0}$. ∎

An Alexandrov-type Principle in the Poincare Ball

$B_{R}(x_{\lambda,\theta})$$H_{\lambda,\theta}$$\left(B_{1}(0),g_{\mathbb{H}^{n}}\right)$.$\Omega_{0}$$\lambda$.$\theta$.$\mathbf{x_{\lambda,\theta}}$$\Rightarrow$$H_{\lambda,\theta}$$B_{R}(x_{\lambda,\theta})$$\left(B_{1}(0),g_{\mathbb{H}^{n}}\right)$.$\Omega_{t}$$\lambda$.$\theta$.$\mathbf{x_{\lambda,\theta}}$

Figure 1. If the part of $\Omega_{0}$ that lies inside the half-space
$H_{\lambda,\theta}$ reflects into the part of $\Omega_{0}$ that lies outside,
then this remains true for each $\Omega_{t}$.

Consider points $x_{1},x_{2}\in B_{1}(0)$ with
$\mathbf{x_{1}}=\rho_{1}\boldsymbol{\theta},\mathbf{x_{2}}=\rho_{2}\boldsymbol{\theta}$
for the same point $\theta\in\partial B_{1}(0)$ and for
$\sup_{\Omega_{0}}\rho\leq\rho_{1}<\rho_{2}$. $x_{1}$ and $x_{2}$ may be
bisected by a sphere $\partial B_{R_{\lambda}}(x_{\lambda,\theta})$ so that
$u(x_{2})=u_{\lambda,\theta}(x_{1})$ and
$H_{\lambda,\theta}\cap\Omega_{0}=\varnothing$. By Proposition 3.1, this
implies that $u(x_{2})\geq u(x_{1})$. In fact, we can compare any pair of
points which are bisected by a sphere that is disjoint from $\Omega_{0}$.

######  Theorem 3.2.

Let $u:(B_{1}(0),g_{\mathbb{H}^{n}})\rightarrow\mathbb{R}$ be a variational
solution to IMCF with initial condition $\Omega_{0}$, and let
$B_{\rho_{+}}(0)\subset B_{1}(0)$ be the smallest ball about $0$ containing
$\Omega_{0}$. Consider points $x_{1},x_{2}\in B_{1}(0)$ with

$\displaystyle\mathbf{x}_{1}$ $\displaystyle=$
$\displaystyle\rho_{1}\boldsymbol{\theta}_{1},$ $\displaystyle\mathbf{x}_{2}$
$\displaystyle=$ $\displaystyle\rho_{2}\boldsymbol{\theta}_{2},$

for $\theta_{1},\theta_{2}\in\partial B_{1}(0)$ and

$\rho_{+}\leq\rho_{1}<\rho_{2}<1.$

Whenever

$\frac{\rho_{2}-\rho_{1}}{|\boldsymbol{\theta}_{2}-\boldsymbol{\theta}_{1}|}\geq\Lambda,\hskip
56.9055pt\text{for }\hskip
28.45274pt\Lambda=\frac{\rho_{+}}{(1-\rho_{+}^{2})^{\frac{1}{2}}},$ (3.8)

we have that $u(x_{2})\geq u(x_{1})$.

As a consequence, $\Sigma_{t}\setminus B_{\rho_{+}}(0)$ is a Lipschitz graph
$\rho_{t}=\rho_{t}(\theta)$ over $\partial B_{1}(0)$ with Lipschitz constant
$\leq\Lambda$.

###### Proof.

Consider the point $x_{0}\in\mathbb{R}^{n}$ given by

$\mathbf{x_{0}}=(1-s_{0})\mathbf{x}_{1}+s_{0}\mathbf{x}_{2}$ (3.9)

for

$s_{0}=\frac{1-\rho_{1}^{2}}{\rho_{2}^{2}-\rho_{1}^{2}}=1+\frac{1-\rho_{2}^{2}}{\rho_{2}^{2}-\rho_{1}^{2}}>1.$
(3.10)

Observe first that

$\displaystyle|\mathbf{x_{0}}|^{2}$ $\displaystyle=$ $\displaystyle
s_{0}^{2}\rho_{2}^{2}+(1-s_{0})^{2}\rho_{1}^{2}+2s_{0}(1-s_{0})\rho_{1}\rho_{2}\langle\boldsymbol{\theta}_{1},\boldsymbol{\theta}_{2}\rangle$
$\displaystyle=$ $\displaystyle
s_{0}^{2}\rho_{2}^{2}+(1-s_{0})^{2}\rho_{1}^{2}+2s_{0}(1-s_{0})\rho_{1}\rho_{2}-2s_{0}(1-s_{0})\rho_{1}\rho_{2}|\boldsymbol{\theta}_{1}-\boldsymbol{\theta}_{2}|^{2}$
$\displaystyle\geq$
$\displaystyle\left(s_{0}\rho_{2}+(1-s_{0})\rho_{1}\right)^{2}=\left(\frac{1-\rho_{1}^{2}}{\rho_{1}+\rho_{2}}+\rho_{1}\right)=\left(\frac{1+\rho_{1}\rho_{2}}{\rho_{1}+\rho_{2}}\right)^{2}$
$\displaystyle>$
$\displaystyle\left(\frac{1}{2}+\frac{\rho_{1}}{1+\rho_{1}\rho_{2}^{-1}}\right)^{2}>1.$

Here, we have used the fact that $\rho_{1}<\rho_{2}<1$ and the identity
$|\boldsymbol{\theta}_{1}-\boldsymbol{\theta}_{2}|^{2}=2-2\langle\boldsymbol{\theta}_{1},\boldsymbol{\theta}_{2}\rangle$
for $\theta_{1},\theta_{2}\in\partial B_{1}(0)$. Next, we consider the ball
$B_{R}(x_{0})\subset\mathbb{R}^{n}$ for

$R=|\mathbf{x_{0}}-\mathbf{x}_{1}|^{\frac{1}{2}}|\mathbf{x_{0}}-\mathbf{x}_{2}|^{\frac{1}{2}}$
(3.12)

Notice for this choice of $R$ that $x_{2}\in B_{R}(x_{0})$, and so
$B_{R}(x_{0})\cap B_{1}(0)\neq\varnothing$. For $x_{1},x_{2}$ satisfying
(3.8), we compute an upper bound on $R$:

$\displaystyle R^{2}$ $\displaystyle=$
$\displaystyle(s_{0}-1)s_{0}|\mathbf{x}_{2}-\mathbf{x}_{1}|^{2}=(s_{0}-1)s_{0}\left(\rho_{2}^{2}+\rho_{1}^{2}-2\rho_{1}\rho_{2}\langle\boldsymbol{\theta}_{1},\boldsymbol{\theta}_{2}\rangle\right)$
$\displaystyle=$
$\displaystyle(s_{0}-1)s_{0}\left(\rho_{2}^{2}+\rho_{1}^{2}-2\rho_{1}\rho_{2}+\rho_{1}\rho_{2}|\boldsymbol{\theta}_{1}-\boldsymbol{\theta}_{2}|^{2}\right)$
$\displaystyle\leq$
$\displaystyle(s_{0}-1)s_{0}(1+\Lambda^{-2})\left(\rho_{2}-\rho_{1}\right)^{2}=\frac{(1-\rho_{1}^{2})(1-\rho_{2}^{2})}{(\rho_{1}+\rho_{2})^{2}}\left(1+\Lambda^{-2}\right)$
$\displaystyle\leq$
$\displaystyle\frac{(1-\rho_{+}^{2})^{2}}{4}\left(1+\Lambda^{-2}\right)=\frac{1}{4}(1-\rho_{+}^{2})^{2}\rho_{+}^{-2}=\frac{1}{4}(\rho_{+}^{-1}-\rho_{+})^{2}.$

Finally, we consider points $\theta\in\partial B_{1}(0)\cap\partial
B_{R}(x_{0})$. For these $\theta$, we have

$R^{2}=\langle\boldsymbol{\theta}-\mathbf{x_{0}},\boldsymbol{\theta}-\mathbf{x_{0}}\rangle=1+|\mathbf{x_{0}}|^{2}-2\langle\boldsymbol{\theta},\mathbf{x_{0}}\rangle.$
(3.14)

Using the first line of (3), the first line of (3), and the identity (3.10)
for $s_{0}$, we have that

$R^{2}-|\mathbf{x_{0}}|^{2}=-s_{0}\rho_{2}^{2}+(s_{0}-1)\rho_{1}^{2}=-1,$

and so from (3.14)

$\langle\mathbf{x_{0}},\boldsymbol{\theta}\rangle=1\hskip
28.45274pt\text{for}\hskip 28.45274pt\theta\in\partial B_{1}(0)\cap\partial
B_{R}(x).$

Thus $\langle\boldsymbol{\theta}-\mathbf{x_{0}},\boldsymbol{\theta}\rangle=0$,
and so $\partial B_{R}(x_{0})$ intersects $\partial B_{1}(0)$ orthogonally. As
a consequence, inversion about $B_{R}(x_{0})$ defines an isometry of the
Poincare Ball, and by (3) the distance $\lambda$ between $B_{R}(x_{0})$ and
$0$ satisfies

$\frac{1}{2}\left(\lambda^{-1}-\lambda\right)=R\leq\frac{1}{2}\left(\rho_{+}^{-1}-\rho_{+}\right).$

This means that $\lambda\geq\rho_{+}$. Let $\theta\in\partial B_{1}(0)$ lie on
the line segment between $x_{0}$ and $0$. Since $B_{\rho_{+}}(0)$ contains
$\Omega_{0}$, we have that $H_{\lambda,\theta}\cap\Omega_{0}=\varnothing$, and
so we may apply the previous propsition in this half-space. It implies that

$u_{\lambda,\theta}(x)\leq u(x),\hskip 56.9055ptx\in H_{\lambda,\theta}.$

Since,

$F_{\lambda,\theta}(\mathbf{x}_{2})=\frac{R^{2}}{|\mathbf{x}_{2}-\mathbf{x}_{0}|^{2}}(\mathbf{x}_{2}-\mathbf{x}_{0})+\mathbf{x}_{0}=\frac{R^{2}}{|\mathbf{x}_{2}-\mathbf{x}_{0}|^{2}}\frac{|\mathbf{x}_{2}-\mathbf{x}_{0}|}{|\mathbf{x}_{1}-\mathbf{x}_{0}|}(\mathbf{x}_{1}-\mathbf{x}_{0})+\mathbf{x}_{0}=\mathbf{x}_{1},$

we have that

$u(x_{2})\geq u(x_{1}),$

as claimed.

