\text{Ric}_{r\mu}^{g}&=\frac{n}{2}\bigg{(}\frac{\operatorname{trace}_{\Omega}(\textbf{m})_{,\mu}}{r^{n+1}}-\frac{\textbf{m}_{\mu,\nu}^{\nu}}{r^{n-1}}\bigg{)}+\frac{n}{2}\bigg{(}\frac{\Gamma^{\nu}_{\rho\mu}\textbf{m}^{\rho}_{\nu}-\Gamma^{\nu}_{\nu\rho}\textbf{m}^{\rho}_{\mu}}{r^{n-1}}\bigg{)}+\mathcal{O}^{\ell-2,\alpha}(r^{-(n+2)}),\\\
\text{Ric}_{\mu\nu}^{g}&=-(n-1)b_{\mu\nu}+\mathcal{O}(1).\end{split}$ (A.2)

In Proposition A.2 we compute the mass vector in Definition 2.4.

######  Proposition A.2.

Let $(M^{n},g,k)$ be asymptotically hyperbolic initial data of type
$(\ell,\alpha,\tau=n,\tau_{0})$ with Wang’s asymptotics as in Definition 2.3.
Then the components of the mass vector $(E,\vec{P})$ are

$E=\frac{1}{(n-1)\omega_{n-1}}\int_{\mathbb{S}^{n-1}}\bigg{(}\operatorname{trace}_{\Omega}(\textbf{p})+\bigg{(}\frac{n-2}{2}\bigg{)}\operatorname{trace}_{\Omega}(\textbf{m})\bigg{)}dS$
(A.3)

and

$P^{i}=\frac{1}{(n-1)\omega_{n-1}}\int_{\mathbb{S}^{n-1}}\bigg{(}\operatorname{trace}_{\Omega}(\textbf{p})+\bigg{(}\frac{n-2}{2}\bigg{)}\operatorname{trace}_{\Omega}(\textbf{m})\bigg{)}x^{i}dS,$
(A.4)

where the $x^{i}$ are the coordinate functions from $\mathbb{R}^{n}$ to
$\mathbb{S}^{n-1}$ and $\omega_{n-1}=|\mathbb{S}^{n-1}|_{\delta}$.

###### Proof.

We first calculate $E$ using Definition 2.4. Clearly the only non-zero
components of $e=g-b$ are

$e_{\mu\nu}=g_{\mu\nu}-b_{\mu\nu}=\frac{\textbf{m}_{\mu\nu}}{r^{n-2}}+R_{\mu\nu},$
(A.5)

where $R_{\mu\nu}$ is a function that falls off as $\mathcal{O}(r^{-\tau})$
and with derivatives falling off as
$\partial^{k}_{r}\partial^{\ell}_{\mu}R_{\mu\nu}=\mathcal{O}(r^{-(\tau+k)})$.
The unit normal with respect to the coordinate sphere of radius $R$ is
$\vec{n}^{r}=\sqrt{1+r^{2}}\partial_{r}$ and so we need only the $r$-component
of the $1$-form appearing in (2.7). The radial component of
$\text{div}^{b}(e)$ is

$\begin{split}\text{div}^{b}(e)_{r}&=b^{ij}(\nabla_{i}e)_{rj}\\\
&=b^{\mu\nu}(\nabla_{\mu}e)_{r\nu}\\\
&=b^{\mu\nu}\bigg{(}e_{r\nu,\mu}-e_{m\mu}\Gamma_{r\nu}^{m}-e_{rm}\Gamma^{m}_{\mu\nu}\bigg{)}\\\
&=-b^{\mu\nu}e_{\sigma\mu}\Gamma_{r\nu}^{\sigma}\\\
&=-b^{\mu\nu}\bigg{(}\frac{\textbf{m}_{\sigma\mu}}{r^{n-2}}+\mathcal{O}(r^{-(n-1)})\bigg{)}\bigg{(}\frac{\delta^{\sigma}_{\nu}}{r}-\frac{n}{2}\frac{\textbf{m}_{\nu}^{\sigma}}{r^{n-1}}+\mathcal{O}(r^{-(n+2)})\bigg{)}\\\
&=-\frac{\operatorname{trace}_{\Omega}(\textbf{m})}{r^{n+1}}+\mathcal{O}(r^{-(n+2)}),\end{split}$
(A.6)

where we used Christoffel symbols from Lemma A.1. Similarly, since $(\langle
b,e\rangle_{b})_{,r}=\langle b,\nabla_{r}b\rangle_{b}$, we have

$\begin{split}d\operatorname{trace}^{b}(e)_{r}&=\operatorname{trace}^{b}(e)_{,r}\\\
&=\langle b,\nabla_{r}b\rangle_{b}\\\ &=b^{ij}(\nabla_{r}b)_{ij}\\\
&=b^{ij}\big{(}e_{ij,r}-\Gamma_{ri}^{\ell}e_{\ell
j}-\Gamma^{\ell}_{rj}e_{i\ell}\big{)}\\\
&=b^{\mu\nu}\big{(}e_{\mu\nu,r}-\Gamma_{r\nu}^{\rho}e_{\rho\mu}-\Gamma_{r\mu}^{\rho}e_{\rho\nu}\big{)}\\\
&=b^{\mu\nu}\big{(}-(n-2)\frac{\textbf{m}_{\mu\nu}}{r^{n-1}}-2\frac{e_{\mu\nu}}{r}+\mathcal{O}(r^{-n})\\\
&=-n\frac{\operatorname{trace}_{\Omega}(\textbf{m})}{r^{n+1}}+\mathcal{O}(r^{-(n+2)}).\end{split}$
(A.7)

It follows that, for $V_{0}=\sqrt{1+r^{2}}$, we have

$\begin{split}V_{0}\big{(}\text{div}^{b}(e)_{r}-d\operatorname{trace}^{b}(e)_{r}\big{)}&=(n-1)\frac{\operatorname{trace}^{b}(\textbf{m})}{r^{n}}+\mathcal{O}(r^{-(n+1)}).\end{split}$
(A.8)

Furthermore, since $dV_{0}=\frac{r}{\sqrt{1+r^{2}}}dr$, we have

$\operatorname{trace}^{b}(e)dV_{0}=\bigg{(}\frac{\operatorname{trace}_{\Omega}(\textbf{m})}{r^{n}}+\mathcal{O}(r^{-(n+1)})\bigg{)}dr.$
(A.9)

Moreover, since $\nabla^{b}V_{0}=r\sqrt{1+r^{2}}\partial_{r}$, we obtain

$\begin{split}(e+2\eta)(\nabla^{b}V_{0},\cdot)_{r}&=(e+2\eta)(\nabla^{b}V_{0},\partial_{r})\\\
&=(e+2\eta)_{rr}r\sqrt{1+r^{2}}\\\ &=2\eta_{rr}r\sqrt{1+r^{2}},\end{split}$
(A.10)

where

$\begin{split}\eta_{rr}&=(k-g)_{rr}-\operatorname{trace}_{g}(k-g)g_{rr}\\\
&=-g^{\mu\nu}(k-g)_{\mu\nu}g_{rr}\\\
&=-\bigg{(}\frac{\operatorname{trace}_{\Omega}(\textbf{p})-\operatorname{trace}_{\Omega}(\textbf{m})}{r^{n}}\bigg{)}\frac{1}{1+r^{2}}+\mathcal{O}(r^{-(n+3)}).\end{split}$
(A.11)

In summary, we obtain

$\begin{split}V_{0}\big{(}\text{div}^{b}(e)_{r}&-d\operatorname{trace}^{b}(e)_{r}\big{)}+\operatorname{trace}^{b}(e)(dV_{0})_{r}-(e+2\eta)(\nabla^{b}V_{0},\cdot)_{r}\\\
&\qquad=2\bigg{(}\frac{\operatorname{trace}_{\Omega}(\textbf{p})}{r^{n}}+\bigg{(}\frac{n-2}{2}\bigg{)}\frac{\operatorname{trace}_{\Omega}(\textbf{m})}{r^{n}}\bigg{)}+\mathcal{O}(r^{-(n+1)}).\end{split}$
(A.12)

Recalling that $\vec{n}^{r}=\sqrt{1+r^{2}}\partial_{r}$ we conclude that

$E=\frac{1}{(n-1)\omega_{n-1}}\int_{\mathbb{S}^{n-1}}\bigg{(}\operatorname{trace}_{\Omega}(\textbf{p})+\bigg{(}\frac{n-2}{2}\bigg{)}\operatorname{trace}_{\Omega}(\textbf{m})\bigg{)}dS\\\
$ (A.13)

as asserted.
We briefly comment on the proof of the expression for $P^{i}$. We have
$V_{i}=x^{i}r$ and so the first terms in the charge integral will become

$V_{i}\bigg{(}\text{div}^{b}(e)_{r}-d\operatorname{trace}^{b}(e)_{r}\bigg{)}=x^{i}\bigg{(}(n-1)\frac{\operatorname{trace}_{\Omega}(\textbf{m})}{r^{n}}+\mathcal{O}(r^{-(n+1)})\bigg{)}.$
(A.14)

Further, we have both $dV_{i}=x^{i}dr+rx^{i}_{,\mu}dx^{\mu}$ and
$\nabla^{b}V_{i}=(1+r^{2})x^{i}\partial_{r}+b^{\mu\nu}rx^{i}_{,\nu}\partial_{\mu}$.
In turn, using (A.11), we obtain

$\begin{split}(e+2\eta)(\nabla^{b}V_{i},\partial_{r})&=2\eta_{rr}(1+r^{2})x^{i}\\\
&=\bigg{(}-2\bigg{(}\frac{\operatorname{trace}_{\Omega}(\textbf{p})-\operatorname{trace}_{\Omega}(\textbf{m})}{r^{n}}\bigg{)}+\mathcal{O}(r^{-(n+1)})\bigg{)}x^{i}.\end{split}$
(A.15)

Conclusively, we get

$\begin{split}V_{i}\big{(}\text{div}^{b}(e)_{r}&-d\operatorname{trace}^{b}(e)_{r}\big{)}+\operatorname{trace}^{b}(e)(dV_{i})_{r}-(e+2\eta)(\nabla^{b}V_{i},\cdot)_{r}\\\
&\qquad=2\bigg{(}\frac{\operatorname{trace}_{\Omega}(\textbf{p})}{r^{n}}+\bigg{(}\frac{n-2}{2}\bigg{)}\frac{\operatorname{trace}_{\Omega}(\textbf{m})}{r^{n}}\bigg{)}x^{i}+\mathcal{O}(r^{-(n+1)})\end{split}$
(A.16)

and so the assertion follows.

∎

##  Appendix B Geometry of a smooth approximate Jang graph

We present some useful properties of the Jang graph obtained in Proposition
5.2 with asymptotics as in Proposition 3.14 under the extra assumption that
the geometry is smooth and asymptotically flat as in Corollary 6.13.
Throughout this section we have an asymptotically hyperbolic initial data
$(M^{n},g,k)$ of type $(\ell=\infty,\alpha,\tau=n,\tau_{0})$ with Wang’s
asymptotics as in Definition 2.3 and a function $f:M^{n}\rightarrow\mathbb{R}$
that is smooth and solves Jang’s equation _approximately_ outside of a compact
set, that is we have $\mathcal{J}(f)=\mathcal{O}(r^{-(n+1-\epsilon)})$. We
recall from Section 3 that such a function has the asymptotics

$f=\sqrt{1+r^{2}}+\frac{\alpha}{r^{n-3}}+q(r,\theta),$ (B.1)

where $q$ is a smooth function such that
$\partial^{k}_{r}\partial^{\ell}_{\mu}q=\mathcal{O}(r^{-(n-2+k-\epsilon)})$.
From Corollary 6.13 we know that the _exact_ solution to Jang’s equation must
also have the asymptotics of (B.1). We let $\hat{M}^{n}$ denote the graph of
$f$ in $M^{n}\times\mathbb{R}$, and similarly we use hatted symbols for the
geometric quantities, so that for instance the induced metric on $\hat{M}^{n}$
is $\hat{g}$ and the Christoffel symbols are denoted by $\hat{\Gamma}$. The
projection diffeomorphism $\Pi:\hat{M}^{n}\rightarrow M^{n}$ pulls back vector
fields $\Pi^{\ast}(\partial_{i})=\partial_{i}+f_{,i}\partial_{t}$. Throughout
the section we mean by $\mathcal{O}_{\infty}(r^{-\tau})$ a smooth function
that falls off as $\mathcal{O}(r^{-\tau})$.

######  Lemma B.1.

The induced metric $\hat{g}$ has the coordinate expression
$\hat{g}_{ij}=g_{ij}+f_{,i}f_{,j}$ with the components:

$\begin{split}\hat{g}_{rr}&=1-2(n-3)\frac{\alpha}{r^{n-2}}+\mathcal{O}_{\infty}(r^{-(n-1-\epsilon)}),\\\
\hat{g}_{\mu
r}&=\frac{\alpha_{,\mu}}{r^{n-3}}+\mathcal{O}_{\infty}(r^{-(n-2-\epsilon)}),\\\
\hat{g}_{\mu\nu}&=\delta_{\mu\nu}+\frac{\textbf{m}_{\mu\nu}}{r^{n-2}}+\mathcal{O}_{\infty}(r^{-(n-1-\epsilon)}).\end{split}$
(B.2)

The components of the inverse metric
$\hat{g}^{ij}=g^{ij}-\frac{f^{,i}f^{,j}}{1+|df|_{g}^{2}}$ are:

$\begin{split}\hat{g}^{rr}&=1+2(n-3)\frac{\alpha}{r^{n-2}}+\mathcal{O}_{\infty}(r^{-(n-1-\epsilon)}),\\\
\hat{g}^{\mu
r}&=-\delta^{\mu\nu}\frac{\alpha_{,\nu}}{r^{n-3}}+\mathcal{O}_{\infty}(r^{-(n-\epsilon)}),\\\
\hat{g}^{\mu\nu}&=\delta^{\mu\nu}+\mathcal{O}_{\infty}(r^{-(n+1-\epsilon)}).\end{split}$
(B.3)

######  Lemma B.2.

The Christoffel symbols of $(\hat{M}^{n},\hat{g})$ are:

$\begin{split}\hat{\Gamma}^{r}_{rr}&=(n-2)(n-3)\frac{\alpha}{r^{n-1}}+\mathcal{O}_{\infty}(r^{-(n-\epsilon)}),\\\
\hat{\Gamma}^{\mu}_{rr}&=\mathcal{O}_{\infty}(r^{-(n+1-\epsilon)}),\\\
\hat{\Gamma}^{r}_{r\mu}&=-(n-2)\frac{\alpha_{,\mu}}{r^{n-2}}+\mathcal{O}_{\infty}(r^{-(n-1-\epsilon)}),\\\
\hat{\Gamma}^{\mu}_{r\nu}&=\frac{\delta^{\mu}_{\nu}}{r}-\bigg{(}\frac{n-2}{2}\bigg{)}\frac{\delta^{\mu\rho}\textbf{m}_{\nu\rho}}{r^{n-1}}+\mathcal{O}_{\infty}(r^{-(n+2-\epsilon)}),\\\
\hat{\Gamma}^{r}_{\mu\nu}&=-\frac{\delta_{\mu\nu}}{r}+\frac{\text{Hess}^{\Omega}_{\mu\nu}(\alpha)}{r^{n-3}}+\bigg{(}\frac{n-2}{2}\bigg{)}\frac{\textbf{m}_{\mu\nu}}{r^{n-1}}\\\
&\qquad-2(n-3)\frac{\alpha}{r^{n-1}}\delta_{\mu\nu}+\mathcal{O}_{\infty}(r^{-(n-\epsilon)}),\\\
\hat{\Gamma}^{\rho}_{\mu\nu}&=\frac{1}{2}\delta^{\rho\sigma}\big{(}\delta_{\mu\sigma,\nu}+\delta_{\nu\sigma,\mu}-\delta_{\mu\nu,\sigma}\big{)}+\mathcal{O}_{\infty}(r^{-(n-1-\epsilon)}).\end{split}$
(B.4)

######  Lemma B.3.

The components of the Ricci tensor of $(\hat{M}^{n},\hat{g})$ are:

$\begin{split}\text{Ric}^{\hat{g}}_{rr}&=-2\frac{\Delta^{\Omega}\alpha}{r^{n}}+n(n-1)(n-3)\frac{\alpha}{r^{n}}+\mathcal{O}_{\infty}(r^{-(n+1-\epsilon)}),\\\
\text{Ric}^{\hat{g}}_{\mu
r}&=-2(n-1)\frac{\alpha_{,\mu}}{r^{n-1}}+\mathcal{O}_{\infty}(r^{-(n-\epsilon)}),\\\
\text{Ric}^{\hat{g}}_{\mu\nu}&=(n-1)\frac{\text{Hess}_{\mu\nu}^{\Omega}\alpha}{r^{n-2}}+\delta_{\mu\nu}\frac{\Delta^{\Omega}\alpha}{r^{n}}-n(n-3)\frac{\alpha}{r^{n}}\delta_{\mu\nu}+\mathcal{O}_{\infty}(r^{-(n-1-\epsilon)}),\\\
\end{split}$ (B.5)

In particular, we have

$R_{\hat{g}}=2(n-2)\frac{\Delta^{\Omega}\alpha}{r^{n}}+\mathcal{O}_{\infty}(r^{-(n+1-\epsilon)}).$
(B.6)

###### Proof.

We assist the reader with estimates on the scalar curvature, as the assertions
in this Lemma are proven by routine computations.
We recall from [SY81b, Equations 2.23-2.25] that, for any function
$f:U_{f}\rightarrow\mathbb{R}$, we have

$\begin{split}R_{\hat{g}}&=2(\mu-J(\omega))+|\hat{A}-k|_{\hat{g}}^{2}+2|q|_{\hat{g}}^{2}-2\text{div}^{\hat{g}}(q)+H_{\hat{M}^{n}}^{2}-(\operatorname{trace}^{\hat{g}}(k))^{2}\\\
&\qquad+2k(\vec{n},\vec{n})\big{(}H_{\hat{M}^{n}}-\operatorname{trace}^{\hat{g}}(k)\big{)}+2\vec{n}\big{(}H_{\hat{M}^{n}}-\operatorname{trace}^{\hat{g}}(k)\big{)}\end{split}$
(B.7)

where $\hat{A}$ is the second fundamental form of the graph of $f$, $k$ is the
symmetric tensor from the initial data $(M^{n},g,k)$ extended trivially to
$(M^{n}\times\mathbb{R},g+dt^{2})$ and

$\omega=\frac{\nabla^{g}f}{\sqrt{1+|df|_{g}^{2}}},\qquad\text{and}\qquad
q_{i}=\frac{f^{,j}}{\sqrt{1+|df|_{g}^{2}}}(\hat{A}_{ij}-k_{ij}).$ (B.8)

If $f$ solves Jang’s equation $\mathcal{J}(f)=0$, the last three terms on the
right hand side vanish, and we recover the _Schoen-Yau_ identity.
The first two of the last three terms have the fall-off rate
$\mathcal{O}(r^{-(n+1-\epsilon)})$ by boundedness of $k(\vec{n},\vec{n})$ and
the equation that $f$ satisfies. For the last term, the same fall-off rate is
obtained after recalling the asymptotics of $\vec{n}^{k}$ from the proof of
Lemma 6.8.
It follows from Definition 2.2 that
$\mu-J(\omega)=\mathcal{O}(r^{-(n+\tau_{0})})$, but we can without loss of
generality assume $\epsilon\leq\tau_{0}$. We estimate the remaining terms;
firstly, we claim

