# Gradient estimate for complex Monge-Ampere equation with continuous right
hand side

Xiuxiong Chen, Jingrui Cheng

###### Abstract.

In this note, we consider complex Monge-Ampere equation posed on a compact
Kähler manifold. We show how to get $L^{p}$($p<\infty$) and $L^{\infty}$
estimate for the gradient of the solution in terms of the continuity of the
right hand side.

##  1\. Introduction

The famous Calabi’s volume conjecture asks: For any compact Kähler manifold
$(M,[\omega_{0}])$ and for any smooth closed $(1,1)$ form $\alpha\in
c_{1}(M)$, can we find a Kähler form
$\omega_{\varphi}:=\omega_{0}+\sqrt{-1}\partial\bar{\partial}\varphi\in[\omega_{0}]$,
such that $Ric(\omega_{\varphi})=\alpha$?

It is not hard to see that this problem can be reduced to a problem of solving
Monge-Amp$\grave{\text{e}}$re equations: Given a function $F$, can we find a
Kähler form
$\omega_{\varphi}=\omega_{0}+\sqrt{-1}\partial\bar{\partial}\varphi\in[\omega_{0}]$
such that

(1.1) $\omega_{\varphi}^{n}=e^{F}\omega_{0}^{n}$

provide that

$\displaystyle\int_{M}\;e^{F}\omega_{0}^{n}=\displaystyle\int_{M}\;\omega_{0}^{n}.$

Yau first solved this equation for any $F\in C^{3}(M)$, in his celebrated
solution of Calabi conjecture[16]. The study of this equation (1.1) has been
very intense over the last few decades with various regularity estimates
obtained when the right hand side is less smooth. Here we list a few results
(by no means complete) to motivate our present study.

  1. (1)

The deep work of Kolodziej which states that if $e^{F}\in L^{p}(M)$ for some
$p>1$, then $\varphi\in C^{\alpha}$ for some $\alpha\in(0,1).$

  2. (2)

$C^{2,\alpha}$ estimate when the right hand side $F\in C^{\alpha}$. For real
Monge Amp$\grave{\text{e}}$re equation, this goes back to Caffarelli[2]. For
complex Monge-Amp$\grave{\text{e}}$re equation, following the theory of Evans-
Krylov(see [15] for details on extension to complex setting), we know
$[\varphi]_{C^{2,\alpha^{\prime}}(M,g)}$ is uniformly bounded, for each
$\alpha^{\prime}<\alpha$, if $\Delta\varphi$ is uniformly bounded.

  3. (3)

The important theorem of Chen-He [5] which proves that if $F\in W^{1,p}$ for
$p>2n$, then we have $n+\Delta\varphi<C.\;$

All the above results either estimate the $C^{0}$ bound(or continuity bound of
$\varphi$), or estimate the complex Hessian of $\varphi$. In this note, we
will obtain estimates which, in a sense, are in between of these results.
Namely we show that it is possible to obtain estimate for the gradient of
$\varphi$, when the right hand side $F$ is only assumed to be have certain
continuity conditions. Let $\omega(r)$ be the modulus of continuity of $F$.
That is, for $r>0$, define $\omega(r)=\sup_{x,y\in M,d(x,y)<r}|F(x)-F(y)|$.
Here $d(x,y)$ is the distance function in terms of the smooth background
metric $\omega_{0}$. The precise results we will obtain are the following
apriori estimates:

######  Theorem 1.1.

Let $(M,\omega_{0})$ be a compact Kähler manifold of complex dimension $n$.
Let $\varphi$ be a $\omega_{0}$-plurisubharmonic function solving
$(\omega_{0}+\sqrt{-1}\partial\bar{\partial}\varphi)^{n}=e^{F}\omega_{0}^{n}.$
Then we have:

  * •

(i)Assume $F$ is continuous, then for any $p<\infty$, we have
$||\nabla\varphi||_{L^{p}(\omega_{0}^{n})}\leq C.$ Here the constant $C$
depends on the background metric $\omega_{0}$, absolute bound of $F$, modulus
of continuity $\omega(r)$ of $F$, $p$ and $n$.

  * •

(ii)When $n=2$, the above constant $C_{1}$ does not depend on the continuity
of $F$.

  * •

(iii)If the modulus of continuity of $F$ satisfies
$\int_{0}^{1}\frac{\omega^{2}(r)}{r}dr<\infty$, then we have
$|\nabla\varphi|\leq C_{2}$ pointwise. Here the constant $C_{2}$ depends on
the background metric $\omega_{0}$, the absolute bound of $F$, an upper bound
for $\int_{0}^{1}\frac{\omega^{2}(r)}{r}dr$ and $n$.

As a direct consequence of the point (iii) above, we can conclude that:

######  Corollary 1.1.

If $F\in C^{\alpha}$ for some $\alpha>0$, then we have $|\nabla\varphi|\leq
C_{3}$ pointwise. Here the constant $C_{3}$ depends on the background metric
$\omega_{0}$, $C^{\alpha}$ norm of $F$ and $n$.

Indeed, this follows immediately from point (iii) if one takes
$\omega(r)=Ar^{\alpha}$.

######  Remark 1.2.

It is straighforward to generalize the results here to the case of Dirichlet
problem on bounded domains, provided that the domain and boundary data is
“reasonable” which allows one to get global $C^{0}$ estimate and boundary
gradient estimate.

There have been some previous works on the gradient estimate for complex
Monge-Amp$\grave{\text{e}}$re. In [1], the author showed how to get $C^{1}$
estimate directly from $C^{0}$ estimate, using a maximum principle argument,
but this estimate will require a Lipschitz bound on the right hand side. This
estimate has been improved in [5], in which the assumption is weakened to
$\nabla F\in L^{p}$ with $p>2n$ and the $C^{1,1}$ bound is obtained.

In this note, we obtain gradient estimates without assuming any bound on the
derivatives of the right hand side, instead, we only impose some continuity
assumptions. Next we briefly explain the ideas.

