# Relative Entropy of Hypersurfaces in Hyperbolic Space

Junfu Yao<EMAIL_ADDRESS>
###### Abstract.

We study a notion of relative entropy for certain hypersurfaces in hyperbolic
space. We relate this quantity to the renormalized area introduced by Graham-
Witten[RW99]. We also obtain a monotonicity formula for relative entropy
applied to mean curvature flows in hyperbolic space.

##  1\. Introduction

Let $n\geq 2$ and $\mathbb{H}^{n+1}$ denote the ($n+1$)-dimensional hyperbolic
space. We recall that the Poincaré half-space model for $\mathbb{H}^{n+1}$ is
the open upper half-space

$\mathbb{R}^{n+1}_{+}=\\{(\textbf{x},y)\in\mathbb{R}^{n}\times\mathbb{R}:\
y>0\\},$

equipped with the metric

$g_{P}=\frac{1}{y^{2}}(d\textbf{x}\otimes d\textbf{x}+dy\otimes
dy)=\frac{1}{y^{2}}g_{\mathbb{R}^{n+1}}.$

In this model, we have the standard compatification: $\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{H}\mkern-1.5mu}\mkern
1.5mu^{n+1}=\mathbb{H}^{n+1}\cup(\mathbb{R}^{n}\times\\{0\\})\cup\\{\infty\\}$.
The set $(\mathbb{R}^{n}\times\\{0\\})\cup\\{\infty\\}=\mathbb{S}^{n+1}$ is
called the ideal boundary of $\mathbb{H}^{n+1}$ and denoted by
$\partial_{\infty}\mathbb{H}^{n+1}$. The ideal boundary of $\mathbb{H}^{n+1}$
is associated to a unique conformal structure. A boundary defining function
$r$ on $\mkern 1.5mu\overline{\mkern-1.5mu\mathbb{H}\mkern-1.5mu}\mkern
1.5mu^{n+1}$ is a function such that the metric $r^{2}g_{P}$ extends smoothly
to $\mkern 1.5mu\overline{\mkern-1.5mu\mathbb{H}\mkern-1.5mu}\mkern
1.5mu^{n+1}$.

Minimal hypersurfaces in $\mathbb{H}^{n+1}$ are critical points of the
functional

(1.1) $\text{Vol}(\Sigma)=\int_{\Sigma}1\cdot
d\mathcal{H}^{n}_{\mathbb{H}^{n+1}}=\int_{\Sigma}\frac{1}{y^{n}}\cdot
d\mathcal{H}^{n}.$

Minimal hypersurfaces in hyperbolic space have been extensively studied, see
[And82], [And83], [Gab97], [dOS98], [MW13], [HL87], [Lin89], [Ton96], [Cos06],
[Cos11], [AM10] and [HW15].

As one may observe, there are no closed minimal hypersurfaces in
$\mathbb{H}^{n+1}$. So, it is natural to consider minimal hypersurfaces that
are asymptotic to the ideal boundary (See Definition 2.1 for the terminology).
However, the volume (1.1) is always divergent for any $\Sigma$ that has
asymptotic boundary. Let $r$ be any boundary defining function for $\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{H}\mkern-1.5mu}\mkern 1.5mu^{n+1}$. In
[RW99], Graham-Witten showed that for a sufficiently regular minimal
hypersurface $\Sigma$ and an even integer $n$, the volume of
$\\{r\geq\epsilon\\}\cap\Sigma$ has an expansion

(1.2)
$\text{Vol}_{\epsilon}(\Sigma;r)=\int_{\Sigma\cap\\{r\geq\epsilon\\}}1\cdot
d\mathcal{H}^{n}_{\mathbb{H}^{n+1}}=\frac{c_{0}}{\epsilon^{n-1}}+\frac{c_{2}}{\epsilon^{n-3}}+\cdots+\frac{c_{n-2}}{\epsilon}+c_{n}+o(1).$

If $n$ is odd, the expansion takes the following form:

(1.3)
$\text{Vol}_{\epsilon}(\Sigma;r)=\int_{\Sigma\cap\\{r\geq\epsilon\\}}1\cdot
d\mathcal{H}^{n}_{\mathbb{H}^{n+1}}=\frac{c_{0}}{\epsilon^{n-1}}+\frac{c_{2}}{\epsilon^{n-3}}+\cdots+\frac{c_{n-3}}{\epsilon^{2}}+c_{n-1}\log\frac{1}{\epsilon}+c_{n}+o(1).$

The coefficient $c_{n}$ is called the renormalized area of $\Sigma$ relative
to $r$, denoted by $\mathcal{A}(\Sigma;r)$. The authors in [RW99] further
showed that when $n$ is even, $\mathcal{A}(\Sigma;r)$ is conformally
invariant. That is, $\mathcal{A}(\Sigma;r)\equiv\mathcal{A}(\Sigma)$ is
independent of the choice of boundary defining functions. In [AM10], Alexakis-
Mazzeo proved that for a $C^{2}$-asymptotic surface $\Sigma$ in
$\mathbb{H}^{3}$, if $\Sigma$ meets $\partial_{\infty}\mathbb{H}^{3}$
orthogonally (which holds for all minimal surfaces), then
$\mathcal{A}(\Sigma)$ can be expressed as

$\mathcal{A}(\Sigma)=-2\pi\chi(\Sigma)-\frac{1}{2}\int_{\Sigma}|\hat{A}_{\Sigma}|^{2}d\mathcal{H}^{2}_{\mathbb{H}},$

where $\chi(\Sigma)$ is the Euler characteristic and $\hat{A}_{\Sigma}$ is the
trace-free second fundamental form of $\Sigma$. In [Ber22], Bernstein proved
an isoperimetric inequality for renormalized area. See also [Ngu21, Theorem 8]
for the same result using a different approach. Recently, Tyrrell [Tyr22]
derived a formula for renormalized area in $\mathbb{H}^{5}$.

We follow [BW22] to define a relative entropy of two minimal hypersurfaces
$\Sigma$ and $\Sigma^{\prime}$ with
$\partial_{\infty}\Sigma=\partial_{\infty}\Sigma^{\prime}$. Let $r$ be any
fixed boundary defining function. We consider

(1.4) $E^{(r)}_{rel}[\Sigma^{\prime},\Sigma]:=\lim_{\epsilon\to
0^{+}}\Big{(}\text{Vol}_{\epsilon}(\Sigma^{\prime};r)-\text{Vol}_{\epsilon}(\Sigma;r)\Big{)}.$

We will show that $E^{(r)}_{rel}[\Sigma^{\prime},\Sigma]$ is independent of
the choice of boundary defining functions, and that the relative entropy
$E^{(r)}_{rel}[\Sigma^{\prime},\Sigma]$ is finite and is equal to the
difference of $\mathcal{A}(\Sigma;r)$ and $\mathcal{A}(\Sigma^{\prime};r)$. It
is then worth noting that, although in odd dimensions, $\mathcal{A}(\Sigma;r)$
is in general dependent on the choice of boundary defining functions, the
difference, $\mathcal{A}(\Sigma^{\prime};r)-\mathcal{A}(\Sigma;r)$, is
conformally invariant.

The first result is the following:

######  Theorem 1.1.

Assume $\alpha\in(0,1)$. Let $M$ be a $C^{n,\alpha}$ submanifold in
$\partial_{\infty}\mathbb{H}^{n+1}$ and $r$ be a boundary defining function
for $\mathbb{H}^{n+1}$. Let $\Sigma$ and $\Sigma^{\prime}$ be two minimal
hypersurfaces in $\mathbb{H}^{n+1}$ that are both $C^{n,\alpha}$-asymptotic to
$M$ $($see $\mathrm{Definition}$ 2.1 for the precise definition$)$. Then
$E^{(r)}_{rel}[\Sigma^{\prime},\Sigma]$ is a finite number and is independent
of the choice of boundary defining function $r$. That is,
$E^{(r)}_{rel}[\Sigma^{\prime},\Sigma]\equiv E_{rel}[\Sigma^{\prime},\Sigma]$.
Moreover,

$E_{rel}[\Sigma^{\prime},\Sigma]=\mathcal{A}(\Sigma^{\prime};r)-\mathcal{A}(\Sigma;r).$

In Theorem 1.1, $C^{n,\alpha}$-asymptotic regularity is to guarantee the
existence of the expansions (1.2) and (1.3). However, when $n=2$, the authors
in [AM10] used the Gauss-Codazzi equations and the Gauss-Bonnet formula to
prove that for all $C^{2}$-asymptotic minimal surfaces, the renormalized area
is well-defined. So, we conjecture that $C^{n}$-asymptotic regularity is
sufficient to guarantee the existence of the renormalized area.

More generally, we shall prove that the relative entropy
$E^{(r)}_{rel}[\Sigma^{\prime},\Sigma]$ is well-defined for $\Sigma^{\prime}$
in a “thin” neighborhood of $\Sigma$. This will be treated in Section 3. Here,
by a “thin” neighborhood, we roughly mean that the volume of the domain
bounded by $\Sigma$ and $\Sigma^{\prime}$ is finite near the ideal boundary.
See Definition 2.2 for the precise statement.

It is also useful to study a weighted analog of the relative entropy for
hypersurfaces in a thin neighborhood of $\Sigma$. This allows us to develop a
variational theory for $E_{rel}$. In particular, a mountain pass theorem for
$E_{rel}$ is proved in [Yao]. We will discuss the weighted relative entropy in
Section 4.

Finally, we prove a monotonicity formula for the relative entropy of a mean
curvature flow.

######  Theorem 1.2.

Let $\\{\Sigma_{t}\\}_{t\in[0,T)}$ be a mean curvature flow in
$\mathbb{H}^{n+1}$, so that each $\Sigma_{t}$ is trapped between
$\Gamma^{\prime}_{-}$ and $\Gamma^{\prime}_{+}$. Assume that $\Sigma_{0}$ is
$C^{2}$-asymptotic to a closed submanifold
$M\subset\partial_{\infty}\mathbb{H}^{n+1}$. Then, there exists an
$E_{0}=E_{0}(\Sigma_{0},\Gamma^{\prime}_{-},\Gamma^{\prime}_{+})>0$, so that
for any $0\leq t\leq s<T$, we have

$E_{rel}[\Sigma_{t},\Sigma]\geq E_{rel}[\Sigma_{s},\Sigma]\geq-E_{0}.$

### Acknowledgement

The author would like to thank his advisor, Jacob Bernstein, who made many
useful comments on various stages of the work and suggested the computation
that appears in the appendix. The author also thanks Letian Chen for
stimulating discussions.

##  2\. Some Preliminaries

Let $M$ be an $(n-1)$-dimensional embedded (smooth) submanifold in
$\partial_{\infty}\mathbb{H}^{n+1}$. If we fix any point
$p_{0}\in\mathbb{H}^{n+1}$, then there is an isometry $i:\
\mathbb{H}^{n+1}\to\mathbb{R}^{n+1}_{+}$, so that $i(p_{0})=(\textbf{0},1)$
and $i^{*}g_{P}=g_{\mathbb{H}^{n+1}}$. Since all such isometries of
$\mathbb{H}^{n+1}$ correspond to the Möbius transformations on
$\partial_{\infty}\mathbb{H}^{n+1}=\mathbb{S}^{n+1}$, we can assume that
$M\subset\mathbb{R}^{n}\times\\{0\\}$ after a suitable transformation.

######  Definition 2.1.

A complete embedded hypersurface $\Sigma\subset\mathbb{H}^{n+1}$ is called
$C^{k,\alpha}$-$asymptotic$ to $M$ if $\Sigma\cup M$ (denoted by $\mkern
1.5mu\overline{\mkern-1.5mu\Sigma\mkern-1.5mu}\mkern 1.5mu$) is a
$C^{k,\alpha}$ hypersurface in $\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{R}_{+}^{n+1}\mkern-1.5mu}\mkern 1.5mu$. $M$
is also denoted by $\partial_{\infty}\Sigma$.

For convenience, all the unspecified notations below are understood in
$\mathbb{R}^{n+1}_{+}$ with metric $g_{P}$. Notations with a bar on the top
are understood in $\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{R}^{n+1}_{+}\mkern-1.5mu}\mkern
1.5mu\cup\\{\infty\\}=\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{H}^{n+1}\mkern-1.5mu}\mkern 1.5mu$ with
metric $r^{2}g_{P}$. For example, for a hypersurface $\Sigma$, $g_{\Sigma}$
(resp. $\mkern 1.5mu\overline{\mkern-1.5mug_{\Sigma}\mkern-1.5mu}\mkern
1.5mu$) is the Riemannian metric induced from $g_{P}$ (resp. $r^{2}g_{P}$),
and $\text{div}_{\Sigma}$ (resp. $\mkern
1.5mu\overline{\mkern-1.5mu\text{div}\mkern-1.5mu}\mkern 1.5mu_{\Sigma}$) is
referred to the divergence taken with respect to $g_{\Sigma}$ (resp. $\mkern
1.5mu\overline{\mkern-1.5mug_{\Sigma}\mkern-1.5mu}\mkern 1.5mu$). To avoid
misunderstanding, for a set $S\subset\mathbb{R}^{n+1}_{+}$, $\mkern
1.5mu\overline{\mkern-1.5muS\mkern-1.5mu}\mkern 1.5mu$ is referred to the
topological closure in $\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{R}^{n+1}_{+}\mkern-1.5mu}\mkern
1.5mu\cup\\{\infty\\}=\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{H}^{n+1}\mkern-1.5mu}\mkern 1.5mu$, and the
closure taken in $\mathbb{R}^{n+1}_{+}$ with metric $g_{P}$ is denoted by
$\text{cl}(S)$. For any two vectors X and Y, we think of them as vectors in
Euclidean space. So, the addition and substraction make sense. The vector norm
in Euclidean space will be denoted by $|\cdot|$ and that in hyperbolic space
will be denoted by $|\cdot|_{\mathbb{H}}$. The inner product
$\left\langle\textbf{X},\textbf{Y}\right\rangle$ is taken with respect to
$g_{P}$, and $\textbf{X}\cdot\textbf{Y}$ is the inner product in Euclidean
space.

######  Definition 2.2.

Given a hypersurface $\Sigma$ which is $C^{2}$-asymptotic to
$M\subset\mathbb{R}^{n}\times\\{0\\}\subset\partial_{\infty}\mathbb{H}^{n+1}$
and meets $M$ orthogonally, a bounded open set
$\Omega\subset\mathbb{R}^{n+1}_{+}$ is called a thin neighborhood of $\Sigma$,
if there are constants $C,\epsilon>0$, such that
$\Omega\cap\\{y=y_{0}\\}\subset\mkern
1.5mu\overline{\mkern-1.5mu\mathcal{N}\mkern-1.5mu}\mkern
1.5mu_{C_{1}y_{0}^{n+1}}(\Sigma)$ for $0<y_{0}<\epsilon$. Here, $\mkern
1.5mu\overline{\mkern-1.5mu\mathcal{N}\mkern-1.5mu}\mkern
1.5mu_{\delta}(\Sigma)$ is the $\delta$-tubular neighborhood of $\Sigma$
considered in Euclidean topology.

Now, let us fix a minimal hypersurface $\Sigma$ with $C^{2}$-asymptotic
boundary $M$. Let $\Omega$ be a thin neighborhood of $\Sigma$. For simplicity,
We choose hypersurfaces $\Gamma^{\prime}_{\pm}$ and open sets
$U_{\Gamma^{\prime}_{\pm}}\subset\mathbb{R}^{n+1}_{+}$ satisfying $\partial
U_{\Gamma^{\prime}_{\pm}}=\Gamma^{\prime}_{\pm}$ and
$U_{\Gamma^{\prime}_{+}}\backslash\text{cl}(U_{\Gamma^{\prime}_{-}})=\Omega$.
In particular, we can choose $\Gamma^{\prime}_{\pm}$ to be $C^{2}$-asymptotic
and orthogonal to $M$. Then, $\Omega$ is quasi-convex in the sense that for
any two points $p_{1}$, $p_{2}$ in $\Omega$, there is a path $\gamma$
connecting them, so that the inequality

$\text{Length}(\gamma)\leq C|\textbf{x}(p_{1})-\textbf{x}(p_{2})|$

holds for some constant $C$ independent of the choice of points. Here, the
length is measured with respect to Euclidean metric, and $\textbf{x}(p_{i})$
is the position vector. We fix this choice of $\Gamma^{\prime}_{\pm}$
throughout the article. With $\Gamma^{\prime}_{-}$ and $\Gamma^{\prime}_{+}$
illustrated above, we define

$\mathcal{C}(\Gamma^{\prime}_{-},\Gamma^{\prime}_{+}):=\\{U:\,U\text{ is a
Caccioppoli set}\text{ and }U_{\Gamma^{\prime}_{-}}\subset U\subset
U_{\Gamma^{\prime}_{+}}\\}.$

We then discuss the space of admissible weights.