To prove graphicality of $\Sigma_{t}\setminus B_{\rho_{+}}(0)$, we proceed by
contradiction: take $x_{1},x_{2}\in\Sigma_{t}\setminus B_{\rho_{+}}(0)$ with
$\rho_{1}<\rho_{2}$, and suppose first that

$\frac{\rho_{2}-\rho_{1}}{|\boldsymbol{\theta}_{2}-\boldsymbol{\theta}_{1}|}\geq\Lambda^{\prime}$

for some $\Lambda^{\prime}>\Lambda$. Consider a sequence
$x^{(i)}_{2}\in\Omega_{t}$ converging in Euclidean norm to
$x_{2}\in\Sigma_{t}$. For $i$ large enough, we must have
$\rho^{(i)}_{2}>\rho_{1}$ and

$\frac{\rho^{(i)}_{2}-\rho_{1}}{|\boldsymbol{\theta}^{(i)}_{2}-\boldsymbol{\theta}_{1}|}\geq\Lambda,$

By the previous theorem, this implies that $u(x^{(i)}_{2})\geq u(x_{1})$. But
on the other hand, we have by definition that $u(x_{1})=t$ and
$u(x^{(i)}_{2})<t$. This is a contradiction, and so we have that

$\frac{\rho_{2}-\rho_{1}}{|\theta_{2}-\theta_{1}|}<\Lambda^{\prime}$

for each $\Lambda^{\prime}>\Lambda$. ∎

######  Remark 3.3.

Determining the correct choice of $p$, $R$, and $\Lambda$ was quite tedious,
but worth it.

Comparison outside $B_{\rho_{+}}(0)$

.$F_{\lambda,\theta}(x_{1})=x_{2}$$\Omega_{0}$.$\mathbf{x_{1}}$.$\mathbf{x_{2}}$.$\lambda$$\mathbf{x}_{0}$$H_{\lambda,\theta}$$\rho_{+}$$B_{R}(x_{0})$$\left(B_{1}(0),g_{\mathbb{H}^{n}}\right)$

Figure 2. One can compare $x_{1}$ and $x_{2}$ if their bisecting sphere does
not contact $\Omega_{0}$. This happens whenever $x_{1}$ and $x_{2}$ lie
outside $B_{\rho_{+}}(0)$ and satisfy (3.8).

######  Corollary 3.4.

Let $n>8$. Then $\Sigma_{t}$ is star-shaped for $t\in(T,\infty)$, where $T$ is
given in (1.7). Moreover, the support function
$\phi_{t}:\Sigma_{t}\rightarrow\mathbb{R}$ obeys the lower bound (1.8).

###### Proof.

Let $B_{\rho_{-}}(0)\subset B_{1}(0)$ be the largest ball about $0$ that is
contained within $\Omega_{0}$. For the IMCF $B_{\rho(t)}(0)$ of
$B_{\rho_{-}}(0)$, the geodesic distance $r(t)$ of $\partial B_{\rho(t)}(0)$
to $0$ is given by

$\sinh\left(r(t)\right)=\sinh\left(r_{-}\right)e^{\frac{t}{n-1}}.$

Therefore, comparing $B_{\rho(t)}$ to the weak IMCF $\Omega_{t}$ of
$\Omega_{0}$ via Theorem 2.3(ii), we have that
$B_{\rho_{+}}(0)=B_{T}(0)\subset\Omega_{T}$ for $T$ defined in (1.7). Thus by
the previous theorem $\Sigma_{t}=\Sigma_{t}\setminus B_{\rho_{+}(0)}$ is a
Lipschitz graph over $\partial B_{1}(0)$ for every $t\in[T,\infty)$.

Since $n<8$, $\Sigma_{t}$ are each $C^{1}$. Thus the support function
$\phi_{t}$ is well-defined, and the Lipschitz constant $\Lambda$ yields a
lower bound on $\phi_{t}$. In addition to (3.1), one may check the relations

$\displaystyle\sinh(r)$ $\displaystyle=$
$\displaystyle\frac{2\rho}{1-\rho^{2}},$ $\displaystyle\cosh(r)$
$\displaystyle=$ $\displaystyle\frac{1+\rho^{2}}{1-\rho^{2}},$
$\displaystyle\operatorname{sech}\left(\frac{r}{2}\right)$ $\displaystyle=$
$\displaystyle(1-\rho^{2})^{\frac{1}{2}}.$

as well as the conformal relation

$\nu(\rho)=\frac{1}{2}(1-\rho^{2})\nu_{\mathbb{R}^{n}}(\rho)=\frac{1}{2}(1-\rho^{2})\frac{1}{\left(|D\rho|^{2}+1\right)^{\frac{1}{2}}}.$

Applying all of these along with the Lipschitz bound on $\rho_{t}(\theta)$, we
have for the support function $\phi_{t}$ that

$\displaystyle\phi_{t}(x)$ $\displaystyle=$ $\displaystyle\langle\nabla
f(\rho),\nu\rangle=\frac{4\rho}{(1-\rho^{2})^{2}}\nu(\rho)$ $\displaystyle=$
$\displaystyle\frac{2\rho}{(1-\rho^{2})}\frac{1}{\left(|D\rho|^{2}+1\right)^{\frac{1}{2}}}$
$\displaystyle\geq$
$\displaystyle\left(1-\rho_{+}^{2}\right)^{\frac{1}{2}}\sinh(r(x))$
$\displaystyle=$
$\displaystyle\operatorname{sech}\left(\frac{r_{+}}{2}\right)\sinh\left(r(x)\right).$

whenever $\rho(x)\geq\rho_{+}$. ∎

To complete the proof of Theorem 1.1, we invoke a result of Li-Wei which
states that for any $C^{1}$ star-shaped $\Sigma_{0}\subset\mathbb{H}^{n}$
there exists a long-time solution
$X:\mathbb{S}^{n-1}\times(0,\infty)\rightarrow\mathbb{H}^{n}$ to (1.2) which
converges to $\Sigma_{0}$ in $C^{0}$ as $t\searrow 0$. The argument, which
originates from Huisken and Ilmanen’s work in $\mathbb{R}^{n}$ from [16], is
based on a Harnack-type inequality for star-shaped IMCFs. The same argument
has also been applied to a general asymptotically hyperbolic background when
$n=3$ in [26].

###### Proof of Theorem 1.1.

We apply Theorem 1.2 of [20] to obtain a smooth, star-shaped IMCF
$\\{\widetilde{\Sigma_{t}}\\}_{T<t<\infty}$ foliating
$\mathbb{H}^{n}\setminus\Omega_{T}$. The function
$\widetilde{u}:\mathbb{H}^{n}\rightarrow\mathbb{R}$ defined by

$\widetilde{u}(x)=\begin{cases}u(x)&x\in\Omega_{T}\\\
t&x\in\widetilde{\Sigma_{t}}\subset\mathbb{H}^{n}\setminus\Omega_{T}\end{cases}$

is a proper weak solution to IMCF with initial condition $\Omega_{0}$. Thus by
Theorem 2.3(iii), we conclude that $\widetilde{u}=u$ on $\mathbb{H}^{n}$ and
hence $\Sigma_{t}$ is a smooth IMCF for $t\in(T,\infty)$.

######  Remark 3.5.

I highly suspect that Theorem 1.1 is true in all dimensions. When $n>7$, the
singular set $Z$ of $\Sigma_{t}$ is possibly non-empty, and therefore we
cannot immediately apply Theorem 1.2 in [20]. However, the weak flow surfaces
$\Sigma_{t}$ are radial Lipschitz graphs for $t>T$ in any dimension (notice
that the comparison argument does not depend on boundary regularity).
Comparing this with the Bernstein problem for minimal graphs, if an entire
minimal graph in $\mathbb{R}^{n}$ is Lipschitz then $Z=\varnothing$ and the
graph must be a plane regardless of dimension, a result due to Moser from
[23]. The functional $J_{u}$ in (2.1) is of the form perimeter-plus-bulk-
energy, so it may be possible to translate the arguments of [23] to show that
$Z=\varnothing$ given that $\Sigma_{t}$ is a Lipschitz graph. However, I am
not an expert on these topics, so I am not aware if this has already been done
or about the degree of difficulty.

∎

##  4\. Brendle’s Inequality for Weak Mean Curvature

In [3], Brendle proved the following inequality for mean-convex domains in
$\mathbb{H}^{n}$.

######  Theorem 4.1 ([3], Theorem 3.5).

Let $\Omega_{0}\subset\mathbb{H}^{n}$ be a bounded domain with smooth $H>0$
boundary $\partial\Omega=\Sigma_{0}$. Then we have the integral inequality

$(n-1)\int_{\Sigma_{0}}\frac{f}{H}d\sigma\geq n\int_{\Omega_{0}}fd\Omega.$
(4.1)

The proof of the Minkowski inequality for star-shaped hypersurfaces by
Brendle-Hung-Wang crucially relies on (4.1), and so a proof involving weak
IMCF will require this as well. The goal of this section is to suitably extend
Brendle’s inequality to a proper weak solution
$\\{\Omega_{t}\\}_{t\in(0,\infty)}$ of IMCF. Our approach here is an
approximation by mean-convex mean curvature flow which that used in [20] and
[16].