Let $\Delta_{\varphi}$ be the Laplace operator defined by the metric
$\omega_{0}+\sqrt{-1}\partial\bar{\partial}\varphi$. Roughly speaking, we will
calculate the term $\Delta_{\varphi}(|\nabla\varphi|^{2})$. In the outcome of
the calculation, the term involving the right hand side is given by
$2\nabla\varphi\cdot\nabla F$. Since we do not wish to assume any bound on
$\nabla F$, we will do integral estimates and integrating by parts in this
term. When $n=2$, this turns out to be sufficient to close the estimate and we
only use the absolute bound of $F$ and not its continuity. As a result, we get
the estimate said in point (ii) of Theorem 1.1. For general dimension, we need
to decompse $F$ as $F=F_{g}+F_{b}$(good plus bad), where $F_{g}$ has bounded
gradient and $F_{b}$ has small $C^{0}$ bound, which uses the continuity of
$F$. Then we need to integrate by parts in the term $\nabla\varphi\cdot\nabla
F_{b}$, and the smallness of $F_{b}$ in $C^{0}$ norm allows us to close the
estimate. These will be done in section 2. In order to get $L^{\infty}$
estimate, we need to use the Nash-Moser iteration, and we need to carefully
keep track of how $|\nabla F_{g}|$ grows when we take $F_{b}$ to have smaller
and smaller $C^{0}$ norm. This accounts for the integrability condition for
the modulus of continuity required in point (iii) of Theorem 1.1. We will
carry out the details in section 3.

This result is still very far from the optimal result one can hope for.
Indeed, from the deep work of Caffarelli [2], [3] on the Dirichlet problem for
real Monge-Amp$\grave{\text{e}}$re equations(or when the solution is strictly
convex), it looks quite reasonable to believe the same results would hold for
complex analogue as well. That is, it may be possible to get $C^{1,\alpha}$
estimate of $\varphi$ when $F$ is only bounded and also $C^{2,\alpha}$
estimate of $\varphi$ when $F\in C^{\alpha}$. This question is already raised
in [5]. There have been some partial results which shows the solution is
$C^{2,\alpha}$ by assuming the solution already has some absolute bound for
the complex Hessian, see the work of [6], [8] and [15]. The work [11] even
shows that it suffices to assume the solution is in $C^{1,\beta}$($\beta$
close to 1) in order for improving the regularity to $C^{2,\alpha}$. However,
the question of getting $C^{2,\alpha}$ estimate without any extra regularity
assumption on the solution remains open and seems out of reach for the moment.

######  Conjecture 1.1.

Do we have $C^{1,\alpha}$ estimate of $\varphi$ when $F$ is only bounded and
also $C^{2,\alpha}$ estimate of $\varphi$ when $F\in C^{\alpha}?$

We would like to emphasize that we are working in compact manifold without
boundary. On local $\mathbb{C}^{n}$, one can construct Pogorelov type
counterexamples. Indeed, one may consider:

$u(z_{1},z_{2},\cdots z_{n})=n^{2\over
n}(1+|z_{1}|^{2}+\cdots|z_{n-1}|^{2})|z_{n}|^{2\over n}$

It is shown in [10] that this function solves the complex Monge-
Amp$\grave{\text{e}}$re equation

(1.2) $\det u_{i\bar{j}}=1$

in the viscosity sense in $\mathbb{C}^{n}.\;$. This example shows that there
is no interior gradient bound at least for $n\geq 3$. Thus, for our conjecture
to be true, the global nature of compact Kähler manifold must come into play
in a crucial way.

##  2\. Estimate of the gradient in $L^{p}$

The goal of this section is to prove point (i) and (ii) in the Theorem 1.1.
Now we set up the notations. In the following $\Delta$ means the Laplace
operator defined by the background metric $\omega_{0}$, namely
$\Delta=g^{i\bar{j}}\partial_{i\bar{j}}$ and
$\omega_{0}=\sqrt{-1}g_{i\bar{j}}dz_{i}\wedge d\bar{z_{j}}$ in local
coordinates. $\Delta_{\varphi}$ means the Laplace operator defined by the
metric $\omega_{\varphi}:=\omega_{0}+\sqrt{-1}\partial\bar{\partial}\varphi$.
$|\nabla\cdot|$ means the gradient is taken under the metric $\omega_{0}$, and
$|\nabla_{\varphi}\cdot|^{2}_{\varphi}$ means the gradient is taken under
$\omega_{\varphi}$.

First we will do some calculations which will work in any dimension, then we
will see how the dimension 2 assumption will help us. Before we start the
calculation, we note that the $C^{0}$ bound of $\varphi$ can be estimated by
the $C^{0}$ bound of $F$, according to the result by Kolodziej(or one can use
Alexandrov maximum principle for an elementary proof.)

Let $H(\varphi)=-C\varphi+\delta\varphi^{2}$. First we compute

(2.1)
$\begin{split}\Delta_{\varphi}\big{(}&e^{H(\varphi)}(|\nabla\varphi|^{2}+K)\big{)}=\Delta_{\varphi}(e^{H(\varphi)})(|\nabla\varphi|^{2}+K)+e^{H(\varphi)}\Delta_{\varphi}(|\nabla\varphi|^{2})\\\
&+2e^{H(\varphi)}H^{\prime}(\varphi)\nabla_{\varphi}\varphi\cdot_{\varphi}\nabla_{\varphi}(|\nabla\varphi|^{2}).\end{split}$

To proceed further, we need a computation and estimate for
$\Delta_{\varphi}(|\nabla\varphi|^{2})$:

######  Lemma 2.1.

There exists a constant $C_{0}$, depending only on the bisectional curvature
of background metric $\omega_{0}$, such that under normal coordinates at a
point $p\in M$, we have

(2.2) $\Delta_{\varphi}(|\nabla\varphi|^{2})(p)\geq-
C_{0}tr_{\omega_{\varphi}}\omega_{0}|\nabla\varphi|^{2}+\frac{|\varphi_{i\alpha}|^{2}}{1+\varphi_{i\bar{i}}}+\frac{\varphi_{i\bar{i}}^{2}}{1+\varphi_{i\bar{i}}}+2\nabla\varphi\cdot\nabla
F|_{p}.$

###### Proof.

We recall that, for any point $p\in M$, we can choose coordinate in a
neighborhood of $p$ so that $g_{i\bar{j}}(p)=\delta_{ij}$,
$\partial_{z_{k}}g_{i\bar{j}}(p)=0$, $\varphi_{i\bar{j}}(p)$ is diagnal. Here
$g_{i\bar{j}}$ is the metric tensor for the Kähler metric $\omega_{0}$, that
is, $\omega_{0}=\sqrt{-1}g_{i\bar{j}}dz_{i}\wedge d\bar{z_{j}}$. Under this
coordinate, we compute:

$\begin{split}\Delta_{\varphi}&(|\nabla\varphi|^{2})=\frac{1}{1+\varphi_{i\bar{i}}}(g^{\alpha\bar{\beta}}\varphi_{\alpha}\varphi_{\bar{\beta}})_{i\bar{i}}=\frac{1}{1+\varphi_{i\bar{i}}}\big{(}(g^{\alpha\bar{\beta}})_{i\bar{i}}\varphi_{\alpha}\varphi_{\bar{\beta}}+g^{\alpha\bar{\beta}}(\varphi_{\alpha}\varphi_{\bar{\beta}})_{i\bar{i}}\\\
&+2Re\big{(}(g^{\alpha\bar{\beta}})_{i}(\varphi_{\alpha}\varphi_{\bar{\beta}})_{\bar{i}}\big{)}.\end{split}$