######  Definition 2.3.

A function $\psi=\psi(p,\textbf{v})$ over
$\text{cl}(\Omega)\times\mathbb{S}^{n}$ is said to be admissible, if

  * •

$\psi$ and $\nabla_{\mathbb{S}^{n}}\psi$ are locally Lipschitz functions over
$\text{cl}(\Omega)\times\mathbb{S}^{n}$;

  * •

$\psi(p,\textbf{v})=\psi(p,-\textbf{v})$;

  * •

$\|\psi\|_{\mathfrak{X}}<\infty$,

where
$\|\psi\|_{\mathfrak{X}}:=\|\psi\|_{\infty}+\|\nabla_{\mathbb{S}^{n}}\psi\|_{\infty}+\|\nabla_{\mathbb{S}^{n}}\nabla_{\mathbb{S}^{n}}\psi\|_{\infty}+\|y\nabla_{\mathbb{R}^{n+1}_{+}}\psi\|_{\infty}+\|y\nabla_{\mathbb{R}^{n+1}_{+}}\nabla_{\mathbb{S}^{n}}\psi\|_{\infty}$.
The space of all admissible weights is denoted by $\mathfrak{X}$.

Since $\Omega$ is quasi-convex, the sup-norm of the gradient is equivalent to
the Lipschitz constant defined by difference quotients. Since
$\psi(p,\textbf{v})=\psi(p,-\textbf{v})$, it can be viewed naturally as a
function over the Grassman $n$-plane bundle of $\Omega$. It can also be
checked that the space $\mathfrak{X}$ is a Banach space and an algebra.

The main tool used in the discussion of the relative entropy is the divergence
theorem applied to an appropriate vector field. Since $\Sigma$ is
$C^{2}$-asymptotic to $M$, we choose a small $\delta>0$, so that the normal
projection $\Pi:\ \mkern
1.5mu\overline{\mkern-1.5mu\mathcal{N}\mkern-1.5mu}\mkern
1.5mu_{\delta}(\mkern 1.5mu\overline{\mkern-1.5mu\Sigma\mkern-1.5mu}\mkern
1.5mu)\rightarrow\mkern 1.5mu\overline{\mkern-1.5mu\Sigma\mkern-1.5mu}\mkern
1.5mu$ is $C^{1}$. We fix a small $\epsilon$, so that
$\text{cl}(\Omega)\cap\\{y<\epsilon\\}$ is contained in $\mkern
1.5mu\overline{\mkern-1.5mu\mathcal{N}\mkern-1.5mu}\mkern
1.5mu_{\delta}(\Sigma)$. Then,

(2.1)
$\textbf{X}(q)=\textbf{n}_{\Sigma}\circ\Pi(q)=y\big{(}\Pi(q)\big{)}\mkern
1.5mu\overline{\mkern-1.5mu\textbf{n}\mkern-1.5mu}\mkern
1.5mu_{\Sigma}\circ\Pi(q)$

is a vector field that is well-defined in
$\text{cl}(\Omega)\cap\\{y<\epsilon\\}$. Let $\mkern
1.5mu\overline{\mkern-1.5mu\textbf{X}\mkern-1.5mu}\mkern
1.5mu=\frac{1}{y}\textbf{X}$. This implies $\mkern
1.5mu\overline{\mkern-1.5mu\text{div}\mkern-1.5mu}\mkern 1.5mu\mkern
1.5mu\overline{\mkern-1.5mu\textbf{X}\mkern-1.5mu}\mkern 1.5mu$ is bounded in
$\Omega\cap\\{y<\epsilon\\}$. Let $X^{(n+1)}=\textbf{X}\cdot\mkern
1.5mu\overline{\mkern-1.5mu\textbf{e}\mkern-1.5mu}\mkern 1.5mu_{n+1}=\mkern
1.5mu\overline{\mkern-1.5mu\textbf{e}\mkern-1.5mu}\mkern
1.5mu_{n+1}\cdot(\textbf{n}_{\Sigma}\circ\Pi)$ be the ($n+1$)-th component of
X. Then, the assumption that $\Sigma$ meets
$\partial_{\infty}\mathbb{H}^{n+1}$ orthogonally gives

(2.2) $\displaystyle|X^{(n+1)}|=y\big{(}\Pi(q)\big{)}|\mkern
1.5mu\overline{\mkern-1.5mu\textbf{n}\mkern-1.5mu}\mkern
1.5mu^{(n+1)}_{\Sigma}\circ\Pi(q)|\leq Cy^{2}\big{(}\Pi(q)\big{)}\leq
C^{\prime}y^{2}(q).$

By a direct computation, we have

(2.3) $|\text{div}\textbf{X}|=|\mkern
1.5mu\overline{\mkern-1.5mu\text{div}\mkern-1.5mu}\mkern 1.5mu(y\mkern
1.5mu\overline{\mkern-1.5mu\textbf{X}\mkern-1.5mu}\mkern
1.5mu)-\frac{n+1}{y}X^{(n+1)}|\leq y|\mkern
1.5mu\overline{\mkern-1.5mu\text{div}\mkern-1.5mu}\mkern 1.5mu\mkern
1.5mu\overline{\mkern-1.5mu\textbf{X}\mkern-1.5mu}\mkern
1.5mu|+\frac{C}{y}|X^{(n+1)}|\leq Cy.$

Since $\Omega$ is a thin neighborhood,

(2.4)
$\mathcal{H}^{n}_{\mathbb{H}^{n+1}}(\Omega\cap\\{y=y_{1}\\})=\frac{1}{y_{1}^{n}}\mathcal{H}^{n}(\Omega\cap\\{y=y_{1}\\})\leq\frac{1}{y_{1}^{n}}\cdot
Cy_{1}^{n+1}=Cy_{1}.$

We shall also consider a family of cut-off functions. Let
$\phi_{t_{1},\delta}(t)$ be a smooth cut-off function that is identically $1$
when $t\geq t_{1}$, and vanishes for $t\leq t_{1}-\delta$. We can further
require that $|\phi^{\prime}_{t_{1},\delta}|\leq\frac{C}{\delta}$, where $C$
is independent of $\delta$ and $t_{1}$. Let
$\phi_{t_{1},t_{2},\delta}:=\phi_{t_{1},\delta}-\phi_{t_{2}+\delta,\delta}$,
for any $t_{1}<t_{2}$ with $0<\frac{t_{1}}{2}<t_{1}-\delta$ and
$t_{2}+\delta<2t_{2}<\epsilon$. Fix a boundary defining function $r$. Define
$\phi^{(r)}_{t_{1},\delta}=\phi_{t_{1},\delta}\circ r$ and
$\phi^{(r)}_{t_{1},t_{2},\delta}=\phi_{t_{1},t_{2},\delta}\circ r$. It is
readily checked that $\frac{1}{C}\leq|\mkern
1.5mu\overline{\mkern-1.5mu\nabla\mkern-1.5mu}\mkern 1.5mur|\leq C$ near
$\partial_{\infty}\mathbb{H}^{n+1}$ for some constant $C>0$. Since $\Sigma$
meets $M$ orthogonally and $\mkern
1.5mu\overline{\mkern-1.5mu\nabla\mkern-1.5mu}\mkern 1.5mur$ is perpendicular
to $\partial_{\infty}\mathbb{H}^{n+1}$ on $M$, we have
$\frac{1}{2C}\leq|\mkern 1.5mu\overline{\mkern-1.5mu\nabla\mkern-1.5mu}\mkern
1.5mu_{\mkern 1.5mu\overline{\mkern-1.5mu\Sigma\mkern-1.5mu}\mkern
1.5mu}r|\leq 2C$ near $M$.

By choosing $\epsilon$ sufficiently small, we can assume that in
$\Omega\cap\\{r\leq\epsilon\\}$, $\frac{y}{C}\leq r(\textbf{x},y)\leq Cy$.
Hence, (2.2) and (2.3) imply that $|\text{div}\textbf{X}|\leq Ct_{1}$ and
$|\textbf{X}^{(n+1)}|\leq Ct_{1}^{2}$ on $\Omega\cap\\{r=t_{1}\\}$ with
$t_{1}\leq\epsilon$. Since $\mkern
1.5mu\overline{\mkern-1.5mu\nabla\mkern-1.5mu}\mkern 1.5mur$ is perpendicular
to $\partial_{\infty}\mathbb{H}^{n+1}$, we also have
$\mathcal{H}^{n}_{\mathbb{H}}(\Omega\cap\\{r=t_{1}\\})\leq Ct_{1}$ on
$\Omega\cap\\{r=t_{1}\\}$.

Let $U_{\Gamma}\in\mathcal{C}(\Gamma^{\prime}_{-},\Gamma^{\prime}_{+})$ with
$\Gamma=\partial^{*}U_{\Gamma}$ being the reduced boundary of $U_{\Gamma}$.
Consider

$E_{rel}[\Gamma,\Sigma;\phi^{(r)}_{t_{1},\delta}]:=\int_{\Gamma}\phi^{(r)}_{t_{1},\delta}d\mathcal{H}^{n}_{\mathbb{H}^{n+1}}-\int_{\Sigma}\phi^{(r)}_{t_{1},\delta}d\mathcal{H}^{n}_{\mathbb{H}^{n+1}}.$

Letting $\delta\to 0^{+}$, it follows from the dominated convergence theorem
that

$E_{rel}[\Gamma,\Sigma;\chi_{\\{r\geq t_{1}\\}}]=\lim_{\delta\to
0^{+}}E_{rel}[\Gamma,\Sigma;\phi^{(r)}_{t_{1},\delta}]=\mathcal{H}^{n}_{\mathbb{H}^{n+1}}(\Gamma\cap\\{r\geq
t_{1}\\})-\mathcal{H}^{n}_{\mathbb{H}^{n+1}}(\Sigma\cap\\{r\geq t_{1}\\}).$

##  3\. The Existence of Relative Entropy

In this section, we prove Theorem 1.1. As a first step, we prove that the
relative entropy is well-defined for all the reduced boundaries of Caccioppoli
sets in $\mathcal{C}(\Gamma^{\prime}_{-},\Gamma^{\prime}_{+})$.

######  Theorem 3.1.

For any $U\in\mathcal{C}(\Gamma^{\prime}_{-},\Gamma^{\prime}_{+})$ and any
boundary defining function $r$,

$E^{(r)}_{rel}[\partial^{*}U,\Sigma]=\lim_{\epsilon\to
0^{+}}\big{(}\mathcal{H}^{n}_{\mathbb{H}}(\partial^{*}U\cap\\{r\geq\epsilon\\})-\mathcal{H}^{n}_{\mathbb{H}}(\Sigma\cap\\{r\geq\epsilon\\})\big{)}$

exists $(\in(-\infty,\infty])$ and $E^{(r)}_{rel}[\partial^{*}U,\Sigma]\equiv
E_{rel}[\partial^{*}U,\Sigma]$ is independent of $r$. Moreover,
$E_{rel}[\partial^{*}U,\Sigma]\geq-E_{0}$ for $E_{0}=E_{0}(\Sigma,\Omega)$.

###### Proof.

Since $\mkern 1.5mu\overline{\mkern-1.5mu\Sigma\mkern-1.5mu}\mkern 1.5mu$ is
an embedded hypersurface in $\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{R}^{n+1}_{+}\mkern-1.5mu}\mkern 1.5mu$, it
is orientable. Thus, we can choose an open set
$U_{\Sigma}\subset\mathbb{R}^{n+1}_{+}$ satisfying
$\Sigma=\text{cl}(U_{\Sigma})\backslash U_{\Sigma}$ and
$U_{\Gamma^{\prime}_{-}}\subset U_{\Sigma}\subset U_{\Gamma^{\prime}_{+}}$.

Fix $t_{1}$, $t_{2}$ and $\delta$ with
$0<\frac{t_{1}}{2}<t_{1}-\delta<t_{1}<t_{2}<t_{2}+\delta<2t_{2}<\epsilon$.
Combining the computation in Section 2 and applying the divergence theorem to
the vector field X, we get

(3.1)
$\displaystyle\int_{\partial^{*}U}\phi^{(r)}_{t_{1},t_{2},\delta}d\mathcal{H}^{n}_{\mathbb{H}}-\int_{\Sigma}\phi^{(r)}_{t_{1},t_{2},\delta}d\mathcal{H}^{n}_{\mathbb{H}}\geq\int_{\partial^{*}U}\left\langle\phi^{(r)}_{t_{1},t_{2},\delta}\textbf{X},\textbf{n}_{\partial^{*}U}\right\rangle
d\mathcal{H}^{n}_{\mathbb{H}}-\int_{\Sigma}\left\langle\phi^{(r)}_{t_{1},t_{2},\delta}\textbf{X},\textbf{n}_{\Sigma}\right\rangle
d\mathcal{H}^{n}_{\mathbb{H}}$ (3.2) $\displaystyle=\int_{U\backslash
U_{\Sigma}}\text{div}\left(\phi^{(r)}_{t_{1},t_{2},\delta}\textbf{X}\right)d\mathcal{H}^{n+1}_{\mathbb{H}}-\int_{U_{\Sigma}\backslash
U}\text{div}\left(\phi^{(r)}_{t_{1},t_{2},\delta}\textbf{X}\right)\mathcal{H}^{n+1}_{\mathbb{H}}$
(3.3)
$\displaystyle\geq-2\int_{\Omega}\left(\phi^{(r)}_{t_{1},t_{2},\delta}|\text{div}\textbf{X}|+|\left\langle\nabla\phi^{(r)}_{t_{1},t_{2},\delta},\textbf{X}\right\rangle|\right)d\mathcal{H}^{n+1}_{\mathbb{H}}$
(3.4)
$\displaystyle\geq-C\int_{\Omega\cap\\{t_{1}-\delta<r<t_{2}+\delta\\}}rd\mathcal{H}^{n+1}_{\mathbb{H}}-C\int_{\Omega\cap\left(\\{t_{1}-\delta<r<t_{1}\\}\cup\\{t_{2}<t<t_{2}+\delta\\}\right)}\frac{1}{\delta}|\mkern
1.5mu\overline{\mkern-1.5mu\nabla\mkern-1.5mu}\mkern
1.5mur\cdot\textbf{X}|d\mathcal{H}^{n+1}_{\mathbb{H}}$

Since $\mkern 1.5mu\overline{\mkern-1.5mu\nabla\mkern-1.5mu}\mkern 1.5mur$ is
perpendicular to $\mathbb{R}^{n}\times\\{0\\}$, we have $|\mkern
1.5mu\overline{\mkern-1.5mu\nabla\mkern-1.5mu}\mkern
1.5mur\cdot\textbf{X}|\leq C|\textbf{X}^{(n+1)}|\leq Cr^{2}$. Using the co-
area formula for (3.4), we get

(3.5)
$\int_{\Omega\cap\\{t_{1}-\delta<r<t_{2}+\delta\\}}rd\mathcal{H}^{n+1}_{\mathbb{H}}=\int_{t_{1}-\delta}^{t_{2}+\delta}dt\int_{\Omega\cap\\{r=t\\}}\frac{r}{y^{n+1}|\mkern
1.5mu\overline{\mkern-1.5mu\nabla\mkern-1.5mu}\mkern
1.5mur|}d\mathcal{H}^{n}\leq Ct^{2}_{2},$

and similarly,

(3.6)
$\int_{\Omega\cap\left(\\{t_{1}-\delta<r<t_{1}\\}\cup\\{t_{2}<t<t_{2}+\delta\\}\right)}\frac{1}{\delta}|\mkern
1.5mu\overline{\mkern-1.5mu\nabla\mkern-1.5mu}\mkern
1.5mur\cdot\textbf{X}|d\mathcal{H}^{n+1}_{\mathbb{H}}\leq Ct^{2}_{2}.$

Hence, (3.4)$\geq-Ct_{2}$. After rewriting (3.1), we see that

$E_{rel}[\partial^{*}U,\Sigma;\phi^{(r)}_{t_{1},\delta}]\geq
E_{rel}[\partial^{*}U,\Sigma;\phi^{(r)}_{t_{2}+\delta,\delta}]-Ct_{2}.$

Since $U$ has locally finite perimeter, $\partial^{*}U$ has finite
$\mathcal{H}^{n}$ measure on each compact set in $\mathbb{R}^{n+1}_{+}$.
Letting $\delta\rightarrow 0^{+}$, the dominated convergence theorem gives

(3.7) $E_{rel}[\partial^{*}U,\Sigma;\chi_{\\{r\geq t_{1}\\}}]\geq
E_{rel}[\partial^{*}U,\Sigma;\chi_{\\{r\geq t_{2}\\}}]-Ct_{2},$

and this implies $\liminf_{t_{1}\rightarrow
0^{+}}E_{rel}[\partial^{*}U,\Sigma;\chi_{\\{r\geq
t_{1}\\}}]\geq\limsup_{t_{2}\rightarrow
0^{+}}E_{rel}[\partial^{*}U,\Sigma;\chi_{\\{r\geq t_{2}\\}}]$. In other words,
$E^{(r)}_{rel}[\partial^{*}U,\Sigma]$ is well-defined.

Fix $t_{2}<\epsilon$ and let $t_{1}\to 0^{+}$. We get

$\limsup_{t_{1}\rightarrow 0^{+}}E_{rel}[\partial^{*}U,\Sigma;\chi_{\\{r\geq
t_{1}\\}}]\geq E_{rel}[\partial^{*}U,\Sigma;\chi_{\\{r\geq
t_{2}\\}}]-Ct_{2}>-\infty.$

That is, $E^{(r)}_{rel}[\partial^{*}U,\Sigma]>-\infty$. Notice that

$E_{rel}[\partial^{*}U,\Sigma;\chi_{\\{r\geq
t_{2}\\}}]-Ct_{2}\geq-\mathcal{H}^{n}_{\mathbb{H}}(\Sigma\cap\\{r\geq
t_{2}\\})-Ct_{2}.$

We can take $E_{0}=\mathcal{H}^{n}_{\mathbb{H}}(\Sigma\cap\\{r\geq
t_{2}\\})+Ct_{2}$.