$|\hat{A}-k|_{\hat{g}}^{2}=\mathcal{O}_{\infty}(r^{-(n+1-\epsilon)})$ (B.9)

Indeed, from Definition 2.3 and Lemmas B.1 and B.4 we get

$\begin{split}\hat{A}_{rr}-k_{rr}&=(n-2)(n-3)\frac{\alpha}{r^{n}}+\mathcal{O}_{\infty}(r^{-(n+1-\epsilon)}),\\\
\hat{A}_{r\mu}-k_{\mu
r}&=-(n-2)\frac{\alpha_{,\mu}}{r^{n-1}}+\mathcal{O}_{\infty}(r^{-(n-\epsilon)}),\\\
\hat{A}_{\mu\nu}-k_{\mu\nu}&=\frac{\text{Hess}_{\mu\nu}^{\Omega}(\alpha)}{r^{n-2}}-\bigg{(}\frac{n-2}{2}\bigg{)}\frac{\textbf{m}_{\mu\nu}}{r^{n-2}}-\frac{\textbf{p}_{\mu\nu}}{r^{n-2}}+\mathcal{O}_{\infty}(r^{-(n-1-\epsilon)}).\end{split}$
(B.10)

From these estimates and Lemma B.1 it is easily seen that
$|\hat{A}-k|_{\hat{g}}^{2}$ has the asserted decay.
As for the $|q|_{\hat{g}}^{2}$-term, we note that from Lemma B.1 we have

$\frac{1}{\sqrt{1+|df|_{g}^{2}}}=\frac{1}{\sqrt{1+r^{2}}}+(n-3)\frac{\alpha}{r^{n-1}}+\mathcal{O}_{\infty}(r^{-(n-\epsilon)}).$
(B.11)

We first compute radial component

$q_{r}=\frac{g^{rr}f_{,r}}{\sqrt{1+|df|_{g}^{2}}}(\hat{A}_{rr}-k_{rr})+\frac{g^{\mu\nu}f_{,\nu}}{\sqrt{1+|df|_{g}^{2}}}(\hat{A}_{r\mu}-k_{r\mu}).\\\
$ (B.12)

The first term is

$\begin{split}\frac{g^{rr}f_{,r}}{\sqrt{1+|df|_{g}^{2}}}(\hat{A}_{rr}-k_{rr})&=(1+r^{2})\bigg{(}\frac{1}{\sqrt{1+r^{2}}}+(n-3)\frac{\alpha}{r^{n-1}}+\mathcal{O}(r^{-(n-\epsilon)})\bigg{)}\\\
&\qquad\times\bigg{(}\frac{r}{\sqrt{1+r^{2}}}-(n-3)\frac{\alpha}{r^{n-2}}+\mathcal{O}(r^{-(n-1-\epsilon)})\bigg{)}\\\
&\qquad\times\bigg{(}(n-2)(n-3)\frac{\alpha}{r^{n}}+\mathcal{O}(r^{-(n+1-\epsilon)})\bigg{)}\\\
&=(n-2)(n-3)\frac{\alpha}{r^{n-1}}+\mathcal{O}(r^{-(n-\epsilon)})\end{split}$
(B.13)

and, similarly, the second term is

$\frac{g^{\mu\nu}f_{,\nu}}{\sqrt{1+|df|_{g}^{2}}}(\hat{A}_{r\mu}-k_{r\mu})=\mathcal{O}_{\infty}(r^{-(2n-1)}),$
(B.14)

so that

$q_{r}=(n-2)(n-3)\frac{\alpha}{r^{n-1}}+\mathcal{O}_{\infty}(r^{-(n-\epsilon)}).$
(B.15)

In a similar fashion, we estimate the tangential component $q_{\mu}$:

$\begin{split}q_{\mu}&=\frac{g^{rr}f_{,r}}{\sqrt{1+|df|_{g}^{2}}}(\hat{A}_{\mu
r}-k_{\mu
r})+\frac{g^{\lambda\rho}f_{,\rho}}{\sqrt{1+|df|_{g}^{2}}}(\hat{A}_{\mu\lambda}-k_{\mu\lambda}),\end{split}$
(B.16)

where the terms are

$\begin{split}\frac{g^{rr}f_{,r}}{\sqrt{1+|df|_{g}^{2}}}(\hat{A}_{\mu
r}-k_{\mu
r})&=-(n-2)\frac{\alpha_{,\mu}}{r^{n-2}}+\mathcal{O}_{\infty}(r^{-(n-1-\epsilon)}),\\\
\frac{g^{\lambda\rho}f_{,\rho}}{\sqrt{1+|df|_{g}^{2}}}(\hat{A}_{\mu\lambda}-k_{\mu\lambda})&=\mathcal{O}_{\infty}(r^{-2(n-1)}),\end{split}$
(B.17)

which gives

$q_{\mu}=-(n-2)\frac{\alpha_{,\mu}}{r^{n-2}}+\mathcal{O}_{\infty}(r^{-(n-1-\epsilon)}).$
(B.18)

The asserted estimate on the norm of $q$ follows:

$\begin{split}|q|_{\hat{g}}^{2}&=\hat{g}^{ij}q_{i}q_{j}\\\
&=\mathcal{O}(r^{-(n+1-\epsilon)}),\end{split}$ (B.19)

using Lemma B.1.
Finally, we estimate $\text{div}^{\hat{g}}(q)$. The Christoffel symbols of the
Jang graph are denoted by $\hat{\Gamma}$, and estimates thereof are found in
Lemma B.2. The covariant derivative of $q$ has components

$(\hat{\nabla}_{i}q)_{j}=q_{j,i}-\hat{\Gamma}_{ij}^{k}q_{k}$ (B.20)

and from (B.15) we see that

$q_{r,r}=-(n-1)(n-2)(n-3)\frac{\alpha}{r^{n}}+\mathcal{O}_{\infty}(r^{-(n+1-\epsilon)}),\\\
$ (B.21)

$q_{r,\mu}=\mathcal{O}_{\infty}(r^{-(n-1)})$,
$q_{\mu,r}=\mathcal{O}_{\infty}(r^{-(n-\epsilon)})$ and
$q_{\mu,\nu}=\mathcal{O}_{\infty}(r^{-(n-1-\epsilon)})$. From Lemma B.2 we
find:

$\begin{split}(\hat{\nabla}_{r}q)_{r}&=q_{r,r}-\big{(}\hat{\Gamma}_{rr}^{r}q_{r}+\hat{\Gamma}_{rr}^{\mu}q_{\mu}\big{)}\\\
&=-(n-1)(n-2)(n-3)\frac{\alpha}{r^{n}}+\mathcal{O}_{\infty}(r^{-(n+1-\epsilon)}).\end{split}$
(B.22)

It follows that

$\hat{g}^{rr}(\hat{\nabla}_{r}q)_{r}=-(n-1)(n-2)(n-3)\frac{\alpha}{r^{n}}+\mathcal{O}_{\infty}(r^{-(n+1-\epsilon)}).$
(B.23)

It is not difficult to see that the mixed terms of $\nabla q$ are

$\begin{split}(\hat{\nabla}_{r}q)_{\mu}&=(n-1)(n-2)\frac{\alpha_{,\mu}}{r^{n-1}}+\mathcal{O}_{\infty}(r^{-(n-\epsilon)})\end{split}$
(B.24)

and similarly

$\begin{split}(\hat{\nabla}_{\mu}q)_{r}&=(n-2)^{2}\frac{\alpha_{,\mu}}{r^{n-1}}+\mathcal{O}_{\infty}(r^{-(n-1)})\end{split}$
(B.25)

so that

$\hat{g}^{r\mu}(\hat{\nabla}_{r}q)_{\mu}=-(n-1)(n-2)\frac{|d\alpha|^{2}_{\Omega}}{r^{-2(n-1)}}+\mathcal{O}_{\infty}(r^{-(2n-1-\epsilon)})$
(B.26)

and

$\hat{g}^{r\mu}(\hat{\nabla}_{\mu}q)_{r}=-(n-2)^{2}\frac{|d\alpha|^{2}_{\Omega}}{r^{-2(n-1)}}+\mathcal{O}_{\infty}(r^{-(2n-1-\epsilon)}).$
(B.27)

Finally the tangential term is

$\begin{split}(\hat{\nabla}_{\mu}q)_{\nu}&=q_{\nu,\mu}-\hat{\Gamma}^{r}_{\mu\nu}q_{r}-\hat{\Gamma}^{\rho}_{\mu\nu}q_{\rho}\\\
&=\delta_{\mu\nu}(n-2)(n-3)\frac{\alpha}{r^{n}}-(n-2)\frac{\text{Hess}^{\Omega}_{\mu\nu}(\alpha)}{r^{n-2}}+\mathcal{O}_{\infty}(r^{-(n-1-\epsilon)})\end{split}$
(B.28)

so that

$\begin{split}\hat{g}^{\mu\nu}(\hat{\nabla}_{\mu}q)_{\nu}&=(n-1)(n-2)(n-3)\frac{\alpha}{r^{n}}-(n-2)\frac{\Delta^{\Omega}(\alpha)}{r^{n}}+\mathcal{O}_{\infty}(r^{-(n+1-\epsilon)}).\end{split}$
(B.29)

The assertion on the fall-off rate of $\text{div}^{\hat{g}}(q)$ follows.
The asserted decay of $R_{\hat{g}}$ now follows.

∎

######  Lemma B.4.

The second fundamental form $\hat{A}$ of $(\hat{M}^{n},\hat{g})$ has
components:

$\begin{split}\hat{A}_{rr}&=\frac{1}{1+r^{2}}+(n-2)(n-3)\frac{\alpha}{r^{n}}+\mathcal{O}_{\infty}(r^{-(n+1-\epsilon)}),\\\
\hat{A}_{\mu
r}&=-(n-2)\frac{\alpha_{,\mu}}{r^{n-1}}+\mathcal{O}_{\infty}(r^{-(n-\epsilon)}),\\\
\hat{A}_{\mu\nu}&=\frac{\text{Hess}_{\mu\nu}^{\Omega}(\alpha)}{r^{n-2}}+\delta_{\mu\nu}-\bigg{(}\frac{n-2}{2}\bigg{)}\frac{\textbf{m}_{\mu\nu}}{r^{n-2}}+\mathcal{O}_{\infty}(r^{-(n-1-\epsilon)}).\end{split}$
(B.30)

######  Lemma B.5.

The covariant derivative $\nabla\hat{g}$ taken with respect to the Euclidean
metric $\delta$ has components:

$\begin{split}(\nabla_{r}\hat{g})_{rr}&=2(n-2)(n-3)\frac{\alpha}{r^{n-1}}+\mathcal{O}_{\infty}(r^{-(n-\epsilon)}),\\\
(\nabla_{r}\hat{g})_{r\mu}&=-(n-2)\frac{\alpha_{,\mu}}{r^{n-2}}+\mathcal{O}_{\infty}(r^{-(n-1-\epsilon)}),\\\
(\nabla_{r}\hat{g})_{\mu\nu}&=\mathcal{O}_{\infty}(r^{-(n-1-\epsilon)})\\\
(\nabla_{\mu}\hat{g})_{rr}&=-2(n-2)\frac{\alpha_{,\mu}}{r^{n-2}}+\mathcal{O}_{\infty}(r^{-(n-1-\epsilon)}),\\\
(\nabla_{\rho}\hat{g})_{r\mu}&=\frac{\alpha_{,\mu\rho}}{r^{n-3}}-2(n-3)\delta_{\rho\mu}\frac{\alpha}{r^{n-1}}-\Gamma^{\sigma}_{\rho\mu}\frac{\alpha_{,\sigma}}{r^{n-3}}+\mathcal{O}_{\infty}(r^{-(n-2-\epsilon)}),\\\
(\nabla_{\rho}\hat{g})_{\mu\nu}&=\frac{\delta_{\mu\rho}\alpha_{,\nu}+\delta_{\rho\nu}\alpha_{,\mu}}{r^{n-2}}+\mathcal{O}_{\infty}(r^{-(1-\epsilon)}).\\\
\end{split}$ (B.31)

######  Lemma B.6.

The covariant derivative $\nabla\hat{g}^{-1}$ taken with respect to the
Euclidean metric $\delta$ has components:

$\begin{split}(\nabla_{r}\hat{g})^{rr}&=-2(n-2)(n-3)\frac{\alpha}{r^{n-1}}+\mathcal{O}_{\infty}(r^{-(n-\epsilon)}),\\\
(\nabla_{r}\hat{g})^{r\mu}&=\mathcal{O}(r^{-(n+1-\epsilon)}),\\\
(\nabla_{r}\hat{g})^{\mu\nu}&=\mathcal{O}(r^{-(n+2-\epsilon)}),\\\
(\nabla_{\sigma}\hat{g})^{rr}&=2(n-2)\frac{\alpha_{,\sigma}}{r^{n-2}}+\mathcal{O}(r^{-(n-1-\epsilon)}),\\\
(\nabla_{\sigma}\hat{g})^{r\mu}&=\mathcal{O}(r^{-(n-1)}),\\\
(\nabla_{\sigma}\hat{g})^{\mu\nu}&=-\delta^{\mu}_{\sigma}\delta^{\nu\beta}\frac{\alpha_{,\beta}}{r^{n-2}}-\delta_{\sigma}^{\nu}\delta^{\mu\beta}\frac{\alpha_{,\beta}}{r^{n-2}}+\mathcal{O}(r^{-(n+1-\epsilon)}).\end{split}$
(B.32)

######  Lemma B.7.

The components of the covariant derivative $\nabla\hat{A}$, taken with respect
to the Euclidean metric $\delta$, are:

$\begin{split}(\nabla_{r}\hat{A})_{rr}&=-\frac{2r}{(1+r^{2})^{2}}-n(n-2)(n-3)\frac{\alpha}{r^{n+1}}+\mathcal{O}_{\infty}(r^{-(n+2-\epsilon)}),\\\
(\nabla_{r}\hat{A})_{\mu r}&=\mathcal{O}_{\infty}(r^{-(n+1-\epsilon)}),\\\
(\nabla_{r}\hat{A})_{\mu\nu}&=-\frac{2}{r}\delta_{\mu\nu}+\mathcal{O}_{\infty}(r^{-(n-1)}),\\\
(\nabla_{\mu}\hat{A})_{rr}&=\mathcal{O}_{\infty}(r^{-n}),\\\
(\nabla_{\mu}\hat{A})_{r\nu}&=-\frac{4}{r}\delta_{\mu\nu}+\mathcal{O}_{\infty}(r^{-(n-1)}),\\\
(\nabla_{\rho}\hat{A})_{\mu\nu}&=\mathcal{O}_{\infty}(r^{-(n-2)}).\end{split}$
(B.33)

##  Appendix C The ADM energy of the Jang graph

We calculate the ADM energy of the Jang graph obtained in Proposition 5.2 with
asymptotics as in Proposition 6.13. The notation used here are as in Section
B.

######  Proposition C.1.

The ADM-mass of the Jang graph $(\hat{M}^{n},\hat{g})$ obtained in Proposition
5.2, with asymptotics as in Corollary 6.13, is

$E_{ADM}=\frac{1}{\omega_{n-1}}\int_{\mathbb{S}^{n-1}}\bigg{(}\operatorname{trace}^{\Omega}(\textbf{p})+\bigg{(}\frac{n-2}{2}\bigg{)}\operatorname{trace}^{\Omega}(\textbf{m})\bigg{)}dS.$
(C.1)

###### Proof.

Throughout this proof the geometric quantities, such as Christoffel symbols
and covariant derivatives, are associated to the Euclidean metric $\delta$.
Similarly to the proof of Proposition A.2, we let the exhaustion of
$\hat{M}^{n}$ be coordinate balls and hence we need the radial components of
the 1-form $\mathbb{U}(\hat{g},\delta)$, the divergence
$\text{div}^{\delta}(\hat{g})$ and the differential of the trace
$d\operatorname{trace}^{\delta}(\hat{g})$. The radial component of the
divergence term is
$\text{div}^{\delta}(\hat{g})_{r}=(\nabla_{r}\hat{g})_{rr}+\delta^{\mu\nu}(\nabla_{\mu}\hat{g})_{r\nu}$.
Using Lemma B.1 and the fact that $\Gamma^{k}_{rr}=0$ we compute

$(\nabla_{r}\hat{g})_{rr}=2(n-1)(n-3)\frac{\alpha}{r^{n-1}}+\mathcal{O}_{\infty}(r^{-(n-\epsilon)}).$
(C.2)

Similarly

$\begin{split}(\nabla_{\mu}\hat{g})_{r\nu}&=\frac{Hess_{\mu\nu}^{\Omega}(\alpha)}{r^{n-3}}-2(n-3)\frac{\alpha}{r^{n-1}}\delta_{\mu\nu}+\mathcal{O}_{\infty}(r^{-(n-2-\epsilon)}).\end{split}$
(C.3)

It follows that

$\begin{split}\delta^{\mu\nu}(\nabla_{\mu}\hat{g})_{r\nu}&=\frac{\Delta^{\Omega}(\alpha)}{r^{n-1}}-2(n-1)(n-3)\frac{\alpha}{r^{n-1}}+\mathcal{O}_{\infty}(r^{-(n-\epsilon)})\\\
\end{split}$ (C.4)

and so in total

$\text{div}^{\delta}(\hat{g})_{r}=\frac{\Delta^{\Omega}(\alpha)}{r^{n-1}}+\mathcal{O}_{\infty}(r^{-(n-\epsilon)}).$
(C.5)

To find the component of the gradient of the trace we first note that

$\operatorname{trace}^{\delta}(\hat{g})=n-2(n-3)\frac{\alpha}{r^{n-2}}+\mathcal{O}_{\infty}(r^{-(n-1-\epsilon)})$
(C.6)

directly from Lemma B.1. Hence,

$d\operatorname{trace}^{\delta}(\hat{g})_{r}=2(n-2)(n-3)\frac{\alpha}{r^{n-1}}+\mathcal{O}_{\infty}(r^{-(n-\epsilon)})$
(C.7)

so that in turn

$\mathbb{U}(\hat{g},\delta)_{r}=\frac{\Delta^{\Omega}(\alpha)}{r^{n-1}}-2(n-1)(n-3)\frac{\alpha}{r^{n-1}}+\mathcal{O}_{\infty}(r^{-(n-\epsilon)}).$
(C.8)

It follows, using $\vec{n}_{r}=\partial_{r}$, that the ADM mass is

$\begin{split}E_{ADM}&=\frac{1}{2(n-1)\omega_{n-1}}\lim_{R\rightarrow\infty}\int_{\\{r=R\\}}\mathbb{U}(\hat{g},\delta)(\vec{n}_{r})d\mu^{\delta}\\\
&=\frac{1}{2(n-1)\omega_{n-1}}\int_{\mathbb{S}^{n-1}}\bigg{(}\Delta^{\Omega}(\alpha)-2(n-1)(n-3)\alpha\bigg{)}dS\\\
&=\frac{1}{\omega_{n-1}}\int_{\mathbb{S}^{n-1}}\bigg{(}-(n-3)\alpha\bigg{)}dS\\\
&=\frac{1}{\omega_{n-1}}\int_{\mathbb{S}^{n-1}}\bigg{(}\operatorname{trace}^{\Omega}(\textbf{p})+\bigg{(}\frac{n-2}{2}\bigg{)}\operatorname{trace}^{\Omega}(\textbf{m})\bigg{)}dS,\end{split}$
(C.9)

where we used (3.4) that $\alpha$ satisfies.