Using our assumption of the chosen coordinate, we see that

$(g^{\alpha\bar{\beta}})_{i\bar{i}}=R_{i\bar{i}\alpha\bar{\beta}},\,\,(g^{\alpha\bar{\beta}})_{i}=0.$

Hence, when evaluated at point $p$,

$\Delta_{\varphi}(|\nabla\varphi|^{2})=\frac{R_{i\bar{i}\alpha\bar{\beta}}\varphi_{\alpha}\varphi_{\bar{\beta}}}{1+\varphi_{i\bar{i}}}+\frac{g^{\alpha\bar{\beta}}(\varphi_{\alpha}\varphi_{\bar{\beta}})_{i\bar{i}}}{1+\varphi_{i\bar{i}}}.$

Moreover,

$\begin{split}\frac{1}{1+\varphi_{i\bar{i}}}&g^{\alpha\bar{\beta}}(\varphi_{\alpha}\varphi_{\bar{\beta}})_{i\bar{i}}=\frac{1}{1+\varphi_{i\bar{i}}}(\varphi_{\alpha}\varphi_{\bar{\alpha}})_{i\bar{i}}=\frac{|\varphi_{\alpha
i}|^{2}}{1+\varphi_{i\bar{i}}}+\frac{|\varphi_{\alpha\bar{i}}|^{2}}{1+\varphi_{i\bar{i}}}\\\
&+2Re\big{(}\frac{\varphi_{\alpha
i\bar{i}}\varphi_{\bar{\alpha}}}{1+\varphi_{i\bar{i}}}\big{)}.\end{split}$

Differentiating the equation in the coordinate $\alpha$ and evaluating at
point $p$, we find that

$\frac{\varphi_{i\bar{i}\alpha}}{1+\varphi_{i\bar{i}}}=F_{\alpha}.$

That is,

$2Re\big{(}\frac{\varphi_{\alpha
i\bar{i}}\varphi_{\bar{\alpha}}}{1+\varphi_{i\bar{i}}}\big{)}=2Re(F_{\alpha}\varphi_{\bar{\alpha}})=2\nabla\varphi\cdot\nabla
F.$

Now we can easily estimate:

$\begin{split}\Delta_{\varphi}(|\nabla\varphi|^{2})&=\frac{R_{i\bar{i}\alpha\bar{\beta}}\varphi_{\alpha}\varphi_{\bar{\beta}}}{1+\varphi_{i\bar{i}}}+\frac{|\varphi_{i\alpha}|^{2}}{1+\varphi_{i\bar{i}}}+\frac{\varphi_{i\bar{i}}^{2}}{1+\varphi_{i\bar{i}}}+2\nabla\varphi\cdot\nabla
F\\\ &\geq-
C_{0}tr_{\varphi}g|\nabla\varphi|^{2}+\frac{|\varphi_{i\alpha}|^{2}}{1+\varphi_{i\bar{i}}}+\frac{\varphi_{i\bar{i}}^{2}}{1+\varphi_{i\bar{i}}}+2\nabla\varphi\cdot\nabla
F.\end{split}$

Here the constant $C_{0}$ depends only on the curvatre bound of the background
metric. ∎

In the following, we will use $tr_{\varphi}g$ to denote
$tr_{\omega_{\varphi}}\omega_{0}$.

Now we continue the calculation:

(2.3)
$\Delta_{\varphi}(e^{H(\varphi)})=e^{H(\varphi)}H^{\prime}(\varphi)\Delta_{\varphi}\varphi+e^{H(\varphi)}(H^{\prime\prime}+(H^{\prime})^{2})|\nabla_{\varphi}\varphi|_{\varphi}^{2}.$

Also

(2.4)
$\begin{split}2\nabla_{\varphi}&\varphi\cdot\nabla_{\varphi}(|\nabla\varphi|^{2})=2Re\bigg{(}\frac{\varphi_{\alpha}\varphi_{\bar{\alpha}\bar{i}}\varphi_{\bar{i}}}{1+\varphi_{i\bar{i}}}\bigg{)}+2Re\bigg{(}\frac{|\varphi_{i}|^{2}\varphi_{i\bar{i}}}{1+\varphi_{i\bar{i}}}\bigg{)}\\\
&=2Re\bigg{(}\frac{\varphi_{\alpha}\varphi_{\bar{\alpha}\bar{i}}\varphi_{\bar{i}}}{1+\varphi_{i\bar{i}}}\bigg{)}+2|\nabla\varphi|^{2}-2|\nabla_{\varphi}\varphi|_{\varphi}^{2}.\end{split}$

After the completion of square:

$(H^{\prime})^{2}|\nabla_{\varphi}\varphi|_{\varphi}^{2}|\nabla\varphi|^{2}+\frac{|\varphi_{i\alpha}|^{2}}{1+\varphi_{i\bar{i}}}+2H^{\prime}Re\bigg{(}\frac{\varphi_{\alpha}\varphi_{\bar{\alpha}\bar{i}}\varphi_{\bar{i}}}{1+\varphi_{i\bar{i}}}\bigg{)}=\frac{1}{1+\varphi_{i\bar{i}}}|H^{\prime}\varphi_{i}\varphi_{\alpha}+\varphi_{i\alpha}|^{2}.$

Dropping this square, and denote $u=e^{H}(|\nabla\varphi|^{2}+K)$, we use
(2.1), (2.2), (2.3) and (2.4) to see that:

$\begin{split}\Delta_{\varphi}&u\geq
e^{H}H^{\prime}\Delta_{\varphi}\varphi(|\nabla\varphi|^{2}+K)+e^{H}(H^{\prime})^{2}K|\nabla_{\varphi}\varphi|_{\varphi}^{2}+e^{H}H^{\prime\prime}|\nabla_{\varphi}\varphi|_{\varphi}^{2}|\nabla\varphi|^{2}\\\
&-C_{0}e^{H}tr_{\varphi}g|\nabla\varphi|^{2}+e^{H}\frac{\varphi_{i\bar{i}}^{2}}{1+\varphi_{i\bar{i}}}+2e^{H}\nabla\varphi\cdot\nabla
F+2e^{H}H^{\prime}(|\nabla\varphi|^{2}-|\nabla_{\varphi}\varphi|_{\varphi}^{2}).\end{split}$