######  Theorem 4.2 ([20], Lemma 4.2, c.f also [16], Lemma 2.6).

Let $\Sigma_{0}=X_{0}(\Sigma^{n-1})\subset\mathbb{H}^{n}$ be an immersion of
class $C^{1}$ with bounded and non-negative measurable weak mean curvature
$H$. Then $\Sigma_{0}$ is of class $C^{1,\alpha}\cap W^{2,p}$, and there
exists a solution
$X:\Sigma^{n-1}\times(0,\epsilon_{0})\rightarrow(\mathbb{H}^{n},g_{\mathbb{H}^{n}})$
to mean curvature flow (MCF),

$\displaystyle\frac{\partial}{\partial\epsilon}X(x,\epsilon)=-H\nu(x,\epsilon),\hskip
56.9055pt(x,\epsilon)\in\Sigma^{n-1}\times(0,\epsilon_{0}),$ (4.2)

converging to $\Sigma_{0}$ locally in $C^{1,\alpha}\cap W^{2,p}$,
$\alpha\in(0,1),p\in(1,\infty)$ and with $H(x,\epsilon)\rightarrow H(x)$
strongly in $L^{p}$ for $p\in[1,\infty)$ as $\epsilon\rightarrow 0$ 111See the
proof of Lemma 2.6 in [16] for the $L^{p}$ convergence statement. Furthermore,
for every $\epsilon\in(0,\epsilon_{0})$ we have $\min_{\Sigma_{\epsilon}}H>0$.

We seek to extend inequality (4.1) to $C^{1}$ hypersurfaces $\Sigma_{0}$ with
non-negative bounded weak mean curvature via approximation by the smooth $H>0$
hypersurfaces $\Sigma_{\epsilon}$. To do so, we must study the reciporical
mean curvature $H^{-1}_{\epsilon}$ of $\Sigma_{\epsilon}$ under MCF as
$\epsilon\searrow 0$. $||H_{\epsilon}^{-1}||_{L^{1}(\Sigma_{\epsilon})}$ is
monotonically decreasing in $\epsilon$ under MCF, and this suggests $L^{1}$
convergence as $\epsilon\searrow 0$. However, we must handle the case where
$H^{-1}$ is integrable and $\text{essinf}_{\Sigma_{0}}H=0$ delicately.

######  Proposition 4.3.

Let $\Sigma_{0}$, $\Sigma_{\epsilon}$ be as in Theorem 4.2, and suppose that
reciprocal $H^{-1}$ of the weak mean curvature of $\Sigma_{0}$ is integrable.
Then there is a subsequence $\epsilon_{j}\searrow 0$ of times such that the
reciporical mean curvature $H^{-1}_{\epsilon_{j}}$ of $\Sigma_{\epsilon_{j}}$
converges to $H^{-1}$ strongly in $L^{1}$ as $\epsilon_{j}\searrow 0$.

###### Proof.

For a fixed constant $\beta\in\mathbb{R}^{+}$, we consider the quantity

$\int_{\Sigma_{\epsilon}}\frac{1}{e^{(n-1)\epsilon}H+\beta}d\sigma.$ (4.3)

under (4.2). The variation formula for mean curvature under MCF in
$\mathbb{H}^{n}$ is

$\displaystyle\frac{\partial}{\partial\epsilon}H$ $\displaystyle=$
$\displaystyle\Delta_{\Sigma_{\epsilon}}H+\left(|h|^{2}+\text{Ric}(\nu,\nu)\right)H$
$\displaystyle=$
$\displaystyle\Delta_{\Sigma_{\epsilon}}H+\left(|h|^{2}-(n-1)\right)H,$

and so

$\frac{\partial}{\partial\epsilon}e^{(n-1)\epsilon}H=\Delta_{\Sigma_{\epsilon}}e^{(n-1)\epsilon}H+|h|^{2}e^{(n-1)\epsilon}H.$

Using this, we compute the evolution of (4.3) under MCF:

$\displaystyle\frac{\partial}{\partial\epsilon}\int_{\Sigma_{\epsilon}}\frac{1}{e^{(n-1)\epsilon}H+\beta}d\sigma$
$\displaystyle=$
$\displaystyle\int_{\Sigma_{\epsilon}}-\frac{1}{\left(e^{(n-1)\epsilon}H+\beta\right)^{2}}\frac{\partial}{\partial\epsilon}\left(e^{(n-1)\epsilon}H\right)-\frac{1}{e^{(n-1)\epsilon}H+\beta}H^{2}d\sigma$
$\displaystyle\leq$
$\displaystyle-\int_{\Sigma_{\epsilon}}\frac{2e^{2(n-1)\epsilon}}{\left(e^{(n-1)\epsilon}H+\beta\right)^{3}}|\nabla
H|^{2}d\sigma\leq 0,$

and so (4.3) is monotonically decreasing in $\epsilon$. By $L^{p}$ convergence
of $H_{\epsilon}$ and uniform boundedness of
$\left(H_{\epsilon}+\beta\right)^{-1}$, we have that

$\int_{\Sigma_{\epsilon_{j}}}\frac{1}{e^{(n-1)\epsilon}H+\beta}d\sigma\rightarrow\int_{\Sigma_{0}}\frac{1}{H+\beta}d\sigma$

along a subsequence of times $\epsilon_{j}$ converging to $0$. Combining this
with (4) yields

$\int_{\Sigma_{\epsilon}}\frac{1}{e^{(n-1)\epsilon}H+\beta}d\sigma\leq\int_{\Sigma_{0}}\frac{1}{H+\beta}d\sigma,\hskip
28.45274pt\beta\in\mathbb{R}^{+},\hskip
28.45274pt\epsilon\in(0,\epsilon_{0}).$ (4.5)

Now, for a fixed $\epsilon\in(0,\epsilon_{0})$ we take the $\beta\searrow 0$
limit of each side of (4.5). By monotone convergence, we get

$\int_{\Sigma_{\epsilon}}\frac{1}{H}d\sigma\leq
e^{(n-1)\epsilon}\int_{\Sigma_{0}}\frac{1}{H}d\sigma.$

Hence

$\displaystyle\limsup_{\epsilon\searrow
0}\int_{\Sigma_{\epsilon}}\frac{1}{H}d\sigma\leq\int_{\Sigma_{0}}\frac{1}{H}d\sigma,$

and once again by $L^{p}$ convergence of $H_{\epsilon}$ we have that that
$H_{\epsilon_{j}}^{-1}\rightarrow H^{-1}$ in $L^{1}$ along a subsequence
$\epsilon_{j}$.

∎

Given this convergence, we may pass inequality (4.1) off to $C^{1}$
$\Sigma_{0}$ with bounded, non-negative $H$.

######  Theorem 4.4 (Brendle’s Inequality for Weak Mean Curvature).

Let $\Omega_{0}\subset\mathbb{H}^{n}$ be a bounded domain. Suppose that
$\partial\Omega_{0}=\Sigma_{0}=X_{0}(\Sigma)$ is $C^{1}$ with bounded and non-
negative measurable weak mean curvature. Then

$(n-1)\int_{\Sigma_{0}}\frac{f}{H}d\mu\geq n\int_{\Omega_{0}}fd\Omega,$ (4.6)

where $H$ is the weak mean curvature of $\Sigma_{0}$.

###### Proof.

We assume $H^{-1}$ is integrable, since otherwise the statement is trivial.
Let $X:\Sigma\times(0,\epsilon_{0})\rightarrow\mathbb{H}^{n}$ be the
approximating flow from Proposition 4.2, and denote by $\Omega_{\epsilon}$ the
region bounded by $\Sigma_{\epsilon}$. Given Proposition 4.3, we consider the
subsequence $\epsilon_{j}\searrow 0$ from Proposition 4.3. By $C^{0}$
convergence of $\Sigma_{\epsilon_{j}}$ and $L^{1}$ convergence of
$H_{\epsilon_{j}}^{-1}$, we have

$\displaystyle\lim_{j}\int_{\Omega_{\epsilon_{j}}}fd\Omega$ $\displaystyle=$
$\displaystyle\int_{\Omega_{0}}fd\Omega,$
$\displaystyle\lim_{j}\int_{\Sigma_{\epsilon_{j}}}\frac{f}{H}d\sigma$
$\displaystyle=$ $\displaystyle\int_{\Sigma_{0}}\frac{f}{H}d\sigma.$

Since $\Sigma_{\epsilon}$ is smooth with $H>0$, we apply Brendle’s inequality
(4.1) over each $\Omega_{\epsilon_{j}}$ to make the conclusion. ∎

In the next section, we will apply Theorem 4.4 to the weak flow surfaces of a
proper weak IMCF. Theorem 2.2(i) gives that $\Sigma_{t}$ is of class $C^{1}$
in dimensions $3\leq n\leq 7$. Unlike with the previous section, I do not
expect the methods of this section to easily carry over to higher dimensions,
since the MCF approximations from [16], [20] use $C^{1}$ regularity in a
critical way. Therefore, extending the results in this paper to arbitrary
dimension presents an interesting challenge.