∎

##  Appendix D Some properties of Fermi coordinates

In this section we provide the proof of Proposition 6.3 used in Section 6:

######  Proposition D.1.

There exists constants $\rho_{0}>0$ and $C\geq 1$ such that
$|\hat{A}_{\rho}|_{\hat{g}_{\rho}}<C$ and $C^{-1}\delta\leq\hat{g}^{\rho}\leq
C\delta$ for any $0\leq\rho\leq\rho_{0}$. Furthermore, all partial derivatives
of $(\hat{g}_{\rho})_{ij}$ and $(\hat{A}_{\rho})^{i}_{j}$ up to order $3$ in
the Fermi coordinates are bounded.

###### Proof.

We follow the proof in Appendix C of [Sak21]. It is well known that the
$(1,1)$-tensor with components $(A_{\rho})_{i}^{j}$ satisfies the Mainardi
equation:

$-(A_{\rho})^{i}_{j,\rho}+(A_{\rho})^{i}_{k}(A_{\rho})_{j}^{k}=\text{Riem}^{i}_{\rho\rho
j}.$ (D.1)

Here indices are raised by $g_{\rho}$. It is convenient to write this on the
form

$-A^{\prime}(\rho)+A^{2}(\rho)=R^{N}(\rho),$ (D.2)

where $R^{N}(\rho)$ acts on vectors $V$ orthogonal to $\partial_{\rho}$ by
$R^{N}(\rho)(V)=\langle R^{N}V,V\rangle=\text{sec}(V,\partial_{\rho})$ and is
nothing but the normal sectional curvature operator. We write $\Lambda(\rho)$
for the largest eigenvalue of $A(\rho)$. The eigenvalues of $A(0)$ are bounded
and we want to show that for some $\rho_{0}>0$ the eigenvalues of $A(\rho)$
are bounded when $\rho<\rho_{0}$. For convenience we will throughout the proof
supress the tangential indices.
Since $\Lambda(\rho)$ is obtained through the Rayleigh quotient it is
Lipschitz continuous, and hence almost everywhere differentiable by
Rademachers theorem. Let $(q,\tilde{\rho})$ be a point where $\Lambda$ is
differentiable. Let $v$ be a unit eigenvector normalized with respect to the
Euclidean metric and extend it parallelly so that
$v(q,\rho)=v(q,\tilde{\rho})$ for all $\rho\in[0,\rho_{0}]$. Let
$\varphi(\rho)=v^{T}A(\rho)v$. Then
$\varphi(\tilde{\rho})=\Lambda(\tilde{\rho})$ and
$\varphi(\rho)\leq\Lambda(\rho)$. Further, there is

$\begin{split}-\Lambda^{\prime}(\tilde{\rho})+\Lambda^{2}(\tilde{\rho})&=-\varphi^{\prime}(\tilde{\rho})+\varphi^{2}(\tilde{\rho})\\\
&=v^{T}\big{(}-A^{\prime}(\tilde{\rho})+A^{2}(\tilde{\rho})\big{)}v\\\
&=v^{T}\big{(}R^{N}(\tilde{\rho})\big{)}v.\end{split}$ (D.3)

The curvature is uniformly bounded and so we conclude that we have

$-C_{1}<-\Lambda^{\prime}(\tilde{\rho})+\Lambda^{2}(\tilde{\rho})<C_{1},$
(D.4)

for some $C_{1}>0$.
We consider, for $C_{0}>|\lambda(0)|$, the initial value problem

$\begin{cases}-\mu^{\prime}(\rho)+\mu^{2}(\rho)&=-C_{1},\\\
\mu(0)&=-C_{0},\end{cases}$ (D.5)

which is solved by
$\mu(\rho)=\sqrt{C_{1}}\tan\big{(}\sqrt{C_{1}}\rho+\arctan(\frac{C_{0}}{\sqrt{C_{1}}})\big{)}$.
By decreasing $\rho_{0}$ if necessary, we may thus assume $\mu(\rho)$ is
bounded on $[0,\rho_{0}]$. Furthermore, we may assert

$\Lambda^{\prime}(\rho)-\Lambda^{2}(\rho)<C_{1}=\mu^{\prime}(\rho)-\mu^{2}(\rho)$
(D.6)

or

$-\big{(}\Lambda^{\prime}(\rho)+\mu^{\prime}(\rho)\big{)}+\big{(}\Lambda^{2}(\rho)+\mu^{2}(\rho)\big{)}<0,$
(D.7)

for almost every $\rho\in[0,\rho]$.
We now show that $\Lambda(\rho)<\mu(\rho)$ for $\rho\in[0,\rho_{0}]$.
$\Lambda$ is again almost everywhere differentiable, and so we can write
$\Lambda(\rho)-\Lambda(0)=\int_{0}^{\rho}\Lambda^{\prime}(\tau)d\tau$ by the
Lebesgue differentiation theorem. Combining this with (D.7) we obtain

$\Lambda(\rho)+\mu(\rho)=\int_{0}^{\rho}\big{(}\Lambda^{\prime}(\tau)+\mu^{\prime}(\tau)\big{)}d\tau+\Lambda(0)+\mu(0)>0$
(D.8)

and so $\Lambda(\rho)>-\mu(\rho)$. Conversely, from (D.6) we find

$\Lambda(\rho)-\mu(\rho)=\int_{0}^{\rho}\big{(}\Lambda^{\prime}(\tau)+\mu^{\prime}(\tau)\big{)}d\tau+\Lambda(0)+\mu(0)>\int_{0}^{\rho}\big{(}\Lambda^{2}(\tau)-\mu^{2}(\tau)\big{)}d\tau.$
(D.9)

Now let $\rho^{\ast}=\inf\\{\rho\>|\>\Lambda(\rho)>\mu(\rho)\\}$. Since
$\Lambda(0)<\mu(0)$ we must have $\rho^{\ast}>0$. It follows that both
$\mu(\rho^{\ast})=\Lambda(\rho^{\ast})$ and $\Lambda(\rho)<\mu(\rho)$ for
$0\leq\rho<\rho^{\ast}$. Since also $\Lambda(\rho)>-\mu(\rho)$ on
$[0,\rho_{0}]$ it follows that $\Lambda^{2}(\rho)-\mu^{2}(\rho)<0$ for
$\rho\in[0,\rho_{0}]$. But this is a contradiction in view of (D.8).
Similarly, one can show that the smallest eigenvalue $\lambda(\rho)$ of
$A(\rho)$ satisfies the differential inequality

$-C_{1}<-\lambda^{\prime}(\rho)+\lambda^{2}(\rho)<C_{1},$ (D.10)

for some $C_{1}>0$ and repeating the arguments as above yields a lower bound
for $\lambda(\rho)$. The asserted estimate on the norm follows.
To get the uniform equivalence of $g_{\rho}$ and $\delta$ we note that from
the proof of Lemma 6.7 we have
$g^{\rho}_{ij,\rho}=-2(\hat{A}_{\rho})^{k}_{i}g^{\rho}_{jk}$. This implies,
for $\Theta(\rho)$ the largest eigenvalue of $g_{\rho}$, that
$\Theta^{\prime}(\rho)\leq C_{\Theta}(\rho)$, where $C_{1}>0$. Hence,
$(\Theta(\rho)e^{-C_{1}\rho})\leq 0$ which is solved by
$\Gamma(\rho)=C_{0}e^{-C_{1}\rho}$ for some $C_{0}$ and we chose
$C_{0}>\Theta(0)$. Then

$(\Theta(\rho)-\Gamma(\rho))e^{-C_{1}\rho}=\int_{0}^{\rho}\big{(}(\Theta(\tau)-\Gamma(\tau))e^{-C_{1}\tau}\big{)}d\tau+\Theta(0)-\Gamma(0)<0$
(D.11)

and so $\Theta(\rho)<\Gamma(\rho)$ for $\rho\in[0,\rho_{0}]$. The lowest
eigenvalue can be estimated similarly and this yields the uniform equivalence
$C^{-1}\delta\leq g_{\rho}\leq C\delta$ for some $C>0$.
We now estimate the derivatives of $(\hat{A}_{\rho})^{i}_{j}$ and
$g^{\rho}_{ij}$. The Mainardi equation gives the bound on the first order
derivative $(\hat{A}_{\rho})^{i}_{j,\rho}$ and
$g^{\rho}_{ij,\rho}=-2(\hat{A}_{\rho})^{k}_{i}g^{\rho}_{jk}$ implies the
estimates of the first order derivative of $g^{\rho}$ in the $\rho$-direction.
Differentiating these equations in the tangential direction and commuting
derivatives yields

$(\hat{A}_{\rho})^{i}_{j,k\rho}=(A_{\rho})^{i}_{\ell,k}(A_{\rho})_{j}^{\ell}+(A_{\rho})^{i}_{\ell}(A_{\rho})_{j,k}^{\ell}-\text{Riem}^{i}_{\rho\rho
j,k}$ (D.12)

and

$g^{\rho}_{ij,k\rho}=-2(\hat{A}_{\rho})^{\ell}_{j,k}g^{\rho}_{\ell
i}-2(\hat{A}_{\rho})^{\ell}_{j}g^{\rho}_{\ell i,k}.$ (D.13)

The formula for the components of the $(1,4)$-tensor $\nabla\text{Riem}$ is

$\text{Riem}^{i}_{\rho\rho j,k}=\nabla_{k}\text{Riem}^{i}_{\rho\rho
j}-\Gamma^{i}_{k\ell}\text{Riem}^{\ell}_{j\rho\rho}+\Gamma^{\ell}_{k\rho}\text{Riem}^{i}_{\ell\rho
j}+\Gamma^{\ell}_{k\rho}\text{Riem}^{i}_{j\rho\ell}+\Gamma^{\ell}_{kj}\text{Riem}^{i}_{\rho\rho\ell}.$
(D.14)

Further, we recall $\Gamma^{\ell}_{k\rho}=(\hat{A}_{\rho})^{\ell}_{k}$ from
the proof of Lemma 6.7. Hence the right hand sides of the above second
derivatives are bounded. We may write these as a the system

$\begin{split}(\partial\hat{A}_{\rho})^{\prime}&=K_{1}(\partial\hat{A}_{\rho})+K_{2}(\partial
g_{\rho})+K_{3},\\\ (\partial
g_{\rho})^{\prime}&=K_{4}(\partial\hat{A}_{\rho})+K_{5}(\partial
g_{\rho}),\end{split}$ (D.15)

where $K_{i}$ are bounded $n^{3}\times n^{3}$-matrices and
$\partial\hat{A}_{\rho}$ and $\partial g_{\rho}$ are treated as vectors in
$\mathbb{R}^{n^{3}}$ with components $(\hat{A}_{\rho})^{i}_{j,k}$ and
$g^{\rho}_{ij,k}$, respectively. We set
$x(\rho)=|\partial_{\rho}\hat{A}_{\rho}|$ and $y(\rho)=|\partial g_{\rho}|$.
From the Cauchy-Schwarz inequality it follows that
$x^{\prime}\leq|(\partial\hat{A}_{\rho})^{\prime}|$ and
$y^{\prime}\leq|(\partial g_{\rho})^{\prime}|$. In turn, there is

$\begin{split}x^{\prime}&\leq c_{1}x+c_{2}y+c_{3},\\\ y^{\prime}&\leq
c_{4}x+c_{5}y+c_{6},\end{split}$ (D.16)

for positive real constants $c_{i}>0$. Thus, there is $x<\tilde{x}$ and
$y<\tilde{y}$ on $[0,\rho_{0}]$, where $(\tilde{x},\tilde{y})$ solves

$\begin{split}\tilde{x}^{\prime}&\leq c_{1}\tilde{x}+c_{2}\tilde{y}+c_{3},\\\
\tilde{y}^{\prime}&\leq c_{4}\tilde{x}+c_{5}\tilde{y}+c_{6},\end{split}$
(D.17)

such that $\tilde{x}(0)>x(0)$ and $\tilde{y}(0)>y(0)$. It follows that the
derivatives $(\hat{A}_{\rho})^{i}_{j,k}$ and $g^{\rho}_{ij,k}$ are bounded and
so all first order derivatives are bounded. Inserting this into Equations D.12
and D.13 we find that the second derivatives $\hat{g}^{\rho}_{ij,k\rho}$ and
$(\hat{A}^{\rho})^{i}_{j,k\rho}$ are bounded. Differentiating the Mainardi
equation and the equation for $\hat{g}^{\rho}_{ij}$ above with respect to
$\rho$ we find that also both $\hat{g}_{ij,\rho\rho}$ and
$(\hat{A}^{\rho})^{i}_{j,\rho\rho}$ are bounded.
Finally, by taking tangential derivatives of Equations D.12 and D.13 we obtain

$\begin{split}(\partial\partial\hat{A}_{\rho})^{\prime}&=K_{1}(\partial\partial\hat{A}_{\rho})+K_{2}(\partial\partial
g_{\rho})+K_{3},\\\ (\partial\partial
g_{\rho})^{\prime}&=K_{4}(\partial\partial\hat{A}_{\rho})+K_{5}(\partial\partial
g_{\rho})+K_{6},\end{split}$ (D.18)

where $K_{i}$, $i=1,\ldots,6$, are $n^{4}\times n^{4}$-matrices with bounded
entries and $\partial\partial\hat{A}_{\rho}$ and $\partial\partial g_{\rho}$
are treated as vectors in $\mathbb{R}^{n^{4}}$ with components
$(\hat{A})^{i}_{j,k\ell}$ and $\hat{g}^{\rho}_{ij,k\ell}$. Repeating the above
arguments yields the boundedness of the second order tangential derivatives.
The same procedure gives boundedness of the third derivatives, as explained in
[Sak21].

∎

##  Appendix E Geometric Measure Theory

In this section we review some preliminaries from Geometric Measure Theory
that are used in Section 5. The reader is assumed to be familiar with basic
notions, such as currents, varifolds and minimizing currents (see for instance
[Sim14] and [KP08]). Our focus will be on the important class of
$\lambda$-minimizing currents $\mathcal{F}_{\lambda}$. Here, we essentially
follow [Eic09, Appendix A].
The following definition, generalizing the notion of a minimizing current, was
first introduced in [DS93].

######  Definition E.1.

Let $T=\partial[[E]]\in\mathcal{D}_{n}(\mathbb{R}^{n+\ell})$ be a current that
is a boundary of an $\mathcal{H}^{n+1}$-measurable set
$E\subset\mathbb{R}^{n+\ell}$ that has locally finite perimeter. Then $T$ is
said to be $\lambda$_-minimizing_ if, for every open subset
$W\subset\subset\mathbb{R}^{n+\ell}$ and every integer multiplicity current
$X\in\mathcal{D}_{n+1}(\mathbb{R}^{n+\ell})$ with compact support in $W$, we
have

$\mathbb{M}_{W}(T)\leq\mathbb{M}_{W}(T+\partial X)+\lambda\mathbb{M}_{W}(X).$
(E.1)

If $N^{n+1}\subset\mathbb{R}^{n+\ell}$ is an embedded, oriented
$C^{2}$-manifold, $E\subset N^{n+1}$, and $T=\partial[[E]]$ satisfies (E.1)
for all $X\in\mathcal{D}_{n+1}(\mathbb{R}^{n+\ell})$ with
$\text{supp}(X)\subset N^{n+1}\cap W$ we say that $T$ is $\lambda$-_minimizing
in_ $N^{n+1}$.
The collection of $\lambda$-minimizing boundaries is denoted by
$\mathcal{F}_{\lambda}$.

Throughout this section it will be assumed, unless stated otherwise, that our
currents $T\in\mathcal{F}_{\lambda}$ are $\lambda$-minimizing in $N^{n+1}$,
which is an orientable $C^{2}$-manifold embedded into $\mathbb{R}^{n+\ell}$.
We note that a current $T$ such that (E.1) holds need not be integer
multiplicity, but the requirement that $T=\partial[[E]]$ where $E$ has locally
finite perimeter together with the local mass bounds
$\mathbb{M}_{W}(T)<\infty$, for any open $W\subset\subset\mathbb{R}^{n+\ell}$,
implies that $T$ is integer multiplicity with multiplicity $1$ (see [Sim14,
Remark 5.2 in Chapter 7]). That the currents in $\mathcal{F}_{\lambda}$ have
locally bounded mass will be shown in the proof of Theorem E.4 below.
It is well-known (see, for instance, [Sim14, Chapter 7]) that the underlying
varifolds of a minimizing current ($\lambda=0$ in (E.1)) is stationary. In the
special case of the current being an oriented $C^{2}$-manifold, the minimizing
property translates into vanishing mean curvature. In the $\lambda$-minimizing
case the underlying varifold has bounded generalized mean curvature, as is
shown in Lemma E.2.

######  Lemma E.2.

Let $T\in\mathcal{F}_{\lambda}$ be $\lambda$-minimizing in $N^{n+1}$ and let
$\vec{H}^{T}$ be the tangential generalized mean curvature vector in
$N^{n+1}$. Then $|\vec{H}^{T}|\leq\lambda$ $\mu_{V}$-almost everywhere, where
$V$ is the associated varifold to $T$.