Now note that if we choose $2\delta||\varphi||_{0}=\frac{C}{2}$, we conclude

$H^{\prime}(\varphi)=-C+2\delta\varphi\leq-\frac{C}{2}.$

Hence

$\begin{split}e^{H}&H^{\prime}\Delta_{\varphi}\varphi(|\nabla\varphi|^{2}+K)-C_{0}e^{H}tr_{\varphi}g|\nabla\varphi|^{2}\geq(\frac{C}{2}-C_{0})e^{H}tr_{\varphi}g|\nabla\varphi|^{2}\\\
&-1.5Cne^{H}(|\nabla\varphi|^{2}+K)+\frac{CK}{2}e^{H}tr_{\varphi}g\\\
&\geq\frac{C}{4}tr_{\varphi}gu-1.5Cnu.\end{split}$

Also if $K>2n$,

$e^{H}\frac{\varphi_{i\bar{i}}^{2}}{1+\varphi_{i\bar{i}}}=e^{H}(n+\Delta\varphi)-2ne^{H}\geq
e^{H}(n+\Delta\varphi)-u.$

Finally

$2e^{H}H^{\prime}|\nabla\varphi|^{2}\geq-3Ce^{H}|\nabla\varphi|^{2}\geq-3Cu.$

So we obtain

(2.5) $\begin{split}\Delta_{\varphi}&u\geq\frac{C}{4}tr_{\varphi}gu+2\delta
e^{H}|\nabla_{\varphi}\varphi|_{\varphi}^{2}|\nabla\varphi|^{2}+e^{H}(n+\Delta\varphi)+2e^{H}\nabla\varphi\cdot\nabla
F\\\
&-(1+4.5C)u+e^{H}\frac{C^{2}K}{4}|\nabla_{\varphi}\varphi|_{\varphi}^{2}.\end{split}$

Now let $q\geq 1$, we can compute

(2.6)
$\begin{split}&\int_{M}(q-1)u^{q-2}|\nabla_{\varphi}u|_{\varphi}^{2}dvol_{\varphi}=\int_{M}u^{q-1}(-\Delta_{\varphi}u)dvol_{\varphi}\leq-\int_{M}\frac{C}{4}u^{q}tr_{\varphi}gdvol_{\varphi}\\\
&-\int_{M}u^{q-1}2\delta
e^{H}|\nabla_{\varphi}\varphi|_{\varphi}^{2}|\nabla\varphi|^{2}dvol_{\varphi}-\int_{M}u^{q-1}e^{H}(n+\Delta\varphi)dvol_{\varphi}\\\
&-\int_{M}2u^{q-1}e^{H}\nabla\varphi\cdot\nabla
Fdvol_{\varphi}+\int_{M}(1+4.5C)u^{q}dvol_{\varphi}\\\
&-\int_{M}u^{q-1}e^{H}\frac{C^{2}K}{4}|\nabla_{\varphi}\varphi|_{\varphi}^{2}dvol_{\varphi}.\end{split}$

The point of this whole calculation is to handle the term
$\nabla\varphi\cdot\nabla F$ above. We will do this next.

###  2.1. When $n=2$

In this case, the situation is much better and we have the following estimate:

######  Proposition 2.2.

When $n=2$, there exists a constant $\varepsilon_{0}>0$, $C_{2.1}>0$,
depending only on the absolute bound of $F$ and the bisectional curvature
bound of background metric $\omega_{0}$, such that

(2.7)
$\int_{M}\exp\big{(}\varepsilon_{0}|\nabla\varphi|^{2}\big{)}\omega_{0}^{n}\leq
C_{2.1}.$

We simply integrate by parts for the term involving $\nabla\varphi\cdot\nabla
F$:

(2.8) $\begin{split}-&\int_{M}2u^{q-1}e^{H}\nabla\varphi\cdot\nabla
Fdvol_{\varphi}=-\int_{M}2u^{q-1}e^{H}\nabla\varphi\cdot\nabla(e^{F})dvol_{g}\\\
&=\int_{M}2(q-1)u^{q-2}e^{H}\nabla u\cdot\nabla\varphi
e^{F}dvol_{g}+\int_{M}2u^{q-1}e^{H}H^{\prime}|\nabla\varphi|^{2}e^{F}dvol_{g}\\\
&+\int_{M}2u^{q-1}e^{H}\Delta\varphi e^{F}dvol_{g}.\end{split}$

We handle the terms on the right hand side above one by one.

$\begin{split}&\int_{M}2(q-1)u^{q-2}e^{H}\nabla u\cdot\nabla\varphi
e^{F}dvol_{g}\leq\int_{M}2(q-1)u^{q-\frac{3}{2}}e^{\frac{1}{2}H}|\nabla
u|dvol_{\varphi}\\\
&\leq\int_{M}2(q-1)u^{q-\frac{3}{2}}e^{\frac{1}{2}H}|\nabla_{\varphi}u|_{\varphi}(n+\Delta\varphi)^{\frac{1}{2}}dvol_{\varphi}\leq\int_{M}\frac{q-1}{2}u^{q-2}|\nabla_{\varphi}u|_{\varphi}^{2}dvol_{\varphi}\\\
&+\int_{M}2(q-1)u^{q-1}e^{H}(n+\Delta\varphi)dvol_{\varphi}\end{split}$

In the first inequality above, we used $e^{\frac{1}{2}H}|\nabla\varphi|\leq
u^{\frac{1}{2}}$, just from the definition of $u$.

For the second term of (2.8), we have

$\int_{M}2u^{q-1}e^{H}H^{\prime}|\nabla\varphi|^{2}e^{F}dvol_{g}\leq\int_{M}2u^{q}\frac{3C}{2}dvol_{\varphi}.$

This is because $|H^{\prime}|\leq\frac{3C}{2}$, and
$e^{H}|\nabla\varphi|^{2}\leq u$.

Finally, for the last term in (2.8), we find,

$\int_{M}2u^{q-1}e^{H}\Delta\varphi
e^{F}dol_{g}\leq\int_{M}2u^{q-1}e^{H}(n+\Delta\varphi)dvol_{\varphi}.$

In conclusion, we have

(2.9) $\begin{split}-\int_{M}&2u^{q-1}e^{H}\nabla\varphi\cdot\nabla
Fdvol_{\varphi}\leq\int_{M}\frac{q-1}{2}u^{q-2}|\nabla_{\varphi}u|_{\varphi}^{2}dvol_{\varphi}\\\
&+\int_{M}2qu^{q-1}e^{H}(n+\Delta\varphi)dvol_{\varphi}+\int_{M}3Cu^{q}dvol_{\varphi}\end{split}$