##  5\. The Volumetric Minkowski Inequality

In this section, we prove the Minkowski inequality (1.9) of Brendle-Hung-Wang
for outer-minimizing domains of $\mathbb{H}^{n}$, $3\leq n\leq 7$ using weak
IMCF. Huisken introduced this approach in his lecture [15], and later Wei [29]
applied this approach in Schwarzschild space. Throughout this section, we will
use the relation

$\nabla^{2}f=fg_{\mathbb{H}^{n}}$ (5.1)

for the Hessian of $f=\cosh\left(r\right)$ in $\mathbb{H}^{n}$. This implies
for a smooth hypersurface $\Sigma_{0}^{n-1}\subset\mathbb{H}^{n}$ that

$H\phi_{0}+\Delta_{\Sigma_{0}}f=(n-1)f,$ (5.2)

and for the domain $\Omega_{0}$ enclosed by $\Sigma_{0}$ that

$\int_{\Sigma_{0}}\phi_{0}d\sigma=\int_{\Omega_{0}}\Delta_{\mathbb{H}^{n}}fd\mu=n\int_{\Omega_{0}}fd\Omega$
(5.3)

via the Divergence Theorem.

In [4], Brendle-Hung-Wang considered the integral quantity

$Q(t)=|\Sigma_{t}|^{\frac{2-n}{n-1}}\left(\int_{\Sigma_{t}}fHd\sigma-n(n-1)\int_{\Omega_{t}}fd\Omega\right)$
(5.4)

under smooth IMCF in $\mathbb{H}^{n}$. To determine the evolution of $Q(t)$,
recall the variation formula for mean curvature with
$\text{Ric}(\nu,\nu)=-(n-1)$,

$\frac{\partial}{\partial
t}H=-\Delta_{\Sigma_{t}}\frac{1}{H}+\left((n-1)-|h|^{2}\right)\frac{1}{H}.$

Now, we compute

$\displaystyle\frac{\partial}{\partial t}Q(t)$ $\displaystyle=$
$\displaystyle\frac{2-n}{n-1}|\Sigma_{t}|^{\frac{2-n}{n-1}}\left(\int_{\Sigma_{t}}fHd\sigma-n(n-1)\int_{\Omega_{t}}fd\Omega\right)$
$\displaystyle+|\Sigma_{t}|^{\frac{2-n}{n-1}}\left(\int_{\Sigma_{t}}\left(\frac{\partial}{\partial
t}f\right)H+f\left(\frac{\partial}{\partial
t}H\right)+fHd\sigma-n(n-1)\int_{\Sigma_{t}}\frac{f}{H}d\sigma\right)$
$\displaystyle=$
$\displaystyle\frac{2-n}{n-1}|\Sigma_{t}|^{\frac{2-n}{n-1}}\left(\int_{\Sigma_{t}}fHd\sigma-n(n-1)\int_{\Omega_{t}}fd\Omega\right)$
$\displaystyle+|\Sigma_{t}|^{\frac{2-n}{n-1}}\left(\int_{\Sigma_{t}}\phi_{t}-f\Delta_{\Sigma_{t}}\frac{1}{H}-\frac{|h|^{2}}{H}f+(n-1)\frac{f}{H}+fHd\sigma-n(n-1)\int_{\Sigma_{t}}\frac{f}{H}d\sigma\right)$
$\displaystyle=$
$\displaystyle\frac{2-n}{n-1}|\Sigma_{t}|^{\frac{2-n}{n-1}}\left(\int_{\Sigma_{t}}fHd\sigma-n(n-1)\int_{\Omega_{t}}fd\Omega\right)$
$\displaystyle+|\Sigma_{t}|^{\frac{2-n}{n-1}}\left(\int_{\Sigma_{t}}2\phi_{t}-(H\phi_{t}+\Delta_{\Sigma_{t}}f)\frac{1}{H}+\frac{n-2}{n-1}fH-\frac{|\mathring{h}|^{2}}{H}f-(n-1)^{2}\frac{f}{H}d\sigma\right)$
$\displaystyle=$
$\displaystyle|\Sigma_{t}|^{\frac{2-n}{n-1}}\left(-\int_{\Sigma_{t}}\frac{|\mathring{h}|^{2}}{H}fd\sigma+n\left(-(n-1)\int_{\Sigma_{t}}\frac{f}{H}d\sigma+n\int_{\Omega_{t}}fd\Omega\right)\right)$
$\displaystyle\leq$
$\displaystyle-|\Sigma_{t}|^{\frac{2-n}{n-1}}\int_{\Sigma_{t}}\frac{|\mathring{h}|^{2}}{H}fd\sigma\leq
0.$

Here, the third equation follows from (5.3) and (5.2) and the inequality
follows from Brendle’s inequality (4.1). The lower bound
$\liminf_{t\nearrow\infty}Q(t)\geq(n-1)w_{n-1}^{\frac{1}{n-1}}$ arises from a
Sobolev inequality, see Section 5 of [4].

To extend (1.10), we must show that $Q(t)$ remains monotone under weak IMCF.
We remark that geometric flows have recently been used to prove families of
inequalities in $\mathbb{H}^{n}$ related to the one in [4]– see for example
[21], [2], [21], and [28]– but the flows used for these are not known to admit
variational formulations, and so a weak flow approach to these inequalities is
currently out of reach. In order to show monotonicity of $Q(t)$ under weak
IMCF, we need to estimate the bulk term and the surface integral term in (5.4)
separately. First we determine the growth of $\int_{\Omega_{t}}fd\Omega$ along
the proper weak solution. This may be accomplished using the robust version of
the co-area formula from [27].

######  Proposition 5.1 (Growth of the Bulk Term).

Let $u:\mathbb{H}^{n}\rightarrow\mathbb{R}$, $3\leq n\leq 7$, be a proper weak
IMCF with initial condition $\Omega_{0}$. Then for every $0\leq t_{1}<t_{2}$,
we have for the potential function $f(r)=\cosh(r)$ that

$\int_{\Omega_{t_{2}}}fd\Omega\geq\int_{\Omega_{t_{1}}}fd\Omega+\frac{n}{n-1}\int_{t_{1}}^{t_{2}}\int_{\Omega_{s}}fd\Omega
ds.$ (5.5)

###### Proof.

Given that $u\in C^{0,1}_{\text{loc}}(\mathbb{H}^{n}\setminus\Omega_{0})$,
$\nabla u(x)$ is defined for a.e. $x\in\mathbb{H}^{n}\setminus\Omega_{0}$.
From this, we define the $\mathcal{H}^{n}$-measurable function
$h:\mathbb{H}^{n}\setminus\Omega_{0}\rightarrow\mathbb{R}$ by

$h(x)=\begin{cases}0&\nabla u(x)=0,\\\ f(x)|\nabla u|^{-1}&\nabla u(x)\neq
0.\end{cases}$

Now, we apply the co-area formula for Lipschitz functions from Theorem 7.3 and
Remark 7.4 in [27] and obtain

$\displaystyle\int_{\Omega_{t_{2}}}fd\Omega-\int_{\Omega_{t_{1}}}fd\Omega\geq\int_{\Omega_{t_{2}}\setminus\Omega_{t_{1}}}h|\nabla
u|d\Omega=\int_{t_{1}}^{t_{2}}\int_{\Sigma_{s}}hd\sigma
ds=\int_{t_{1}}^{t_{2}}\int_{\Sigma_{s}}\frac{f}{H}d\sigma ds.$

Here, we used the fact that $\Sigma_{t}=\Sigma^{+}_{t}=\\{u=t\\}$ for a.e.
$t\in(0,\infty)$. Likewise, the last equation follows from the fact that for
a.e. $t\in(0,\infty)$ we have $|\nabla u(x)|=H(x)$ for $\mathcal{H}^{n-1}$
a.e. $x\in\Sigma_{t}$. The conclusion follows from Theorem 4.4. ∎

Note that Proposition 5.1 is equivalent to

$\int_{\Omega_{t_{2}}}fd\Omega\geq
e^{\frac{n}{n-1}(t_{2}-t_{1})}\int_{\Omega_{t_{1}}}fd\Omega$ (5.6)

for $t_{2}>t_{1}$ by Gronwall’s lemma– we will use this explicit lower bound
on the bulk term in Section 6.

Next, we estimate the surface integral term of (5.4), and for this it is more
convenient to work with the smooth approximating functions $u_{\epsilon}$ from
(2.5). Our approach here is similar to the one of Yong Wei in Section 4 of
[29], but with a different underlying potential equation. We first need an
inequality for the regular level sets of $u^{i}=u_{\epsilon_{i}}$.

######  Lemma 5.2.

Let $\Omega_{0}\subset\mathbb{H}^{n}$ be a smooth bounded domain, and let
$u\in C^{+}(\mathbb{H}^{n})$ a proper smooth function with
$u|_{\Sigma_{0}}\equiv 0$. Define the domain $\Omega_{t}=\\{u\leq t\\}\subset
M^{n}$, and let $\Psi:(0,t)\rightarrow\mathbb{R}^{+}$ be a compactly-supported
Lipschitz function. Then for the function

$\psi(x)=(\Psi\circ u)(x),\hskip 56.9055ptx\in\Omega_{t},$

we have the integral identity

$-\int_{\Omega_{t}}fH\frac{\partial\psi}{\partial\nu}d\Omega=\int_{\Omega_{t}}\psi[2H\phi_{s}+f(H^{2}-|h|^{2})]d\Omega,$
(5.7)

where $\nu$, $H$, $h$, and $\phi$ are the unit normal, mean curvature, second
fundamental form, and support function of the regular level sets
$\Sigma_{s}=\\{u=s\\}$ of $u$.