###### Proof.

The proof given in a more general case can be found in [DS93]. Here, we follow
[Eic09, Remark A.2 in Appendix A]. Take any $X\in
C^{1}_{c}(W,\mathbb{R}^{n+\ell})$ such that $X(p)\subset T_{p}N^{n+1}$ for
each $p$, where $W\subset\subset\mathbb{R}^{n+\ell}$ is open and let
$\varphi:[0,1]\times\mathbb{R}^{n+\ell}\rightarrow\mathbb{R}^{n+\ell}$ be the
associated flow generated by $X$. We denote the current
$\big{[}\big{[}[0,1]\big{]}\big{]}$ equipped with the standard orientation by
$[[0,1]]$. From the homotopy formula [Sim14, (2.25) in Chapter 6] we have

$\varphi^{t}_{\\#}(T)-T=\partial\varphi_{\\#}\big{(}[[0,t]]\times T\big{)},\\\
$ (E.2)

since $\partial T=0$ and $\varphi^{0}_{\\#}(T)=T$. The $\lambda$-minimizing
property in (E.1) implies, since $\varphi^{t}_{\\#}(T)=T$ outside of the
compact support of $X$, that

$\begin{split}\mathbb{M}_{W}(T)&\leq\mathbb{M}_{W}\big{(}T+\partial\varphi_{\\#}\big{(}[[0,t]]\times
T\big{)}\big{)}+\lambda\mathbb{M}_{W}(\varphi_{\\#}\big{(}[[0,t]]\times
T\big{)})\\\
&=\mathbb{M}_{W}(\varphi^{t}_{\\#}(T))+\lambda\mathbb{M}_{W}(\varphi_{\\#}\big{(}[[0,t]]\times
T\big{)}),\end{split}$ (E.3)

for any $W\subset\subset\mathbb{R}^{n+\ell}$ open. For $t$ close to zero it
follows that

$0\leq\big{(}\mathbb{M}_{W}(\varphi^{t}_{\\#}(T))-\mathbb{M}_{W}(T)\big{)}t^{-1}+\lambda
t^{-1}\mathbb{M}_{W}\big{(}\varphi_{\\#}([[0,t]]\times T)\big{)}.$ (E.4)

We now take the $\limsup$ as $t\rightarrow 0$. In the first term nothing but
the first variation of the associated varifold $V=(M,\theta)$, where $M$ is
the rectifiable set and $\theta$ the multiplicity function of the integer
multiplicity current $T=(M,\theta,\xi)$. For the second term we can assert
that

$\limsup_{t\downarrow 0}\lambda
t^{-1}\mathbb{M}_{W}\big{(}\varphi_{\\#}([[0,t]]\times
T)\big{)}\leq\lambda\mathbb{M}_{W}(T)\sup|X|,$ (E.5)

(cf. the discussion leading up to [Sim14, (2.27) in Chapter 6] for details).
It is standard that the first variation for a varifold under any flow is
related to the generalized mean curvature vector $\vec{H}$ in
$\mathbb{R}^{n+\ell}$ of $V$ via

$\frac{d}{dt}\bigg{|}_{t=0}\mathbb{M}_{W}(\varphi^{t}_{\\#}(T))=\int_{M}\text{div}^{M}(X)d\mu_{V}=-\int_{M}X\cdot\vec{H}d\mu_{V}.$
(E.6)

We decompose $\vec{H}=\vec{H}^{N}+\vec{H}^{T}$ where $\vec{H}^{N}$ is the mean
curvature vector of $N^{n+1}$ and $\vec{H}^{T}$ is tangential to $N^{n+1}$.
Since $X=X^{T}$ by assumption we obtain conclusively

$\int_{W}X\cdot\vec{H}^{T}d\mu_{V}\leq\lambda\mathbb{M}_{W}(T)\sup|X|,$ (E.7)

from (E.4), keeping (E.5) and (E.6) in mind. We assume further that $|X|\leq
1$ and view (E.7) as an inequality of measures:
$\mu_{X}(W)\leq\lambda\mu_{V}(W)$, where we $\mu_{X}$ is signed. By changing
sign of $X$ if necessary, we see that $\mu_{X}<<\mu_{V}$ and so the Radon-
Nikodym theorem implies that

$\mu_{X}(W)=\int_{W}fd\mu_{V},$ (E.8)

where $f$ is integrable with respect to $\mu_{V}$. Further, we must have
$|f|\leq\lambda$ $\mu_{V}$-almost everywhere, as otherwise the inequality
$\mu_{X}\leq\lambda\mu_{V}$ would be violated. From the uniqueness of $f$, we
identify $f=X\cdot\vec{H}^{T}$, and since again $|X|\leq 1$ it follows that
$|\vec{X}^{T}|\leq\lambda$ $\mu_{V}$-almost everywhere, as asserted.

∎

The following example shows that any graph with bounded mean curvature is a
$\lambda$-minimizing current.

######  Example E.3.

We study a real-valued function $f\in C^{2}(\mathbb{R}^{n})$ where the
associated graph in $\mathbb{R}^{n}\times\mathbb{R}$ has bounded mean
curvature $H_{\hat{g}}$ at each point
$\vec{x}=(x^{1},\ldots,x^{n})\in\mathbb{R}^{n}$. With the above notation we
set

$E=\\{(\vec{x},t)\>|\>f(\vec{x})\leq t\\}$ (E.9)

and let $T=\partial[[E]]$, which is precisely the graph of $f$. Clearly, $E$
has locally finite perimeter and $T$ is integer multiplicity one.
We claim that $T$ has the $\lambda$-minimizing property. Let
$W\subset\subset\mathbb{R}^{n+1}$ and
$X\in\mathcal{D}_{n+1}(\mathbb{R}^{n+1})$ with $\text{supp}(X)\subset W$ be as
in Definition E.1. Let

$\begin{split}\vec{n}&=\frac{\nabla^{\delta}f-\partial_{t}}{\sqrt{1+|\nabla^{\delta}f|_{\delta}^{2}}}\\\
\end{split}$ (E.10)

be the downward pointing unit normal of the graph of $f$. Extend $\vec{n}$
trivially to all of $\mathbb{R}^{n+1}$. From (4.9), we know that the mean
curvature of the graph is $H_{f}=\text{div}_{\delta}(\vec{n})$. Let
$\omega=dx^{1}\wedge\ldots\wedge dx^{n}\wedge dt$ be the top-form and
$\sigma=\vec{n}\lrcorner\omega$ be the area form of $T$. We introduce the
notation

$\begin{split}d\hat{x}^{k}&=dx^{1}\wedge\ldots\wedge dx^{k-1}\wedge
dx^{k+1}\wedge\ldots\wedge dt,\\\
\hat{\partial}_{k}&=\partial_{1}\wedge\ldots\wedge\partial_{k-1}\wedge\partial_{k+1}\wedge\ldots\wedge\partial_{t}.\end{split}$
(E.11)

From the definition of the ”elbow” operation $\lrcorner$ we find, for any
$1\leq k\leq n$:

$\begin{split}\langle\sigma,\hat{\partial}_{k}\rangle&=\langle\vec{n}\lrcorner\omega,\hat{\partial}_{k}\rangle\\\
&=\langle\omega,\vec{n}\wedge\hat{\partial}_{k}\rangle\\\
&=\bigg{\langle}\omega,\frac{(-1)^{k-1}f^{,k}}{\sqrt{1+|\nabla^{\delta}f|^{2}_{\delta}}}\omega\bigg{\rangle}\\\
&=\frac{(-1)^{k-1}f^{,k}}{\sqrt{1+|\nabla^{\delta}f|^{2}_{\delta}}}.\end{split}$
(E.12)

Similarly, for $\hat{\partial}_{t}$ we have

$\begin{split}\langle\sigma,\hat{\partial}_{t}\rangle&=\langle\vec{n}\lrcorner\omega,\hat{\partial}_{t}\rangle\\\
&=\langle\omega,\vec{n}\wedge\hat{\partial}_{t}\rangle\\\
&=\bigg{\langle}\omega,\frac{(-1)^{n}}{\sqrt{1+|\nabla^{\delta}f|^{2}_{\delta}}}\omega\bigg{\rangle}\\\
&=\frac{(-1)^{n}}{\sqrt{1+|\nabla^{\delta}f|^{2}_{\delta}}}.\end{split}$
(E.13)

Thus, we have

$\sigma=\frac{(-1)^{k-1}f^{,k}}{\sqrt{1+|\nabla^{\delta}f|^{2}_{\delta}}}d\hat{x}_{k}+\frac{(-1)^{n}}{\sqrt{1+|\nabla^{\delta}f|^{2}_{\delta}}}d\hat{t},$
(E.14)

where $k=1,\ldots,n$.
Similarly, we obtain $d\sigma=\text{div}_{\delta}(\vec{n})\omega$. Indeed,
since $dx^{k}\wedge d\hat{x}^{\ell}=\delta_{k\ell}(-1)^{k-1}\omega$, we see
that

$\begin{split}d\sigma&=\bigg{(}\frac{(-1)^{k-1}f^{,k}}{\sqrt{1+|\nabla^{\delta}f|^{2}_{\delta}}}\bigg{)}_{,k}dx^{k}\wedge
d\hat{x}^{k}+\bigg{(}\frac{(-1)^{n}}{\sqrt{1+|\nabla^{\delta}f|^{2}_{\delta}}}\bigg{)}_{,t}dt\wedge
d\hat{t}\\\
&=\bigg{(}\frac{f^{,k}}{\sqrt{1+|\nabla^{\delta}f|^{2}_{\delta}}}\bigg{)}_{,k}\omega\\\
&=\text{div}_{\delta}(\vec{n})\omega.\end{split}$ (E.15)

Thus,

$d\sigma(\vec{x},t)=H_{f}(\vec{x},t)dx^{1}\wedge\ldots\wedge dx^{k}\wedge dt.$
(E.16)

Let $\mathcal{D}_{X}(W)$ denote all smooth functions on $\mathbb{R}^{n+1}$
with support in $W$ that are equal to one in a neighbourhood of the support of
$X$. Then

$\begin{split}\mathbb{M}_{W}(T+\partial
X)&=\sup_{\omega\in\mathcal{D}^{n}(W),|\omega|\leq 1}(T+\partial X)(\omega)\\\
&\geq\sup_{\varphi\in\mathcal{D}_{X}(W),|\omega|\leq 1}(T+\partial
X)(\varphi\sigma)\\\ &\geq\sup_{\varphi\in\mathcal{D}_{X}(W),|\omega|\leq
1}T(\varphi\sigma)-\sup_{\varphi\in\mathcal{D}_{X}(W),|\omega|\leq 1}\partial
X(\varphi\sigma)\\\ &=\mathbb{M}_{W}(T)-|X(d\sigma)|\\\
&\geq\mathbb{M}_{W}(T)-\lambda\mathbb{M}_{W}(X),\end{split}$ (E.17)

where the first inequality is due to inclusion, the second inequality comes
from the triangle inequality and the last inequality follows from (E.16) and
$|H_{\hat{g}}|\leq\lambda$. Hence, $T$ is $\lambda$-minimizing.

In the following theorem we prove the compactness of $\mathcal{F}_{\lambda}$.

######  Theorem E.4.

Let $N^{n+1}\subset\mathbb{R}^{n+\ell}$ be an embedded, orientable
$C^{2}$-manifold and let $\\{T_{k}\\}\subset\mathcal{F}_{\lambda}$ be
$\lambda$-minimizing in $N^{n+1}$. Then there exists a subsequence
$\\{T_{k^{\prime}}\\}$ such that $T_{k^{\prime}}\rightharpoonup
T\in\mathcal{F}_{\lambda}$, where $T$ is $\lambda$-minimizing in $N^{n+1}$.
Furthermore, we have the convergence of the indicator functions
$\chi_{E_{k}}\rightarrow\chi_{E}$ in $L^{1}_{loc}(\mathcal{H}^{n+1})$, and
$\mu_{T_{k}}\rightarrow\mu_{T}$ as Radon measures.

###### Proof.

We follow the proof of [Eic09, Lemma A.2], which is very similar to the proofs
of [Sim14, Theorems 2.4 and 5.3 in Chapter 7]. We start by proving the local
boundedness the mass for every $T_{k}$ (by suitable modifications of the proof
of [Sim14, Theorem 5.3 in Chapter 7]). For a fixed $q\in N^{n+1}$ we define
$r(p)=|p-q|$ to be the Euclidean distance to $q$. For $\rho>0$ we may slice

$\partial[[E_{k}\cap B_{\rho}(q)]]=T_{k}\llcorner
B_{\rho}(q)+\langle[[E_{k}]],r,\rho\rangle$ (E.18)

and note that these currents are compactly supported in $\bar{B}_{\rho}(q)$,
so that for any open set $W$ such that $B_{\rho}(q)\subset
W\subset\subset\mathbb{R}^{n+\ell}$ the $\lambda$-minimizing property implies

$\mathbb{M}(T_{k}\llcorner
B_{\rho}(q))\leq\mathbb{M}(\langle[[E_{k}]],r,\rho\rangle)+\lambda\mathbb{M}([[E_{k}\cap
B_{\rho}(q)]]).$ (E.19)

Now define $\tilde{E}_{k}=N^{n+1}\setminus E_{k}$ and
$\tilde{T}_{k}=\partial[[\tilde{E}_{k}]]$. Then $\tilde{T}_{k}=-T_{k}$ is also
$\lambda$-minimizing in $N^{n+1}$ and we have

$\begin{split}\mathbb{M}(T_{k}\llcorner
B_{\rho}(q))&\leq\min\bigg{\\{}\mathbb{M}\langle[[E_{k}]],r,\rho\rangle+\lambda\mathbb{M}([[E_{k}\cap
B_{\rho}(q)]]),\\\
&\qquad\mathbb{M}\langle[[\tilde{E}_{k}]],r,\rho\rangle+\lambda\mathbb{M}([[\tilde{E}_{k}\cap
B_{\rho}(q)]])\bigg{\\}},\end{split}$ (E.20)

for Lebesgue almost every $\rho>0$. Since
$[[E_{k}]]+[[\tilde{E}_{k}]]=[[N^{n+1}]]$ we also have

$\langle[[E_{k}]],r,\rho\rangle+\langle[[\tilde{E}_{k}]],r,\rho\rangle=\langle
N^{n+1},r,\rho\rangle$ (E.21)

for Lebesgue almost every $\rho>0$, which in turn implies

$\mathbb{M}\big{(}\langle[[E_{k}]],r,\rho\rangle\big{)}+\mathbb{M}\big{(}\langle[[\tilde{E}_{k}]],r,\rho\rangle\big{)}=\mathbb{M}\big{(}\langle
N^{n+1},r,\rho\rangle\big{)},$ (E.22)

since $E_{k}$ and $\tilde{E}_{k}$ are disjoint as sets. Further, it is clear
that

$\mathbb{M}\big{(}\langle
N^{n+1},r,\rho\rangle\big{)}\leq\mathcal{H}^{n}\big{(}N^{n+1}\cap\partial
B_{\rho}(q)\big{)},$ (E.23)

and

$\mathbb{M}\big{(}[[E_{k}\cap
B_{\rho}(q)]]\big{)}\leq\mathcal{H}^{n+1}\big{(}N^{n+1}\cap
B_{\rho}(q)\big{)}.$ (E.24)

The same estiates hold for $\tilde{E}_{k}$. Combining (E.20) - (E.24) we
obtain

$\mathbb{M}\big{(}T_{k}\llcorner
B_{\rho}(q)\big{)}\leq\frac{1}{2}\mathcal{H}^{n}\big{(}N^{n+1}\cap\partial
B_{\rho}(q)\big{)}+\frac{\lambda}{2}\mathcal{H}^{n+1}\big{(}N^{n+1}\cap
B_{\rho}(q)\big{)}$ (E.25)

for Lebesgue almost every $\rho>0$. The local boundedness follows.
We now prove the statement about the indicator functions $\chi_{E_{k}}$. From
(E.25) and [Sim14, Remark 5.2 in Chapter 7] together with the compactness
results for $BV_{loc}$-functions [Sim14, Theorem 2.6 in Chapter 2] the
sequence $\\{\chi_{E_{k}}\\}$ has a convergent subsequence
$\\{\chi_{E_{k^{\prime}}}\\}$ that converges in $L^{1}_{loc}$ to an indicator
function $\chi_{E}\in BV_{loc}$, where $E$ is some
$\mathcal{H}^{n+1}$-measurable set. The $L^{1}$-convergence implies the
current convergence $[[E_{k^{\prime}}]]\rightharpoonup[[E]]$ and, in turn,
also $T_{k^{\prime}}\rightharpoonup T$.
Our next aim is to show that $T$ is $\lambda$-almost minimizing. For this,
following [Eic09, Lemma A.2], we modify the proof of [Sim14, Theorem 2.4 in
Chapter 7]. For simplicity, we only consider the setting when $T_{k}$ are
$\lambda$-minimizing in $\mathbb{R}^{n+1}$. The argument extends to the
general of $\lambda$-minimizing boundaries in a submanifold $N^{n+1}$ by the
same techniques as mentioned in [Sim14, Remark 2.5 (2) in Chapter 7].
Let $K\subset\mathbb{R}^{n+1}$ be an arbitrary compact set and let
$\varphi:\mathbb{R}^{n+1}\rightarrow[0,1]$ a smooth function such that
$\varphi\equiv 1$ in a neighbourhood of $K$, support inside an
$\epsilon$-neighbourhood $U_{\epsilon}=\\{p\>|\>\text{dist}(p,K)<\epsilon\\}$
of $K$. For $\gamma\in(0,1)$ we denote the superlevel set

$W_{\gamma}=\\{p\in\mathbb{R}^{n+1}\>|\>\varphi(p)>\gamma\\}.$ (E.26)

We define the current $R_{k}=[[E]]-[[E_{k}]]$ and observe that
$\mathbb{M}_{W_{0}}(R_{k^{\prime}})\rightarrow 0$ as
$k^{\prime}\rightarrow\infty$, where $k^{\prime}$ is the index of the
subsequence of indicator functions $\chi_{k^{\prime}}$ that converges in
$L^{1}_{loc}$.
We now slice the currents $\\{R_{k}\\}$ with respect to $\varphi$. From
slicing theory [Sim14, Section 4 in Chapter 6] we may choose $\alpha\in(0,1)$
and a subsequence of $\\{R_{k}\\}$, still denoted by $\\{R_{k}\\}$, so that

$P_{k}=\partial(R_{k}\llcorner W_{\alpha})-(\partial R_{k})\llcorner
W_{\alpha}$ (E.27)

is integer multiplicity with support in $\partial W_{\alpha}$ and such that
$\mathbb{M}(P_{k})\rightarrow 0$. Furthermore, $\alpha$ can be chosen so that
both

$\mathbb{M}_{W_{0}}(T_{k}\llcorner\partial W_{\alpha})=0$ (E.28)

for all $k$ and $\mathbb{M}_{W_{0}}(T\llcorner\partial W_{\alpha})=0$. By
taking restriction to $W_{\alpha}$, we have

$T\llcorner W_{\alpha}=T_{k}\llcorner W_{\alpha}+\partial(R_{k}\llcorner
W_{\alpha})-P_{k}$ (E.29)

where both $P_{k}$ and $\partial(R_{k}\llcorner W_{\alpha})$ are integer
multiplicity with support in $\overline{W}_{\alpha}$ and whose masses tends to
zero in the limit.
Consider a compactly supported $X\in\mathcal{D}_{n+1}(\mathbb{R}^{n+1})$ with
$\text{supp}(X)\subset K$ and take $\gamma\in(0,\alpha)$. Then the
$\lambda$-minimizing property implies

$\begin{split}\mathbb{M}_{W_{\gamma}}(T_{k}\llcorner
W_{\alpha})&\leq\mathbb{M}_{W_{\gamma}}(T_{k}\llcorner
W_{\alpha}-P_{k})+\mathbb{M}_{W_{\gamma}}(P_{k})\\\
&\leq\mathbb{M}_{W_{\gamma}}(T_{k}\llcorner
W_{\alpha}-P_{k}+\partial(R_{k}\llcorner W_{\alpha})+\partial X)\\\
&\qquad+\lambda\mathbb{M}_{W_{\gamma}}(X)+\lambda\mathbb{M}_{W_{\gamma}}(R_{k}\llcorner
W_{\alpha})+\mathbb{M}_{W_{\gamma}}(P_{k})\\\
&=\mathbb{M}_{W_{\gamma}}(T\llcorner W_{\alpha}+\partial
X)+\lambda\mathbb{M}_{W_{\gamma}}(X)\\\
&\qquad+\lambda\mathbb{M}_{W_{\gamma}}(R_{k}\llcorner
W_{\alpha})+\mathbb{M}_{W_{\gamma}}(P_{k}),\end{split}$ (E.30)

since both $X$ and $R_{k}\llcorner W_{\alpha}$ are compactly supported. Taking
the limit $\gamma\rightarrow 0$ we obtain