Finally, given another boundary defining function $\widetilde{r}$, we show
$E^{(r)}_{rel}[\partial^{*}U,\Sigma]=E^{(\widetilde{r})}_{rel}[\partial^{*}U,\Sigma]$.
We have shown that both $E^{(r)}_{rel}[\partial^{*}U,\Sigma]$ and
$E^{(\widetilde{r})}_{rel}[\partial^{*}U,\Sigma]$ are well-defined. We pick a
sequence $\\{t_{i}\\}_{i=1}^{\infty}$ converging to $0$ and appropriately
choose another sequence $\\{s_{i}\\}_{i=1}^{\infty}$ satisfying $s_{i}\to 0$
and $\Omega\cap\\{r\geq t_{i}\\}\subset\Omega\cap\\{\widetilde{r}\geq
s_{i}\\}$. For $i$ sufficiently large, we choose $\delta$ sufficiently small,
so that
$\psi_{i}^{\delta}=\phi^{(\widetilde{r})}_{s_{i},\delta}-\phi^{(r)}_{t_{i}+\delta,\delta}$
is well-defined. Then, using a computation similar to (3.1)-(3.6),

$\displaystyle\int_{\partial^{*}U}\psi_{i}^{\delta}d\mathcal{H}^{n}_{\mathbb{H}}-\int_{\Sigma}\psi_{i}^{\delta}d\mathcal{H}^{n}_{\mathbb{H}}\geq-2\int_{\Omega}\left(\psi_{i}^{\delta}|\text{div}\textbf{X}|+|\left\langle\nabla\psi_{i}^{\delta},\textbf{X}\right\rangle|\right)d\mathcal{H}^{n+1}_{\mathbb{H}}$
$\displaystyle\geq$
$\displaystyle-C\mathcal{H}^{n+1}_{\mathbb{H}}(\Omega\cap\\{r<t_{i}+\delta\\})-\frac{C}{\delta}\mathcal{H}^{n+1}_{\mathbb{H}}(\Omega\cap\\{t_{i}<r<t_{i}+\delta\\})-\frac{C}{\delta}\mathcal{H}^{n+1}_{\mathbb{H}}(\Omega\cap\\{s_{i}-\delta<\widetilde{r}<s_{i}\\})$
$\displaystyle\geq$ $\displaystyle-C(t_{i}+s_{i}).$

This implies
$E_{rel}[\partial^{*}U,\Sigma;\phi^{(\widetilde{r})}_{s_{i},\delta}]\geq
E_{rel}[\partial^{*}U,\Sigma;\phi^{(r)}_{t_{i}+\delta,\delta}]-C(t_{i}+s_{i})$.
Letting $\delta\to 0$,

$E_{rel}[\partial^{*}U,\Sigma;\chi_{\\{\widetilde{r}\geq s_{i}\\}}]\geq
E_{rel}[\partial^{*}U,\Sigma;\chi_{\\{r\geq t_{i}\\}}]-C(t_{i}+s_{i}).$

Letting $i\to\infty$, we set
$E^{(\widetilde{r})}_{rel}[\partial^{*}U,\Sigma]\geq
E^{(r)}_{rel}[\partial^{*}U,\Sigma]$. Interchanging $r$ and $\widetilde{r}$
gives the reverse inequality. Since $r$ and $\widetilde{r}$ are arbitrary, we
have $E^{(r)}_{rel}[\partial^{*}U,\Sigma]\equiv E_{rel}[\partial^{*}U,\Sigma]$
for any boundary defining function $r$. ∎

######  Remark 3.2.

We see from the proof of Theorem 3.1 that $E_{rel}[\partial^{*}U,\Sigma]$ is
still well-defined, even if we replace the condition
$\Omega\cap\\{y=y_{0}\\}\subset\mkern
1.5mu\overline{\mkern-1.5mu\mathcal{N}\mkern-1.5mu}\mkern
1.5mu_{C_{1}y_{0}^{n+1}}(\Sigma)$ by $\Omega\cap\\{y=y_{0}\\}\subset\mkern
1.5mu\overline{\mkern-1.5mu\mathcal{N}\mkern-1.5mu}\mkern
1.5mu_{C_{1}y_{0}^{n}}(\Sigma)$. However, the thin neighborhood allows us to
define the weighted relative entropy for a broader class of weights.

Let $M$ be an ($n-1$)-dimensional $C^{m}$ closed submanifold in
$\mathbb{R}^{n}\times\\{0\\}$ with $1\leq m\leq n$. After a suitable
transformation, we may assume that $\textbf{0}\in M$ and that the normal
vector of $M$ at 0 in $\mathbb{R}^{n}\times\\{0\\}$ is equal to $\mkern
1.5mu\overline{\mkern-1.5mu\textbf{e}_{n}\mkern-1.5mu}\mkern
1.5mu=(0,\cdots,0,1,0)$. Let $\Sigma$ be a minimal hypersurface with
$C^{m}$-asymptotic boundary $M$. Since $\Sigma$ meets
$\mathbb{R}^{n}\times\\{0\\}$ orthogonally, we see that for a small
$\delta>0$, there is a function $u=u(\textbf{x}^{\prime},y)$ in
$C^{m}\Big{(}(-\delta,\delta)^{n-1}\times[0,\delta)\Big{)}$, where
$\textbf{x}^{\prime}$ is the first $(n-1)$ coordinates and $y$ is the
$(n+1)$-th coordinate, so that

(3.8)
$\displaystyle\Sigma\cap\Big{(}(-\delta,\delta)^{n}\times[0,\delta)\Big{)}=\\{(\textbf{x}^{\prime},x_{n},y):\
x_{n}=u(\textbf{x}^{\prime},y),\
(\textbf{x}^{\prime},y)\in(-\delta,\delta)^{n-1}\times[0,\delta)\\}.$

The orthogonality condition implies
$\partial_{y}u(\textbf{x}^{\prime},0)=\textbf{0}$. Since $\Sigma$ is minimal
in $\mathbb{H}^{n+1}$, $u$ satisfies the equation

(3.9)
$\displaystyle\partial_{yy}u+\sum_{i=1}^{n-1}\partial_{ii}u-\sum_{j,k\in\\{1,\cdots,n-1,y\\}}\frac{\partial_{j}u\,\partial_{k}u\,\partial_{jk}u}{1+(\partial_{y}u)^{2}+\sum_{i=1}^{n-1}(\partial_{i}u)^{2}}-\frac{n}{y}\partial_{y}u=0.$

######  Lemma 3.3.

Assume $M$ is $C^{m}$ with $1\leq m\leq n$ and $\Sigma$ is a minimal
hypersurface that is $C^{m}$-asymptotic to $M$. If we write $\Sigma$ locally
as a graph as in $(\ref{44})$, then
$\partial^{(i)}_{y}u(\mathbf{x}^{\prime},0)$ is determined by $M$ for $0\leq
i\leq k$.

###### Proof.

Assume $m\geq 2$, otherwise there is nothing to prove. Fix $1\leq k\leq m-1$.
Assume that $\partial^{(i)}_{y}u(\textbf{x}^{\prime},0)$ is determined by the
boundary data, $u(\textbf{x}^{\prime},0)$, for $1\leq i\leq k$, which is true
for $k=1$. We prove by induction that the $(k+1)$-th $y$-derivative at $y=0$
is also determined by the boundary data. Multiply both sides of (3.9) by $y$
and take the $(k-1)$-th $y$-derivative on both sides. We get

$\displaystyle
y\partial^{(k+1)}_{y}u(\textbf{x}^{\prime},y)+(k-1-n)\partial^{(k)}_{y}u(\textbf{x}^{\prime}y)+\frac{y\partial_{y}u\,\partial_{y}u\,\partial^{(k+1)}_{y}u}{1+(\partial_{y}u)^{2}+\sum_{i=1}^{n-1}(\partial_{i}u)^{2}}+f+g\partial^{(k)}_{y}u=0,$

where $f$ and $g$ are rational functions of
$\partial^{\alpha_{1}}_{x_{1}}\cdots\partial^{\alpha_{n-1}}_{x_{n-1}}\partial^{(i)}_{y}u$
($1\leq i\leq k-1$) with $g\big{|}_{y=0}=0$. As a result,

$\displaystyle(k-n)\partial^{(k+1)}_{y}u(\textbf{x}^{\prime},0)+\partial_{y}f+\partial_{y}g\,\partial^{(k)}_{y}u=0,\text{
for }\textbf{x}^{\prime}\in(-\delta,\delta)^{n-1}.$

Hence, by induction, $\partial^{(i)}_{y}u(\textbf{x}^{\prime},0)$ is
determined by the boundary data when $1\leq i\leq m$. ∎

Let $\Gamma_{1}$ and $\Gamma_{2}$ be two minimal hypersurfaces that are both
$C^{3}$-asymptotic to $M\subset\mathbb{R}^{n}\times\\{0\\}$. In the next
lemma, we show that $\Gamma_{2}$ is in a thin neighborhood of $\Gamma_{1}$.

######  Lemma 3.4.

Suppose $\Gamma_{1}$, $\Gamma_{2}$ are two minimal hypersurfaces in
$\mathbb{H}^{n+1}$ that are both $C^{3}$-asymptotic to
$M\subset\mathbb{R}^{n}\times\\{0\\}\subset\partial_{\infty}\mathbb{H}^{n+1}$.
Then, there exist an $\epsilon_{1}>0$ and a function $h$, so that

(3.10)
$\Gamma_{2}\cap\\{y\leq\frac{\epsilon_{1}}{2}\\}\subset\big{\\{}p+h(p)\mkern
1.5mu\overline{\mkern-1.5mu\mathbf{n}\mkern-1.5mu}\mkern
1.5mu_{\Gamma_{1}}:p\in\Gamma_{1}\cap\\{y\leq\epsilon_{1}\\}\big{\\}}\subset\Gamma_{2}\cap\\{y\leq
2\epsilon_{1}\\},$

where $h$ satisfies the estimate $|h(p)|\leq Cy^{n+1}(p)$ for $C>0$ and
$p\in\Gamma_{1}\cap\\{y\leq\epsilon_{1}\\}$.

###### Proof.

Since both $\Gamma_{1}$ and $\Gamma_{2}$ meet $M$ orthogonally and the second
fundamental form, $|\mkern
1.5mu\overline{\mkern-1.5muA_{\Gamma_{i}}\mkern-1.5mu}\mkern 1.5mu|$, is
bounded near the ideal boundary, it follows that there are an $\epsilon_{1}>0$
and a function $h$ satisfying (3.10). Moreover, $h$ is $C^{2}$ up to $M$. By
Lemma 3.3, for a small $\epsilon>0$, when writing $\Gamma_{i}$ as graphs over
$M\times[0,\epsilon)$ near $\\{y=0\\}$, the $y$-derivatives are determined by
$M$ up to the second order. So,

$h\big{|}_{M}=0,\ \mkern 1.5mu\overline{\mkern-1.5mu\nabla\mkern-1.5mu}\mkern
1.5muh\big{|}_{M}=0\text{ and }\mkern
1.5mu\overline{\mkern-1.5mu\nabla\mkern-1.5mu}\mkern 1.5mu^{2}h\big{|}_{M}=0.$

Set $\Gamma_{1,\epsilon_{1}}:=\Gamma_{1}\cap\\{y\leq\epsilon_{1}\\}$. By a
direct computation, $f=\frac{h}{y}$ satisfies the equation

$\Delta f+(|A_{\Gamma_{1}}|^{2}_{\mathbb{H}}-n)f=a(p,h,\mkern
1.5mu\overline{\mkern-1.5mu\nabla\mkern-1.5mu}\mkern 1.5muh,\mkern
1.5mu\overline{\mkern-1.5mu\nabla\mkern-1.5mu}\mkern
1.5mu^{2}h)f+y\left\langle\textbf{b}(p,h,\mkern
1.5mu\overline{\mkern-1.5mu\nabla\mkern-1.5mu}\mkern 1.5muh,\mkern
1.5mu\overline{\mkern-1.5mu\nabla\mkern-1.5mu}\mkern 1.5mu^{2}h),\nabla
f\right\rangle,$

where $a\in
C^{0}(\Gamma_{1,\epsilon_{1}}\times\mathbb{R}\times\mathbb{R}^{n+1}\times\mathbb{R}^{(n+1)^{2}};\mathbb{R})$,
$b\in
C^{0}(\Gamma_{1,\epsilon_{1}}\times\mathbb{R}\times\mathbb{R}^{n+1}\times\mathbb{R}^{(n+1)^{2}};\mathbb{R}^{n+1})$
satisfy

$|a|+|\textbf{b}|\leq C(|h|+|\mkern
1.5mu\overline{\mkern-1.5mu\nabla\mkern-1.5mu}\mkern 1.5muh|+|\mkern
1.5mu\overline{\mkern-1.5mu\nabla\mkern-1.5mu}\mkern 1.5mu^{2}h|).$

So, $a\big{|}_{M}=0$ and $\textbf{b}\big{|}_{M}=\textbf{0}$.

Since $|A_{\Gamma_{1}}|^{2}_{\mathbb{H}}=\mathcal{O}(y^{2})$, we can shrink
$\epsilon_{1}$ if necessary, so that
$|A_{\Gamma_{1}}|^{2}_{\mathbb{H}}-n-a\leq 0$ for $y(p)\leq\epsilon_{1}$. That
is, the operator
$\mathcal{L}=\Delta-y\left\langle\textbf{b},\nabla\right\rangle+(|A_{\Gamma_{1}}|^{2}_{\mathbb{H}}-n-a)$
satisfies the maximum principle in $\Gamma_{1,\epsilon_{1}}$. Let
$g(p)=y^{n}(p)-Ky^{n+1}(p)$($K$ to be determined). Then, near the ideal
boundary,

(3.11) $\displaystyle\mathcal{L}(y^{n}(p))$ $\displaystyle=\Delta
y^{n}(p)-y\left\langle\textbf{b},\nabla
y^{n}(p)\right\rangle+(|A_{\Gamma_{1}}|^{2}_{\mathbb{H}}-n-a)y^{n}(p)$
$\displaystyle=y^{2}\partial_{yy}(y^{n})+(2-n)y\partial_{y}(y^{n})-ny^{n}+\mathcal{O}(y^{n+1})=\mathcal{O}(y^{n+1}).$

Similarly, we have $\mathcal{L}(y^{n+1})=(n+2)y^{n+1}+\mathcal{O}(y^{n+2})$.
So, by further shrinking $\epsilon_{1}$ and choosing $K$ appropriately, we
have $\mathcal{L}g=\mathcal{L}(y^{n}-Ky^{n+1})\leq 0$. Let
$\lambda=\sup_{\\{y(p)=\epsilon_{1}\\}}f^{+}\geq 0$ and
$g_{1}=\frac{\lambda}{\epsilon_{1}^{n}-K\epsilon_{1}^{n+1}}g$. Then,

$\mathcal{L}(f-g_{1})\geq 0,\ f\big{|}_{y=0}=0=g_{1}\big{|}_{y=0}\text{ and
}f\big{|}_{y=\epsilon_{1}}\leq g_{1}\big{|}_{y=\epsilon_{1}}.$

By the maximum principle, $f\leq g_{1}$ in
$\Gamma_{1}\cap\\{y\leq\epsilon_{1}\\}$. This proves the upper bound.

The lower bound follows in a similar manner if we let
$\lambda^{\prime}=\sup_{\\{y=\epsilon_{1}\\}}f^{-}\geq 0$ and consider
$\mathcal{L}(f+\frac{\lambda^{\prime}}{\epsilon_{1}^{n}-K\epsilon_{1}^{n+1}}g)\leq
0$. ∎

###### Proof of Theorem 1.1.

By Lemma 3.4, $\Sigma^{\prime}$ is in a thin neighborhood of $\Sigma$ when
$n\geq 3$. If $n=2$, by Remark 3.2, $E_{rel}[\Sigma^{\prime},\Sigma]$ is still
well-defined. It follows from Theorem 3.1 that
$E^{(r)}_{rel}[\Sigma^{\prime},\Sigma]=E_{rel}[\Sigma^{\prime},\Sigma]$ for
any boundary defining function $r$. Moreover, we have
$E_{rel}[\Sigma^{\prime},\Sigma]>-\infty$. Similarly,
$-E_{rel}[\Sigma^{\prime},\Sigma]=E_{rel}[\Sigma,\Sigma^{\prime}]>-\infty$.
This implies $E_{rel}[\Sigma^{\prime},\Sigma]$ is a finite value.