###### Proof.

We will initially assume $\Psi\in C^{1}_{0}(0,t)$. By Sard’s theorem, a.e.
level set $\Sigma_{s}=\\{u=s\\}$, $s\in[0,t)$, is regular. For a regular level
set $\Sigma_{s}$, we consider the variation field

$\frac{\nabla u}{|\nabla u|^{2}}=|\nabla u|^{-1}\nu.$

Taking the first variation formula for mean curvature with
$\text{Ric}(\nu,\nu)=-(n-1)$, we find

$-|\nabla u|^{-1}\frac{\partial H}{\partial\nu}=\Delta_{\Sigma_{s}}|\nabla
u|^{-1}+(|h|^{2}+-(n-1))|\nabla u|^{-1}.$

Multiplying by $f$ and integrating yields

$\displaystyle\int_{\Sigma_{s}}-f|\nabla u|^{-1}\frac{\partial
H}{\partial\nu}d\sigma$ $\displaystyle=$
$\displaystyle\int_{\Sigma_{s}}f\Delta_{\Sigma_{s}}|\nabla
u|^{-1}+(|h|^{2}+-(n-1))f|\nabla u|^{-1}d\sigma$ $\displaystyle=$
$\displaystyle\int_{\Sigma_{s}}(\Delta_{\Sigma_{s}}f-(n-1)f+|h|^{2}f)|\nabla
u|^{-1}d\sigma$ $\displaystyle=$
$\displaystyle\int_{\Sigma_{s}}-H\phi_{s}|\nabla u|^{-1}+|h|^{2}f|\nabla
u|^{-1}d\sigma,$

where we used the potential equation (5.2). Since $\Sigma_{s}=\\{u=s\\}$ is
regular for a.e. $s\in(0,t)$, we apply (5) to the co-area formula

$\displaystyle\int_{\Omega_{t}}f\psi\frac{\partial H}{\partial\nu}d\Omega$
$\displaystyle=$
$\displaystyle\int_{0}^{t}\int_{\Sigma_{s}}f\psi\frac{\partial
H}{\partial\nu}|\nabla u|^{-1}d\sigma
ds=\int_{0}^{t}\Psi(s)\int_{\Sigma_{s}}f\frac{\partial H}{\partial\nu}|\nabla
u|^{-1}d\sigma ds$ $\displaystyle=$
$\displaystyle\int_{0}^{t}\Psi(s)\int_{\Sigma_{s}}H\phi_{s}|\nabla
u|^{-1}-f|h|^{2}|\nabla u|^{-1}d\sigma ds$ $\displaystyle=$
$\displaystyle\int_{\Omega_{t}}\psi(H\phi_{s}-f|h|^{2})d\Omega,$

Now,

$\displaystyle\text{div}(f\psi H\nu)$ $\displaystyle=$ $\displaystyle\psi
H\phi+fH\frac{\partial\psi}{\partial\nu}+f\psi\frac{\partial
H}{\partial\nu}+f\psi H\text{div}\nu$ (5.10)

Because $\Psi$ is compactly supported, $\psi$ vanishes over
$\partial\Omega_{t}=\\{u=t\\}\cup\partial M$. Therefore, the integral of the
left-hand side of (5.10) vanishes, and we are left with

$\displaystyle-\int_{\Omega_{t}}fH\frac{\partial\psi}{\partial\nu}d\Omega$
$\displaystyle=$ $\displaystyle\int_{\Omega_{t}}\psi
H\phi_{s}+f\psi\frac{\partial H}{\partial\nu}+f\psi H\text{div}\nu d\Omega$
$\displaystyle=$
$\displaystyle\int_{\Omega_{t}}\psi(2H\phi_{s}-f|h|^{2})d\Omega+\int_{\Omega_{t}}f\psi
H\text{div}\nu d\Omega.$

For the divergence term, we once again see from the co-area formula

$\displaystyle\int_{\Omega_{t}}f\psi H\text{div}\nu d\Omega$ $\displaystyle=$
$\displaystyle\int_{0}^{t}\int_{\Sigma_{s}}f\psi H\text{div}(\nu)|\nabla
u|^{-1}d\sigma ds$ $\displaystyle=$
$\displaystyle\int_{0}^{t}\int_{\Sigma_{s}}f\psi
H\text{div}_{\Sigma_{s}}(\nu)|\nabla u|^{-1}d\sigma ds$ $\displaystyle=$
$\displaystyle\int_{0}^{t}\int_{\Sigma_{s}}f\psi H^{2}|\nabla u|^{-1}d\sigma
ds$ $\displaystyle=$ $\displaystyle\int_{\Omega_{t}}f\psi H^{2}d\Omega.$

Substituting this into (5) yields (5.7). ∎

######  Remark 5.3.

The formula

$\int_{\Omega}fd\Omega=\int_{0}^{t}\int_{\\{u=s\\}}\frac{f}{|\nabla u|}d\sigma
ds$

for $f\in C^{\infty}_{+}(\mathbb{H}^{n})$ and locally Lipschitz $u$ is
sometimes applied erroneously in the literature. The above formula is always
true as an inequality, such as in the proof of Proposition 5.1. For exact
identity, the critical set of $u$ must have measure $0$. This is indeed the
case for $u^{i}$ solving (2.5) (check the Laplacian at critical points). This
is clarified in [5].

(5.7) implies an integral inequality for the level sets of a smooth, proper
$u$. We may pass this inequality to level sets of a proper weak IMCF, which
yields a growth estimate for the first term in (5.4).

######  Lemma 5.4 (Growth of the Total Mean Curvature).

Let $u:\mathbb{H}^{n}\rightarrow\mathbb{R}$ be a proper weak IMCF with initial
condition $\Omega_{0}$. Then for any $0\leq t_{1}<t_{2}$ we have

$\int_{\Sigma_{t_{2}}}fHd\sigma\leq\int_{\Sigma_{t_{1}}}fHd\sigma+\int_{t_{1}}^{t_{2}}\int_{\Sigma_{s}}\left(\frac{n-2}{n-1}fH+2\phi_{s}\right)d\sigma
ds.$ (5.12)

###### Proof.

For the approximating sequence $u_{\epsilon_{i}}=u^{i}$ in Theorem 2.4, we
have by (5.7) and the inequality $H^{2}\leq(n-1)|h|^{2}$ that

$-\int_{\Omega^{i}_{t}}fH_{i}\frac{\partial\psi_{i}}{\partial\nu_{i}}d\Omega\leq\int_{\Omega^{i}_{t}}\psi_{i}\left(2H_{i}\phi^{i}_{s}+\frac{n-2}{n-1}H_{i}^{2}\right)d\Omega.$

Thus the convergence (2.6) implies

$-\int_{\Omega_{t}}fH\frac{\partial\psi}{\partial\nu}d\Omega\leq\int_{\Omega_{t}}\psi\left(2H\phi_{s}+\frac{n-2}{n-1}fH^{2}\right)d\Omega,$
(5.13)

where $H$ and $\phi_{s}$ are the support function and weak mean curvature of
the weak flow surfaces $\Sigma_{t}=\partial\\{u<t\\}$. Applying (5.13) along
with the co-area formula, we find for any $\Psi\in C^{0,1}_{0}((0,t))$ that

$\displaystyle-\int_{0}^{t}\Psi^{\prime}(s)\int_{\Sigma_{s}}fHd\sigma_{s}ds$
$\displaystyle=$
$\displaystyle-\int_{\Omega_{t}}\frac{\partial\psi}{\partial\nu}fHd\Omega$
$\displaystyle\leq$
$\displaystyle\int_{\Omega_{t}}\psi\left(2\phi_{s}H+\frac{n-2}{n-1}fH^{2}\right)d\Omega$
$\displaystyle=$
$\displaystyle\int_{\Omega_{t}}\psi\left(2\phi_{s}+\frac{n-2}{n-1}fH\right)|\nabla
u|d\Omega$ $\displaystyle=$
$\displaystyle\int_{0}^{t}\Psi(s)\int_{\Sigma_{s}}\left(\frac{n-2}{n-1}fH+2\phi_{s}\right)d\sigma
ds$

for $\Psi\in C^{0,1}_{0}((0,t))$. Fix $0\leq t_{1}<t_{2}$, and for
$0<\delta<\frac{1}{2}(t_{2}-t_{1})$, choose

$\Psi(s)=\begin{cases}0&s\in(0,t_{1})\\\
\frac{1}{\delta}(s-t_{1})&s\in(t_{1},t_{1}+\delta)\\\
1&s\in(t_{1}+\delta,t_{2}-\delta)\\\
\frac{1}{\delta}(t_{2}-s)&s\in(t_{2}-\delta,t_{2})\\\
0&s\in(t_{2},t).\par\end{cases}$

Then

$\frac{1}{\delta}\int_{t_{2}-\delta}^{t_{2}}\int_{\Sigma_{s}}fHd\sigma
ds-\frac{1}{\delta}\int_{t_{1}}^{t_{1}+\delta}\int_{\Sigma_{s}}fHd\sigma
ds\leq\int_{t_{1}}^{t_{2}}\int_{\Sigma_{s}}\left(\frac{n-2}{n-1}fH+2\phi_{s}\right)d\sigma
ds.$

The limit of the left-hand side as $\delta\rightarrow 0$ is well-defined for
a.e. $0<t_{1}<t_{2}$, and so for a.e. $0<t_{1}<t_{2}$ we have

$\displaystyle\int_{\Sigma_{t_{2}}}fHd\sigma-\int_{\Sigma_{t_{1}}}fHd\sigma$
$\displaystyle\leq$
$\displaystyle\int_{t_{1}}^{t_{2}}\int_{\Sigma_{s}}\left(\frac{n-2}{n-1}fH+2\phi_{s}\right)d\sigma
ds.$ (5.14)

To extend this inequality to all times, for any $0<t_{1}<t_{2}$ we choose
sequences $t_{j}\nearrow t_{1}$ and $\tilde{t}_{j}\nearrow t_{2}$ such that
(5.14) holds for the times $t_{j},\tilde{t}_{j}$. By the second property in
Theorem 2.2, we have that