$\begin{split}\mathbb{M}_{W_{\alpha}}(T_{k})&\leq\mathbb{M}_{W_{\alpha}}(T+\partial
X)+\lambda\mathbb{M}_{W_{\alpha}}(X)\\\
&\qquad+\lambda\mathbb{M}_{W_{\alpha}}(R_{k})+\mathbb{M}(P_{k}).\end{split}$
(E.31)

If we now let $X\equiv 0$ and take the superior limit in (E.31), then
recalling that the masses of $P_{k}$ and $R_{k}$ tend to zero we obtain
$\limsup_{k}\mathbb{M}_{W_{\alpha}}(T_{k})\leq\mathbb{M}_{W_{\alpha}}(T)$.
From the lower semi-continuity of the mass it then follows that

$\mathbb{M}_{W_{\alpha}}(T_{k})\rightarrow\mathbb{M}_{W_{\alpha}}(T).$ (E.32)

In other words, no mass is lost under the weak convergence. Thus, taking the
limit $k\rightarrow\infty$ in (E.31) and recalling that $K$ was arbitrary we
conclude that $T\in\mathcal{F}_{\lambda}$ as asserted.
Finally, we verify the Radon measure convergence. For this, we again follow
the proof of [Sim14, Theorem 2.4 in Chapter 7]. We let $X\equiv 0$ in (E.30)
and since by construction $K\subset W_{\gamma}\subset U_{\epsilon}$ we get

$\begin{split}\limsup_{k}\mu_{T_{k}}(K)&\leq\limsup_{k}\mathbb{M}_{W_{\gamma}}(T_{k})\\\
&\leq\mathbb{M}_{U_{\epsilon}}(T).\end{split}$ (E.33)

In the limit $\epsilon\rightarrow 0$ we thus get

$\limsup_{k}\mu_{T_{k}}(K)\leq\mu_{T}(K).$ (E.34)

Summing up, we see that the Radon measures $\\{\mu_{T_{k}}\\}$ are upper semi-
continuous when restricted to compact sets and from the lower semi-continuity
of the mass we know that they are lower semi-continuous on when restricted to
open sets. As explanied in the end of the proof in [Sim14, Theorem 2.4 in
Chapter 7], using an approximation argument we can show that this implies
Radon measure convergence, that is for $f\in C_{c}(\mathbb{R}^{n+1})$ we have

$\int_{\mathbb{R}^{n+1}}fd\mu_{T_{k}}\rightarrow\int_{\mathbb{R}^{n+1}}fd\mu_{T}.$
(E.35)

∎

At this point it is convenient to state the approximate monotonicity formula
for currents in $\mathcal{F}_{\lambda}$ (see [Sim14, Theorem 3.17 in Chapter
4]):

$F(\rho)\frac{\mu_{T}(B_{\rho}(q))}{\omega_{n}\rho^{n}}-F(\sigma)\frac{\mu_{T}(B_{\sigma}(q))}{\omega_{n}\sigma^{n}}=G(\sigma,\rho)\int_{B_{\rho}(q)-B_{\sigma}(q)}\frac{|\nabla^{\perp}r|^{2}}{r^{n}}d\mu_{V},$
(E.36)

where $V$ is the varifold associated to $T$, $0<\sigma<\rho$,
$F(\rho)\in[e^{-\Lambda\rho},e^{\Lambda\rho}]$ and $G\geq 0$ is continuous and
bounded for small $\rho$. It follows that the function $\rho\rightarrow
F(\rho)\frac{\mu_{T}(B_{\rho}(q))}{\omega_{n}\rho^{n}}$ has a limit as
$\rho\rightarrow 0$ and since $\lim_{\rho\rightarrow 0}F(\rho)=1$ it also
follows that the density

$\Theta^{n}(\mu_{T},q)=\lim_{\rho\rightarrow
0}\frac{\mu_{T}(B_{\rho}(q))}{\omega_{n}\rho^{n}}$ (E.37)

is defined at every point $q$. This will be $\mu_{T}$-almost everywhere equal
to the multiplicity function: $\Theta^{n}(\mu_{T},q)=\theta(q)$.
The following Lemma will be useful.

######  Lemma E.5.

Let $T,T_{k}\in\mathcal{F}_{\lambda}$, $T_{k}\rightharpoonup T$ and
$q_{k}\rightarrow q$ with $q_{k}\in\text{supp}(T_{k})$ and
$q\in\text{supp}(T)$. Then

$\limsup_{k}\Theta^{n}(\mu_{T_{k}},q_{k})\leq\Theta^{n}(\mu_{T},q).$ (E.38)

###### Proof.

From the approximate monotonicity formula (E.36) it follows that for $\rho>0$
sufficiently small so that $F(\rho)<1+\epsilon_{1}$ and for $\epsilon_{2}>0$
fixed we have

$\Theta^{n}(\mu_{T_{k}},q_{k})\leq(1+\epsilon_{1})\frac{\mu_{T_{k}}(B_{\rho}(q_{k}))}{\omega_{n}\rho^{n}}\leq(1+\epsilon_{1})\frac{\mu_{T_{k}}(B_{\rho+\epsilon_{2}}(q))}{\omega_{n}\rho^{n}},$
(E.39)

for sufficiently large $k$. Further, from the proof of Proposition E.4 we know
that mass is not lost under current convergence,
$\mathbb{M}_{W}(T_{k})\rightarrow\mathbb{M}_{W}(T)$. Taking the superior limit
of both sides we obtain

$\limsup_{k}\Theta^{n}(T_{k},q_{k})\leq(1+\epsilon_{1})\frac{\mu_{T}(B_{\rho+\epsilon_{2}}(q))}{\omega_{n}\rho^{n}}.$
(E.40)

Taking the limit $\epsilon_{2}\rightarrow 0$ followed by $\rho\rightarrow 0$
and finally $\epsilon_{1}\rightarrow 0$ the assertion follows.

∎

We define the map
$\eta_{q,\gamma}:\mathbb{R}^{n+\ell}\rightarrow\mathbb{R}^{n+\ell}$ by

$\eta_{q,\gamma}(p)=\frac{p-q}{\gamma}$ (E.41)

and note the following property for the pushforward with respect to
$\eta_{q,\gamma}$.

######  Lemma E.6.

If $T\in\mathcal{F}_{\lambda}$, then
$\eta_{q,\gamma\>\\#}T\in\mathcal{F}_{\lambda\gamma}$.

###### Proof.

Let
$\omega=\sum_{\alpha}\omega_{\alpha}dx^{\alpha}\in\mathcal{D}^{n}(\mathbb{R}^{n+\ell})$.
Then for $\eta_{q,\gamma}$ as in (E.41) we have

$\eta_{q,\gamma}^{\\#}\omega=\sum_{\alpha}(\omega_{\alpha}\circ\eta_{q,\gamma})\>\frac{dx^{\alpha}}{\gamma^{n}}.$
(E.42)

Consequently, for any open $W\subset\subset\mathbb{R}^{n+\ell}$ and any
$T\in\mathcal{D}_{n}(\mathbb{R}^{n+\ell})$ we have

$\begin{split}\mathbb{M}_{W}(\eta_{q,\gamma\>\\#}T)&=\sup_{\omega\in\mathcal{D}^{n}(\mathbb{R}^{n+\ell}),|\omega|\leq
1,\text{supp}(\omega)\subset W}(\eta_{q,\gamma\>\\#}T)(\omega)\\\
&=\sup_{\omega\in\mathcal{D}^{n}(\mathbb{R}^{n+\ell}),|\omega|\leq
1,\text{supp}(\omega)\subset W}T(\eta_{q,\gamma}^{\\#}\omega)\\\
&=\sup_{\omega\in\mathcal{D}^{n}(\mathbb{R}^{n+\ell}),|\omega|\leq
1,\text{supp}(\omega)\subset
W}T\bigg{(}\sum_{\alpha}(\omega_{\alpha}\circ\eta_{q,\gamma})\>\frac{dx^{\alpha}}{\gamma^{n}}\bigg{)}\\\
&=\frac{\mathbb{M}_{\eta^{-1}_{q,\gamma}(W)}(T)}{\gamma^{n}}.\end{split}$
(E.43)

Hence, for a $\lambda$-minimizing $T$ and
$X\in\mathcal{D}_{n+1}(\mathbb{R}^{n+\ell})$ compactly supported in $W$, and
$Y\in\mathcal{D}_{n+1}(\mathbb{R}^{n+\ell})$ such that
$\eta_{q,\gamma\>\\#}Y=X$ we have

$\begin{split}\mathbb{M}_{W}(\eta_{q,\gamma\>\\#}T)&=\frac{\mathbb{M}_{\eta^{-1}_{q,\gamma}(W)}(T)}{\gamma^{n}}\\\
&\leq\bigg{(}\mathbb{M}_{\eta^{-1}_{q,\gamma}(W)}(T+\partial
Y)+\lambda\mathbb{M}_{\eta^{-1}_{q,\gamma}(W)}(Y)\bigg{)}\gamma^{-n}\\\
&=\mathbb{M}_{W}(\eta_{q,\gamma\>\\#}T+\partial
X)+\frac{\lambda\gamma}{\gamma^{n+1}}\mathbb{M}_{\eta^{-1}_{q,\gamma}(W)}(Y)\\\
&=\mathbb{M}_{W}(\eta_{q,\gamma\>\\#}T+\partial
X)+\lambda\gamma\mathbb{M}_{W}(X)\end{split}$ (E.44)

as asserted.

∎

In particular, Lemma E.6 implies that if $\gamma<1$ and
$T\in\mathcal{F}_{\lambda}$, then
$\eta_{q,\gamma\>\\#}T\in\mathcal{F}_{\lambda}$.
The following theorem shows that the tangent cones of currents in
$\mathcal{F}_{\lambda}$ are minimizing.

######  Theorem E.7.

Suppose $T\in\mathcal{F}_{\lambda}$ is $\lambda$-minimizing in an embedded
orientable submanifold $N^{n+1}$. Then, for each $p\in\emph{supp}(T)$ and each
sequence of positive real numbers $\\{\lambda_{k}\\}$ tending to zero there
exists a subsequence $\\{\lambda_{k^{\prime}}\\}$ and a minimizing integer
multiplicity current $\mathcal{C}\in\mathcal{D}_{n}(\mathbb{R}^{n+\ell})$ with
$0\in\text{supp}(\mathcal{C})\subset T_{p}N^{n+1}$ such that

$\mu_{\eta_{p,\lambda_{k^{\prime}}\>\\#}T}\rightarrow\mu_{C}$ (E.45)

as Radon measures. Further, there exists an $\mathcal{H}^{n+1}$-measurable set
$F$ in $T_{p}N^{n+1}$ such that $\mathcal{C}=\partial[[F]]$ and

$\chi_{\text{pr}_{T_{p}N^{n+1}}}(\eta_{p,\lambda_{k^{\prime}}}(E))\rightarrow\chi_{F}$
(E.46)

in the $L^{1}_{loc}(\mathcal{H}^{n+1})$ sense, where
$\text{pr}_{T_{p}N^{n+1}}$ is the orthogonal projection onto $T_{p}N^{n+1}$.
Finally

$\eta_{0,\gamma\\#}\mathcal{C}=\mathcal{C}$ (E.47)

and $\eta_{0,\gamma}(F)=F$ as sets for any $\gamma>0$.

###### Proof.

We write for brevity $\eta_{p,\lambda_{k}\>\\#}T=T_{k}$ and observe that
$\eta_{p,\lambda_{k}}N^{n+1}\rightarrow T_{p}N^{n+1}$ smoothly. It is not
difficult to modify the proof of Theorem E.4 to the case where the
$E_{k}\subset N^{n+1}_{k}$ and where $N^{n+1}_{k}$ converge to $N^{n+1}$
smoothly using nearest point projection and the homotopy formula. From Lemma
E.6 we know that $T_{k}\in\mathcal{F}_{\lambda\gamma_{k}}$ and so by Theorem
E.4 we obtain the assertions about subconvergence.
It only remains to prove the minimizing property. We know from Lemma E.6 that

$\mathbb{M}_{W}(T_{k})\leq\mathbb{M}_{W}(T_{k}+\partial
X)+\lambda\gamma_{k}\mathbb{M}_{W}(X),$ (E.48)

where $W$ and $X$ are as in Definition E.1. Since
$\mathcal{F}_{\lambda\gamma_{k}}\subset\mathcal{F}_{\lambda}$ and since we
know from the proof of Theorem E.4 that mass in not lost under current
convergence we obtain
$\mathbb{M}_{W}(\mathcal{C})\leq\mathbb{M}_{W}(\mathcal{C}+\partial X)$ after
taking $\liminf$ on both sides.

∎

We now discuss the regularity of currents in $\mathcal{F}_{\lambda}$.

######  Definition E.8.

For $T\in\mathcal{F}_{\lambda}$ the set

$\emph{reg}(T)=\\{p\in\text{supp}(T)\>|\>T\llcorner B_{\rho}(p)\>\text{is a
connected}\>C^{1,\alpha}-\text{graph}\\}$ (E.49)

for some $\alpha\in(0,1)$ and some $\rho>0$, is called the _regular set_ and
the set

$\emph{sing}(T)=\emph{supp}(T)-\emph{reg}(T)$ (E.50)

is called the _singular set_.

The following theorem shows that for $n\leq 6$ any current in
$\mathcal{F}_{\lambda}$ has $\text{sing}(T)=\emptyset$.

######  Theorem E.9.

Let $T\in\mathcal{F}_{\lambda}$ be $\lambda$-minimizing in $N^{n+1}$. Then
$\emph{sing}(T)=\emptyset$ for $n\leq 6$, $\emph{sing}(T)$ consists of
isolated points if $n=7$ and $\mathcal{H}^{n-7+\alpha}(\emph{sing}(T))=0$ for
$\alpha>0$.

###### Proof.

We perform the tangent cone analysis and the abstract dimension reduction
argument as in [Eic09, Theorem A.1] and [Sim14, Theorem 5.8 in Chapter 7].
Since the argument is rather well-known we provide only a sketch.
We take any $q\in\text{sing}(T)$ and recall that the density
$\Theta^{n}(\mu_{T},q)$ will exist everywhere from the approximate
monotonicity formula in (E.36). If $q$ is in the singular set, then some
critera in Allard’s theorem must be violated. The mean curvature is bounded
and so there must be some $\delta_{0}>0$ such that $\Theta^{n}(\mu_{T},q)\geq
1+\delta_{0}$.
We define the set of weak limits

$\mathcal{T}=\\{S\>|\>\eta_{q_{k},\lambda_{k}\>\\#}T\rightharpoonup S\\},$
(E.51)

for some convergent sequences $\\{q_{k}\\}$ and $\\{\lambda_{k}\\}$ with
limits $q=\lim_{k}q_{k}$ and $\lambda=\lim_{k}\lambda_{k}$ where
$0<\lambda_{k}<1$ and $0\leq\lambda<1$. We note that
$\limsup_{k}\mathbb{M}_{W}(S_{k})<\infty$ in view of the current convergence
and the facts that $T$ is integer multiplicity and that the limits are
$\lambda$-minimizing from Theorem E.4. Moreover, it is not difficult to see
that $\mathcal{T}_{p,\tau}=\mathcal{T}$ whenever $0<\tau<1$.
We now construct a function $\varphi_{S}$ for any $S\in\mathcal{T}$ that will
satisfy the criteria of the reduction argument in [Sim14, Appendix A]. We let
$\varphi_{S}:\mathbb{R}^{n+\ell}\rightarrow\mathbb{R}^{n+1}$ be defined by

$\varphi_{S}^{0}(p)=\theta_{S}(p),\qquad\varphi_{S}^{k}(p)=\theta_{S}(p)\xi_{S}^{k}(p),\qquad
k=1,\ldots,n+1,$ (E.52)

where the $\xi_{S}^{k}$ is the $k$:th component of the orientation vector
$\vec{S}(p)$ of $S$, and let
$\mathcal{F}=\\{\varphi_{S}\>:\>S\in\mathcal{T}\\}$. It follows by the theory
in [Sim14] that either $\text{sing}(S)=\emptyset$ or

$\text{dim}\big{(}B_{1}(0)\cap\text{sing}(S)\big{)}\leq d,$ (E.53)

where $d\in\\{0,\ldots,n-1\\}$, for all $S\in\mathcal{T}$. Furthermore it also
follows that there is some $S\in\mathcal{T}$ and some $d$-dimensional linear
subspace $L$ of $\mathbb{R}^{n+\ell}$ such that $\text{sing}(S)=L$ and

$\eta_{q,\lambda\>\\#}S=S$ (E.54)

for all $q\in L$ and all $\lambda>0$. Without loss of generality we may assume
that $L=\mathbb{R}^{d}\times\\{0\\}$ so that $S=[[R^{d}]]\times S_{0}$, where
$\partial S_{0}=0$ and $\text{sing}(S_{0})=\\{0\\}$ and that $S_{0}$ is
minimizing in $\mathbb{R}^{n+\ell-d}$. Further, the assumption that $T\subset
N^{n+1}$ implies that the rescaling gives that
$\text{supp}(S)\subset\mathbb{R}^{n+1}$ (after some orthogonal transformation,
if necessary) and so we may assume that $S_{0}$ is an $(n-d)$-dimensional
minimizing cone in $\mathbb{R}^{n-d+1}$. The singular of this minimizing cone
is the origin. The assertion follows from the non-existence of stable minimal
hypercones by [Sim68] in dimension $\leq 6$, so that
$\text{sing}(T)=\emptyset$ in case $n\leq 6$. If $n=7$ we obtain that
$\text{sing}(T)$ consists of isolated points from the theory in [Sim14,
Theorem 5.8 in Chapter 7].

∎

We state an important result concerning convergence in the case of smooth
hypersurfaces.

######  Lemma E.10.

Let $\\{T_{k}\\}\subset\mathcal{F}_{\lambda}$ be a sequence of integer
multiplicity currents that are $\lambda$-minimizing in $N^{n+1}$ and suppose
$T_{k}\rightharpoonup T\in\mathcal{F}_{\lambda}$, where $T$ is also
$\lambda$-minimizing in $N^{n+1}$. If $T$ and $T_{k}$ have empty singular sets
then there exists a subsequence $\\{T_{k^{\prime}}\\}$ that converges to $T$
in $C^{1,\alpha}_{loc}$.