Fix any boundary defining function $r$ and consider
$\mathcal{H}^{n}_{\mathbb{H}}(\Sigma\cap\\{r\geq\epsilon\\})$. We choose an
$\epsilon_{1}>0$, so that $\frac{1}{C}\leq|\mkern
1.5mu\overline{\mkern-1.5mu\nabla\mkern-1.5mu}\mkern 1.5mu^{\mkern
1.5mu\overline{\mkern-1.5mu\Sigma\mkern-1.5mu}\mkern 1.5mu}r|\leq C$ for
$r\leq\epsilon_{1}$. By the coarea formula,

(3.12)
$\displaystyle\mathcal{H}^{n}_{\mathbb{H}}(\Sigma\cap\\{r\geq\epsilon\\})$
$\displaystyle=\mathcal{H}^{n}_{\mathbb{H}}(\Sigma\cap\\{r\geq\epsilon_{1}\\})+\mathcal{H}^{n}_{\mathbb{H}}(\Sigma\cap\\{\epsilon\leq
r<\epsilon_{1}\\})$
$\displaystyle=\mathcal{H}^{n}_{\mathbb{H}}(\Sigma\cap\\{r\geq\epsilon_{1}\\})+\int_{\epsilon}^{\epsilon_{1}}\int_{\Sigma\cap\\{r=s\\}}\frac{1}{|\nabla^{\Sigma}r|}d\mathcal{H}^{n-1}_{\mathbb{H}}ds,$

where $|\nabla^{\Sigma}r|=r|\mkern
1.5mu\overline{\mkern-1.5mu\nabla\mkern-1.5mu}\mkern 1.5mu_{r}^{\mkern
1.5mu\overline{\mkern-1.5mu\Sigma\mkern-1.5mu}\mkern 1.5mu}r|$. Here, $\mkern
1.5mu\overline{\mkern-1.5mu\nabla\mkern-1.5mu}\mkern 1.5mu_{r}$ is the
gradient taken in $(\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{H}\mkern-1.5mu}\mkern
1.5mu^{n+1},r^{2}g_{P})$. So,

(3.13)
$\int_{\Sigma\cap\\{r=s\\}}\frac{1}{|\nabla^{\Sigma}r|}d\mathcal{H}^{n-1}_{\mathbb{H}}=\frac{1}{s^{n}}\int_{\Sigma\cap\\{r=s\\}}\frac{1}{|\mkern
1.5mu\overline{\mkern-1.5mu\nabla\mkern-1.5mu}\mkern 1.5mu_{r}^{\mkern
1.5mu\overline{\mkern-1.5mu\Sigma\mkern-1.5mu}\mkern
1.5mu}r|}d\mathcal{H}_{r}^{n-1},$

where $\mathcal{H}_{r}^{n-1}$ is the ($n-1$)-dimensional Hausdorff measure in
($\mkern 1.5mu\overline{\mkern-1.5mu\mathbb{H}\mkern-1.5mu}\mkern
1.5mu^{n+1}$,$r^{2}g_{P}$). Since $\Sigma$ is $C^{n,\alpha}$-asymptotic to
$M$, the quantity $\int_{\Sigma\cap\\{r=s\\}}\frac{1}{|\mkern
1.5mu\overline{\mkern-1.5mu\nabla\mkern-1.5mu}\mkern 1.5mu_{r}^{\mkern
1.5mu\overline{\mkern-1.5mu\Sigma\mkern-1.5mu}\mkern
1.5mu}r|}d\mathcal{H}_{r}^{n-1}$ is a $C^{n-1,\alpha}$ function in
$s\in[0,\epsilon_{1})$. Thus, taking the Taylor’s expansion at $s=0$,

(3.14) $\int_{\Sigma\cap\\{r=s\\}}\frac{1}{|\mkern
1.5mu\overline{\mkern-1.5mu\nabla\mkern-1.5mu}\mkern 1.5mu_{r}^{\mkern
1.5mu\overline{\mkern-1.5mu\Sigma\mkern-1.5mu}\mkern
1.5mu}r|}d\mathcal{H}_{r}^{n-1}=a_{0}+a_{1}s+\cdots+a_{n-1}s^{n-1}+\mathcal{O}(s^{n-1+\alpha}).$

Combining (3.13) and (3.14), we see that

(3.15)
$\displaystyle\int_{\Sigma\cap\\{r=s\\}}\frac{1}{|\nabla^{\Sigma}r|}d\mathcal{H}^{n-1}_{\mathbb{H}}=\frac{1}{s^{n}}\big{(}a_{0}+a_{1}s+\cdots+a_{n-1}s^{n-1}+\mathcal{O}(s^{n-1+\alpha})\big{)}.$

Plugging (3.15) into (3.12), we get an expansion for
$\mathcal{H}^{n}_{\mathbb{H}}(\Sigma\cap\\{r\geq\epsilon\\})$:

$\mathcal{H}^{n}_{\mathbb{H}}(\Sigma\cap\\{r\geq\epsilon\\})=\frac{b_{0}}{\epsilon^{n-1}}+\cdots+\frac{b_{n-2}}{\epsilon}+b_{n-1}\log\left(\frac{1}{\epsilon}\right)+b_{n}+o(1).$

This proves the existence of the renormalized area $\mathcal{A}(\Sigma;r)$
(equal to $b_{n}$) when $M$ is $C^{n,\alpha}$. Similarly, we have

$\mathcal{H}^{n}_{\mathbb{H}}(\Sigma^{\prime}\cap\\{r\geq\epsilon\\})=\frac{b^{\prime}_{0}}{\epsilon^{n-1}}+\cdots+\frac{b^{\prime}_{n-2}}{\epsilon}+b^{\prime}_{n-1}\log\left(\frac{1}{\epsilon}\right)+b^{\prime}_{n}+o(1).$

By definition, $E_{rel}[\Sigma^{\prime},\Sigma]=\lim_{\epsilon\to
0}\Big{(}\mathcal{H}^{n}_{\mathbb{H}}(\Sigma^{\prime}\cap\\{r\geq\epsilon\\})-\mathcal{H}^{n}_{\mathbb{H}}(\Sigma\cap\\{r\geq\epsilon\\})\Big{)}$.
Since $E_{rel}[\Sigma^{\prime},\Sigma]$ is finite, we see that
$b_{i}=b^{\prime}_{i}$ for $0\leq i\leq n-1$ and that
$E_{rel}[\Sigma^{\prime},\Sigma]=b^{\prime}_{n}-b_{n}=\mathcal{A}(\Sigma^{\prime};r)-\mathcal{A}(\Sigma;r)$.
∎

######  Remark 3.5.

In Appendix A, we will give an explicit formula of the relative entropy for
geodesics in hyperbolic plane and verify that it is indeed independent of the
choice of boundary defining functions.

##  4\. Weighted Relative Entropy

In this section, we study a weighted version of the relative entropy. Since
the thin neighborhood $\Omega$ is bounded in $\mathbb{R}^{n+1}_{+}$, we can
pick a boundary defining function $r_{1}$ satisfying $r_{1}(\textbf{x},y)=y$
in a compact subset of $\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{R}^{n+1}_{+}\mkern-1.5mu}\mkern 1.5mu$
containing $\Omega$. Thus,

$E_{rel}[\Sigma^{\prime},\Sigma]=E^{(r_{1})}_{rel}[\Sigma^{\prime},\Sigma]=\lim_{\epsilon\to
0^{+}}\big{(}\mathcal{H}^{n}_{\mathbb{H}}(\Sigma^{\prime}\cap\\{y\geq\epsilon\\})-\mathcal{H}^{n}_{\mathbb{H}}(\Sigma\cap\\{y\geq\epsilon\\})\big{)}.$

For simplicity, we will always use this specific boundary defining function
when computing the relative entropy. For any function $\psi$, let

(4.1) $\displaystyle E_{rel}[\partial^{*}U,\Sigma;\psi]:=\lim_{\epsilon\to
0^{+}}E_{rel}[\partial^{*}U,\Sigma;\psi\chi_{\\{y\geq\epsilon\\}}]=\lim_{\epsilon\to
0^{+}}\Big{(}\int_{\partial^{*}U\cap\\{y\geq\epsilon\\}}\psi
d\mathcal{H}^{n}_{\mathbb{H}}-\int_{\Sigma\cap\\{y\geq\epsilon\\}}\psi
d\mathcal{H}^{n}_{\mathbb{H}}\Big{)}$

whenever it exists. Observe that (4.1) is always well-defined for any
compactly supported continuous function whose support is away from
$\mathbb{R}^{n}\times\\{0\\}$.

The main theorem of this section is the following:

######  Theorem 4.1.

Assume that $U\in\mathcal{C}(\Gamma^{\prime}_{-},\Gamma^{\prime}_{+})$
satisfies $E_{rel}[\partial^{*}U,\Sigma]<\infty$. Then, for any
$\psi(p,\mathbf{v})\in\mathfrak{X}$, the limit $(\ref{56})$ exists. If we
denote it by $E_{rel}[\partial^{*}U,\Sigma;\psi]$, then

(4.2) $|E_{rel}[\partial^{*}U,\Sigma;\psi]|\leq
C(|E_{rel}[\partial^{*}U,\Sigma]|+1)\|\psi\|_{\mathfrak{X}},$

where the constant $C$ only depends on $\Sigma$ and $\Omega$. Moreover, for
$0<\frac{y_{1}}{2}<y_{1}-\delta<y_{1}<y_{2}<y_{2}+\delta<2y_{2}<\epsilon$,

(4.3) $|E_{rel}[\Gamma,\Sigma;\phi_{y_{1},y_{2},\delta}\psi]|\leq
C(|E_{rel}[\Gamma,\Sigma;\phi_{y_{1},y_{2},\delta}]|+y_{2})\|\psi\|_{\mathfrak{X}}.$

where $\epsilon$ is made so small that the vector field $\mathbf{X}$ defined
in $(\ref{39})$ exists.

The arguments in [BW22, Section 4] can be carried out in our setting. In
[BW22], the authors first proved that if the weights are quadratic forms, then
the weighted relative entropy is well-defined. Then, for a general weight,
they showed that by subtracting off an appropriate quadratic form from it, the
error term has a reasonably good estimate. We follow the same scheme.

We first prove a weak version of Theorem 4.1 that will be used later on.

######  Lemma 4.2.

Let $U_{\Gamma}\in\mathcal{C}(\Gamma^{\prime}_{-},\Gamma^{\prime}_{+})$ with
$\Gamma=\partial^{*}U_{\Gamma}$. If $\psi$ is a locally Lipschitz function
over $\mathbb{R}^{n+1}_{+}$ with $\|\psi\|_{\infty}+\|y\mkern
1.5mu\overline{\mkern-1.5mu\nabla\mkern-1.5mu}\mkern 1.5mu\psi\|_{\infty}\leq
1$, then for
$0<\frac{y_{1}}{2}<y_{1}-\delta<y_{1}<y_{2}<y_{2}+\delta<2y_{2}<\epsilon$,

(4.4)
$\big{|}E_{rel}[\Gamma,\Sigma;\psi\phi_{y_{1},y_{2},\delta}]\big{|}\leq\big{|}E_{rel}[\Gamma,\Sigma;\phi_{y_{1},y_{2},\delta}]\big{|}+Cy_{2},$

where $\epsilon$ is the same as in Theorem 4.1.

###### Proof.

We first assume $\psi$ is nonnegative. Then, similar to the proof of Theorem
3.1, we have

$\displaystyle\int_{\Gamma}\psi\phi_{y_{1},y_{2},\delta}d\mathcal{H}^{n}_{\mathbb{H}^{n+1}}-\int_{\Sigma}\psi\phi_{y_{1},y_{2},\delta}d\mathcal{H}^{n}_{\mathbb{H}^{n+1}}$
$\displaystyle\geq$
$\displaystyle-2\int_{\Omega}|\text{div}\left(\psi\phi_{y_{1},y_{2},\delta}\textbf{X}\right)|d\mathcal{H}^{n+1}_{\mathbb{H}^{n+1}}$
$\displaystyle\geq$
$\displaystyle-2\int_{\Omega}\psi\phi_{y_{1},y_{2},\delta}|\text{div}\textbf{X}|+\psi|\left\langle\nabla\phi_{y_{1},y_{2},\delta},\textbf{X}\right\rangle|+\phi_{y_{1},y_{2},\delta}|\left\langle\nabla\psi,\textbf{X}\right\rangle|d\mathcal{H}^{n+1}_{\mathbb{H}^{n+1}}.$

Direct computations show that
$\psi\phi_{y_{1},y_{2},\delta}|\text{div}\textbf{X}|\leq
C\phi_{y_{1},y_{2},\delta}$ and

$\displaystyle|\left\langle\nabla\psi,\textbf{X}\right\rangle|=|\mkern
1.5mu\overline{\mkern-1.5mu\nabla\mkern-1.5mu}\mkern
1.5mu\psi\cdot\textbf{X}|=|\big{(}y\mkern
1.5mu\overline{\mkern-1.5mu\nabla\mkern-1.5mu}\mkern
1.5mu\psi\big{)}\cdot\big{(}\frac{\textbf{X}}{y}\big{)}|\leq C,$

Thus, applying the divergence theorem to the vector field X in a similar way
as in the proof of Theorem 3.1,

$\displaystyle\int_{\Gamma}\psi\phi_{y_{1},y_{2},\delta}d\mathcal{H}^{n}_{\mathbb{H}^{n+1}}-\int_{\Sigma}\psi\phi_{y_{1},y_{2},\delta}d\mathcal{H}^{n}_{\mathbb{H}^{n+1}}$
$\displaystyle\geq$
$\displaystyle-C\mathcal{H}^{n}_{\mathbb{H}^{n+1}}\Big{(}\Omega\cap\\{y_{1}-\delta\leq
y\leq y_{2}+\delta\\}\Big{)}-C\int_{\Omega\cap\big{(}\\{y_{1}-\delta\leq y\leq
y_{1}\\}\cup\\{y_{2}\leq y\leq
y_{2}+\delta\\}\big{)}}\frac{1}{\delta}d\mathcal{H}^{n}_{\mathbb{H}^{n+1}}$
$\displaystyle\geq$ $\displaystyle-Cy_{2}.$

This gives $E_{rel}[\Gamma,\Sigma;\psi\phi_{y_{1},y_{2},\delta}]\geq-Cy_{2}$.
If we replace $\psi$ by $1-\psi$ and notice that $1-\psi\geq 0$, then

$\displaystyle
E_{rel}[\Gamma,\Sigma;\psi\phi_{y_{1},y_{2},\delta}]=E_{rel}[\Gamma,\Sigma;\phi_{y_{1},y_{2},\delta}]-E_{rel}[\Gamma,\Sigma;(1-\psi)\phi_{y_{1},y_{2},\delta}]\leq
E_{rel}[\Gamma,\Sigma;\phi_{y_{1},y_{2},\delta}]+Cy_{2}.$

For a general $\psi\in\mathfrak{X}$ with $\|\psi\|_{\mathfrak{X}}\leq 1$, we
have $\frac{1+\psi}{2}\geq 0$ and $\|\frac{1+\psi}{2}\|_{\mathfrak{X}}\leq 1$.
So, (4.4) follows if we apply the preceding argument to $\frac{1+\psi}{2}$. ∎

It is then immediate from the next proposition that Theorem 4.1 holds for
weights $\psi(p,\textbf{v})\in\mathfrak{X}$ that is independent of v.

######  Proposition 4.3.

Let $U_{\Gamma}\in\mathcal{C}(\Gamma^{\prime}_{-},\Gamma^{\prime}_{+})$ with
$\Gamma=\partial^{*}U_{\Gamma}$. Assume that $E_{rel}[\Gamma,\Sigma]$ is
finite and that $\psi$ is a locally Lipschitz function over
$\mathbb{R}^{n+1}_{+}$ with $\|\psi\|_{w}:=\|\psi\|_{\infty}+\|y\mkern
1.5mu\overline{\mkern-1.5mu\nabla\mkern-1.5mu}\mkern
1.5mu\psi\|_{\infty}<\infty$. Then,

(4.5) $\displaystyle\big{|}E_{rel}[\Gamma,\Sigma,\psi]\big{|}\leq
C\Big{(}\big{|}E_{rel}[\Gamma,\Sigma]\big{|}+1\Big{)}\|\psi\|_{w},$

where $C$ is independent of $\Gamma=\partial^{*}U$ and $\psi$.