$\Sigma_{t_{j}}\rightarrow\Sigma_{t_{1}}\hskip 56.9055pt\text{ in }\hskip
28.45274ptC^{1,\alpha},\\\ $

resp. for $\tilde{t}_{j},t_{2}$, away from a small singular set $Z$. Equation
(1.13) in [17] then implies $L^{1}$ convergence of the weak mean curvature,
and so altogether we find that

$\displaystyle\lim_{j\rightarrow\infty}\int_{\Sigma_{t_{j}}}fHd\sigma$
$\displaystyle=$ $\displaystyle\int_{\Sigma_{t_{1}}}fHd\sigma,$
$\displaystyle\lim_{j\rightarrow\infty}\int_{\Sigma_{\tilde{t}_{j}}}fHd\sigma$
$\displaystyle=$ $\displaystyle\int_{\Sigma_{t_{2}}}fHd\sigma.$

Therefore, applying (5.14) to $t_{j},\tilde{t}_{j}$, we may conclude

$\displaystyle\int_{\Sigma_{t_{2}}}fHd\sigma-\int_{\Sigma_{t_{1}}}fHd\sigma$
$\displaystyle\leq$
$\displaystyle\lim_{j\rightarrow\infty}\int_{t_{j}}^{\tilde{t}_{j}}\int_{\Sigma_{s}}\left(\frac{n-2}{n-1}fH+2\phi_{s}\right)d\sigma
ds$ $\displaystyle=$
$\displaystyle\int_{t_{1}}^{t_{2}}\int_{\Sigma_{s}}\left(\frac{n-2}{n-1}fH+2\phi_{s}\right)d\sigma
ds.$

For any $0<t_{1}<t_{2}$. If $t_{1}=0$, then the inequality follows from the
fact that $\int_{\Sigma_{0}}fHd\sigma\geq\int_{\Sigma^{+}_{0}}fHd\sigma$ for
$\Sigma_{0}^{+}=\partial\\{u>0\\}$ and the $C^{1,\alpha}$ convergence
$\Sigma_{t_{j}}\rightarrow\Sigma^{+}_{0}$ as $t\searrow 0$. ∎

###### Proof of Theorem 1.2, I..

Consider the quantity

$q(t)=\int_{\Sigma_{t}}fHd\sigma-n(n-1)\int_{\Omega_{t}}fd\Omega.$

Combining (5.12) and (5.5), we find for $0\leq t_{1}<t_{2}$ that

$\displaystyle q(t_{1})-q(t_{2})$ $\displaystyle\leq$
$\displaystyle\int_{t_{1}}^{t_{2}}\left(\frac{n-2}{n-1}\int_{\Sigma_{s}}fHd\sigma+2\int_{\Sigma_{s}}\phi_{s}d\sigma-n^{2}\int_{\Omega_{s}}fd\Omega\right)ds$
$\displaystyle=$
$\displaystyle\int_{t_{1}}^{t_{2}}\left(\frac{n-2}{n-1}\int_{\Sigma_{s}}fHd\sigma-n(n-2)\int_{\Omega_{s}}fd\Omega\right)ds=\frac{n-2}{n-1}\int_{t_{1}}^{t_{2}}q(s)ds,$

where we once again applied (5.3) over the $C^{1}$ domain $\Omega_{s}$.
Therefore, if $\Sigma_{0}$ is outer-minimizing then the area growth formula
from Theorem 2.2 and Gronwall’s Lemma yield that $Q(t_{2})\leq Q(t_{1})$ for
$t_{2}>t_{1}$. By Theorem 1.2 in [4], 222Theorem 1.2 in the ArXiv version we
have that

$Q(t)\geq(n-1)w_{n-1}^{\frac{1}{n-1}}\hskip 28.45274pt\text{ for all}\hskip
28.45274ptt>T,$

and so $Q(0)\geq(n-1)w_{n-1}^{\frac{1}{n-1}}$.

It remains to address the case of equality. Suppose that $Q(0)=Q(t)$ for every
$t$, and let

$t_{0}=\inf\\{t\in(0,\infty)|\Sigma_{t}\text{ is star-shaped}\\}$

It follows once again by Theorem 1.2 in [20] that
$\\{\Sigma_{t}\\}_{t\in(t_{0},\infty)}$ is a smooth solution to (1.2). By the
equality statement in [4], this means that $\Sigma_{t}$ is a slice
$\mathbb{S}^{n-1}\times\\{r(t)\\}$ of $\mathbb{H}^{n}$ for each
$t\in(t_{0},\infty)$. If $Q(t)=Q(0)$ for each $t$, then as in Section 8 of
[17] we must have that $\Sigma_{t}=\Sigma^{+}_{t}$ for each $t\in(0,\infty)$,
i.e. the flow has no jumps. The $C^{1}$ convergence
$\Sigma_{t}\rightarrow\Sigma^{+}_{t_{0}}=\Sigma_{t_{0}}$ as $t\searrow t_{0}$
from Theorem 2.2(ii) then implies that
$\Sigma_{t_{0}}=\\{r(t_{0})\\}\times\mathbb{S}^{n-1}$ is also a slice of
(1.1).

Now suppose that $t_{0}>0$. Since
$\Sigma_{t_{0}}=\\{r(t_{0})\\}\times\mathbb{S}^{n-1}$, we must have that the
support function $\phi_{t_{0}}(x)=\sinh(r(t_{0}))$ is a positive constant. By
the $C^{1}$ convergence $\Sigma_{t}\rightarrow\Sigma_{t_{0}}$ as $t\nearrow
t_{0}$, we get that $\phi_{t}>0$ on $\Sigma_{t}$ for
$t\in(t_{0}-\epsilon,t_{0})$. This is a contradiction, and so we conclude that
$t_{0}=0$ and hence $\Sigma_{0}$ is a slice of $\mathbb{H}^{n}$. ∎

######  Remark 5.5.

The estimate (5) for $q(t)$ is true regardless of the choice of $\Omega_{0}$,
but the area growth formula of Theorem 2.2(iv) is only valid for outer-
minimizing $\Omega_{0}$. This restriction is not removable, since if $Q$ may
increase across a jump at time $t=0$.

##  6\. The Pure Area Minkowski Inequality

In this Section, we extend the inequality of De Lima and Girao [10] to outer-
minimizing $\Omega_{0}$. In their paper, these authors considered the
functional

$P(t)=|\Sigma_{t}|^{\frac{2-n}{n-1}}\left(\int_{\Sigma_{t}}fHd\sigma-(n-1)w_{n-1}^{-\frac{1}{n-1}}|\Sigma_{t}|^{\frac{n}{n-1}}\right)$
(6.1)

under smooth star-shaped IMCF. Notice that if

$\frac{n}{w_{n-1}}\int_{\Omega_{0}}fd\Omega\geq\left(\frac{|\Sigma_{0}|}{w_{n-1}}\right)^{\frac{n}{n-1}},$
(6.2)

over a star-shaped, $H>0$ $\Sigma_{0}$, then $P(0)\geq Q(0)$ and so inequality
(1.10) is implied by Brendle-Hung-Wang’s inequality. In fact, condition (6.2)
is preserved under smooth IMCF due to Brendle’s inequality, so it suffices to
show that the quantity $P(t)$ is monotone non-increasing over the largest
interval $(0,\tilde{t})$ on which

$\frac{n}{w_{n-1}}\int_{\Omega_{t}}fd\Omega<\left(\frac{|\Sigma_{t}|}{w_{n-1}}\right)^{\frac{n}{n-1}}.$

Under this condition, we compute

$\displaystyle\frac{\partial}{\partial t}P(t)$ $\displaystyle=$
$\displaystyle\frac{2-n}{n-1}|\Sigma_{t}|^{\frac{2-n}{n-1}}\left(\int_{\Sigma_{t}}fHd\mu-(n-1)w_{n-1}^{-\frac{1}{n-1}}|\Sigma_{t}|^{\frac{n}{n-1}}\right)$
$\displaystyle+|\Sigma_{t}|^{\frac{2-n}{n-1}}\left(\frac{\partial}{\partial
t}\int_{\Sigma_{s}}fHd\sigma-
nw_{n-1}^{-\frac{1}{n-1}}|\Sigma_{t}|^{\frac{n}{n-1}}\right)$ $\displaystyle=$
$\displaystyle\frac{2-n}{n-1}|\Sigma_{t}|^{\frac{2-n}{n-1}}\left(\int_{\Sigma_{t}}fHd\mu-(n-1)w_{n-1}^{-\frac{1}{n-1}}|\Sigma_{t}|^{\frac{n}{n-1}}\right)$
$\displaystyle+|\Sigma_{t}|^{\frac{2-n}{n-1}}\left(\int_{\Sigma_{t}}\frac{n-2}{n-1}fH-\frac{|\mathring{h}|^{2}}{H}fd\sigma+2n\int_{\Omega_{t}}fd\Omega-
nw_{n-1}^{-\frac{1}{n-1}}|\Sigma_{t}|^{\frac{n}{n-1}}\right)$
$\displaystyle\leq$
$\displaystyle|\Sigma_{t}|^{\frac{2-n}{n-1}}\left(\int_{\Sigma_{t}}-\frac{|\mathring{h}|^{2}}{H}fd\sigma+2n\int_{\Omega_{t}}fd\Omega-2w_{n-1}^{-\frac{1}{n-1}}|\Sigma_{t}|^{\frac{n}{n-1}}\right)\leq
0.$

Suppose $\tilde{t}<+\infty$. Since $\tilde{t}$ satisfies (6.2) we have
$P(0)\geq P(\tilde{t})\geq Q(\tilde{t})\geq(n-1)w_{n-1}^{\frac{1}{n-1}}$. De
Lima and Girao carry out the asymptotic analysis for the case
$\tilde{t}=+\infty$ in Appendix A of [10]. Their result may be straight-
forwardly extended using results from the previous sections.