###### Proof.

Let $p\in\text{supp}(T)$ and let $p_{k}\in\text{supp}(T_{k})$ converge to $p$.
Such a sequence $\\{p_{k}\\}$ must exist by the current convergence. By
assumption, the currents $T$ and $T_{k}$ are locally the graphs of
$C^{1,\alpha}$-functions $f$ and $f_{k}$, defined on the tangent planes at $p$
and $p_{k}$, respectively. We can without loss of generality assume that $p=0$
and $\nabla^{\mathbb{R}^{n}}f=\vec{0}$. Since the generalized mean curvatures
$H_{k}^{T}$ of $T_{k}$ are locally bounded it follows from Allard’s Regularity
Theorem [Sim14, Theorem 5.2 in Chapter 5] that there exist $0<\rho<1$,
$\gamma(n,\ell,2n)\in(0,1)$ and $\delta\in(0,1/2)$ such that the weighted
Hölder norms are uniformly bounded:

$\begin{split}&\rho^{-1}\sup_{q\in B_{\gamma\rho}^{n}(0)}|f_{k}(q)|+\sup_{q\in
B_{\gamma\rho}^{n}(0)}|\nabla^{\delta}f_{k}(q)|\\\
&\qquad+\rho^{1-n/p}\sup_{p,q\in
B_{\gamma\rho}^{n}(0),p\not=q}\frac{|\nabla^{\delta}f_{k}(p)-\nabla^{\delta}f_{k}(q)|}{|p-q|^{\frac{1}{2}}}\leq
C(n,\ell,2n)\delta^{\frac{1}{2(n+1)}}.\end{split}$ (E.55)

Here $B^{n+\ell}_{R}(0)$ is the ball in $R^{n+\ell}$ and $B^{n}_{R}(0)$ the
ball in $\mathbb{R}^{n}\times\\{p^{n+1}=0,\ldots,p^{n+\ell}=0\\}$. Since the
tangent spaces converge,
$T_{p_{k}}\text{graph}(f_{k})\rightarrow\mathbb{R}^{n}\times\\{p^{n+1}=0,\ldots,p^{n+\ell}=0\\}$,
it is clear that we may write $\text{supp}T_{k}\cap
B^{N}_{\rho\gamma}(0)=\text{graph}(f_{k})$ for large enough $k$, where now the
domain of $f_{k}$ is
$\mathbb{R}^{n}\times\\{p^{n+1}=0,\ldots,p^{n+\ell}=0\\}$. The assertion now
follows from the Arzela-Ascoli Theorem.

∎

We end this section by recalling why the regularity results above do not in
general hold when $n\geq 7$:

######  Example E.11.

The following is an example given by [Sim68]. Consider the set

$C=\bigg{\\{}(x,y)\in\mathbb{R}^{4}\times\mathbb{R}^{4}\>\bigg{|}\>||x||=||y||\bigg{\\}},$
(E.56)

with the induced Euclidean metric $\delta$. Then $C$ is a stable minimal
hypercone.

## References

  * [ACG08] Lars Andersson, Mingliang Cai, and Gregory Galloway, _Rigidity and positivity of mass for asymptotically hyperbolic manifolds_ , Ann. Henri Poincaré 9 (2008), no. 1, 1–33.
  * [ADM59] Richard Arnowitt, Stanley Deser, and Charles Misner, _Dynamical structure and definition of energy in general relativity_ , Phys. Rev. (2) 116 (1959), 1322–1330. MR 113667
  * [AEM11] Lars Andersson, Michael Eichmair, and Jan Metzger, _Jang’s equation and its applications to marginally trapped surfaces_ , Complex analysis and dynamical systems IV. Part 2, Contemp. Math., vol. 554, Amer. Math. Soc., Providence, RI, 2011, pp. 13–45.
  * [AMO21] Virginia Agostiniani, Lorenzo Mazzieri, and Francesca Oronzio, _A green’s function proof of the positive mass theorem_ , 2021, http://arxiv.org/abs/1010.4256.
  * [Bes08] Arthur Besse, _Einstein manifolds_ , Classics in Mathematics, Springer-Verlag, Berlin, 2008, Reprint of the 1987 edition.
  * [BK11] Hubert Bray and Marcus Khuri, _P.D.E.’s which imply the Penrose conjecture_ , Asian J. Math. 15 (2011), no. 4, 557–610.
  * [BKKS22] Hubert Bray, Demetre Kazaras, Marcus Khuri, and Daniel Stern, _Harmonic functions and the mass of 3-dimensional asymptotically flat Riemannian manifolds_ , J. Geom. Anal. 32 (2022), no. 6, Paper No. 184, 29.
  * [BKS21] Edward Bryden, Marcus Khuri, and Christina Sormani, _Stability of the spacetime positive mass theorem in spherical symmetry_ , J. Geom. Anal. 31 (2021), no. 4, 4191–4239.
  * [CCS16] Carla Cederbaum, Julien Cortier, and Anna Sakovich, _On the center of mass of asymptotically hyperbolic initial data sets_ , Ann. Henri Poincaré 17 (2016), no. 6, 1505–1528.
  * [CGNP18] Piotr Chrusciel, Gregory Galloway, Luc Nguyen, and Tim-Torben Paetz, _On the mass aspect function and positive energy theorems for asymptotically hyperbolic manifolds_ , Classical and Quantum Gravity 35 (2018).
  * [CH03] Piotr Chrusciel and Marc Herzlich, _The mass of asymptotically hyperbolic Riemannian manifolds_ , Pacific J. Math. 212 (2003), no. 2, 231–264.
  * [CJL04] Piotr Chrusciel, Jacek Jezierski, and Szymon Leski, _The trautman-bondi mass of hyperboloidal initial data sets_ , Adv. Theor. Math. Phys. 8 (2004), no. 1, 83–139.
  * [CM06] Piotr Chrusciel and Daniel Maerten, _Killing vectors in asymptotically flat space-times. II. Asymptotically translational Killing vectors and the rigid positive energy theorem in higher dimensions_ , J. Math. Phys. 47 (2006), no. 2, 022502, 10.
  * [CMT06] Piotr Chrusciel, Daniel Maerten, and Paul Tod, _Rigid upper bounds for the angular momentum and centre of mass on non-singular asymptotically anti-de Sitter space-times_ , J. High Energy Phys. (2006), no. 11, 084, 42.
  * [CN01] Piotr Chrusciel and Gabriel Nagy, _The mass of spacelike hypersurfaces in asymptotically anti-de Sitter space-times_ , Adv. Theor. Math. Phys. 5 (2001), no. 4, 697–754.
  * [DS93] Frank Duzaar and Klaus Steffen, _$\lambda$ minimizing currents_, Manuscripta Math. 80 (1993), no. 4, 403–447.
  * [DS21] Mattias Dahl and Anna Sakovich, _A density theorem for asymptotically hyperbolic initial data satisfying the dominant energy condition_ , Pure Appl. Math. Q. 17 (2021), no. 5, 1669–1710.
  * [EHLS16] Michael Eichmair, Lan-Hsuan Huang, Dan Lee, and Richard Schoen, _The spacetime positive mass theorem in dimensions less than eight_ , J. Eur. Math. Soc. (JEMS) 18 (2016), no. 1, 83–121.
  * [Eic09] Michael Eichmair, _The Plateau problem for marginally outer trapped surfaces_ , J. Differential Geom. 83 (2009), no. 3, 551–583.
  * [Eic13] by same author, _The Jang equation reduction of the spacetime positive energy theorem in dimensions less than eight_ , Comm. Math. Phys. 319 (2013), no. 3, 575–593.
  * [Eld13] Jaap Eldering, _Normally hyperbolic invariant manifolds_ , Atlantis Studies in Dynamical Systems, vol. 2, Atlantis Press, Paris, 2013, The noncompact case. MR 3098498
  * [EM16] Michael Eichmair and Jan Metzger, _Jenkins-Serrin-type results for the Jang equation_ , J. Differential Geom. 102 (2016), no. 2, 207–242.
  * [GT01] David Gilbarg and Neil Trudinger, _Elliptic partial differential equations of second order_ , Classics in Mathematics, Springer-Verlag, Berlin, 2001, Reprint of the 1998 edition.
  * [Har02] Philip Hartman, _Ordinary differential equations_ , Classics in Applied Mathematics, vol. 38, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2002, Corrected reprint of the second (1982) edition [Birkhäuser, Boston, MA; MR0658490 (83e:34002)], With a foreword by Peter Bates.
  * [HKK22] Sven Hirsch, Demetre Kazaras, and Marcus Khuri, _Spacetime harmonic functions and the mass of 3-dimensional asymptotically flat initial data for the Einstein equations_ , J. Differential Geom. 122 (2022), no. 2, 223–258.
  * [HL20] Lan-Hsuan Huang and Dan Lee, _Equality in the spacetime positive mass theorem_ , Comm. Math. Phys. 376 (2020), no. 3, 2379–2407. MR 4104553
  * [HL23] Lan-Hsuan Huang and Dan A. Lee, _Equality in the spacetime positive mass theorem ii_ , 2023.
  * [Jan78] Pong Soo Jang, _On the positivity of energy in general relativity_ , J. Math. Phys. 19 (1978), no. 5, 1152–1155.
  * [KP08] Steven Krantz and Harold Parks, _Geometric integration theory_ , Cornerstones, Birkhäuser Boston, Inc., Boston, MA, 2008.
  * [LUY21] Martin Lesourd, Ryan Unger, and Shing-Tung Yau, _The positive mass theorem with arbitrary ends_ , 2021.
  * [Mae06] Daniel Maerten, _Positive energy-momentum theorem for AdS-asymptotically hyperbolic manifolds_ , Ann. Henri Poincaré 7 (2006), no. 5, 975–1011.
  * [Mey63] Norman Meyers, _An expansion about infinity for solutions of linear elliptic equations_ , J. Math. Mech. 12 (1963), 247–264.
  * [Mic11] Benoit Michel, _Geometric invariance of mass-like asymptotic invariants_ , J. Math. Phys. 52 (2011), no. 5, 052504, 14.
  * [MOM04] Edward Malec and Niall Ó Murchadha, _The Jang equation, apparent horizons and the Penrose inequality_ , Classical Quantum Gravity 21 (2004), no. 24, 5777–5787.
  * [Pet16] Peter Petersen, _Riemannian geometry_ , third ed., Graduate Texts in Mathematics, vol. 171, Springer, Cham, 2016.
  * [Sak21] Anna Sakovich, _The Jang equation and the positive mass theorem in the asymptotically hyperbolic setting_ , Comm. Math. Phys. 386 (2021), no. 2, 903–973.
  * [Sch89] Richard Schoen, _Variational theory for the total scalar curvature functional for Riemannian metrics and related topics_ , Topics in calculus of variations (Montecatini Terme, 1987), Lecture Notes in Math., vol. 1365, Springer, Berlin, 1989, pp. 120–154.
  * [Sim68] James Simons, _Minimal varieties in riemannian manifolds_ , Ann. of Math. (2) 88 (1968), 62–105.
  * [Sim14] Leon Simon, _Introduction to geometric measure theory_ , 2014, https://web.stanford.edu/class/math285/ts-gmt.pdf.
  * [Spr07] Joel Spruck, _Interior gradient estimates and existence theorems for constant mean curvature graphs in $M^{n}\times\mathbb{R}$_, Pure Appl. Math. Q. 3 (2007), no. 3, Special Issue: In honor of Leon Simon. Part 2, 785–800.
  * [SY79a] Richard Schoen and Shing-Tung Yau, _Complete manifolds with nonnegative scalar curvature and the positive action conjecture in general relativity_ , Proc. Nat. Acad. Sci. U.S.A. 76 (1979), no. 3, 1024–1025.
  * [SY79b] by same author, _On the proof of the positive mass conjecture in general relativity_ , Comm. Math. Phys. 65 (1979), no. 1, 45–76.
  * [SY81a] by same author, _The energy and the linear momentum of space-times in general relativity_ , Comm. Math. Phys. 79 (1981), no. 1, 47–51.
  * [SY81b] by same author, _Proof of the positive mass theorem. II_ , Comm. Math. Phys. 79 (1981), no. 2, 231–260.
  * [SY22] by same author, _Positive scalar curvature and minimal hypersurface singularities_ , Surveys in differential geometry 2019. Differential geometry, Calabi-Yau theory, and general relativity. Part 2, Surv. Differ. Geom., vol. 24, Int. Press, Boston, MA, [2022] ©2022, pp. 441–480.
  * [Wan01] Xiaodong Wang, _The mass of asymptotically hyperbolic manifolds_ , J. Differential Geom. 57 (2001), no. 2, 273–299.
  * [Wit81] Edward Witten, _A new proof of the positive energy theorem_ , Comm. Math. Phys. 80 (1981), no. 3, 381–402.
  * [WX15] Yaohua Wang and Xu Xu, _Hyperbolic positive energy theorem with electromagnetic fields_ , Classical Quantum Gravity 32 (2015), no. 2, 025007, 20.
  * [XZ08] Naqing Xie and Xiao Zhang, _Positive mass theorems for asymptotically AdS spacetimes with arbitrary cosmological constant_ , Internat. J. Math. 19 (2008), no. 3, 285–302.
  * [Zha99] Xiao Zhang, _Angular momentum and positive mass theorem_ , Comm. Math. Phys. 206 (1999), no. 1, 137–155.
  * [Zha04] by same author, _A definition of total energy-momenta and the positive mass theorem on asymptotically hyperbolic 3-manifolds. I_ , Comm. Math. Phys. 249 (2004), no. 3, 529–548.


# Some $q$-congruences involving central $q$-binomial coefficients

He-Xia Ni  Department of Applied Mathematics, Nanjing Audit University
Nanjing 211815, People’s Republic of China<EMAIL_ADDRESS>
###### Abstract.

Suppose that $p$ is an odd prime and $m$ is an integer not divisible by $p$.
Sun and Tauraso [Adv. in Appl. Math., 45(2010), 125–148] gave
$\sum_{k=0}^{n-1}\binom{2k}{k+d}/m^{k}$ and
$\sum_{k=0}^{n-1}\binom{2k}{k+d}/(km^{k})$ modulo $p$ for all $d=0,1,\ldots n$
and $n=p^{a}$, where $a$ is a positive integer. In this paper, we present some
$q$-analogues of these congruences in the cases $m=2,4$ for any positive
integer $n$.

###### Key words and phrases:

Congruence; $q$-Binomial coefficient; Cyclotomic polynomial

###### 2010 Mathematics Subject Classification:

Primary 11B65; Secondary 05A10, 05A30, 11A07

The work is supported by the Natural Science Foundation of the Higher
Education Institutions of Jiangsu Province (20KJB110023) and the National
Natural Science Foundation of China (Nos. 12001279 ).

##  1\. Introduction

A central binomial coefficient has the form $\binom{2n}{n}$ with
$n\in\mathbb{N}=\\{0,1,\ldots\\}$. The well-known Fibonacci sequence
${F}_{n\in\mathbb{N}}$ is defined by

$F_{0}=0,F_{1}=1,{\rm and}\ F_{n+1}=F_{n}+F_{n-1}\ {\rm for}\ n=1,2,3,\ldots.$

Let $p$ be an odd prime. Sun and Tauraso [24] proved for any $m\in\mathbb{Z}$
with $p\nmid m$,

$\displaystyle\sum_{k=0}^{p^{a}-1}(-1)^{k}\binom{2k}{k+d}$
$\displaystyle\equiv(-1)^{d-1+\delta_{p,5}}F_{2(|d|-(\frac{p^{a}}{5}))}\
({\rm{mod}}\ p),$ (1.1)
$\displaystyle\sum_{k=0}^{p^{a}-1}\frac{\binom{2k}{k+d}}{m^{k}}$
$\displaystyle\equiv u_{p^{a}-|d|}(m-2)\ ({\rm{mod}}\ p),$ (1.2)
$\displaystyle|d|\sum_{k=0}^{p^{a}-1}\frac{\binom{2k}{k+d}}{km^{k-1}}$
$\displaystyle\equiv 2(-1)^{d}+v_{p^{a}-|d|}(m-2)\ ({\rm{mod}}\ p),$ (1.3)

where $|d|\in\\{0,\ldots p^{a}\\}$ with $a\in\mathbb{Z}^{+}$, $\delta_{i,j}$
is the Kronecker delta, and the two linear recurrences
$\\{u_{n}(x)\\}_{n\in\mathbb{N}}$ and $\\{v_{n}(x)\\}_{n\in\mathbb{N}}$ of
polynomials are defined as follows:

$u_{0}(x)=0,\ u_{1}(x)=1,{\rm and}\ u_{n+1}(x)=xu_{n}(x)-u_{n-1}(x)\
(n=1,2,\ldots),$ $v_{0}(x)=2,\ v_{1}(x)=x,{\rm and}\
v_{n+1}(x)=xv_{n}(x)-v_{n-1}(x)\ (n=1,2,\ldots),$

and

$\ u_{-1}(x)=xu_{0}(x)-u_{1}(x)=-1,v_{-1}(x)=xv_{0}(x)-v_{1}(x)=x.$

Later Sun and Tauraso [25] also proved

$\displaystyle\sum_{k=0}^{p^{a}-1}\binom{2k}{k+d}$
$\displaystyle\equiv\left(\frac{p^{a}-|d|}{3}\right)\ ({\rm{mod}}\ p),$ (1.4)

where $\big{(}\frac{.}{p}\big{)}$ denotes the Legendre symbol. The $a=1$ case
of this congruence was first proved by Pan and Sun [23, Theorem 1.2] via a
sophisticated combinatorial identity.

It is well known that binomial identities or congruences usually have nice
$q$-analogues (see [1, 3, 2, 15, 7, 10, 11, 9, 8, 12, 13, 14, 16, 17, 19, 20,
21, 22, 26, 27]). John Green [6] proved a conjecture made by Daan Krammer. The
identity was served as an inspiration for searching $q$-analogues of some
identities in [24, 25]. In particular, Tauraso [15] proved the following
generalization of (1.4):

$\displaystyle\sum_{k=0}^{n-1}q^{k}{\genfrac{[}{]}{0.0pt}{}{2k}{k+d}}\equiv\left(\frac{n-|d|}{3}\right)q^{3r(r+1)/2+|d|(2r+1)}\
({\rm{mod}}\ \Phi_{n}(q))$

with $r=\lfloor 2(n-|d|)/3\rfloor.$ Liu and Petrov [21] proved that, for any
positive integer $n$,

$\displaystyle\sum_{k=0}^{n-1}q^{k}{\genfrac{[}{]}{0.0pt}{}{2k}{k}}\equiv\left(\frac{n}{3}\right)q^{(n^{2}-1)/3}\
({\rm{mod}}\ \Phi_{n}(q)^{2}),$

which was originally conjectured by Guo [7]. Here and in what follows, the
$n$-th cyclotomic polynomial is defined as

$\Phi_{n}(q):=\prod_{\begin{subarray}{c}1\leq k\leq n\\\
\gcd(n,k)=1\end{subarray}}(q-e^{2\pi\sqrt{-1}\cdot\frac{k}{n}}).$

It is well known that $\Phi_{d}(q)$ is an irreducible polynomial with integral
coefficients, and

$[n]=\prod_{\begin{subarray}{c}d\geq 2\\\ d\mid n\end{subarray}}\Phi_{d}(q).$

So, for $d\geqslant 2$, $\Phi_{d}(q)$ divides $[n]$ if and only if $d$ divides
$n$.