###### Proof.

For simplicity, we assume that $\|\psi\|_{w}=1$ if $\psi$ is nonzero.
Otherwise, the conclusion follows trivially. In Lemma 4.2, by letting
$\delta\to 0^{+}$, we get

(4.6) $\displaystyle\Big{|}E_{rel}[\Gamma,\Sigma;\psi\chi_{\\{y\geq
y_{1}\\}}]-E_{rel}[\Gamma,\Sigma;\psi\chi_{\\{y\geq y_{2}\\}}]\Big{|}\leq
Cy_{2}+\Big{|}E_{rel}[\Gamma,\Sigma;\chi_{\\{y\geq
y_{1}\\}}]-E_{rel}[\Gamma,\Sigma;\chi_{\\{y\geq y_{2}\\}}]\Big{|}.$

Since $E_{rel}[\Gamma,\Sigma]=\lim_{\epsilon\to
0^{+}}E_{rel}[\Gamma,\Sigma;\chi_{\\{y\geq\epsilon\\}}]$ exists and is a
finite number, the right hand side of (4.6) can be made arbitrarily small. So,
Cauchy’s criterion implies that

$\displaystyle E_{rel}[\Gamma,\Sigma;\psi]=\lim_{\epsilon\to
0^{+}}E_{rel}[\Gamma,\Sigma;\psi\chi_{\\{y\geq\epsilon\\}}]$

is well-defined and is finite valued. Taking $r=r_{1}$ in (3.7), we see that

$E_{rel}[\Gamma,\Sigma;\chi_{\\{y\geq y_{1}\\}}]\geq
E_{rel}[\Gamma,\Sigma;\chi_{\\{y\geq y_{2}\\}}]-Cy_{2}$

holds for $y_{1}<y_{2}<\epsilon$. We choose $y_{1}<\epsilon$ so small that
$\big{|}E_{rel}[\Gamma,\Sigma;\chi_{\\{y\geq y_{1}\\}}]\big{|}\leq
2\big{|}E_{rel}[\Gamma,\Sigma]\big{|}$. Note that $y_{1}$ may depend on
$\Gamma$. However, for any $y_{2}<\epsilon$,

$E_{rel}[\Gamma,\Sigma;\chi_{\\{y\geq y_{2}\\}}]\leq
Cy_{2}+E_{rel}[\Gamma,\Sigma;\chi_{\\{y\geq y_{1}\\}}]\leq
Cy_{2}+2\big{|}E_{rel}[\Gamma,\Sigma]\big{|}.$

Letting $y_{1}\to 0$ in (4.6), we get

(4.7) $\displaystyle\big{|}E_{rel}[\Gamma,\Sigma;\psi]\big{|}$
$\displaystyle\leq
Cy_{2}+\Big{|}E_{rel}[\Gamma,\Sigma]-E_{rel}[\Gamma,\Sigma;\chi_{\\{y\geq
y_{2}\\}}]\Big{|}+|E_{rel}[\Gamma,\Sigma;\psi\chi_{\\{y\geq y_{2}\\}}]|$
$\displaystyle\leq
Cy_{2}+3|E_{rel}[\Gamma,\Sigma]|+|E_{rel}[\Gamma,\Sigma;\psi\chi_{\\{y\geq
y_{2}\\}}]|.$

Notice that

(4.8) $\displaystyle|E_{rel}[\Gamma,\Sigma;\psi\chi_{\\{y\geq y_{2}\\}}]|=$
$\displaystyle\Big{|}\int_{\Gamma\cap\\{y\geq y_{2}\\}}\psi
d\mathcal{H}^{n}_{\mathbb{H}}-\int_{\Sigma\cap\\{y\geq y_{2}\\}}\psi
d\mathcal{H}^{n}_{\mathbb{H}}\Big{|}$ (4.9) $\displaystyle\leq$
$\displaystyle\max\Big{\\{}\big{|}\int_{\Gamma\cap\\{y\geq y_{2}\\}}\psi
d\mathcal{H}^{n}_{\mathbb{H}}\big{|},\big{|}\int_{\Sigma\cap\\{y\geq
y_{2}\\}}\psi d\mathcal{H}^{n}_{\mathbb{H}}\big{|}\Big{\\}}$ (4.10)
$\displaystyle\leq$
$\displaystyle\mathcal{H}^{n}_{\mathbb{H}}(\Sigma\cap\\{y\geq
y_{2}\\})+|E_{rel}[\Gamma,\Sigma;\chi_{\\{y\geq y_{2}\\}}|$ (4.11)
$\displaystyle\leq$ $\displaystyle
2\mathcal{H}^{n}_{\mathbb{H}}(\Sigma\cap\\{y\geq
y_{2}\\})+2|E_{rel}[\Gamma,\Sigma]|.$

Since $y_{2}<\epsilon$ is fixed and
$\mathcal{H}^{n}_{\mathbb{H}}(\Sigma\cap\\{y\geq y_{2}\\})<\infty$, combining
(4.7)-(4.11) gives the desired upper bound. Since $y_{2}$ is independent of
$\Gamma$, so is the constant $C$ in (4.5). ∎

Now, we follow [BW22] to establish the estimate for quadratic forms.

######  Lemma 4.4.

Let $\mathbf{Y}_{1}$ and $\mathbf{Y}_{2}$ be two vector fields in
$\mathbb{R}^{n+1}_{+}$ with

$\|\mathbf{Y}_{i}\|_{\infty}+\|y\mkern
1.5mu\overline{\mkern-1.5mu\nabla\mkern-1.5mu}\mkern
1.5mu\mathbf{Y}_{i}\|_{\infty}\leq 1.$

Then, for
$0<\frac{y_{1}}{2}<y_{1}-\delta<y_{1}<y_{2}<y_{2}+\delta<2y_{2}<\epsilon$, we
have

(4.12)
$|E_{rel}[\Gamma,\Sigma;\phi_{y_{1},y_{2},\delta}(\mathbf{Y}_{1}\cdot\mathbf{v})(\mathbf{Y}_{2}\cdot\mathbf{v})]|\leq
C(y_{2}+|E_{rel}[\Gamma,\Sigma;\phi_{y_{1},y_{2},\delta}]|).$

###### Proof.

Let Y be a locally Lipschitz vector field on $\mathbb{R}^{n+1}_{+}$ with
$\|\textbf{Y}\|_{\infty}+\|y\mkern
1.5mu\overline{\mkern-1.5mu\nabla\mkern-1.5mu}\mkern
1.5mu\textbf{Y}\|_{\infty}\leq 1$. We first show that Lemma 4.4 holds for
$\textbf{Y}_{1}=\textbf{Y}_{2}=\textbf{Y}$. By a direct computation,

$|(\textbf{Y}\cdot\textbf{X})\textbf{Y}^{(n+1)}|\leq Cy,\ \left|\mkern
1.5mu\overline{\mkern-1.5mu\text{div}\mkern-1.5mu}\mkern
1.5mu\Big{(}(\textbf{Y}\cdot\textbf{X})\textbf{Y}\Big{)}\right|\leq C.$

Applying the divergence theorem to the vector field
$(\textbf{X}\cdot\textbf{Y})\textbf{Y}$ in a similar way as in the proof of
Theorem 3.1,

$\displaystyle\int_{\Gamma}\phi_{y_{1},y_{2},\delta}(\textbf{Y}\cdot\textbf{X})\left\langle\textbf{Y},\textbf{n}_{\Gamma}\right\rangle
d\mathcal{H}^{n}_{\mathbb{H}^{n+1}}-\int_{\Sigma}\phi_{y_{1},y_{2},\delta}(\textbf{Y}\cdot\textbf{X})\left\langle\textbf{Y},\textbf{n}_{\Sigma}\right\rangle
d\mathcal{H}^{n}_{\mathbb{H}^{n+1}}$
$\displaystyle\geq-2\int_{\Omega}\left|\text{div}\Big{(}\phi_{y_{1},y_{2},\delta}(\textbf{X}\cdot\textbf{Y})\textbf{Y}\Big{)}\right|d\mathcal{H}^{n+1}_{\mathbb{H}^{n+1}}$
$\displaystyle\geq-Cy_{2}.$

Using the inequality
$(\textbf{Y}\cdot\textbf{X})\left\langle\textbf{Y},\textbf{n}_{\Gamma}\right\rangle=(\textbf{Y}\cdot\mkern
1.5mu\overline{\mkern-1.5mu\textbf{X}\mkern-1.5mu}\mkern
1.5mu)(\textbf{Y}\cdot\mkern
1.5mu\overline{\mkern-1.5mu\textbf{n}\mkern-1.5mu}\mkern
1.5mu_{\Gamma})\leq\frac{1}{2}(\textbf{Y}\cdot\mkern
1.5mu\overline{\mkern-1.5mu\textbf{X}\mkern-1.5mu}\mkern
1.5mu)^{2}+\frac{1}{2}(\textbf{Y}\cdot\mkern
1.5mu\overline{\mkern-1.5mu\textbf{n}\mkern-1.5mu}\mkern 1.5mu_{\Gamma})^{2}$
and noticing that $\textbf{Y}\cdot\mkern
1.5mu\overline{\mkern-1.5mu\textbf{X}\mkern-1.5mu}\mkern
1.5mu=\textbf{Y}\cdot\mkern
1.5mu\overline{\mkern-1.5mu\textbf{n}\mkern-1.5mu}\mkern 1.5mu_{\Gamma}$ over
$\Sigma\cap\\{y\leq\epsilon\\}$, one gets

(4.13)
$\displaystyle\int_{\Gamma}\phi_{y_{1},y_{2},\delta}(\textbf{Y}\cdot\textbf{X})\left\langle\textbf{Y},\textbf{n}_{\Gamma}\right\rangle
d\mathcal{H}^{n}_{\mathbb{H}^{n+1}}-\int_{\Sigma}\phi_{y_{1},y_{2},\delta}(\textbf{Y}\cdot\textbf{X})\left\langle\textbf{Y},\textbf{n}_{\Sigma}\right\rangle
d\mathcal{H}^{n}_{\mathbb{H}^{n+1}}$ (4.14) $\displaystyle\leq$
$\displaystyle\int_{\Gamma}\phi_{y_{1},y_{2},\delta}\left(\frac{1}{2}(\textbf{Y}\cdot\mkern
1.5mu\overline{\mkern-1.5mu\textbf{X}\mkern-1.5mu}\mkern
1.5mu)^{2}+\frac{1}{2}(\textbf{Y}\cdot\mkern
1.5mu\overline{\mkern-1.5mu\textbf{n}\mkern-1.5mu}\mkern
1.5mu_{\Gamma})^{2}\right)d\mathcal{H}^{n}_{\mathbb{H}^{n+1}}-\int_{\Sigma}\phi_{y_{1},y_{2},\delta}\left(\frac{1}{2}(\textbf{Y}\cdot\mkern
1.5mu\overline{\mkern-1.5mu\textbf{X}\mkern-1.5mu}\mkern
1.5mu)^{2}+\frac{1}{2}(\textbf{Y}\cdot\mkern
1.5mu\overline{\mkern-1.5mu\textbf{n}\mkern-1.5mu}\mkern
1.5mu_{\Sigma})^{2}\right)d\mathcal{H}^{n}_{\mathbb{H}^{n+1}}$ (4.15)
$\displaystyle=$ $\displaystyle
E_{rel}[\Gamma,\Sigma;\frac{1}{2}\phi_{y_{1},y_{2},\delta}(\textbf{Y}\cdot\mkern
1.5mu\overline{\mkern-1.5mu\textbf{X}\mkern-1.5mu}\mkern
1.5mu)^{2}]+E_{rel}[\Gamma,\Sigma;\frac{1}{2}\phi_{y_{1},y_{2},\delta}(\textbf{Y}\cdot\textbf{v})^{2}].$

Notice that $\frac{1}{2}(\textbf{Y}\cdot\mkern
1.5mu\overline{\mkern-1.5mu\textbf{X}\mkern-1.5mu}\mkern 1.5mu)^{2}$ satisfies

$\|(\textbf{Y}\cdot\mkern
1.5mu\overline{\mkern-1.5mu\textbf{X}\mkern-1.5mu}\mkern
1.5mu)^{2}\|_{\infty}+\|y\mkern
1.5mu\overline{\mkern-1.5mu\nabla\mkern-1.5mu}\mkern
1.5mu\big{(}\textbf{Y}\cdot\mkern
1.5mu\overline{\mkern-1.5mu\textbf{X}\mkern-1.5mu}\mkern
1.5mu\big{)}^{2}\|_{\infty}\leq C.$

So, applying Lemma 4.2 to the first term in (4.15), it is bounded by
$Cy_{2}+C|E_{rel}[\Gamma,\Sigma;\phi_{y_{1},y_{2},\delta}]|$. Hence,

(4.16)
$E_{rel}[\Gamma,\Sigma;\phi_{y_{1},y_{2},\delta}(\textbf{Y}\cdot\textbf{v})^{2}]\geq-
Cy_{2}-C|E_{rel}[\Gamma,\Sigma;\phi_{y_{1},y_{2},\delta}]|.$

If we are given two vector fields $\textbf{Y}_{1}$ and $\textbf{Y}_{2}$ with

$\displaystyle\|\textbf{Y}_{i}\|_{\infty}+\|y\mkern
1.5mu\overline{\mkern-1.5mu\nabla\mkern-1.5mu}\mkern
1.5mu\textbf{Y}_{i}\|_{\infty}\leq 1\text{ for }i=1,2,$

then, using (4.16) we have

$\displaystyle
E_{rel}[\Gamma,\Sigma;\phi_{y_{1},y_{2},\delta}\big{(}(\textbf{Y}_{1}+\textbf{Y}_{2})\cdot\textbf{v}\big{)}^{2}]$
$\displaystyle=$ $\displaystyle
2\sum_{i=1}^{2}E_{rel}[\Gamma,\Sigma;\phi_{y_{1},y_{2},\delta}(\textbf{Y}_{1}\cdot\textbf{v})^{2}]-4E_{rel}[\Gamma,\Sigma;\phi_{y_{1},y_{2},\delta}\big{(}\frac{\textbf{Y}_{1}-\textbf{Y}_{2}}{2}\cdot\textbf{v}\big{)}^{2}]$
$\displaystyle\leq$ $\displaystyle
2\sum_{i=1}^{2}E_{rel}[\Gamma,\Sigma;\phi_{y_{1},y_{2},\delta}(\textbf{Y}_{1}\cdot\textbf{v})^{2}]+4\Big{(}Cy_{2}+C|E_{rel}[\Gamma,\Sigma;\phi_{y_{1},y_{2},\delta}]|\Big{)}.$

Now, assume $\textbf{Y}_{i}$($i=1,\cdots,k$) are vector fields with
$\|\textbf{Y}_{i}\|_{\infty}+\|y\mkern
1.5mu\overline{\mkern-1.5mu\nabla\mkern-1.5mu}\mkern
1.5mu\textbf{Y}_{i}\|_{\infty}\leq 1$. By induction, it is immediate that

(4.17) $\displaystyle
E_{rel}[\Gamma,\Sigma;\phi_{y_{1},y_{2},\delta}\big{(}(\sum_{i=1}^{k}\textbf{Y}_{i})\cdot\textbf{v}\big{)}^{2}]\leq
2^{k}\sum_{i=1}^{k}E_{rel}[\Gamma,\Sigma;\phi_{y_{1},y_{2},\delta}\big{(}\textbf{Y}_{i}\cdot\textbf{v}\big{)}^{2}]+C_{k}(|E_{rel}[\Gamma,\Sigma;\phi_{y_{1},y_{2},\delta}]|+y_{2}).$

If we write $\textbf{Y}=\sum_{i=1}^{n+1}Y_{i}\mkern
1.5mu\overline{\mkern-1.5mu\textbf{e}\mkern-1.5mu}\mkern 1.5mu_{i}$, then

$\big{(}(Y_{i}\mkern 1.5mu\overline{\mkern-1.5mu\textbf{e}\mkern-1.5mu}\mkern
1.5mu_{i})\cdot\textbf{v}\big{)}^{2}=Y_{i}^{2}-\sum_{m\neq
i}\big{(}(Y_{i}\mkern 1.5mu\overline{\mkern-1.5mu\textbf{e}\mkern-1.5mu}\mkern
1.5mu_{m})\cdot\textbf{v}\big{)}^{2}.$

If we take $\psi=Y_{i}^{2}$ in Lemma 4.2 and use (4.16) with
$(\textbf{Y}\cdot\textbf{v})^{2}$ replaced by
$((Y_{i}\textbf{e}_{m})\cdot\textbf{v})^{2}$, then

$\displaystyle
E_{rel}[\Gamma,\Sigma;\phi_{y_{1},y_{2},\delta}\big{(}(Y_{i}\mkern
1.5mu\overline{\mkern-1.5mu\textbf{e}\mkern-1.5mu}\mkern
1.5mu_{i})\cdot\textbf{v}\big{)}^{2}]\leq
C(y_{2}+|E_{rel}[\Gamma,\Sigma;\phi_{y_{1},y_{2},\delta}]|).$

Together with (4.17), we get

$\displaystyle
E_{rel}[\Gamma,\Sigma;\phi_{y_{1},y_{2},\delta}\big{(}\textbf{Y}\cdot\textbf{v}\big{)}^{2}]=E_{rel}[\Gamma,\Sigma;\phi_{y_{1},y_{2},\delta}\big{(}(\sum_{i=1}^{n+1}Y_{i}\mkern
1.5mu\overline{\mkern-1.5mu\textbf{e}\mkern-1.5mu}\mkern
1.5mu_{i})\cdot\textbf{v}\big{)}^{2}]\leq
C(y_{2}+|E_{rel}[\Gamma,\Sigma;\phi_{y_{1},y_{2},\delta}]|).$

Combining this with (4.16), we see that

$\displaystyle\big{|}E_{rel}[\Gamma,\Sigma;\phi_{y_{1},y_{2},\delta}\big{(}\textbf{Y}\cdot\textbf{v}\big{)}^{2}]\big{|}\leq
C(y_{2}+\big{|}E_{rel}[\Gamma,\Sigma;\phi_{y_{1},y_{2},\delta}]\big{|}).$

Then, (4.12) follows from the following observation:

$(\textbf{Y}_{1}\cdot\textbf{v})(\textbf{Y}_{2}\cdot\textbf{v})=\left(\frac{\textbf{Y}_{1}+\textbf{Y}_{2}}{2}\cdot\textbf{v}\right)^{2}-\left(\frac{\textbf{Y}_{1}-\textbf{Y}_{2}}{2}\cdot\textbf{v}\right)^{2}.$