Plug this back to (2.6), we find

(2.10)
$\begin{split}&\int_{M}\frac{q-1}{2}u^{q-2}|\nabla_{\varphi}u|_{\varphi}^{2}dvol_{\varphi}\leq-\int_{M}\frac{C}{4}tr_{\varphi}gu^{q}dvol_{\varphi}-\int_{M}2\delta
u^{q-1}e^{H}|\nabla_{\varphi}\varphi|_{\varphi}^{2}|\nabla\varphi|^{2}dvol_{\varphi}\\\
&+\int_{M}(2q-2)u^{q-1}e^{H}(n+\Delta\varphi)dvol_{\varphi}+\int_{M}(1+7.5C)u^{q}dvol_{\varphi}\end{split}$

Now it is time to use the assumption $n=2$. Then we can write
$tr_{\varphi}g=e^{-F}(n+\Delta\varphi)$. Then we can estimate

$\begin{split}&\int_{M}(2q-2)u^{q-1}e^{H}(n+\Delta\varphi)dvol_{\varphi}=\int_{M}(2q-2)e^{qH}(|\nabla\varphi|^{2}+K)^{q-1}(n+\Delta\varphi)dvol_{\varphi}\\\
&\leq\int_{M}e^{qH}(|\nabla\varphi|^{2}+K)^{q}(n+\Delta\varphi)dvol_{\varphi}+\int_{M}(2q-2)^{q}e^{qH}(n+\Delta\varphi)dvol_{\varphi}.\end{split}$

On the other hand,

$\int_{M}\frac{C}{4}tr_{\varphi}gu^{q}dvol_{\varphi}=\int_{M}\frac{C}{4}e^{-F}e^{qH}(|\nabla\varphi|^{2}+K)^{q}(n+\Delta\varphi)dvol_{\varphi}.$

Therefore, if we choose $C$ sufficiently large so that $\frac{C}{4}e^{-F}>2$,
we then have

$\begin{split}&\int_{M}\frac{q-1}{2}u^{q-2}|\nabla_{\varphi}u|_{\varphi}^{2}dvol_{\varphi}\leq-\int_{M}e^{qH}(n+\Delta\varphi)(|\nabla\varphi|^{2}+K)^{q}dvol_{\varphi}\\\
&-\int_{M}2\delta
u^{q-1}e^{H}|\nabla_{\varphi}\varphi|_{\varphi}^{2}|\nabla\varphi|^{2}dvol_{\varphi}+\int_{M}(1+7.5C)u^{q}dvol_{\varphi}\\\
&+\int_{M}(2q)^{q}e^{qH}(n+\Delta\varphi)dvol_{\varphi}.\end{split}$

Now we have

$\begin{split}-\int_{M}&e^{qH}\frac{1}{2}(n+\Delta\varphi)(|\nabla\varphi|^{2}+K)^{q}dvol_{\varphi}-\int_{M}2\delta
e^{qH}|\nabla\varphi|^{2q}|\nabla_{\varphi}\varphi|_{\varphi}^{2}dvol_{\varphi}\\\
&\leq-\int_{M}\delta
e^{qH}|\nabla\varphi|^{2q+1}dvol_{\varphi}-\int_{M}e^{qH}e^{\frac{F}{n}}(|\nabla\varphi|^{2}+K)^{q}dvol_{\varphi}\\\
&\leq-\int_{M}c_{1}e^{qH}(|\nabla\varphi|^{2}+K)^{q+\frac{1}{2}}dvol_{\varphi}.\end{split}$

Here $c_{1}$ is a constant depending only on the $C^{0}$ bound of $F$ and
$\varphi$, but does not depend on $q$. Hence

$\begin{split}\int_{M}&\frac{q-1}{2}u^{q-2}|\nabla_{\varphi}u|_{\varphi}^{2}dvol_{\varphi}+\int_{M}c_{1}e^{qH}(|\nabla\varphi|^{2}+K)^{q+\frac{1}{2}}dvol_{\varphi}\\\
&+\int_{M}\frac{e^{qH}}{2}(n+\Delta\varphi)(|\nabla\varphi|^{2}+K)^{q}dvol_{\varphi}\leq\int_{M}(1+7.5C)u^{q}dvol_{\varphi}\\\
&+\int_{M}(2q)^{q}e^{qH}(n+\Delta\varphi)dvol_{\varphi}.\end{split}$

For the last term above, we have

$\int_{M}(2q)^{q}e^{qH}(n+\Delta\varphi)dvol_{\varphi}\leq
e^{q||H||_{0}}e^{||F||_{0}}(2q)^{q}Vol(M).$

Also we can estimate

$\begin{split}\int_{M}&(1+7.5C)u^{q}dvol_{\varphi}=\int_{M}(1+7.5C)e^{qH}(|\nabla\varphi|^{2}+K)^{q}dvol_{\varphi}\\\
&\leq\int_{M}\frac{c_{1}}{2}e^{qH}(|\nabla\varphi|^{2}+K)^{q+\frac{1}{2}}dvol_{\varphi}+(1+7.5C)^{2q+1}\big{(}\frac{2}{c_{1}}\big{)}^{2q}Vol(M).\end{split}$

Hence

$\begin{split}\int_{M}&\frac{c_{1}}{2}e^{qH}(|\nabla\varphi|^{2}+K)^{q+\frac{1}{2}}dvol_{\varphi}\leq
e^{q||H||_{0}}e^{||F||_{0}}(2q)^{q}Vol(M)\\\
&+(1+7.5C)^{2q+1}\big{(}\frac{2}{c_{1}}\big{)}^{2q}Vol(M).\end{split}$

Now if we put $k=q+\frac{1}{2}$, with $k$ an integer $\geq 2$, we then find
that for some $L>1$, $c_{2}>0$ sufficiently large, we have

$\int_{M}(|\nabla\varphi|^{2}+K)^{k}dvol_{\varphi}\leq c_{2}L^{k}k^{k}.$

Now recall the Sterling formula $k^{k}\approx k!e^{k}\sqrt{2\pi k}$, we see
that we have an integral bound for
$\int_{M}\exp(\varepsilon(|\nabla\varphi|^{2}+K))dvol_{\varphi}$ for some
small $\varepsilon>0$.

###  2.2. General dimension

If the dimension is higher than 2, then we can no longer control the term
involving $\Delta\varphi$ in terms of $tr_{\varphi}g$ and we will need to use
the continuity of $F$. The idea is to decompose $F$ into two parts, namely
$F=F_{g}+F_{b}$(“good” plus “bad”). Here $\nabla F_{g}$ is bounded and $F_{b}$
has small $C^{0}$ bound. For later use, we need a lemma which shows
quantitatively how the gradient bound for $F_{g}$ and smallness of $F_{b}$
relate to the continuity of $F$.