Furthermore, the $q$-binomial coefficient ${\genfrac{[}{]}{0.0pt}{}{n}{k}}$ is
defined as

$\displaystyle{\genfrac{[}{]}{0.0pt}{}{n}{k}}={\genfrac{[}{]}{0.0pt}{}{n}{k}}_{q}=\begin{cases}\dfrac{(q;q)_{n}}{(q;q)_{k}(q,q)_{n-k}}&\text{if
}0\leqslant k\leqslant n,\\\\[10.0pt] 0&\text{otherwise },\end{cases}$

where

$(x;q)_{n}=\begin{cases}(1-x)(1-xq)\cdots(1-xq^{n-1}),&\text{if }n\geq
1,\\\\[5.0pt] 1,&\text{if }n=0.\end{cases}$

Guo and Zeng [15] proved that if $n\geq 1$ and $d=0,1,\ldots,n-1,$ then

$\displaystyle\sum_{k=0}^{n-1}q^{-k(k+3)/2}{\genfrac{[}{]}{0.0pt}{}{2k}{k+d}}(-q^{k+1};q)_{n-1-k}$
$\displaystyle\equiv\begin{cases}(-1)^{(n+d-2)/2}q^{(5-(n-d+1)^{2})/4}(1-q^{n-d})\
({\rm{mod}}\ \Phi_{n}(q))&\text{if }n\equiv d\ ({\rm{mod}}\ 2),\\\\[5.0pt]
(-1)^{(n+d-1)/2}q^{(5-(n-d)^{2})/4}\ ({\rm{mod}}\ \Phi_{n}(q))&\text{if
}n\not\equiv d\ ({\rm{mod}}\ 2),\end{cases}$ (1.5)

and

$\displaystyle\sum_{k=0}^{n-1}q^{-k(k+3)/2}{\genfrac{[}{]}{0.0pt}{}{2k}{k+d}}(-q^{k};q)_{n-1-k}$
$\displaystyle\equiv\begin{cases}(-1)^{(n+d-2)/2}q^{(9-(n-d+1)^{2})/4}\frac{1-q^{n-d}}{1+q}\
({\rm{mod}}\ \Phi_{n}(q))&\text{if }n\equiv d\ ({\rm{mod}}\ 2),\\\\[5.0pt]
(-1)^{(n+d-1)/2}q^{(5-(n-d)^{2})/4}\ ({\rm{mod}}\ \Phi_{n}(q))&\text{if
}n\not\equiv d\ ({\rm{mod}}\ 2).\end{cases}$ (1.6)

Clearly, (1) and (1) are partial $q$-analogues of (1.2).

In this paper, we shall consider some $q$-analogues of (1.2) and (1.3) as well
as some variations of them, which are different from (1) and (1). Our main
results can be stated as follows.

######  Theorem 1.1.

For $n\geqslant 1$ and $0\leqslant d\leqslant n-1$, we have

$\displaystyle\sum_{k=0}^{n-1}q^{k}{\genfrac{[}{]}{0.0pt}{}{2k}{k+d}}(-q^{k+1};q)_{n-1-k}$
$\displaystyle\quad=\sum_{\begin{subarray}{c}k=0\\\ k\equiv n-d\bmod
2\end{subarray}}^{n-d}(-1)^{(n-d-k)/2}q^{3(n^{2}+k^{2}-d^{2})/4-3nk/2+n-k}\frac{(1-q^{k})(1+q^{n-k+1})}{(1-q^{2n-k+1})(1+q^{n})}{\genfrac{[}{]}{0.0pt}{}{2n}{k}},$
(1.7)
$\displaystyle\sum_{k=0}^{n-1}q^{k}{\genfrac{[}{]}{0.0pt}{}{2k}{k+d}}(-q^{k+1};q)_{n-1-k}^{2}$
$\displaystyle\quad=\sum_{k=d+1}^{n}\sum_{j=k+1}^{n+1}\frac{q^{(j^{2}-3j+k^{2}-k)/2-d^{2}+1}1+q^{j-1}}{1+q^{n}}{\genfrac{[}{]}{0.0pt}{}{2n}{n-j+1}}.$
(1.8)

######  Corollary 1.1.

Let $n\geqslant 1$ and $0\leqslant d\leqslant n-1$. Then, modulo
$\Phi_{n}(q)$, we have

$\displaystyle\sum_{k=0}^{n-1}q^{k}{\genfrac{[}{]}{0.0pt}{}{2k}{k+d}}(-q^{k+1};q)_{n-1-k}\equiv\begin{cases}0&\\!\\!\\!\text{if
}n\equiv d\ ({\rm{mod}}\ 2),\\\ q^{3(n-1)^{2}/4-3d^{2}/4-1}&\\!\\!\\!\text{if
}n\equiv d+1\ ({\rm{mod}}\ 4),\\\
-q^{3(n-1)^{2}/4-3d^{2}/4-1}&\\!\\!\\!\text{if }n\equiv d-1\ ({\rm{mod}}\
4),\end{cases}$ (1.9)

and

$\displaystyle\sum_{k=0}^{n-1}q^{k}{\genfrac{[}{]}{0.0pt}{}{2k}{k+d}}(-q^{k+1};q)_{n-1-k}^{2}\equiv\sum_{k=d+1}^{n}q^{\binom{n}{2}+\binom{k}{2}-d^{2}}.$
(1.10)

Note that (1.9) was proved by Guo and Zeng [15, Corollary 4.2] by using
Andrews’s $q$-analogue of Gauss’s second theorem (see [1, 2]). (1.9) and
(1.10) may be deemed partial $q$-analogues of (1.2) for the cases $m=2$ and
$m=4$, respectively. Thus, when $n=p^{a},q\rightarrow 1$, the $q$-congruences
(1.9) and (1.10) reduce to the following consequences [24, Corollary 1.1].

######  Corollary 1.2.

For any odd prime $p$ and $0\leqslant d\leqslant p^{a}-1$ with
$a\in\mathbb{Z}^{+}$, modulo $p$, we have

$\displaystyle\sum_{k=0}^{p^{a}-1}\binom{2k}{k+d}\frac{1}{2^{k}}\equiv\begin{cases}0&\text{if
}p^{a}\equiv d\ ({\rm{mod}}\ 2),\\\ 1&\text{if }p^{a}\equiv d+1\ ({\rm{mod}}\
4),\\\ -1&\text{if }p^{a}\equiv d-1\ ({\rm{mod}}\ 4),\end{cases}$ (1.11)

and

$\displaystyle\sum_{k=0}^{p^{a}-1}\binom{2k}{k+d}\frac{1}{4^{k}}\equiv-d.$
(1.12)

######  Theorem 1.2.

For $n\geqslant 1$ and $1\leqslant d\leqslant n-1$, we have

$\displaystyle\sum_{k=1}^{n-1}q^{k}{\genfrac{[}{]}{0.0pt}{}{2k}{k+d}}\frac{(-q^{k+1};q)_{n-k}}{[k]}$
$\displaystyle=-\sum_{k=1}^{\lfloor(n+1-d)/2\rfloor}(-1)^{k}q^{3k^{2}+(3d-5)k-2d+2}\frac{[2d+4k-2]}{[d][n]}{\genfrac{[}{]}{0.0pt}{}{2n}{n-d-2k+1}},$
(1.13)
$\displaystyle\sum_{k=1}^{n}q^{k}{\genfrac{[}{]}{0.0pt}{}{2k}{k+d}}\frac{(-q^{k+1};q)_{n-k}^{2}}{[k]}$
$\displaystyle=\sum_{k=d}^{n-1}q^{\binom{k+1}{2}-\binom{d}{2}}\frac{{\genfrac{[}{]}{0.0pt}{}{2n+1}{n-k}}}{[d]}+\frac{q^{\binom{n+1}{2}-\binom{d}{2}}}{[d]}.$
(1.14)

######  Corollary 1.3.

For $n\geqslant 1$ and $1\leqslant d\leqslant n-1$, modulo $\Phi_{n}(q)$, we
have

$\displaystyle[d]\sum_{k=1}^{n-1}q^{k}{\genfrac{[}{]}{0.0pt}{}{2k}{k+d}}\frac{(-q^{k+1};q)_{n-k}}{[k]}$
$\displaystyle\equiv\begin{cases}-2q^{-\binom{n+1}{2}-\binom{d+1}{2}}\sum_{k=2}^{(n+1-d)/2-1}(-1)^{k}q^{k^{2}+(d-2)k+1}(1+q^{d+2k-1})\\\
-2(-1)^{(n+1-d)/2}q^{-(n^{2}+3d^{2}-2d-1)/4}+2q^{-\binom{n}{2}-\binom{d}{2}}(1+q^{d+1})&\text{if
}n\not\equiv d\ ({\rm{mod}}\ 2),\\\\[5.0pt]
2q^{-\binom{n+1}{2}-\binom{d+1}{2}}\sum_{k=2}^{(n-d)/2-1}(-1)^{k}q^{k^{2}+(d-2)k+1}(1+q^{d+2k-1})\\\
+2(-1)^{(n-d)/2}q^{(-n^{2}-2n-3d^{2}+2d)/4}(1+q)-2q^{-\binom{n}{2}-\binom{d}{2}}(1+q^{d+1})&\text{if
}n\equiv d\ ({\rm{mod}}\ 2),\\\ \end{cases}$ (1.15)

and

$\displaystyle[d]\sum_{k=1}^{n}q^{k}{\genfrac{[}{]}{0.0pt}{}{2k}{k+d}}\frac{(-q^{k+1};q)_{n-k}^{2}}{[k]}\equiv\begin{cases}2q^{-\binom{d}{2}}&\text{if
}n\equiv 1\ ({\rm{mod}}\ 2),\\\ -2q^{-\binom{d}{2}}&\text{otherwise
}.\end{cases}$ (1.16)

Once again, the congruences (1.3) and (1.16) may be deemed partial
$q$-analogues of (1.3) for the cases $m=2,4$ which imply the following
conclusion.

######  Corollary 1.4.

For any odd prime $p$ and $1\leqslant d\leqslant p^{a}-1$ with
$a\in\mathbb{Z}^{+}$, modulo $p$, we have

$\displaystyle
d\sum_{k=1}^{p^{a}-1}\frac{\binom{2k}{k+d}}{k2^{k}}-(-1)^{d}\equiv\begin{cases}0&\text{if
}p^{a}\not\equiv d\ ({\rm{mod}}\ 2),\\\ 1&\text{if }p^{a}\equiv d\
({\rm{mod}}\ 4),\\\ -1&\text{if }p^{a}\equiv d+2\ ({\rm{mod}}\ 4),\\\
\end{cases}$ (1.17)

and

$\displaystyle
d\sum_{k=1}^{p^{a}}\frac{\binom{2k}{k+d}}{k4^{k}}\equiv\begin{cases}1/2&\text{if
}p^{a}\not\equiv d\ ({\rm{mod}}\ 2),\\\ -1/2&\text{if }p^{a}\equiv d\
({\rm{mod}}\ 2).\\\ \end{cases}$ (1.18)

The rest of this paper is organized as follows. In the next section, we are
going to present some auxiliary lemmas and prove Theorem 1.1 and Corollary
1.1. Finally, Theorem 1.2 and Corollary 1.3 will be proved in Section 3.

##  2\. Proofs of Theorem 1.1 and Corollary 1.1

Firstly, we give few properties that we will need later.

######  Lemma 2.1.

[4, 5, 18] Let $a,b,c,d\in\mathbb{Z}$ and $0\leq b,d\leq n-1$. Then, modulo
$\Phi_{n}(q)$,

$\displaystyle{\genfrac{[}{]}{0.0pt}{}{an+b}{cn+d}}\equiv\binom{a}{c}{\genfrac{[}{]}{0.0pt}{}{b}{d}}.$
(2.1)

Note that (2.1) is a generation of the well-known Lucas congruence. From (2.1)
we can easily deduce the following result.

######  Lemma 2.2.

[26] Let $n>1$, and $a\geqslant 1$. Then, modulo $\Phi_{n}(q)$,

$\displaystyle{\genfrac{[}{]}{0.0pt}{}{an}{k}}\equiv\begin{cases}\binom{a}{k/n}&\text{if
}n\mid k,\\\ 0&\text{otherwise },\end{cases}$ (2.2)

and

$\displaystyle{\genfrac{[}{]}{0.0pt}{}{2n+1}{k}}\equiv\begin{cases}1&\text{if
}k=0,1,2n,2n+1,\\\ 2&\text{if }k=n,n+1,\\\ 0&\text{otherwise }.\end{cases}$
(2.3)

###### Proof of Theorem 1.1.

Denote the left-hand side and right-hand side of (1.7) by $S(n,d)$ and
$T(n,d)$, respectively.

For $n\geq 1,$ we have

$\displaystyle S(n,n)=T(n,n)=0,S(n,n-1)=T(n,n-1)=q^{n-1},$ $\displaystyle
S(n,n-2)=T(n,n-2)=\begin{cases}q^{n-1}[2n-2]+q^{n-2}(1+q^{n-1})&\text{if
}n>1,\\\ 0&\text{if }n=1.\end{cases}$

Next, we shall prove that for $0\leq d\leq n-1,$ both $S(n,d)$ and $T(n,d)$
satisfy the following recurrence equation

$\displaystyle
X(n,d)+q^{3d+3}X(n,d+2)=\frac{(q^{d}-q^{n})(1+q^{d+1})}{(1-q^{n+d+1})(1+q^{n})}{\genfrac{[}{]}{0.0pt}{}{2n}{n+d}}.$
(2.4)

We start with $S(n,d),$ and shall prove (2.4) by induction on $n$. Clearly, it
holds for $n=1,$ and we will prove that, for $n\geq 1,$

$S(n+1,d)+q^{3d+3}S(n+1,d+2)=\frac{(q^{d}-q^{n+1})(1+q^{d+1})}{(1-q^{n+d+2})(1+q^{n+1})}{\genfrac{[}{]}{0.0pt}{}{2n+2}{n+d+1}}.$

Since the left-hand side is equal to

$(1+q^{n})S(n,d)+q^{n}{\genfrac{[}{]}{0.0pt}{}{2n}{n+d}}+q^{3d+3}\left((1+q^{n})S(n,d+2)+q^{n}{\genfrac{[}{]}{0.0pt}{}{2n}{n+d+2}}\right),$

by induction, we only need to check that

$\displaystyle\frac{(q^{d}-q^{n})(1+q^{d+1})}{1-q^{n+d+1}}{\genfrac{[}{]}{0.0pt}{}{2n}{n+d}}+q^{n}{\genfrac{[}{]}{0.0pt}{}{2n}{n+d}}+q^{3d+3+n}{\genfrac{[}{]}{0.0pt}{}{2n}{n+d+2}}$
$\displaystyle\quad=\frac{(q^{d}-q^{n+1})(1+q^{d+1})}{(1-q^{n+d+2})(1+q^{n+1})}{\genfrac{[}{]}{0.0pt}{}{2n+2}{n+d+1}},$

which is indeed true.

For $T(n,d)$, there holds

$\displaystyle q^{3d+3}T(n,d+2)$ $\displaystyle=$ $\displaystyle
q^{3d+3}\sum_{\begin{subarray}{c}k=0\\\ k\equiv n-d\bmod
2\end{subarray}}^{n-d-2}(-1)^{(n-d-k-2)/2}q^{3(n^{2}+k^{2}-(d+2)^{2})/4-3nk/2+n-k}\frac{(1-q^{k})(1+q^{n-k+1})}{(1-q^{2n-k+1})(1+q^{n})}{\genfrac{[}{]}{0.0pt}{}{2n}{k}}$
$\displaystyle=$ $\displaystyle-\sum_{\begin{subarray}{c}k=0\\\ k\equiv
n-d\bmod
2\end{subarray}}^{n-d-2}(-1)^{(n-d-k)/2}q^{3(n^{2}+k^{2}-d^{2})/4-3nk/2+n-k}\frac{(1-q^{k})(1+q^{n-k+1})}{(1-q^{2n-k+1})(1+q^{n})}{\genfrac{[}{]}{0.0pt}{}{2n}{k}}$
$\displaystyle=$
$\displaystyle-T(n,d)+\frac{(q^{d}-q^{n})(1+q^{d+1})}{(1-q^{n+d+1})(1+q^{n})}{\genfrac{[}{]}{0.0pt}{}{2n}{n+d}}.$

Hence, the identity (1.7) is concluded.

Now, let $S(n,d)$ and $T(n,d)$ denote the left-hand side and right-hand side
of (1.8), respectively. Likewise, we need to check that for $0\leq d\leq
n-1,S(n,d)=T(n,d).$ For $n\geq 1$ we have

$\displaystyle S(n,n)=T(n,n)=0,S(n,n-1)=T(n,n-1)=q^{n-1},$ $\displaystyle
S(n,n-2)=T(n,n-2)=\begin{cases}q^{n-1}[2n-2]+q^{n-2}(1+q^{n-1})^{2}&\text{if
}n>1,\\\ 0&\text{if }n=1.\end{cases}$

Then, we will show that both $S(n,d)$ and $T(n,d)$ satisfy the following
recurrence equation

$X(n,d)-q^{d}(1+q^{d+1})X(n,d+1)+q^{3d+3}X(n,d+2)=\frac{q^{d}(1-q^{n})(1+q^{d+1})(q;q)_{2n-1}}{(q;q)_{n-d-1}(q;q)_{n+d+1}}.$

We start with $S(n,d)$. Clearly, the above identity holds for $n=1$. We need
to prove that

$\displaystyle S(n+1,d)-q^{d}(1+q^{d+1})S(n+1,d+1)+q^{3d+3}X(n+1,d+2)$ (2.5)
$\displaystyle\quad=\frac{q^{d}(1-q^{n+1})(1+q^{d+1})(q;q)_{2n+1}}{(q;q)_{n-d}(q;q)_{n+d+2}}$

for $n\geq 2$. The left-hand side of (2.5) is equal to

$\displaystyle(1+q^{n})^{2}S(n,d)+q^{n}{\genfrac{[}{]}{0.0pt}{}{2n}{n+d}}-q^{d}(1+q^{d+1})\left((1+q^{n})^{2}S(n,d+1)+q^{n}{\genfrac{[}{]}{0.0pt}{}{2n}{n+d+1}}\right)$
$\displaystyle\quad+q^{3d+3}\left((1+q^{n})^{2}S(n,d+2)+q^{n}{\genfrac{[}{]}{0.0pt}{}{2n}{n+d+2}}\right).$

By induction, we only need to verify that

$\displaystyle(1+q^{n})^{2}\frac{q^{d}(1-q^{n})(1+q^{d+1})(q;q)_{2n-1}}{(q;q)_{n-d-1}(q;q)_{n+d+1}}+q^{n}{\genfrac{[}{]}{0.0pt}{}{2n}{n+d}}-q^{d+n}(1+q^{d+1}){\genfrac{[}{]}{0.0pt}{}{2n}{n+d+1}}$
$\displaystyle\quad+q^{3d+3+n}{\genfrac{[}{]}{0.0pt}{}{2n}{n+d+2}}=\frac{q^{d}(1-q^{n+1})(1+q^{d+1})(q;q)_{2n+1}}{(q;q)_{n-d}(q;q)_{n+d+2}},$

which is clearly true.