∎

Now, we discuss the quadratic approximation for a general weight
$\psi\in\mathfrak{X}$ and establish the estimate for the error term. Fix a
small $\epsilon>0$. Let $\mkern
1.5mu\overline{\mkern-1.5mu\psi\mkern-1.5mu}\mkern
1.5mu(p,\textbf{v}):=\psi(p,\textbf{v})-(1-\phi_{\frac{\epsilon}{2},\epsilon})(\textbf{Y}_{\psi}(p)\cdot\textbf{v})(\mkern
1.5mu\overline{\mkern-1.5mu\textbf{X}\mkern-1.5mu}\mkern
1.5mu(p)\cdot\textbf{v})$, where
$\textbf{Y}_{\psi}(p):=(\nabla_{\mathbb{S}^{n}}\psi)(p,\mkern
1.5mu\overline{\mkern-1.5mu\textbf{X}\mkern-1.5mu}\mkern 1.5mu(p))$. Notice
that if $\epsilon$ is sufficiently small, then $\mkern
1.5mu\overline{\mkern-1.5mu\textbf{X}\mkern-1.5mu}\mkern 1.5mu$ is well-
defined. Fix a point $p$ in $\\{y<\frac{\epsilon}{2}\\}$. Notice that
$\nabla_{\mathbb{S}^{n}}\mkern
1.5mu\overline{\mkern-1.5mu\psi\mkern-1.5mu}\mkern
1.5mu(p,\textbf{v})\big{|}_{\textbf{v}=\mkern
1.5mu\overline{\mkern-1.5mu\textbf{X}\mkern-1.5mu}\mkern 1.5mu(p)}=0$. Let
$\gamma(t)$ ($t\in[0,\ell]$) be a geodesic on $\mathbb{S}^{n}$ connecting
$\mkern 1.5mu\overline{\mkern-1.5mu\textbf{X}\mkern-1.5mu}\mkern 1.5mu(p)$ and
v, where $\ell$ is the length of $\gamma$. If we let $\varphi(t)=\mkern
1.5mu\overline{\mkern-1.5mu\psi\mkern-1.5mu}\mkern 1.5mu(p,\gamma(t))$, then

$\displaystyle\varphi^{\prime}(0)=\nabla_{\mathbb{S}^{n}}\mkern
1.5mu\overline{\mkern-1.5mu\psi\mkern-1.5mu}\mkern
1.5mu\big{|}_{\textbf{v}=\mkern
1.5mu\overline{\mkern-1.5mu\textbf{X}\mkern-1.5mu}\mkern
1.5mu(p)}\cdot\gamma^{\prime}(0)=0.$

Hence,

(4.18) $\displaystyle|\mkern
1.5mu\overline{\mkern-1.5mu\psi\mkern-1.5mu}\mkern 1.5mu(p,\textbf{v})-\mkern
1.5mu\overline{\mkern-1.5mu\psi\mkern-1.5mu}\mkern 1.5mu(p,\mkern
1.5mu\overline{\mkern-1.5mu\textbf{X}\mkern-1.5mu}\mkern 1.5mu(p))|=$
$\displaystyle|\varphi(\ell)-\varphi(0)|=|\int_{0}^{\ell}\varphi^{\prime}(t)dt|=|\int_{0}^{\ell}\int_{0}^{t}\varphi^{\prime\prime}(s)dsdt|$
(4.19) $\displaystyle=$
$\displaystyle|\int_{0}^{\ell}\int_{0}^{t}\nabla_{\mathbb{S}^{n}}\nabla_{\mathbb{S}^{n}}\psi(\gamma^{\prime}(s),\gamma^{\prime}(s))dsdt|$
(4.20) $\displaystyle\leq$
$\displaystyle\|\nabla_{\mathbb{S}^{n}}\nabla_{\mathbb{S}^{n}}\psi\|_{\infty}\ell^{2}=\|\nabla_{\mathbb{S}^{n}}\nabla_{\mathbb{S}^{n}}\psi\|_{\infty}d^{2}_{\mathbb{S}^{n}}(\textbf{v},\mkern
1.5mu\overline{\mkern-1.5mu\textbf{X}\mkern-1.5mu}\mkern 1.5mu(p)).$

Since $\psi$ is an even function in v, so is $\mkern
1.5mu\overline{\mkern-1.5mu\psi\mkern-1.5mu}\mkern 1.5mu$. Thus,

(4.21) $\displaystyle|\mkern
1.5mu\overline{\mkern-1.5mu\psi\mkern-1.5mu}\mkern 1.5mu(p,\textbf{v})-\mkern
1.5mu\overline{\mkern-1.5mu\psi\mkern-1.5mu}\mkern 1.5mu(p,\mkern
1.5mu\overline{\mkern-1.5mu\textbf{X}\mkern-1.5mu}\mkern 1.5mu(p))|=|\mkern
1.5mu\overline{\mkern-1.5mu\psi\mkern-1.5mu}\mkern 1.5mu(p,-\textbf{v})-\mkern
1.5mu\overline{\mkern-1.5mu\psi\mkern-1.5mu}\mkern 1.5mu(p,\mkern
1.5mu\overline{\mkern-1.5mu\textbf{X}\mkern-1.5mu}\mkern
1.5mu(p))|\leq\|\nabla_{\mathbb{S}^{n}}\nabla_{\mathbb{S}^{n}}\psi\|_{\infty}d^{2}_{\mathbb{S}^{n}}(-\textbf{v},\mkern
1.5mu\overline{\mkern-1.5mu\textbf{X}\mkern-1.5mu}\mkern 1.5mu(p)).$

Combining (4.20) and (4.21), we get

$\displaystyle|\mkern 1.5mu\overline{\mkern-1.5mu\psi\mkern-1.5mu}\mkern
1.5mu(p,\textbf{v})-\mkern 1.5mu\overline{\mkern-1.5mu\psi\mkern-1.5mu}\mkern
1.5mu(p,\mkern 1.5mu\overline{\mkern-1.5mu\textbf{X}\mkern-1.5mu}\mkern
1.5mu(p))|\leq$
$\displaystyle\|\nabla_{\mathbb{S}^{n}}\nabla_{\mathbb{S}^{n}}\psi\|_{\infty}\min\Big{\\{}d^{2}_{\mathbb{S}^{n}}(\textbf{v},\mkern
1.5mu\overline{\mkern-1.5mu\textbf{X}\mkern-1.5mu}\mkern
1.5mu(p)),d^{2}_{\mathbb{S}^{n}}(-\textbf{v},\mkern
1.5mu\overline{\mkern-1.5mu\textbf{X}\mkern-1.5mu}\mkern 1.5mu(p))\Big{\\}}$
(4.22) $\displaystyle\leq$ $\displaystyle
C_{n}\|\nabla_{\mathbb{S}^{n}}\nabla_{\mathbb{S}^{n}}\psi\|_{\infty}\big{(}1-(\textbf{v}\cdot\mkern
1.5mu\overline{\mkern-1.5mu\textbf{X}\mkern-1.5mu}\mkern 1.5mu)^{2}\big{)}.$

By a direct computation, we also have

(4.23) $\displaystyle\|\mkern
1.5mu\overline{\mkern-1.5mu\psi\mkern-1.5mu}\mkern 1.5mu(p,\mkern
1.5mu\overline{\mkern-1.5mu\textbf{X}\mkern-1.5mu}\mkern 1.5mu(p))\|_{w}\leq
C(\|\psi\|_{\infty}+\|y\nabla_{\mathbb{R}^{n+1}_{+}}\psi\|_{\infty}+\|\nabla_{\mathbb{S}^{n}}\psi\|_{\infty}+\|y\nabla_{\mathbb{R}^{n+1}_{+}}\nabla_{\mathbb{S}^{n}}\psi\|_{\infty}),$
(4.24)
$\displaystyle\|\textbf{Y}_{\psi}\|_{\infty}+\|y\nabla_{\mathbb{R}^{n+1}_{+}}\textbf{Y}_{\psi}\|_{\infty}\leq
C(\|\nabla_{\mathbb{S}^{n}}\psi\|_{\infty}+\|y\nabla_{\mathbb{R}^{n+1}_{+}}\nabla_{\mathbb{S}^{n}}\psi\|_{\infty}+\|\nabla_{\mathbb{S}^{n}}\nabla_{\mathbb{S}^{n}}\psi||_{\infty}).$

Now, we are ready to prove Theorem 4.1.

###### Proof of Theorem 4.1.

Fix $\psi\in\mathfrak{X}$. Without loss of generality, we can assume that
$\|\psi\|_{\mathfrak{X}}\leq 1$. Let $\epsilon$ be the same number that
appears in the definition of $\mkern
1.5mu\overline{\mkern-1.5mu\psi\mkern-1.5mu}\mkern 1.5mu$. For positive
constants $y_{1}$, $y_{2}$ and $\delta$ with

$\displaystyle
0<\frac{y_{1}}{2}<y_{1}-\delta<y_{1}<y_{2}<y_{2}+\delta<2y_{2}<\frac{\epsilon}{2},$

we prove that

(4.25)
$\displaystyle|E_{rel}[\Gamma,\Sigma;\phi_{y_{1},y_{2},\delta}\psi]|\leq
C(|E_{rel}[\Gamma,\Sigma;\phi_{y_{1},y_{2},\delta}]|+y_{2})\|\psi\|_{\mathfrak{X}}.$

This gives the existence of $E_{rel}[\Gamma,\Sigma,\psi]$. Observe that
$E_{rel}[\Gamma,\Sigma;\phi_{y_{1},y_{2},\delta}\psi]$ can be written as

$\displaystyle E_{rel}[\Gamma,\Sigma;\phi_{y_{1},y_{2},\delta}\psi]=$
$\displaystyle
E_{rel}[\Gamma,\Sigma;\phi_{y_{1},y_{2},\delta}(\textbf{Y}_{\psi}(p)\cdot\textbf{v})(\mkern
1.5mu\overline{\mkern-1.5mu\textbf{X}\mkern-1.5mu}\mkern
1.5mu(p)\cdot\textbf{v})]+E_{rel}[\Gamma,\Sigma;\phi_{y_{1},y_{2},\delta}\mkern
1.5mu\overline{\mkern-1.5mu\psi\mkern-1.5mu}\mkern 1.5mu(p,\mkern
1.5mu\overline{\mkern-1.5mu\textbf{X}\mkern-1.5mu}\mkern 1.5mu(p))]$
$\displaystyle+E_{rel}[\Gamma,\Sigma;\phi_{y_{1},y_{2},\delta}\left(\mkern
1.5mu\overline{\mkern-1.5mu\psi\mkern-1.5mu}\mkern 1.5mu(p,\textbf{v})-\mkern
1.5mu\overline{\mkern-1.5mu\psi\mkern-1.5mu}\mkern 1.5mu(p,\mkern
1.5mu\overline{\mkern-1.5mu\textbf{X}\mkern-1.5mu}\mkern 1.5mu(p))\right)].$

By (4.12) and (4.24),we have

$\displaystyle|E_{rel}[\Gamma,\Sigma;\phi_{y_{1},y_{2},\delta}(\textbf{Y}_{\psi}(p)\cdot\textbf{v})(\mkern
1.5mu\overline{\mkern-1.5mu\textbf{X}\mkern-1.5mu}\mkern
1.5mu(p)\cdot\textbf{v})]|\leq
C(y_{2}+|E_{rel}[\Gamma,\Sigma;\phi_{y_{1},y_{2},\delta}]|)\|\psi\|_{\mathfrak{X}}.$

Proposition 4.3 and (4.23) gives

$\displaystyle|E_{rel}[\Gamma,\Sigma;\phi_{y_{1},y_{2},\delta}\mkern
1.5mu\overline{\mkern-1.5mu\psi\mkern-1.5mu}\mkern 1.5mu(p,\mkern
1.5mu\overline{\mkern-1.5mu\textbf{X}\mkern-1.5mu}\mkern 1.5mu(p))]|\leq
C(y_{2}+|E_{rel}[\Gamma,\Sigma;\phi_{y_{1},y_{2},\delta}]|)\|\psi\|_{\mathfrak{X}}.$

Finally, by (4.22) and noticing that both $\left(\mkern
1.5mu\overline{\mkern-1.5mu\psi\mkern-1.5mu}\mkern 1.5mu(p,\textbf{v})-\mkern
1.5mu\overline{\mkern-1.5mu\psi\mkern-1.5mu}\mkern 1.5mu(p,\mkern
1.5mu\overline{\mkern-1.5mu\textbf{X}\mkern-1.5mu}\mkern 1.5mu(p))\right)$ and
$\left(1-(\textbf{v}\cdot\mkern
1.5mu\overline{\mkern-1.5mu\textbf{X}\mkern-1.5mu}\mkern 1.5mu)^{2}\right)$
vanish on $\Sigma$,

$\displaystyle\big{|}E_{rel}[\Gamma,\Sigma;\phi_{y_{1},y_{2},\delta}\left(\mkern
1.5mu\overline{\mkern-1.5mu\psi\mkern-1.5mu}\mkern 1.5mu(p,\textbf{v})-\mkern
1.5mu\overline{\mkern-1.5mu\psi\mkern-1.5mu}\mkern 1.5mu(p,\mkern
1.5mu\overline{\mkern-1.5mu\textbf{X}\mkern-1.5mu}\mkern
1.5mu(p))\right)]\big{|}=\left|\int_{\Gamma}\phi_{y_{1},y_{2},\delta}\left(\mkern
1.5mu\overline{\mkern-1.5mu\psi\mkern-1.5mu}\mkern 1.5mu(p,\textbf{v})-\mkern
1.5mu\overline{\mkern-1.5mu\psi\mkern-1.5mu}\mkern 1.5mu(p,\mkern
1.5mu\overline{\mkern-1.5mu\textbf{X}\mkern-1.5mu}\mkern
1.5mu(p))\right)d\mathcal{H}^{n}_{\mathbb{H}^{n+1}}\right|$ $\displaystyle\leq
C\|\psi\|_{\mathfrak{X}}\left|\int_{\Gamma}\phi_{y_{1},y_{2},\delta}\big{(}1-(\mkern
1.5mu\overline{\mkern-1.5mu\textbf{n}\mkern-1.5mu}\mkern
1.5mu_{\Gamma}\cdot\mkern
1.5mu\overline{\mkern-1.5mu\textbf{X}\mkern-1.5mu}\mkern
1.5mu)^{2}\big{)}d\mathcal{H}^{n}_{\mathbb{H}^{n+1}}\right|=C\|\psi\|_{\mathfrak{X}}\,\big{|}E_{rel}[\Gamma,\Sigma;\phi_{y_{1},y_{2},\delta}\big{(}1-(\textbf{v}\cdot\mkern
1.5mu\overline{\mkern-1.5mu\textbf{X}\mkern-1.5mu}\mkern
1.5mu)^{2}\big{)}]\big{|}$ $\displaystyle\leq
C\|\psi\|_{\mathfrak{X}}(y_{2}+|E_{rel}[\Gamma,\Sigma;\phi_{y_{1},y_{2},\delta}]|).$

This proves (4.25). It follows from Cauchy’s criterion that
$E_{rel}[\Gamma,\Sigma;\psi]$ is well-defined. We use an argument similar to
(4.7)-(4.11) in the proof of Proposition 4.3 to conclude that the inequality
(4.2) holds. ∎

##  5\. Monotonicity Formula for Relative Entropy

In this section, we prove Theorem 1.2. As a starting point, we show that the
flow is $C^{1,1}$-asymptotic to the ideal boundary as long as it exists.

######  Lemma 5.1.

Let $\\{\Sigma_{t}\\}_{t\in[0,T)}$ be a mean curvature flow in
$\mathbb{H}^{n+1}$ trapped between $\Gamma^{\prime}_{-}$ and
$\Gamma^{\prime}_{+}$. Assume $\Sigma_{0}$ is $C^{2}$-asymptotic to a closed
submanifold $M\subset\partial_{\infty}\mathbb{H}^{n+1}$. Then, for all
$t\in[0,T)$, $\Sigma_{t}$ is $C^{1,1}$-asymptotic to $M$. Moreover, there is a
constant $\epsilon_{0}>0$, so that

(5.1) $\sup_{0\leq t<T,\,p\in\Sigma_{t}\cap\\{y\leq\epsilon_{0}\\}}|\mkern
1.5mu\overline{\mkern-1.5muA_{\Sigma_{t}}\mkern-1.5mu}\mkern 1.5mu|<\infty.$

###### Proof.

Since $\Sigma_{0}$ is $C^{2}$-asymptotic to $M$, we have
$|A_{\Sigma_{0}}|_{\mathbb{H}}=\mathcal{O}(y)$ as $y\to 0$. So, by the pseudo-
locality theorem [CY07, Theorem 7.5], for any $T_{0}<T$, there are
$\epsilon_{0}>0$ and $C_{0}>0$, so that $|A_{\Sigma_{t}}|_{\mathbb{H}}(p)\leq
C_{0}$ for all $t\in[0,T_{0}]$ and
$p\in\Sigma_{t}\cap\\{y\leq\epsilon_{0}\\}$. This implies that $|\mkern
1.5mu\overline{\mkern-1.5muA_{\Sigma_{t}}\mkern-1.5mu}\mkern
1.5mu|\leq\frac{C}{y}$ for $t\in[0,T_{0}]$ and
$p\in\Sigma_{t}\cap\\{y\leq\epsilon_{0}\\}$.