######  Definition 2.3.

Let $\omega:[0,1)\rightarrow\mathbb{R}^{+}$ be a continuous and strictly
increasing function with $\omega(0)=0$, such that
$|F(x)-F(y)|\leq\omega(d_{\omega_{0}}(x,y))$, for any $x,\,y\in M$ and
$d_{\omega_{0}}(x,y)\leq 1$, then we say $\omega(r)$ is a modulus of
continuity for the function $F$. Here $d_{\omega_{0}}(x,y)$ is the distance
function defined by the metric $\omega_{0}$.

Now we formulate the approximation lemma in terms of the modulus of continuity
defined above.

######  Lemma 2.4.

Let $F$ be a continuous function on $M$ and let $\omega(r)$ be a modulus of
continuity for $F$. Then there exists a constant $C_{2.2}>0$, depending only
on the background metric $\omega_{0}$ , the manifold $M$, and absolute bound
for $F$, such that for any $0<r<1$, there exists a function $F_{r}$ which
satisfies the estimates:

$|\nabla F_{r}|\leq\frac{C_{2.2}}{r},\,\,|F-F_{r}|\leq C_{2.2}(\omega(r)+r).$

###### Proof.

The proof of this lemma should be quite standard, but we were unable to find
an exact reference. First we can cover the manifold by finitely many
coordinate balls $\\{B(p_{i},2)\\}_{i=1}^{N}$, such that
$\cup_{i=1}^{N}B(p_{i},1)$ covers $M$. Let $\\{\zeta_{i}\\}_{i=1}^{N}$ be a
partition of unity for the open cover $B(p_{i},1)$, namely $\zeta_{i}$ has
compact support in $B(p_{i},1)$, nonnegative, and $\sum_{i=1}^{N}\zeta_{i}=1$.
For each $i$, $F\zeta_{i}$ has compact support in $B(p_{i},1)$ and for $0<r<1$
we define:

$F_{i,r}=(F\zeta_{i})*\rho_{r},\,\,\,F_{r}=\sum_{i=1}^{N}F_{i,r}.$

Here $\rho_{r}=\rho(\frac{\cdot}{r})$ and $\rho$ is the standard smoothing
kernel supported in the unit ball of $\mathbb{C}^{n}$. Note that each
$F_{i,r}$ has compact support in $B(p_{i},2)$, hence can be extended smoothly
by 0 to $M$. From the standard properties of convolution, we know that for
each $i$,

$|\nabla
F_{i,r}|\leq||F\zeta_{i}||_{0}\frac{1}{r}\leq\frac{||F||_{0}}{r},\,\,|F_{i,r}-F\zeta_{i}|\leq\omega_{F\zeta_{i}}(r).$

Here $\omega_{F\zeta_{i}}(r)$ is a modulus of continuity for the function
$F\zeta_{i}$ and it is clear that

$\omega_{F\zeta_{i}}(r)\leq\omega(r)+C_{2.15}||F||_{0}r.$

Here the constant $C_{2.15}$ is related to the gradient bound for the cut-off
function $\zeta_{i}$. Therefore, we have that:

$\begin{split}&|\nabla F|\leq\sum_{i=1}^{N}|\nabla
F_{i,r}|\leq\frac{N||F||_{0}}{r},\\\
&|F-F_{r}|\leq\sum_{i=1}^{N}|F\zeta_{i}-F_{i,r}|\leq\sum_{i=1}^{N}\omega_{F\zeta_{i}}(r)\leq
N\omega(r)+NC_{2.15}||F||_{0}r.\end{split}$

∎

Now we go back to (2.6), and we need to use the continuity of $F$ to handle
the term involving $\nabla F$. Let $0<r<1$, and $F_{r}$ be the function given
by the above lemma, then

(2.11) $\begin{split}&\int_{M}-2u^{q-1}e^{H}\nabla\varphi\cdot\nabla
Fdvol_{\varphi}=-\int_{M}2u^{q-1}e^{H}\nabla\varphi\cdot\nabla(e^{F}-e^{F_{r}})dvol_{g}\\\
&-\int_{M}2u^{q-1}e^{H}\nabla\varphi\cdot\nabla(e^{F_{r}})dvol_{g}.\end{split}$

The handling of the second term is easy. Indeed, choosing $0<r<1$, we then
have

(2.12)
$\begin{split}-&\int_{M}2u^{q-1}e^{H}\nabla\varphi\cdot\nabla(e^{F_{r}})dvol_{g}=-\int_{M}2u^{q-1}e^{H}\nabla\varphi\cdot\nabla
F_{r}e^{F_{r}-F}dvol_{\varphi}\\\
&\leq\int_{M}2u^{q-1}e^{H}|\nabla\varphi|\frac{c_{3}}{r}e^{\omega(r)}dvol_{\varphi}\leq\frac{2c_{3}e}{r}\int_{M}e^{\frac{1}{2}H}u^{q-\frac{1}{2}}dvol_{\varphi}.\end{split}$

Here and in the following, the constants $c_{i}$ will have the dependence as
described in point (ii) of the Theorem 1.1 and not on $q$. For the first term
in (2.11), we integrate by parts in the same way as before.

(2.13)
$\begin{split}-&\int_{M}2u^{q-1}e^{H}\nabla\varphi\cdot\nabla(e^{F}-e^{F_{r}})dvol_{g}=\int_{M}2u^{q-1}e^{H}\Delta\varphi(e^{F}-e^{F_{r}})dvol_{g}\\\
&+\int_{M}2(q-1)u^{q-2}e^{H}\nabla
u\cdot\nabla\varphi(e^{F}-e^{F_{r}})dvol_{g}+\int_{M}2u^{q-1}e^{H}H^{\prime}|\nabla\varphi|^{2}(e^{F}-e^{F_{r}})dvol_{g}.\end{split}$

There is no loss of generality to assume that $\omega(r)\geq r$. Hence for the
first term on the right hand side of (2.13),

(2.14)
$\begin{split}-&\int_{M}2u^{q-1}e^{H}\Delta\varphi(e^{F}-e^{F_{r}})dvol_{g}=-\int_{M}2u^{q-1}e^{H}\Delta\varphi(1-e^{F_{r}-F})dvol_{\varphi}\\\
&\leq\int_{M}2u^{q-1}e^{H}c_{4}\omega(r)(n+\Delta\varphi)dvol_{\varphi}+\int_{M}2nc_{5}\omega(r)u^{q-1}e^{H}dvol_{\varphi}\end{split}$