Now, let us turn to consider $T(n,d),$

$\displaystyle-q^{d}(1+q^{d+1})T(n,d+1)$ $\displaystyle=$
$\displaystyle-(1+q^{d+1})\sum_{k=d+2}^{n}\sum_{j=k+1}^{n+1}q^{(j^{2}-3j+k^{2}-k)/2-d^{2}-d}\frac{1+q^{j-1}}{1+q^{n}}{\genfrac{[}{]}{0.0pt}{}{2n}{n-j+1}}$
$\displaystyle=$
$\displaystyle-\sum_{k=d+2}^{n}\sum_{j=k+1}^{n+1}q^{(j^{2}-3j+k^{2}-k)/2-d^{2}-d}\frac{1+q^{j-1}}{1+q^{n}}{\genfrac{[}{]}{0.0pt}{}{2n}{n-j+1}}$
$\displaystyle\quad-q^{d+1}\sum_{k=d+2}^{n+1}\sum_{j=k+1}^{n+1}q^{(j^{2}-3j+k^{2}-k)/2-d^{2}-d}\frac{1+q^{j-1}}{1+q^{n}}{\genfrac{[}{]}{0.0pt}{}{2n}{n-j+1}}$
$\displaystyle=$
$\displaystyle-q^{3d+3}T(n,d+2)-q^{3d+3}\sum_{j=d+3}^{n+1}q^{(j^{2}-3j+(d+2)^{2}-(d+2))/2-(d+2)^{2}+1}\frac{1+q^{j-1}}{1+q^{n}}{\genfrac{[}{]}{0.0pt}{}{2n}{n-j+1}}$
$\displaystyle\quad-T(n,d)+\sum_{j=d+2}^{n+1}q^{(j^{2}-3j+(d+1)^{2}-(d+1))/2-d^{2}+1}\frac{1+q^{j-1}}{1+q^{n}}{\genfrac{[}{]}{0.0pt}{}{2n}{n-j+1}}$
$\displaystyle=$
$\displaystyle-q^{3d+3}T(n,d+2)-T(n,d)+q^{d}\frac{(1-q^{n})(1+q^{d+1})(q;q)_{2n-1}}{(q;q)_{n-d-1}(q;q)_{n+d+1}}.$

Therefore, the identity (1.8) follows. ∎

###### Proof of Corollary 1.1.

For $d=0,1,2\ldots,$ via (2.2), we have

$\displaystyle\frac{1-q^{k}}{1-q^{2n-k+1}}{\genfrac{[}{]}{0.0pt}{}{2n}{k}}=\frac{[2n]}{[2n-k+1]}{\genfrac{[}{]}{0.0pt}{}{2n-1}{2n-k}}={\genfrac{[}{]}{0.0pt}{}{2n}{k-1}},$

and

$\displaystyle\sum_{k=0}^{n-1}q^{k}{\genfrac{[}{]}{0.0pt}{}{2k}{k+d}}(-q^{k+1};q)_{n-1-k}$
$\displaystyle\equiv
1/2\sum_{k=1}^{n-d}(-1)^{(n-d-k)/2}q^{3(n^{2}+k^{2}-d^{2})/4-3nk/2-k}[2\mid
n-d+k](1+q^{1-k}){\genfrac{[}{]}{0.0pt}{}{2n}{k-1}}$
$\displaystyle\equiv\begin{cases}0&\text{if }n\equiv d\ ({\rm{mod}}\ 2)\\\
(-1)^{(n-1-d)/2}q^{3(n-1)^{2}/4-3d^{2}/4-1}&\text{if }n\not\equiv d\
({\rm{mod}}\ 2)\end{cases}\ ({\rm{mod}}\ \Phi_{n}(q)).$

Similarly, since

${\genfrac{[}{]}{0.0pt}{}{2n}{n-j+1}}\equiv 0\ ({\rm{mod}}\ \Phi_{n}(q)),$

for $d+1\leq k<j<n+1,$ we are led to

$\displaystyle\sum_{k=0}^{n-1}q^{k}{\genfrac{[}{]}{0.0pt}{}{2k}{k+d}}(-q^{k+1};q)_{n-1-k}^{2}\equiv\sum_{k=d+1}^{n}q^{\binom{n}{2}+\binom{k}{2}-d^{2}}\
({\rm{mod}}\ \Phi_{n}(q)).$

The proof of the theorem is complete. ∎

##  3\. Proofs of Theorem 1.2 and Corollary 1.3

###### Proof of Theorem 1.2.

Let $S(n,d)$ and $T(n,d)$ be the left-hand side and the right-hand side of the
claimed identity. It is very easy to verify that $S(n,d)=T(n,d)$ holds for
$d=n,n-1,n-2.$ Along the same lines of the proof of Theorem 1.1, we can prove
that

$\displaystyle\frac{q^{2+3d}(1-q^{d+2})}{1-q^{d}}T(n,d+2)=-T(n,d)+q^{d}\frac{(1+q^{n})(1-q^{2d+2})(q;q)_{2n-1}}{(1-q^{d})(q;q)_{n+d+1}(q^{2};q)_{n-d-2}},$

where $\lfloor x\rfloor$ denotes the largest integer less than or equal to
$x$.

Moreover, for $n\geq 1,$ we shall prove that, for $1\leq d\leq n,$ $S(n,d)$
satisfy the same recurrence equation. Clearly, it holds for $n=1$. We will
prove that, for $n\geq 1$,

$\displaystyle\frac{q^{2+3d}(1-q^{d+2})}{1-q^{d}}S(n+1,d+2)+S(n+1,d)=q^{d}\frac{(1+q^{n+1})(1-q^{2d+2})(q;q)_{2n+1}}{(1-q^{d})(q;q)_{n+d+2}(q^{2};q)_{n-d-1}}.$
(3.1)

It is clear to see that the left-hand side is equal to

$\displaystyle(1+q^{n+1})S(n,d)+q^{n}\frac{(1+q^{n+1}){\genfrac{[}{]}{0.0pt}{}{2n}{n+d}}}{[n]}$
$\displaystyle\quad+\frac{q^{2+3d}(1+q^{n+1})(1-q^{d+2})}{1-q^{d}}S(n,d+2)+\frac{q^{2+3d+n}(1-q^{d+2})(1+q^{n+1}){\genfrac{[}{]}{0.0pt}{}{2n}{n+d+2}}}{(1-q^{d})[n]}.$

By induction and using the following identity

$\displaystyle
q^{d}\frac{(1+q^{n})(1-q^{2d+2})(q;q)_{2n-1}}{(1-q^{d})(q;q)_{n+d+1}(q^{2};q)_{n-d-2}}+\frac{q^{n+2+3d}(1-q^{d+2}){\genfrac{[}{]}{0.0pt}{}{2n}{n+d+2}}}{(1-q^{d})[n]}+q^{n}\frac{{\genfrac{[}{]}{0.0pt}{}{2n}{n+d}}}{[n]}$
$\displaystyle=$ $\displaystyle
q^{d}\frac{(1-q^{2d+2})(q;q)_{2n+1}}{(1-q^{d})(q;q)_{n+d+2}(q^{2};q)_{n-d-1}},$

we complete the proof of (3.1).

Similarly, we denonte the left-hand side and the right-hand side of (1.14) by
$S(n,d)$ and $T(n,d)$, respectively. Clearly, $S(n,d)=T(n,d)$ holds for $d=n$.
Moreover,

$\displaystyle T(n,d+1)=$
$\displaystyle\sum_{k=d+1}^{n-1}q^{\binom{k+1}{2}-\binom{d+1}{2}}\frac{{\genfrac{[}{]}{0.0pt}{}{2n+1}{n-k}}}{[d+1]}+\frac{q^{\binom{n+1}{2}-\binom{d+1}{2}}}{[d+1]}$
$\displaystyle=$
$\displaystyle-\frac{1-q^{-d}}{1-q^{d+1}}T(n,d)-\frac{{\genfrac{[}{]}{0.0pt}{}{2n+1}{n-d}}}{[d+1]}.$

It remains to show that, for $n\geq 1$ and $1\leq d\leq n$,

$\displaystyle
S(n,d+1)+\frac{1-q^{-d}}{1-q^{d+1}}S(n,d)=-\frac{{\genfrac{[}{]}{0.0pt}{}{2n+1}{n-d}}}{[d+1]}.$

We proceed by induction on $n$. Clearly, it holds for $n=1$. Suppose that it
holds for $n$. We will prove that

$\displaystyle
S(n+1,d+1)+\frac{1-q^{-d}}{1-q^{d+1}}S(n+1,d)=-\frac{{\genfrac{[}{]}{0.0pt}{}{2n+3}{n+1-d}}}{[d+1]}.$
(3.2)

The left-hand side of the above identity is equal to

$\displaystyle\frac{1-q^{-d}}{1-q^{d+1}}(1+q^{n+1})^{2}S(n,d)+(1+q^{n+1})^{2}S(n,d+1)$
$\displaystyle\quad+\frac{q^{n+1}(1-q^{-d})}{(1-q^{d+1})[n+1]}{\genfrac{[}{]}{0.0pt}{}{2n+2}{n+1+d}}+q^{n+1}\frac{{\genfrac{[}{]}{0.0pt}{}{2n+2}{n+2+d}}}{[n+1]}.$

By the induction hypothesis and applying the following identity

$\displaystyle(1+q^{n+1})^{2}\frac{{\genfrac{[}{]}{0.0pt}{}{2n+1}{n-d}}}{[d+1]}-\frac{{\genfrac{[}{]}{0.0pt}{}{2n+3}{n+1-d}}}{[d+1]}-q^{n+1}\frac{(1-q^{-d}){\genfrac{[}{]}{0.0pt}{}{2n+2}{n+1+d}}}{(1-q^{d+1})[n+1]}=q^{n+1}\frac{{\genfrac{[}{]}{0.0pt}{}{2n+2}{n+2+d}}}{[n+1]},$

we are led to (3.2). This completes the proof. ∎

###### Proof of Corollary 1.3.

Notice that for $1\leq j<n,$

${\genfrac{[}{]}{0.0pt}{}{an}{j}}=\frac{[an]}{[j]}{\genfrac{[}{]}{0.0pt}{}{an-1}{j-1}}\equiv
0\ ({\rm{mod}}\ \Phi_{n}(q)),$

and

${\genfrac{[}{]}{0.0pt}{}{an-1}{j-1}}_{q}\equiv{\genfrac{[}{]}{0.0pt}{}{-1}{j-1}}_{q}=(-1)^{j-1}q^{-\binom{j}{2}}\
({\rm{mod}}\ \Phi_{n}(q)).$

When $n-d\equiv 0\ ({\rm{mod}}\ 2)$, we have

$\displaystyle[d]\sum_{k=1}^{n-1}q^{k}{\genfrac{[}{]}{0.0pt}{}{2k}{k+d}}\frac{(-q^{k+1};q)_{n-k}}{[k]}$
$\displaystyle\quad=-\sum_{k=1}^{(n-d)/2}(-1)^{k}q^{3k^{2}+(3d-5)k-2d+2}\frac{[2d+4k-2]}{[n]}{\genfrac{[}{]}{0.0pt}{}{2n}{n-d-2k+1}}$
$\displaystyle\quad\equiv-(-1)^{(n-d)/2}q^{3(n-d)^{2}/4+(3d-5)(n-d)/2-2d+2}(1+q^{n})[2n-2]+q^{d}{\genfrac{[}{]}{0.0pt}{}{2n}{n-d-1}}\frac{[2d+2]}{[n]}$
$\displaystyle\qquad+\sum_{k=2}^{(n-d)/2-1}(-1)^{k}q^{3k^{2}+(3d-3)k-d+1}(1+q^{n})(1+q^{d+2k-1}){\genfrac{[}{]}{0.0pt}{}{-1}{n-d-2k}}$
$\displaystyle\quad\equiv-2(-1)^{(n-d)/2}q^{(3n^{2}-10n-3d^{2}+2d+8)/4}[2n-2]-2q^{-(n^{2}-2nd-3n+d^{2}-d)/2}(1+q^{d+1})$
$\displaystyle\qquad+2\sum_{k=2}^{(n-d)/2-1}(-1)^{k}q^{-\binom{n+1}{2}-\binom{d+1}{2}+k^{2}+(d-2)k+1}(1+q^{d+2k-1})$
$\displaystyle\quad\equiv
2q^{-\binom{n+1}{2}-\binom{d+1}{2}}\sum_{k=2}^{(n-d)/2-1}(-1)^{k}q^{k^{2}+(d-2)k+1}(1+q^{d+2k-1})$
$\displaystyle\qquad+2(-1)^{(n-d)/2}q^{(-n^{2}-2n-3d^{2}+2d)/4}(1+q)-2q^{-\binom{n}{2}-\binom{d}{2}}(1+q^{d+1})\
({\rm{mod}}\ \Phi_{n}(q)).$

When $n-d\equiv 1\ ({\rm{mod}}\ 2)$, we obtain

$\displaystyle[d]\sum_{k=1}^{n-1}q^{k}{\genfrac{[}{]}{0.0pt}{}{2k}{k+d}}\frac{(-q^{k+1};q)_{n-k}}{[k]}$
$\displaystyle\quad=-\sum_{k=1}^{(n-d+1)/2}(-1)^{k}q^{3k^{2}+(3d-5)k-2d+2}\frac{[2d+4k-2]}{[n]}{\genfrac{[}{]}{0.0pt}{}{2n}{n-d-2k+1}}$
$\displaystyle\quad\equiv-2(-1)^{(n-d+1)/2}q^{3(n-d+1)^{2}/4+(3d-5)(n-d+1)/2-2d+2}+q^{d}{\genfrac{[}{]}{0.0pt}{}{2n}{n-d-1}}\frac{[2d+2]}{[n]}$
$\displaystyle\qquad+2\sum_{k=2}^{(n-d+1)/2-1}(-1)^{k}q^{3k^{2}+(3d-3)k-d+1}(1+q^{d+2k-1}){\genfrac{[}{]}{0.0pt}{}{-1}{n-d-2k}}$
$\displaystyle\quad\equiv-2q^{-\binom{n+1}{2}-\binom{d+1}{2}}\sum_{k=2}^{(n-d+1)/2-1}(-1)^{k}q^{k^{2}+(d-2)k+1}(1+q^{d+2k-1})$
$\displaystyle\qquad-2(-1)^{(n-d+1)/2}q^{-(n^{2}+3d^{2}-2d-1)/4}+2q^{-\binom{n}{2}-\binom{d}{2}}(1+q^{d+1})\
({\rm{mod}}\ \Phi_{n}(q)).$

Thus, the $q$-congruence (1.3) is concluded.

Similarly as before, by Lemma (2.2), we have

$\displaystyle[d]\sum_{k=1}^{n}q^{k}{\genfrac{[}{]}{0.0pt}{}{2k}{k+d}}\frac{(-q^{k+1};q)_{n-k}^{2}}{[k]}=$
$\displaystyle\sum_{k=d}^{n-1}q^{\binom{k+1}{2}-\binom{d}{2}}{\genfrac{[}{]}{0.0pt}{}{2n+1}{n-k}}+q^{\binom{n+1}{2}-\binom{d}{2}}$
$\displaystyle\equiv$ $\displaystyle
q^{\binom{n}{2}-\binom{d}{2}}+q^{\binom{n+1}{2}-\binom{d}{2}}$
$\displaystyle\equiv$ $\displaystyle\begin{cases}2q^{-\binom{d}{2}}\
({\rm{mod}}\ \Phi_{n}(q))&\text{if $n$ is odd},\\\ -2q^{-\binom{d}{2}}\
({\rm{mod}}\ \Phi_{n}(q))&\text{otherwise }.\end{cases}$

This completes the proof of Corollary (1.3). ∎

## References

  * [1] G.E. Andrews, On the $q$-analog of Kummer’s theorem and applications, Duke Math. J. 40 (1973), 525–528.
  * [2] G.E. Andrews, Applications of basic hypergeometric functions, SIAM Rev. 16 (1974), 441–484.
  * [3] G.E. Andrews, On the Greene-Krammer theorem and related identities, Ganita 43 (1992), 69–73.
  * [4] T.-X. Cai, A generalization of Lucas’ congruence for $q$-binomial coefficients, Publ. Math. 59 (2001), 51–56.
  * [5] R. D. Fray, Congruence properties of ordinary and $q$-binomial coefficients, Duke Math. J. 34 (1967), 467–480.
  * [6] J. Greene, On a conjecture of Krammer, J. Combin. Theory Ser.A 56 (1991), 309–311.
  * [7] V.J.W. Guo, Proof of a $q$-congruence conjectured by Tauraso, Int. J. Number Theory 15 (2019), 37–41.
  * [8] V.J.W. Guo, Proof of some $q$-supercongruences modulo the fourth power of a cyclotomic polynomial, Results Math. 75 (2020), Art. 77.
  * [9] V.J.W. Guo, Proof of a generalization of the (C.2) supercongruence of Van Hamme, Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat. 115 (2021), Art. 45.
  * [10] V.J.W. Guo, A further $q$-analogue of Van Hamme’s (H.2) supercongruence for primes $p\equiv 3\ ({\rm{mod}}\ 4)$, Int. J. Number Theory, in press; https://doi.org/10.1142/S1793042121500329
  * [11] V.J.W. Guo, Some variations of a ‘divergent’ Ramanujan-type $q$-supercongruence, J. Difference Equ. Appl., in press; https://doi.org/10.1080/10236198.2021.1900140
  * [12] V.J.W. Guo and J.-C. Liu, $q$-Analogues of two Ramanujan-type formulas for $1/\pi$, J. Difference Equ. Appl. 24 (2018), 1368–1373.
  * [13] V.J.W. Guo and M.J. Schlosser, Proof of a basic hypergeometric supercongruence modulo the fifith power of a cyclotomic polynomial, J. Difference Equ. Appl. 25 (2019), 921–929.
  * [14] V.J.W. Guo and M.J. Schlosser, A new family of $q$-supercongruences modulo the fourth power of a cyclotomic polynomial, Results Math. 75 (2020), Art. 155.
  * [15] V.J.W. Guo and J. Zeng, Some congruences involving central $q$-binomial coefficients , Adv. in Appl. Math. 45 (2010), 303–316.
  * [16] V.J.W. Guo and W. Zudilin, A $q$-microscope for supercongruences, Adv. Math. 346 (2019), 329–358.
  * [17] V.J.W. Guo and W. Zudilin, Dwork-type supercongruences through a creative $q$-microscope, J. Combin. Theory, Ser. A 178 (2021), Art. 105362.
  * [18] H. Hu and Z.-W. Sun, An extension of Lucas’ theorem , Proc. Amer. Math. Soc. 129 (2001), 3471–3478.
  * [19] L. Li, Some $q$-supercongruences for truncated forms of squares of basic hypergeometric series, J. Difference Equ. Appl. 27 (2021), 16–25.
  * [20] L. Li and S.-D. Wang, Proof of a $q$-supercongruence conjectured by Guo and Schlosser, Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat. 114 (2020), Art. 190.
  * [21] J.-C. Liu and F. Petrov, Congruences on sums of $q$-binomial coefficients, Adv. in Appl. Math. 116 (2020), Art. 102003.
  * [22] H.-X. Ni and H. Pan, Some symmertric $q$-congruences modulo the square of a cyclotomic polynomial, J. Math. Anal. Appl. 481 (2020), Art. 123372.
  * [23] H. Pan and Z.-W. Sun, A combinatorial identity with application to Catalan numbers , Discrete Math. 306 (2006), 1921–1940.
  * [24] Z.-W. Sun and R.Tauraso, New congruences for central binomial coefficients, Adv. in Appl. Math. 45 (2010), no.1, 125–148.
  * [25] Z.-W. Sun and R.Tauraso, On some new congruences for binomial coefficients, Int. J. Number Theory 7 (2011), no.3, 645–662.
  * [26] R. Tauraso, $q$-Analogs of some congruences involving Catalan numbers, Adv. in Appl. Math. 48 (2010), 603–614.
  * [27] X. Wang and M. Yue, Some $q$-supercongruences from Watson’s ${}_{8}\phi_{7}$ transformation formula, Results Math. 75 (2020), Art. 71.