###### Proof of Theorem 1.2, II.

Given an outer-minimizing initial domain $\Omega_{0}\subset\mathbb{H}^{n}$, we
consider two cases separately:

$(i)$
$\frac{n}{w_{n-1}}\int_{\Omega_{0}}fd\Omega\geq\left(\frac{|\Sigma_{0}|}{w_{n-1}}\right)^{\frac{n}{n-1}}$:
since (1.9) holds on $\Omega_{0}$ according to the previous section, we
conclude that (1.10) holds as well.

$(ii)$
$\frac{n}{w_{n-1}}\int_{\Omega_{0}}fd\Omega<\left(\frac{|\Sigma_{0}|}{w_{n-1}}\right)^{\frac{n}{n-1}}$:
first, observe that if

$\frac{n}{w_{n-1}}\int_{\Omega_{t_{0}}}fd\Omega\geq\left(\frac{|\Sigma_{t_{0}}|}{w_{n-1}}\right)^{\frac{n}{n-1}}$

for some $t_{0}>0$, then the formula
$|\Sigma_{t}|=e^{(t_{-}t_{0})}||\Sigma_{t_{0}}|$ and the growth estimate (5.6)
on $\int_{\Omega_{t}}fd\Omega$ imply that

$\frac{n}{w_{n-1}}\int_{\Omega_{t}}fd\Omega\geq\left(\frac{|\Sigma_{t}|}{w_{n-1}}\right)^{\frac{n}{n-1}},\hskip
28.45274ptt\in[t_{0},\infty).$

Therefore, we let $(0,\widetilde{t})$, $\widetilde{t}\leq\infty$, be the
largest interval over which

$\frac{n}{w_{n-1}}\int_{\Omega_{t}}fd\Omega<\left(\frac{|\Sigma_{t}|}{w_{n-1}}\right)^{\frac{n}{n-1}},$

and we show that $P(t)$ is monotone non-increasing over $(0,\widetilde{t})$.
As in the proof of the volumetric Minkowski inequality, for any $0\leq
t_{1}<t_{2}<\widetilde{t}$ we have

$\displaystyle\int_{\Sigma_{t_{2}}}fHd\sigma-\int_{\Sigma_{t_{1}}}fHd\sigma$
$\displaystyle\leq$
$\displaystyle\int_{t_{1}}^{t_{2}}\int_{\Sigma_{s}}\left(2\phi_{s}+\frac{n-2}{n-1}fH\right)d\sigma
ds$ $\displaystyle=$
$\displaystyle\int_{t_{1}}^{t_{2}}\left(2n\int_{\Omega_{s}}fd\Omega+\int_{\Sigma_{s}}\frac{n-2}{n-1}fHd\sigma\right)ds$
$\displaystyle<$
$\displaystyle\int_{t_{1}}^{t_{2}}\left(2w_{n-1}^{-\frac{1}{n-1}}|\Sigma_{s}|^{\frac{n}{n-1}}+\int_{\Sigma_{s}}\frac{n-2}{n-1}fHd\sigma\right)ds,$

where the last line comes from the volume assumption. On the other hand, we
can write

$(n-1)w_{n-1}^{-\frac{1}{n-1}}|\Sigma_{t_{2}}|^{\frac{n}{n-1}}-(n-1)w_{n-1}^{-\frac{1}{n-1}}|\Sigma_{t_{1}}|^{\frac{n}{n-1}}=\int_{t_{1}}^{t_{2}}nw_{n-1}^{-\frac{1}{n-1}}|\Sigma_{s}|^{\frac{n}{n-1}}ds.$
(6.4)

Combining (6) and (6.4), we get for
$p(t)=\int_{\Sigma_{t}}fHd\sigma-(n-1)w_{n-1}^{-\frac{1}{n-1}}|\Sigma_{t}|^{-\frac{n}{n-1}}$
that

$\displaystyle p(t_{2})-p(t_{1})$ $\displaystyle\leq$
$\displaystyle\int_{t_{1}}^{t_{2}}\left((2-n)w_{n}^{-\frac{1}{n-1}}|\Sigma_{s}|^{\frac{n}{n-1}}+\int_{\Sigma_{s}}\frac{n-2}{n-1}fHd\sigma\right)ds$
$\displaystyle=$ $\displaystyle\frac{n-2}{n-1}\int_{t_{1}}^{t_{2}}p(s)ds.$

Using Gronwall’s Lemma once again, we conclude that $P(t_{2})\leq P(t_{1})$
for $P(t)=\left(|\Sigma_{t}|\right)^{\frac{2-n}{n-1}}p(t)$, and so $P$ is
monotone over $(0,\widetilde{t})$. If $\widetilde{t}<+\infty$, then from the
previous section

$\displaystyle P(0)\geq P(\widetilde{t})\geq
Q(\widetilde{t})\geq(n-1)w_{n-1}^{\frac{1}{n-1}}.$

If $\widetilde{t}=+\infty$, then since $\Sigma_{t}$ solves the smooth flow for
$t>T$, the asymptotic situation reduces to the one in [10]. Altogether, we
conclude that

$P(0)\geq(n-1)w_{n-1}^{\frac{1}{n-1}},$ (6.5)

which yields (1.10). The argument from the previous section for equality also
applies here. ∎

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# Neutron scattering investigations of the boson peak

U. Buchenau

(Received June 6, 2019, in final form July 31, 2019)

###### Abstract

Inelastic neutron scattering is not only capable of determining a generalized
vibrational density of states around the boson peak of a glass, but can also
be used to get information on the eigenvectors. The eigenvectors determine the
dynamic structure factor, which is $Q^{2}S(Q)$ ($Q$ wavevector) for long
wavelength sound waves. This enables the determination of the sound wave
fraction below and at the boson peak, done for SiO2, B2O3, polybutadiene and
amorphous germanium. The temperature dependence of the boson peak in silica
and glycerol is shown. X-ray Brillouin scattering data show that the damping
of the longitudinal sound waves in silica and glycerol follows the soft
potential model $\omega^{4}$ prediction in the limit of low frequency.

Key words: glasses, inelastic neutron scattering, boson peak

PACS: 61.20.Lc, 61.43.Fs, 61.12.Ex

###### Abstract

Íåïðóæíå ðîçñþâàííÿ íåéòðîíâ íå ëèøå äîçâîëÿ âèçíàèòè óçàãàëüíåíó ãóñòèíó
êîëèâíèõ ñòàíâ áëÿ áîçîííîãî ïêó â ñêëîâèäíîìó ñòàí, àëå òàêîæ ìîæå áóòè
âèêîðèñòàíå äëÿ îòðèìàííÿ íôîðìàö¿ ïðî âëàñí âåêòîðè. Âëàñí âåêòîðè âèçíààþòü
äèíàìíèé ñòðóêòóðíèé ôàêòîð, ÿêèé $Q^{2}S(Q)$ ($Q$ õâèëüîâèé âåêòîð) äëÿ
äîâãîõâèëüîâèõ çâóêîâèõ õâèëü. Öå äîçâîëÿ âñòàíîâëåííÿ àñòêè çâóêîâèõ õâèëü
íèæå òà íà áîçîííîìó ïêó, ùî áóëî çðîáëåíî äëÿ SiO2, B2O3, ïîëáóòàäíó òà
àìîðôíîãî ãåðìàíþ. Ïîêàçàíî òåìïåðàòóðíó çàëåæíñòü áîçîííîãî ïêó â êðåìíåçåì
òà ãëöåðîë. Äàí áðëþåíâñüêîãî ðîçñþâàííÿ ðåíòãåíâñüêèõ ïðîìåíâ âêàçóþòü, ùî
çàãàñàííÿ ïîâçäîâæíèõ õâèëü â êðåìíåçåì òà ãëöåðîë ñëäó ïåðåäáàåííÿì ìîäåë
ì’ÿêèõ ïîòåíöàëâ $\omega^{4}$ â ãðàíèö íèçüêî¿ àñòîòè.

Ключов слова: ñêëîâèäí ñèñòåìè, íåïðóæíå ðîçñþâàííÿ íåéòðîíâ, áîçîííèé ïê

##  1 Introduction

An important part of Giancarlo Ruocco’s scientific work was devoted to the
riddle of the low frequency vibrations in glasses. Glasses have not only the
low frequency longitudinal and transverse sound waves which one expects in an
elastic solid on the basis of the Debye model, but also additional
excitations, tunneling states and soft vibrations, which are clearly visible
in the heat capacity, the thermal conductivity and the sound absorption at low
temperatures [1].

In scattering techniques (Raman, neutron and x-ray scattering), the additional
soft vibrations appear as a broad peak between the frequency zero and the
lowest Van Hove singularity of the corresponding crystal. At lower
temperatures, it shows the linear temperature increase expected for harmonic
vibrations, so it got the name boson peak. In the heat capacity
($c_{\text{p}}$) measurements, it appears as a peak in $c_{\text{p}}/T^{3}$
around 5 to 10 K. Thus, the boson peak is not a peak in the density of states
$g(\omega)$, but rather a peak in $g(\omega)/\omega^{2}$, which is a constant
in the Debye model.

The present paper summarizes the neutron scattering studies of the boson peak.
As will be seen, neutrons are a good tool to study the vibrational density of
states of a glass and are capable of providing information on the mode
eigenvectors. However, if it comes to Brillouin scattering studies of the
longitudinal sound absorption in the THz range, x-rays are much better than
neutrons, because their energy loss or gain in the Brillouin scattering
process is negligible. This advantage was established by the first x-ray
Brillouin measurements in vitreous silica [2, 3], for Giancarlo Ruocco the
start of an impressive scientific career.