Now for the second term of (2.13), we have

(2.15) $\begin{split}&\int_{M}2(q-1)u^{q-2}e^{H}\nabla
u\cdot\nabla\varphi(e^{F}-e^{F_{r}})dvol_{g}=\int_{M}2(q-1)u^{q-2}e^{H}\nabla
u\cdot\nabla\varphi(1-e^{F_{r}-F})dvol_{\varphi}\\\
&\leq\int_{M}2(q-1)u^{q-2}|\nabla
u|e^{H}|\nabla\varphi|c_{4}\omega(r)dvol_{\varphi}\leq\int_{M}2(q-1)u^{q-\frac{3}{2}}|\nabla
u|e^{\frac{1}{2}H}c_{4}\omega(r)dvol_{\varphi}\\\
&\leq\int_{M}2(q-1)u^{q-\frac{3}{2}}|\nabla_{\varphi}u|_{\varphi}(n+\Delta\varphi)^{\frac{1}{2}}e^{\frac{1}{2}H}c_{4}\omega(r)dvol_{\varphi}\leq\int_{M}\frac{q-1}{2}u^{q-2}|\nabla_{\varphi}u|_{\varphi}^{2}dvol_{\varphi}\\\
&+\int_{M}2(q-1)u^{q-1}e^{H}(n+\Delta\varphi)c_{4}^{2}\omega^{2}(r)dvol_{\varphi}.\end{split}$

For the last term in (2.13), we see that

(2.16)
$\int_{M}2u^{q-1}e^{H}H^{\prime}|\nabla\varphi|^{2}(e^{F}-e^{F_{r}})dvol_{g}\leq\int_{M}2u^{q}\frac{3C}{2}c_{5}\omega(r)dvol_{\varphi}.$

Combining above calculations (2.14)-(2.16), we obtain from (2.13)

(2.17)
$\begin{split}-&\int_{M}2u^{q-1}e^{H}\nabla\varphi\cdot\nabla(e^{F}-e^{F_{r}})dvol_{g}\leq\int_{M}\frac{q-1}{2}u^{q-2}|\nabla_{\varphi}u|_{\varphi}^{2}dvol_{\varphi}\\\
&+\int_{M}u^{q-1}e^{H}(n+\Delta\varphi)(2c_{4}\omega(r)+2(q-1)c_{4}^{2}\omega^{2}(r))dvol_{\varphi}\\\
&+\int_{M}3C\omega(r)u^{q}dvol_{\varphi}+\int_{M}2nc_{5}\omega(r)u^{q-1}e^{H}dvol_{\varphi}.\end{split}$

Combining this with (2.11) and (2.12), we obtain that for any $0<r<1$,

(2.18) $\begin{split}&-\int_{M}2u^{q-1}e^{H}\nabla\varphi\cdot\nabla
Fdvol_{\varphi}\leq\int_{M}\frac{q-1}{2}u^{q-2}|\nabla_{\varphi}u|_{\varphi}^{2}dvol_{\varphi}\\\
&+\int_{M}u^{q-1}e^{H}(n+\Delta\varphi)\big{(}2c_{4}\omega(r)+2(q-1)c_{4}^{2}\omega^{2}(r)\big{)}dvol_{\varphi}+\int_{M}3C\omega(r)u^{q}dvol_{\varphi}\\\
&+\int_{M}2nc_{5}\omega(r)u^{q-1}e^{H}dvol_{\varphi}+\frac{2c_{3}e}{r}\int_{M}e^{\frac{1}{2}H}u^{q-\frac{1}{2}}dvol_{\varphi}.\end{split}$

Plug (2.18) back to (2.6), we obtain that

(2.19)
$\begin{split}&\int_{M}\frac{q-1}{2}u^{q-2}|\nabla_{\varphi}u|_{\varphi}^{2}dvol_{\varphi}+\int_{M}\frac{C}{4}u^{q}tr_{\varphi}gdvol_{\varphi}+\int_{M}u^{q-1}2\delta
e^{H}|\nabla_{\varphi}\varphi|_{\varphi}^{2}|\nabla\varphi|^{2}dvol_{\varphi}\\\
&+\int_{M}u^{q-1}e^{H}(n+\Delta\varphi)\big{(}1-2c_{4}\omega(r)-2(q-1)c_{4}^{2}\omega^{2}(r)\big{)}dvol_{\varphi}\\\
&+\int_{M}u^{q-1}e^{H}\frac{C^{2}K}{4}|\nabla_{\varphi}\varphi|_{\varphi}^{2}dvol_{\varphi}\leq\int_{M}(1+7.5C)u^{q}dvol_{\varphi}+\int_{M}2nc_{5}\omega(r)u^{q-1}e^{H}dvol_{\varphi}\\\
&+\frac{2c_{3}e}{r}\int_{M}e^{\frac{1}{2}H}u^{q-\frac{1}{2}}dvol_{\varphi}.\end{split}$

Now we observe that in the above,

(2.20) $\begin{split}\int_{M}&u^{q-1}2\delta
e^{H}|\nabla_{\varphi}\varphi|_{\varphi}^{2}|\nabla\varphi|^{2}dvol_{\varphi}+\int_{M}u^{q-1}e^{H}\frac{C^{2}K}{4}|\nabla_{\varphi}\varphi|_{\varphi}^{2}dvol_{\varphi}\\\
&\geq\int_{M}c_{6}u^{q}|\nabla_{\varphi}\varphi|_{\varphi}^{2}dvol_{\varphi}.\end{split}$

Also

$tr_{\varphi}g+|\nabla_{\varphi}\varphi|_{\varphi}^{2}\geq
e^{-\frac{F}{n-1}}(n+\Delta\varphi)^{\frac{1}{n-1}}+|\nabla_{\varphi}\varphi|_{\varphi}^{2}\geq\frac{1}{2}e^{-\frac{F}{n}}(|\nabla\varphi|^{\frac{2}{n}}+1).$

Combining the right hand side of (2.20) with the term in (2.19) involving
$tr_{\varphi}g$, we can conclude that

(2.21)
$\begin{split}\int_{M}&\frac{q-1}{2}u^{q-2}|\nabla_{\varphi}u|_{\varphi}^{2}dvol_{\varphi}+\int_{M}c_{7}u^{q+\frac{1}{n}}dvol_{\varphi}+\int_{M}u^{q-1}e^{H}(n+\Delta\varphi)A_{q}(r)dvol_{\varphi}\\\
&\leq\int_{M}(1+7.5C)u^{q}dvol_{\varphi}+\int_{M}2nc_{5}\omega(r)u^{q-1}e^{H}dvol_{\varphi}+\frac{2c_{3}e}{r}\int_{M}e^{\frac{1}{2}H}u^{q-\frac{1}{2}}dvol_{\varphi}.\end{split}$

In the above

$A_{q}(r)=1-2c_{4}\omega(r)-2(q-1)c_{4}^{2}\omega^{2}(r).$

For any fixed $q>1$, we can choose $r$ sufficiently small, so that
$A_{q}(r)>0$. Then we conclude from (2.21) that $|\nabla\varphi|^{2}\in
L^{q+\frac{1}{n}}$.