For $p_{0}\in\Sigma_{t}$, consider the ball $B_{\delta y_{0}}(p_{0})$, where
$\delta<1$ is a small constant and $y_{0}=y(p_{0})$. We have $|\mkern
1.5mu\overline{\mkern-1.5muA_{\Sigma_{t}}\mkern-1.5mu}\mkern
1.5mu|\leq\frac{C}{y_{0}}$ in $B_{\delta y_{0}}(p_{0})\cap\Sigma_{t}$. Since
$\Omega$ is a thin neighborhood and both $\Gamma^{\prime}_{-}$ and
$\Gamma^{\prime}_{+}$ meet $M$ orthogonally, by shrinking $\epsilon_{0}$ if
necessary, we can assume that $|\mkern
1.5mu\overline{\mkern-1.5mue\mkern-1.5mu}\mkern 1.5mu_{n+1}\cdot\mkern
1.5mu\overline{\mkern-1.5mun\mkern-1.5mu}\mkern
1.5mu_{\Sigma_{t}}|\leq\frac{1}{2}$ and that
$\Sigma_{t}\cap\\{y\leq\epsilon_{0}\\}$ is a graph over
$M\times(0,\epsilon_{0})$. Otherwise, $B_{\delta y_{0}}(p_{0})\cap\Sigma_{t}$
would intersect the boundary of $\Omega$.

By a suitable transformation, we may assume $\textbf{0}\in M$ and that the
unit normal of $M$ at 0 is equal to $\mkern
1.5mu\overline{\mkern-1.5mue\mkern-1.5mu}\mkern 1.5mu_{n}=(0,\cdots,0,1,0)$.
Thus, for a sufficiently small $\delta>0$, the mean curvature flow
$\\{\Sigma_{t}\\}_{t\in[0,T_{0}]}$ in
$(-\delta,\delta)^{n}\times(0,\epsilon_{0})$ can be described by

(5.2)
$\Sigma_{t}\cap\Big{(}(-\delta,\delta)^{n}\times(0,\epsilon_{0})\Big{)}=\\{(\textbf{x}^{\prime},u,y):\,u=u(t,\textbf{x}^{\prime},y),\,\textbf{x}^{\prime}\in(-\delta,\delta)^{n-1},\,y\in(0,\epsilon_{0})\\},$

where $u$ satisfies the equation

(5.3)
$\partial_{t}u=y^{2}\sqrt{1+|Du|^{2}}\text{div}\left(\frac{Du}{\sqrt{1+|Du|^{2}}}\right)-ny\partial_{y}u.$

The equation (5.3) is scaling invariant in the sense that
$u_{\lambda}(t,\textbf{x}^{\prime},y):=\frac{1}{\lambda}u(t,\lambda\textbf{x}^{\prime},\lambda
y)$ satisfies the same equation. Since $\Sigma_{t}$ is trapped between
$\Gamma^{\prime}_{-}$ and $\Gamma^{\prime}_{+}$ and $\Sigma$ meets $M$
orthogonally, we have the estimate $|u(t,\textbf{x}^{\prime},y)|\leq Cy^{2}$.
Let $k\in\mathbb{N}$ and $\lambda_{k}=\epsilon_{0}^{k}$. Then,
$u_{\lambda_{k}}$ solves (5.3) in
$[0,T_{0}]\times(-\frac{\delta}{\lambda_{k}},\frac{\delta}{\lambda_{k}})^{n-1}\times[\epsilon_{0},1)$.
It follows from the gradient estimates for quasi-linear equations that $Du$ is
Hölder continuous (see, e.g., [LSU88, Theorem 6.1.1] for a proof). Then, we
can apply the standard Schauder theory to $u_{\lambda_{k}}$ to conclude that,
when rescaling back to $u$, $|Du|\leq Cy$ and $|D^{2}u|\leq C$. Since $T_{0}$
is arbitrary, this implies each $\Sigma_{t}$ is $C^{1,1}$-asymptotic to $M$.

Now, fix a specific choice of $T_{0}<T$ and the corresponding $\epsilon_{0}$.
Let

$S=\big{\\{}s\in(0,T):\Sigma_{t}\cap\\{y\leq\epsilon_{0}\\}\text{ is a graph
over }M\times(0,\epsilon_{0}]\text{ for all }t\in[0,s]\big{\\}}$

and $T^{\prime}=\sup S$. Then, $T^{\prime}\geq T_{0}$. We claim that
$T^{\prime}=T$. If, on the contrary, $T^{\prime}<T$, then by definition,
$\Sigma_{t}\cap\\{y\leq\epsilon_{0}\\}$ is a graph over
$M\times(0,\epsilon_{0}]$ for $t\in[0,T^{\prime})$. So, locally we can write
$\Sigma_{t}\cap\Big{(}(-\delta,\delta)^{n}\times(0,\epsilon_{0})\Big{)}$ as a
graph of $u$ as in (5.2) for $t\in[0,T)$, where $u$ satisfies the equation in
(5.3) in $[0,T)\times\Big{(}(-\delta,\delta)^{n}\times(0,\epsilon_{0})$. Then,
the estimates on $u$ in the preceding paragraph implies that

$\sup_{t\in[0,T^{\prime}),p\in\Sigma_{t}\cap\\{y\leq\epsilon_{0}\\}}|A_{\Sigma_{t}}|<\infty.$

This implies the graph can be extended to $t=T^{\prime}$, and hence
$T^{\prime}\in S$. Finally, as $|Du|$ remains bounded for
$t\in[0,T^{\prime}]$, we can see that $T^{\prime}+\epsilon\in S$ for some
$\epsilon$. This contradicts our assumption.

So, there is a $\epsilon_{0}>0$, so that
$\Sigma_{t}\cap\\{y\leq\epsilon_{0}\\}$ can be expressed as graphs over
$M\times(0,\epsilon_{0}]$ for all $t<T$. Then, (5.1) follows from the
estimates on the equation (5.3). ∎

######  Lemma 5.2.

Let $\widetilde{\Sigma}$ be a hypersurface that is $C^{1,1}$-asymptotic to $M$
and is trapped between $\Gamma^{\prime}_{-}$ and $\Gamma^{\prime}_{+}$. If
$\widetilde{\Sigma}$ satisfies

$\sup_{p\in\widetilde{\Sigma}\cap\\{y\leq\epsilon_{0}\\}}|\mkern
1.5mu\overline{\mkern-1.5muA\mkern-1.5mu}\mkern
1.5mu_{\widetilde{\Sigma}}|(p)\leq C_{0}$

for some $C_{0},\ \epsilon_{0}>0$, then there exist constants
$\epsilon_{1}=\epsilon_{1}(C_{0},\epsilon_{0},\Gamma^{\prime}_{-},\Gamma^{\prime}_{+})$
and $C_{1}=C_{1}(C_{0},\epsilon_{0},\Gamma^{\prime}_{-},\Gamma^{\prime}_{+})$,
so that for any
$0<\frac{s_{1}}{2}<s_{1}-\delta<s_{1}<s_{2}<s_{2}+\delta<2s_{2}<\epsilon_{1}$,

$\big{|}E_{rel}[\widetilde{\Sigma},\Sigma;\phi_{s_{1},s_{2},\delta}]\big{|}\leq
C_{1}s_{2}.$

###### Proof.

Since $\Gamma^{\prime}_{-}$ and $\Gamma^{\prime}_{+}$ are both orthogonal to
$\partial_{\infty}\mathbb{H}^{n+1}$, so is $\widetilde{\Sigma}$. Together with
the upper bound for $|\mkern 1.5mu\overline{\mkern-1.5muA\mkern-1.5mu}\mkern
1.5mu_{\widetilde{\Sigma}}|$, we have the existence of constant
$\epsilon_{1}$, so $\widetilde{\Sigma}\cap\\{y\leq\epsilon_{1}\\}$ can be
written as a graph over $\Sigma$. That is, for some $C^{1,1}$ function $f$
defined on $\Sigma\cap\\{y\leq 2\epsilon_{1}\\}$,

$\widetilde{\Sigma}\cap\\{y\leq\epsilon_{1}\\}\subset\big{\\{}p+f(p)\mkern
1.5mu\overline{\mkern-1.5mu\textbf{v}\mkern-1.5mu}\mkern
1.5mu(p):\,\textup{p}\in\Sigma\cap\\{y\leq
2\epsilon_{1}\\}\big{\\}}\subset\widetilde{\Sigma},$

where $\mkern 1.5mu\overline{\mkern-1.5mu\textbf{v}\mkern-1.5mu}\mkern
1.5mu(p)$ is a $C^{1,1}$ transverse section222See [BW21, Section 2.4] for the
precise definition. over $\mkern
1.5mu\overline{\mkern-1.5mu\Sigma\mkern-1.5mu}\mkern 1.5mu$ satisfying $\mkern
1.5mu\overline{\mkern-1.5mu\textbf{v}\mkern-1.5mu}\mkern 1.5mu(p)=\mkern
1.5mu\overline{\mkern-1.5mu\textbf{n}\mkern-1.5mu}\mkern 1.5mu_{M}$ on $M$.
The assumptions on $\widetilde{\Sigma}$ immediately give

$|f(p)|\leq Cy^{n+1}(p),\ |\mkern
1.5mu\overline{\mkern-1.5mu\nabla\mkern-1.5mu}\mkern 1.5mu_{\Sigma}f(p)|\leq
Cy(p),\ |\mkern 1.5mu\overline{\mkern-1.5mu\nabla\mkern-1.5mu}\mkern
1.5mu^{2}_{\Sigma}f(p)|\leq C.$

For $0\leq s\leq 1$, we let
$\hat{\Sigma}_{s}=\big{\\{}\textbf{f}_{s}(p)=p+sf(p)\mkern
1.5mu\overline{\mkern-1.5mu\textbf{v}\mkern-1.5mu}\mkern
1.5mu(p):\,p\in\Sigma\cap\\{y\leq 2\epsilon_{1}\\}\big{\\}}$. Then, by the
first variation formula,

(5.4)
$\displaystyle\frac{d}{ds}\int_{\hat{\Sigma}_{s}}\phi_{s_{1},s_{2},\delta}d\mathcal{H}^{n}_{\mathbb{H}}=\int_{\hat{\Sigma}_{s}}\left\langle\nabla\phi_{s_{1},s_{2},\delta},\textbf{Y}_{s}^{\perp}\right\rangle-\phi_{s_{1},s_{2},\delta}\left\langle\textbf{Y}_{s},\textbf{H}_{s}\right\rangle
d\mathcal{H}^{n}_{\mathbb{H}},$

where $\textbf{H}_{s}$ is the mean curvature of $\hat{\Sigma}_{s}$ and
$\textbf{Y}_{s}=(f\mkern 1.5mu\overline{\mkern-1.5mun\mkern-1.5mu}\mkern
1.5mu_{\Sigma})\circ\textbf{f}_{s}^{-1}$. By shrinking $\epsilon_{1}$ if
needed, we can assume that $y(p)$ is comparable to $y(\textbf{f}_{s}^{-1}(p))$
for $0\leq s\leq 1$ and $p\in\hat{\Sigma}_{s}$. We also have
$\textbf{H}_{s}=y^{2}\mkern
1.5mu\overline{\mkern-1.5mu\textbf{H}\mkern-1.5mu}\mkern 1.5mu_{s}+ny(\mkern
1.5mu\overline{\mkern-1.5mue\mkern-1.5mu}\mkern 1.5mu_{n+1}\cdot\mkern
1.5mu\overline{\mkern-1.5mun\mkern-1.5mu}\mkern
1.5mu_{\hat{\Sigma}_{s}})\mkern
1.5mu\overline{\mkern-1.5mun\mkern-1.5mu}\mkern
1.5mu_{\hat{\Sigma}_{s}}=\mathcal{O}(y^{2})$, where $\mathcal{O}(y^{2})$ is
uniform in $s$. Hence,

(5.5)
$\displaystyle\left|\left\langle\nabla\phi_{s_{1},s_{2},\delta},\textbf{Y}_{s}^{\perp}\right\rangle\right|\leq\frac{C}{\delta}y^{n+1}(p)(\chi_{\\{s_{1}-\delta\leq
y\leq s_{1}\\}}+\chi_{\\{s_{2}\leq y\leq s_{2}+\delta\\}});$ (5.6)
$\displaystyle\left|\phi_{s_{1},s_{2},\delta}\left\langle\textbf{Y}_{s},\textbf{H}_{s}\right\rangle\right|\leq
Cy^{n+1}(p)\chi_{\\{y\leq s_{2}+\delta\\}}.$

Then, applying the coarea formula to the right hand side of (5.4),

$\displaystyle(\ref{20})=\int_{s_{1}-\delta}^{s_{2}+\delta}\frac{1}{r^{n}}dr\int_{\hat{\Sigma}_{s}\cap\\{y=r\\}}\left(\left\langle\nabla\phi_{s_{1},s_{2},\delta},\textbf{Y}_{s}^{\perp}\right\rangle-\phi_{s_{1},s_{2},\delta}\left\langle\textbf{Y}_{s},\textbf{H}_{s}\right\rangle\right)\frac{1}{|\mkern
1.5mu\overline{\mkern-1.5mu\nabla\mkern-1.5mu}\mkern
1.5mu_{\hat{\Sigma}_{s}}y|}d\mathcal{H}^{n-1}.$

Combining this with (5.5) and (5.6), we get (5.4) is bounded by $C_{1}s_{2}$.
Finally, if we take the integral in (5.4) from $s=0$ to $s=1$, the desired
estimates follows. ∎

Now, let’s turn to the proof of the monotonicity formula in Theorem 1.2. In
fact, we prove the following theorem, of which Theorem 1.2 is a special case.

######  Theorem 5.3.

Let $\\{\Sigma_{t}\\}_{t\in[0,T)}$ be a mean curvature flow in
$\mathbb{H}^{n+1}$. We assume that each $\Sigma_{t}$ is trapped between
$\Gamma^{\prime}_{-}$ and $\Gamma^{\prime}_{+}$ and that $\Sigma_{0}$ is
$C^{2}$-asymptotic to $M$. Then, $E_{rel}[\Sigma_{t},\Sigma]$ exists and is
finite for all $0\leq t<T$. There is a constant
$E_{0}=E_{0}(\Sigma_{0},\Gamma^{\prime}_{-},\Gamma^{\prime}_{+})>0$, so that
$E_{rel}[\Sigma_{t},\Sigma]\geq-E_{0}$ for all $t\in[0,T)$. Furthermore, for
any $0\leq t\leq s<T$ and non-negative $f=f(x,t)$ defined for
$(x,t)\in\mathbb{R}^{n+1}_{+}\times[0,T)$ with

$\sum_{i=0}^{3}\|(y\nabla_{\mathbb{R}^{n+1}_{+}})^{i}f\|_{\infty}+\sum_{i=0}^{1}\|(y\nabla_{\mathbb{R}^{n+1}_{+}})^{i}\partial_{t}f\|_{\infty}<\infty,$

we have

(5.7) $\displaystyle E_{rel}[\Sigma_{t},\Sigma;f]=$ $\displaystyle
E_{rel}[\Sigma_{s},\Sigma;f]+\int_{t}^{s}dr\int_{\Sigma_{r}}f\left\langle\mathbf{H},\mathbf{H}\right\rangle
d\mathcal{H}^{n}_{\mathbb{H}}$
$\displaystyle-\int_{t}^{s}E_{rel}[\Sigma_{r},\Sigma;\partial_{t}f-\Delta
f+\left\langle\nabla_{\mathbf{n}_{\Sigma_{r}}}\nabla
f,\mathbf{n}_{\Sigma_{r}}\right\rangle]dr.$

###### Proof.

From Lemma 5.1 and Lemma 5.2, we know that there is $\epsilon_{1}>0$, so that
for any $0<s_{1}\leq s_{2}<\epsilon_{1}$,

(5.8) $\big{|}E_{rel}[\Sigma_{t},\Sigma;\chi_{\\{y\geq
s_{1}\\}}]-E_{rel}[\Sigma_{t},\Sigma;\chi_{\\{y\geq s_{2}\\}}]\big{|}\leq
C_{1}s_{2}.$

This implies the existence and finiteness of $E_{rel}[\Sigma_{t},\Sigma]$. In
(5.8), letting $s_{1}\to 0^{+}$, we have

$E_{rel}[\Sigma_{t},\Sigma]\geq E_{rel}[\Sigma_{t},\Sigma;\chi_{\\{y\geq
s_{2}\\}}]-C_{1}s_{2}.$

On the other hand,

$E_{rel}[\Sigma_{t},\Sigma;\chi_{\\{y\geq
s_{2}\\}}]=\int_{\Sigma_{t}}\chi_{\\{y\geq
s_{2}\\}}d\mathcal{H}^{n}_{\mathbb{H}}-\int_{\Sigma}\chi_{\\{y\geq
s_{2}\\}}d\mathcal{H}^{n}_{\mathbb{H}}\geq-\mathcal{H}^{n}_{\mathbb{H}}(\Sigma\cap\\{y\geq
s_{2}\\}).$

Thus, we can take $E_{0}=\mathcal{H}^{n}_{\mathbb{H}}(\Sigma\cap\\{y\geq
s_{2}\\})+C_{1}s_{2}$.