The paper focuses on the neutron results in section 2 and the low frequency
limit of neutron and x-ray data in section 3. Section 4 contains the
conclusions.

##  2 Neutron scattering at the boson peak

The theoretical basis for neutron scattering studies of the boson peak was
supplied by Carpenter and Pelizzari [4], who treated the scattering from low
frequency sound waves in an elastically isotropic solid. They separated the
Brillouin scattering from longitudinal sound waves at low wavevector $Q$ (that
is, the one where x-rays are much better suited than neutrons) from the
$Q^{2}S(Q)$-signal at higher wavevectors, the Umklapp scattering, which
contains contributions from all sound waves.

A nice feature of the Umklapp scattering from sound waves is that its
intensity at a known temperature is given by the Debye density of states and
$S(Q)$, more precisely the elastic scattering $S(Q)\exp(-2W)$, which has the
same Debye-Waller factor as the one-phonon scattering.

If the low frequency vibrations are not sound waves, their scattering and its
wavevector dependence can be calculated from their density of states and their
eigenvector. This property has been used to identify the modes at the boson
peak of vitreous silica as coupled librations of SiO4-tetrahedra (see figure
1).

Figure 1: Neutron scattering from vitreous silica at higher temperatures as a
function of the wavevector $Q$ [5] (a) Elastic scattering (b) Inelastic
scattering in the boson peak window. There is a $Q$-independent multiple
scattering background, which can be disregarded. The sound wave scattering is
low (the continuous line). When multiplied with 6, one sees that it still does
not describe the data, which have the dynamic structure factor of coupled
rigid SiO4-tetrahedra.

Vitreous silica happens to have a very impressive boson peak, with an
intensity which is several times higher than the Debye level. Figure 1 (b)
compares the measured data [5] in the boson peak frequency window to the
predictions of Carpenter and Pelizzari [4] (the continuous line) for the sound
wave scattering. It is clearly seen that the measured signal is not only a
factor of four to five higher than the sound wave prediction, but that it also
has a strongly different wavevector dependence (the dashed line shows the
sound wave expectation multiplied by the factor six). It turns out that the
measured wavevector dependence is well described by coupled librations of
rigid SiO4-tetrahedra, a finding which has been corroborated in later
experimental [6] and numerical [7] work.

The dynamic structure factor of the boson peak vibrations has been also
measured in boron trioxide [8] and in polybutadiene [9]. In these two cases,
one finds a mixture of the sound wave structure factor with a simple
$Q^{2}$-dependence, with the sound wave structure factor fraction
extrapolating to the Debye density of states at zero frequency.

An interesting case is amorphous germanium, which does not have a boson peak
and markedly less tunneling states than other glasses [10]. Instead of a boson
peak, it has a low frequency tail of the lowest Van Hove singularity of the
crystal at 2 THz. In this tail, neutron data [11] find a mixture of sound
waves and of the bond bending motion of the lowest Van Hove singularity.

Figure 2: The boson peak in silica at 51 and 1673 K [5] (the glass transition
is at 1473 K).

The boson peak in vitreous silica has an unusual temperature dependence (see
figure 2), with a strong hardening at increasing temperature [5], being
continuous even through the glass transition at 1473 K. Figure 2 compares the
measured vibrational density of states [5] at 51 and 1673 K.

Figure 3: The boson peak in glycerol at 120, 170 and 212 K [12] (the glass
transition is at 187 K).

The usual temperature dependence of the boson peak, namely a softening with
increasing temperature which becomes even more marked above the glass
transition temperature $T_{\text{g}}$, is demonstrated in figure 3 for
glycerol [12].

##  3 Low frequency limit of neutron and x-ray Brillouin data

Though both neutron and x-ray inelastic scattering data are limited by the
instrumental resolution at low frequency, it has been possible to obtain
information in substances with a relatively high boson peak frequency. Figure
2 and figure 3 show that for low enough temperature the density of states
extrapolates to the Debye density of states, i.e., that the density
$g_{\text{s}}(\omega)$ of the additional soft modes tends to zero.

The way in which $g_{\text{s}}(\omega)$ disappears is better seen in heat
capacity data [13, 14], which show a $T^{5}$-increase, compatible with
$g_{\text{s}}(\omega)\propto\omega^{4}$.

This conclusion has been recently corroborated by numerical results [15, 16,
17]. In the first of these [15], it was observed that all these localized soft
modes have a positive fourth order term in their displacement potential. Both
findings, a $g_{\text{s}}(\omega)\propto\omega^{4}$ and a positive fourth
order term in the mode potential, are predictions of the soft potential model
[18] for the low temperature anomalies of glasses, a model which considers
tunneling states and soft vibrations as members of a common distribution.

For the x-ray Brillouin data, the soft potential model predicts a damping
$\Gamma\propto\omega^{4}$, which results from the rise of
$g_{\text{s}}(\omega)$

$\Gamma(\omega)={\rm v}_{l}l^{-1}_{\text{res,vib}}=\frac{\piup\omega
C_{l}}{8}\left(\frac{\hbar\omega}{W}\right)^{3}=\frac{\omega^{4}}{2\piup\omega_{\text{IR}}^{3}}\,,$
(3.1)

where $\Gamma(\omega)$ is the half width at half maximum of the Brillouin
line, ${\rm v}_{l}$ is the longitudinal sound velocity,
$l^{-1}_{\text{res,vib}}$ is the mean free path of the longitudinal waves
under the resonant interaction with the soft vibrational modes, $C_{l}$ is a
dimensionless constant, and $W$ is the crossover energy between tunneling
states and soft vibrations. Equation (3.1) defines the Ioffe-Regel frequency
$\omega_{\text{IR}}$, where $\Gamma=\omega/2\piup$ and the mean free path is
equal to the wavelength. Table 1 shows that the soft potential fits of the
plateau in the thermal conductivity for silica [18] and for glycerol [14]
predict the Ioffe-Regel limit at the boson peak for both glasses.

Table 1: Soft potential parameters and Ioffe-Regel limit for vitreous silica [18] and glycerol [14]. substance |  $W/k_{\text{B}}$, K |  $C$, $\times 10^{4}$ |  $v_{l}$, km/s |  $\hbar\omega_{\text{IR}}$, meV
---|---|---|---|---
silica | 3.9 | 2.6 | 5.8 | 3.9
glycerol | 4.3 | 1.9 | 3.6 | 4.8
Figure 4: Comparison of measured x-ray Brillouin damping [24] in vitreous
silica at 320 and 1620 K with the soft potential expectation of equation
(3.1).

An alternative explanation of a damping $\Gamma\propto\omega^{4}$ is Rayleigh
scattering from the static disorder. However, this explanation was found to be
too weak to explain the strong sound wave damping in glasses [19].

What one usually observes in x-ray Brillouin data, however, is not an
$\omega^{4}$-increase (or $Q^{4}$-increase, where $Q$ is the Brillouin
wavevector) of $\Gamma$, but rather $\Gamma\propto Q^{2}$, a behavior which
was explained by Walter Schirmacher, Giancarlo Ruocco and Tullio Scopigno [20]
in terms of a gaussian distribution of local shear restoring forces around the
average one, based on Schirmachers theory [21] of the plateau in the thermal
conductivity.

However, if one goes low enough in frequency, one begins to see deviations
from the $Q^{2}$-dependence. The first indications for an
$\omega^{4}$-dependence of $\Gamma$ were found in densified silica [22] and
lithium borate [23], where one does not have soft potential fits. However,
recent x-ray Brillouin scattering data [24] managed to find clear indications
for a $Q^{4}$-dependence of the damping $\Gamma$ of the longitudinal sound
waves in silica at the lowest accessible frequencies at 300 and 1620 K. These
data are compared with the soft potential prediction of equation (3.1) in
figure 4, taking the soft potential parameters from table 1.

The soft potential paper [18] does only determine an average coupling constant
for longitudinal and transverse sound waves, but for vitreous silica one knows
from measurements [25] below 1 K that $C$ is the same for longitudinal and
transverse waves.

In the case of glycerol, the x-ray Brillouin data [26] show a clearly defined
$Q^{4}$-damping, which extrapolates to the Ioffe-Regel limit at 5.2 meV, in
close agreement with the soft potential value of 4.8 meV in table 1.

##  4 Conclusions

In the foregoing, it was demonstrated that x-ray and neutron scattering can
provide ample and complementary information on the vibrational soft modes of
the boson peak. While x-rays are better capable of measuring the width and the
position of the Brillouin peaks, neutrons suit well to measure the vibrational
density of states as well as its temperature dependence. In addition, they are
capable of providing information on the eigenvectors, as demonstrated in
section 2 for vitreous silica.

The ongoing improvement of the x-ray Brillouin technique has made it possible
to measure the deviations from a damping proportional to $\omega^{2}$ at lower
frequencies in vitreous silica [24]. In the present paper, it was shown that
these deviations extrapolate to the low-frequency prediction of the soft
potential model. In glycerol, the measured $Q^{4}$-dependence of the x-ray
Brillouin damping is close to the soft potential prediction. The agreement
supports the hypothesis that the $Q^{4}$-damping in glasses is connected to
the $\omega^{4}$-rise of the density of vibrational non-phonon modes, seen in
the low-temperature heat capacity and in recent numerical work, related to the
tunneling states as postulated in the soft potential model.

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Äîñëäíèöüêèé öåíòð Þëõ, Þëõñüêèé öåíòð íåéòðîííèõ äîñëäæåíü (JCNS-1) òà
íñòèòóò ñêëàäíèõ ñèñòåì (ICS-1), 52425 Þëõ, Íìåèíà