Now, let us take a smooth cut-off function
$\eta_{\epsilon}(y)=\eta_{\epsilon}(y(p))$ that is $1$ when $y\geq\epsilon$
and vanishes when $y\leq\frac{\epsilon}{2}$. We can further assume that
$\big{|}\epsilon^{k}\eta^{(k)}_{\epsilon}\big{|}\leq C_{k}$ for any $k$. Then,
computing the first variation, we get

(5.9)
$\displaystyle\frac{d}{dt}E_{rel}[\Sigma_{t},\Sigma;\eta_{\epsilon}f]=E_{rel}[\Sigma_{t},\Sigma;\partial_{t}(\eta_{\epsilon}f)+\left\langle\nabla(\eta_{\epsilon}f),\textbf{H}\right\rangle]-\int_{\Sigma_{t}}\eta_{\epsilon}f\left\langle\textbf{H},\textbf{H}\right\rangle
d\mathcal{H}^{n}_{\mathbb{H}}.$

Here, we have used the minimality of $\Sigma$. Integrating (5.9) from $t$ to
$s$, we get

$\displaystyle
E_{rel}[\Sigma_{s},\Sigma;\eta_{\epsilon}f]-E_{rel}[\Sigma_{t},\Sigma;\eta_{\epsilon}f]$
$\displaystyle=$
$\displaystyle-\int_{t}^{s}dr\int_{\Sigma_{r}}\eta_{\epsilon}f\left\langle\textbf{H},\textbf{H}\right\rangle
d\mathcal{H}^{n}_{\mathbb{H}}+\int_{t}^{s}\left(E_{rel}[\Sigma_{r},\Sigma;\partial_{t}(\eta_{\epsilon}f)+\left\langle\nabla(\eta_{\epsilon}f),\textbf{H}\right\rangle]\right)dr$
$\displaystyle=$
$\displaystyle-\int_{t}^{s}dr\int_{\Sigma_{r}}\eta_{\epsilon}f\left\langle\textbf{H},\textbf{H}\right\rangle
d\mathcal{H}^{n}_{\mathbb{H}}+\int_{t}^{s}E_{rel}[\Sigma_{r},\Sigma;\partial_{t}(\eta_{\epsilon}f)-\Delta(\eta_{\epsilon}f)+\left\langle\nabla_{\textbf{n}_{\Sigma_{r}}}\nabla(\eta_{\epsilon}f),\textbf{n}_{\Sigma_{r}}\right\rangle]dr.$

From the assumptions on $f$ and $\eta_{\epsilon}$, we know that
$\partial_{t}(\eta_{\epsilon}f)-\Delta(\eta_{\epsilon}f)+\left\langle\nabla_{\textbf{n}_{\Sigma_{r}}}\nabla(\eta_{\epsilon}f),\textbf{n}_{\Sigma_{r}}\right\rangle$
is uniformly bounded in $\mathfrak{X}$ and pointwisely converges to
$\partial_{t}f-\Delta f+\left\langle\nabla_{\textbf{n}_{\Sigma_{r}}}\nabla
f,\textbf{n}_{\Sigma_{r}}\right\rangle$. The lemma below shows that the
corresponding relative entropy also converges. By the same reasoning, we have

$E_{rel}[\Sigma_{s},\Sigma;\eta_{\epsilon}f]\to E_{rel}[\Sigma_{s},\Sigma;f],\
E_{rel}[\Sigma_{t},\Sigma;\eta_{\epsilon}f]\to E_{rel}[\Sigma_{t},\Sigma;f]$

as $\epsilon\to 0^{+}$. The monotone convergence theorem gives

$\int_{t}^{s}dr\int_{\Sigma_{r}}\eta_{\epsilon}f\left\langle\textbf{H},\textbf{H}\right\rangle
d\mathcal{H}^{n}_{\mathbb{H}}\to\int_{t}^{s}dr\int_{\Sigma_{r}}f\left\langle\textbf{H},\textbf{H}\right\rangle
d\mathcal{H}^{n}_{\mathbb{H}}.$

Putting them together, we get (5.7). ∎

######  Lemma 5.4.

If $f$ and $f_{i}$ are functions in $\mathfrak{X}$ with uniformly bounded
$\mathfrak{X}$-norms and $f_{i}$ converges pointwise to $f$, then we have
$E_{rel}[\partial^{*}U,\Sigma;f]=\lim_{i\rightarrow\infty}E_{rel}[\partial^{*}U,\Sigma;f_{i}]$
for any $U\in\mathcal{C}(\Gamma^{\prime}_{-},\Gamma^{\prime}_{+})$ with
$E_{rel}[\partial^{*}U,\Sigma]<\infty$.

###### Proof.

Assume $\|f\|_{\mathfrak{X}}\leq C$ and $\|f_{i}\|_{\mathfrak{X}}\leq C$ for
all $i$. For any $y_{1},y_{2}$ small with
$0<\frac{y_{1}}{2}<y_{1}-\delta<y_{1}<y_{2}<y_{2}+\delta<2y_{2}<\epsilon$,
where $\epsilon$ is the same as in Theorem 4.1,

$\displaystyle|E_{rel}[\partial^{*}U,\Sigma;(f_{i}-f)\phi_{y_{1},y_{2},\delta}]|$
$\displaystyle\leq
C(|E_{rel}[\partial^{*}U,\Sigma;\phi_{y_{1},y_{2},\delta}]|+y_{2})\|f_{i}-f\|_{\mathfrak{X}}$
$\displaystyle\leq
C(|E_{rel}[\partial^{*}U,\Sigma;\phi_{y_{1},y_{2},\delta}]|+y_{2}).$

Letting $\delta\to 0^{+}$ and $y_{1}\to 0^{+}$,

$\displaystyle|E_{rel}[\partial^{*}U,\Sigma;(f_{i}-f)]-E_{rel}[\partial^{*}U,\Sigma;(f_{i}-f)\chi_{\\{y\geq
y_{2}\\}}]|$ $\displaystyle\leq
C(|E_{rel}[\partial^{*}U,\Sigma]-E_{rel}[\partial^{*}U,\Sigma;\chi_{\\{y\geq
y_{2}\\}}]|+y_{2}).$

This implies

$\displaystyle\big{|}E_{rel}[\partial^{*}U,\Sigma;(f_{i}-f)]\big{|}\leq$
$\displaystyle\big{|}E_{rel}[\partial^{*}U,\Sigma;(f_{i}-f)\chi_{\\{y\geq
y_{2}\\}}]\big{|}$
$\displaystyle+C(|E_{rel}[\partial^{*}U,\Sigma]-E_{rel}[\partial^{*}U,\Sigma;\chi_{\\{y\geq
y_{2}\\}}]|+y_{2}).$

For any $\epsilon^{\prime}>0$, we fix
$y_{2}<\min\\{\epsilon,\epsilon^{\prime}\\}$ so small that
$|E_{rel}[\partial^{*}U,\Sigma]-E_{rel}[\partial^{*}U,\Sigma;\chi_{\\{y\geq
y_{2}\\}}]|<\epsilon^{\prime}$. By the dominated convergence theorem, for any
fix $y_{2}$, $\big{|}E_{rel}[\partial^{*}U,\Sigma;(f_{i}-f)\chi_{\\{y\geq
y_{2}\\}}]\big{|}<\epsilon^{\prime}$ for $i$ sufficiently large. So, putting
these together, we see that for any $\epsilon^{\prime}>0$,
$\big{|}E_{rel}[\partial^{*}U,\Sigma;(f_{i}-f)]\big{|}\leq C\epsilon^{\prime}$
for $i$ sufficienly large. This concludes the proof. ∎

##  Appendix A A computation for geodesics in hyperbolic place

We verify the relative entropy is conformally invariant for $n=1$ by computing
out the explicit formula of the relative entropy for geodesics in hyperbolic
plane. Specifically, consider the upper half space model,
($\mathbb{R}^{2}_{+},\frac{1}{y^{2}}(dx\otimes dx+dy\otimes dy)$), of
$\mathbb{H}^{2}$. Let $a_{1}<a_{2}<a_{3}<a_{4}$ be four points on the real
line. Let $\gamma_{1}$ be the union of the geodesic joining $a_{1}$ and
$a_{2}$ (denoted by $\gamma_{1,1}$) and the geodesic joining $a_{3}$ and
$a_{4}$ (denoted by $\gamma_{1,2}$). Let $\gamma_{2}$ be the union of the
geodesic joining $a_{1}$ and $a_{3}$ (denoted by $\gamma_{2,1}$) and the
geodesic joining $a_{2}$ and $a_{4}$ (denoted by $\gamma_{2,2}$). We will give
an explicit formula for $E_{rel}[\gamma_{1},\gamma_{2}]$.

It is easy to check that $\gamma_{1,1}$ is the upper half circle defined
implicitly by the equation

(A.1) $y^{2}+(x-a_{1})(x-a_{2})=0,\ y\geq 0.$

Then,

$\displaystyle\mathcal{H}^{1}_{\mathbb{H}}(\gamma_{1,1}\cap\\{y\geq\epsilon\\})=\int_{\gamma_{1,1}\cap\\{y\geq\epsilon\\}}\frac{1}{y}d\mathcal{H}^{1}_{\mathbb{H}}=\int_{x_{1}(\epsilon)}^{x_{2}(\epsilon)}\frac{1}{y}\sqrt{1+\big{(}y^{\prime}(x)\big{)}^{2}}dx,$

where $y$ is thought of as a function in $x$ and
$x_{1}(\epsilon)<x_{2}(\epsilon)$ are the roots to the equation
$\epsilon^{2}+(x-a_{1})(x-a_{2})=0$. Differentiating (A.1), we get
$yy^{\prime}+(x-\frac{a_{1}+a_{2}}{2})=0$. Hence,

$\sqrt{1+\big{(}y^{\prime}(x)\big{)}^{2}}=\frac{1}{y}\sqrt{y^{2}+\left(x-\frac{a_{1}+a_{2}}{2}\right)^{2}}=\frac{a_{2}-a_{1}}{2y}.$

As a result,

$\displaystyle\mathcal{H}^{1}_{\mathbb{H}}(\gamma_{1,1}\cap\\{y\geq\epsilon\\})=\int_{x_{1}(\epsilon)}^{x_{2}(\epsilon)}\frac{a_{2}-a_{1}}{2y^{2}}dx=\int_{x_{1}(\epsilon)}^{x_{2}(\epsilon)}\frac{a_{2}-a_{1}}{2(a_{2}-x)(x-a_{1})}dx.$

Notice that
$x_{i}(\epsilon)=\frac{1}{2}(a_{1}+a_{2}\pm\sqrt{(a_{2}-a_{1})^{2}-4\epsilon^{2}})$.
So,

$\mathcal{H}^{1}_{\mathbb{H}}(\gamma_{1,1}\cap\\{y\geq\epsilon\\})=\ln\left(\frac{x_{2}(\epsilon)-a_{1}}{x_{1}(\epsilon)-a_{1}}\right)=2\ln\left(\frac{a_{2}-a_{1}+\sqrt{(a_{2}-a_{1})^{2}-4\epsilon^{2}}}{2\epsilon}\right).$

Then, we do similar computations for $\gamma_{1,2}$, $\gamma_{2,1}$ and
$\gamma_{2,2}$. As explained in Section 4, we can pick a boundary defining
function $r_{1}$ satisfying $r_{1}(\textbf{x},y)=y$ in a compact subset of
$\mkern 1.5mu\overline{\mkern-1.5mu\mathbb{R}^{n+1}_{+}\mkern-1.5mu}\mkern
1.5mu$ containing $\Omega$. So,

$\displaystyle E_{rel}[\gamma_{1},\gamma_{2}]$
$\displaystyle=\lim_{\epsilon\to
0}\sum_{i=1}^{2}\mathcal{H}^{1}_{\mathbb{H}}(\gamma_{1,i}\cap\\{y\geq\epsilon\\})-\sum_{i=1}^{2}\mathcal{H}^{1}_{\mathbb{H}}(\gamma_{2,i}\cap\\{y\geq\epsilon\\})$
$\displaystyle=\lim_{\epsilon\to
0}2\ln\left(\frac{(a_{2}-a_{1}+\sqrt{(a_{2}-a_{1})^{2}-4\epsilon^{2}})(a_{4}-a_{3}+\sqrt{(a_{4}-a_{3})^{2}-4\epsilon^{2}})}{(a_{3}-a_{1}+\sqrt{(a_{3}-a_{1})^{2}-4\epsilon^{2}})(a_{4}-a_{2}+\sqrt{(a_{4}-a_{2})^{2}-4\epsilon^{2}})}\right)$
$\displaystyle=2\ln\left(\frac{(a_{2}-a_{1})(a_{4}-a_{3})}{(a_{3}-a_{1})(a_{4}-a_{2})}\right)=2\ln[a_{1},a_{4};a_{2},a_{3}],$

where $[a_{1},a_{4};a_{2},a_{3}]$ is the cross ratio. It is preserved under
the Möbius transformations of $\mathbb{S}^{1}$.

Let $\gamma_{3,1}$ be geodesic joining $a_{1}$ and $a_{3}$, and let
$\gamma_{3,2}$ be the geodesic joining $a_{2}$ and $a_{4}$. Let
$\gamma_{3}=\gamma_{3,1}\sqcup\gamma_{3,2}$. Then, $\gamma_{3}$ is stationary
in $\mathbb{H}^{2}$ but has a singular point. Consider
$E_{rel}[\gamma_{i},\gamma_{j}]$ with $1\leq i,j\leq 3$ and $i\neq j$. This
gives $6$ different values of $E_{rel}$ in general. So, the relative entropy
gives every permutation of the cross ratio of $a_{1}$, $a_{2}$, $a_{3}$ and
$a_{4}$.

## References

  * [AM10] Spyridon Alexakis and Rafe Mazzeo.  Renormalized area and properly embedded minimal surfaces in hyperbolic 3-manifolds.  Communications in Mathematical Physics, 297(3):621–651, 2010.
  * [And82] Michael T. Anderson.  Complete minimal varieties in hyperbolic space.  Inventiones mathematicae, 69(3):477–494, Oct 1982.
  * [And83] Michael T. Anderson.  Complete minimal hypersurfaces in hyperbolic n-manifolds.  Commentarii Mathematici Helvetici, 58(1):264–290, Dec 1983.
  * [Ber22] Jacob Bernstein.  A sharp isoperimetric property of the renormalized area of a minimal surface in hyperbolic space.  Proc. Amer. Math. Soc., 2022.
  * [BW21] Jacob Bernstein and Lu Wang.  The space of asymptotically conical self-expanders of mean curvature flow.  Math. Ann., 380:175–230, 2021.
  * [BW22] Jacob Bernstein and Lu Wang.  Relative expander entropy in the presence of a two-sided obstacle and applications.  Advances in Mathematics, 399:108284, 2022.
  * [Cos06] Baris Coskunuzer.  Generic uniqueness of least area planes in hyperbolic space.  Geometry & Topology, 10:401–412, 2006.
  * [Cos11] Baris Coskunuzer.  On the number of solutions to the asymptotic plateau problem.  Journal of Gökova Geometry Topology, 5:1–19, 2011.
  * [CY07] Binglong Chen and Le Yin.  Uniqueness and pseudolocality theorems of the mean curvature flow.  Communications in Analysis and Geometry, 15:435–490, 2007.
  * [dOS98] Geraldo de Oliveira and Marc Soret.  Complete minimal surfaces in hyperbolic space.  Mathematische Annalen, 311(3):397–419, Jul 1998.
  * [Gab97] David Gabai.  On the geometric and topological rigidity of hyperbolic 3-manifolds.  Journal of the American Mathematical Society, 10(1):37–74, 1997\.
  * [Giu84] Enrico Giusti.  Minimal Surfaces and Functions of Bounded Variation.  Monographs in Mathematics. Birkhäuser Boston, 1984.
  * [HL87] Robert Hardt and Fang-Hua Lin.  Regularity at infinity for area-minimizing hypersurfaces in hyperbolic space.  Inventiones mathematicae, 88(1):217–224, 1987.
  * [HW15] Zheng Huang and Biao Wang.  Counting minimal surfaces in quasi-fuchsian three-manifolds.  Transactions of the American Mathematical Society, 367(9):6063–6083, 2015.
  * [Lin89] Fang-Hua Lin.  On the dirichlet problem for minimal graphs in hyperbolic space.  Inventiones mathematicae, 96(3):593–612, 1989.
  * [LSU88] O.A. Ladyženskaja, V.A. Solonnikov, and N.N. Ural’ceva.  Linear and Quasi-linear Equations of Parabolic Type.  Translations of mathematical monographs. American Mathematical Society, 1988.
  * [MW13] Francisco Martin and Brian White.  Properly embedded, area-minimizing surfaces in hyperbolic 3-space.  arXiv: Differential Geometry, 2013.
  * [Ngu21] Manh Tien Nguyen.  Weighted monotonicity theorems and applications to minimal surfaces in hyperbolic space.  https://arxiv.org/abs/2105.12625, 2021.  Preprint.
  * [RW99] C. Robin Graham and Edward Witten.  Conformal anomaly of submanifold observables in ads/cft correspondence.  Nuclear Physics B, 546(1):52–64, 1999.
  * [Ton96] Yoshihiro Tonegawa.  Existence and regularity of constant mean curvature hypersurfaces in hyperbolic space.  Mathematische Zeitschrift, 221(1):591–615, 1996.
  * [Tyr22] Aaron J. Tyrrell.  Renormalized Area for Minimal Hypersurfaces of 5D Poincaré-Einstein Spaces.  https://arxiv.org/abs/2204.06663, 2022.  Preprint.
  * [Yao] Junfu Yao.  A mountain-pass theorem in hyperbolic space and its application.  In preparation.