##  3\. Estimate of the gradient in $L^{\infty}$

In order to get a bound for $\nabla\varphi$ in $L^{\infty}$, we need to
estimate more carefully how the right hand side of (2.21) grows as $q$
increases. By doing approximation, we can assume without loss of generality
that $\omega:\mathbb{R}_{+}\rightarrow\mathbb{R}_{+}$ satisfies $\omega(0)=0$,
$\omega(r)\in C^{1}((0,1])$ and $\omega^{\prime}(r)>0$. Define for any $q\geq
9$,

(3.1) $r_{q}=\sup\\{0<r\leq 1:c_{4}\omega(r)\leq\frac{1}{\sqrt{8(q-1)}}\\}.$

It is easy to verify that for any $r\leq r_{q}$, we have
$2c_{4}\omega(r)\leq\frac{1}{4}$,
$2(q-1)c_{4}^{2}\omega^{2}(r)\leq\frac{1}{4}$. Hence
$A_{q}(r)\geq\frac{1}{2}$. Choosing $r=r_{q}$ in (2.21), we find that

(3.2)
$\begin{split}&\int_{M}\frac{q-1}{2}u^{q-2}|\nabla_{\varphi}u|_{\varphi}^{2}dvol_{\varphi}+\int_{M}c_{7}u^{q+\frac{1}{n}}dvol_{\varphi}+\int_{M}\frac{1}{2}u^{q-1}e^{H}(n+\Delta\varphi)dvol_{\varphi}\\\
&\leq\int_{M}(1+7.5C)u^{q}dvol_{\varphi}+\int_{M}\frac{2nc_{5}u^{q-1}}{c_{4}\sqrt{8(q-1)}}e^{H}dvol_{\varphi}+\frac{2c_{3}e}{r_{q}}\int_{M}e^{\frac{1}{2}H}u^{q-\frac{1}{2}}dvol_{\varphi}\\\
&\leq\int_{M}(1+7.5C)u^{q}dvol_{\varphi}+\int_{M}\frac{c_{8}}{r_{q}}e^{\frac{1}{2}H}u^{q-\frac{1}{2}}dvol_{\varphi}.\end{split}$

Again $c_{8}$ depends on $C^{0}$ bound of $F$ and $\varphi$ but not on $q$.

On the other hand, if we look at the left hand side of (3.2), we see

(3.3)
$\begin{split}\int_{M}&\frac{q-1}{2}u^{q-2}|\nabla_{\varphi}u|_{\varphi}^{2}dvol_{\varphi}+\int_{M}\frac{1}{2}u^{q-1}e^{H}(n+\Delta\varphi)dvol_{\varphi}\\\
&\geq\int_{M}\sqrt{q-1}u^{q-\frac{3}{2}}e^{\frac{1}{2}H}|\nabla
u|dvol_{\varphi}\geq c_{8}\int_{M}\sqrt{q-1}u^{q-\frac{3}{2}}|\nabla
u|dvol_{g}.\end{split}$

Also we have

(3.4)
$\int_{M}(1+7.5C)u^{q}dvol_{\varphi}\leq\int_{M}\frac{c_{7}}{2}u^{q+\frac{1}{n}}dvol_{\varphi}+\int_{M}c_{9}u^{q-\frac{1}{2}}dvol_{\varphi}.$

Therefore

(3.5) $c_{8}\int_{M}\sqrt{q-1}u^{q-\frac{3}{2}}|\nabla
u|dvol_{g}\leq\int_{M}\big{(}c_{9}+\frac{c_{8}}{r_{q}}\big{)}dvol_{\varphi}.$

Hence, using $r_{q}\leq 1$, we conclude by combining (3.2)-(3.5):

(3.6) $\int_{M}\sqrt{q-1}u^{q-\frac{3}{2}}|\nabla
u|dvol_{g}\leq\int_{M}\frac{c_{10}}{r_{q}}u^{q-\frac{1}{2}}dvol_{g}.$

The constants $c_{7}$, $c_{8}$, $c_{9}$ and $c_{10}$ above depnd only on the
$C^{0}$ bound of $F$ and $\varphi$, but not on $q$. Putting
$\chi=\frac{2n}{2n-1}$, we observe that by Sobolev imbedding, we have

$\bigg{(}\int_{M}u^{(q-\frac{1}{2})\chi}dvol_{g}\bigg{)}^{\frac{1}{\chi}}\leq
c_{11}\bigg{(}\int_{M}|\nabla(u^{q-\frac{1}{2}})|dvol_{g}+\int_{M}u^{q-\frac{1}{2}}dvol_{g}\bigg{)}.$

Hence we conclude from (3.6) that for $q\geq 9$,

$\begin{split}\bigg{(}&\int_{M}u^{(q-\frac{1}{2})\chi}dvol_{g}\bigg{)}^{\frac{1}{\chi}}\leq\big{(}\frac{c_{11}c_{10}\sqrt{q-\frac{1}{2}}}{r_{q}}+c_{11}\big{)}\int_{M}u^{q-\frac{1}{2}}dvol_{g}\\\
&\leq\frac{c_{12}\sqrt{q}}{r_{q}}\int_{M}u^{q-\frac{1}{2}}dvol_{g}.\end{split}$

Put $q-\frac{1}{2}=\chi^{k}$, we see that for $k\geq k_{0}$ (here $k_{0}$ is
the smallest integer for which $\chi^{k_{0}}-\frac{1}{2}\geq 9$),

$||u||_{L^{\chi^{k+1}}}\leq
c_{12}^{\frac{1}{\chi^{k}}}(2\chi^{k})^{\frac{1}{\chi^{k}}}(r_{\chi^{k}+\frac{1}{2}})^{\frac{1}{\chi^{k}}}||u||_{L^{\chi^{k}}}.$

In order to show $L^{\infty}$ bound of $u$, we just need

$\sum_{k=k_{0}}^{\infty}\log\big{(}c_{12}^{\frac{1}{\chi^{k}}}(2\chi^{k})^{\frac{1}{\chi^{k}}}(r^{-1}_{\chi^{k}+\frac{1}{2}})^{\frac{1}{\chi^{k}}}\big{)}<\infty.$

Above is easily seen to be equivalent to showing:

(3.7)
$\sum_{k=k_{0}}^{\infty}\frac{1}{\chi^{k}}\log(r_{\chi^{k}+\frac{1}{2}}^{-1})<\infty.$

Here $r_{q}$ is defined by (3.1). We show that this is guaranteed by the
condition: $\int_{0}^{1}\frac{\omega(r)^{2}}{r}dr<\infty$. This follows from
the following elementary lemma:

######  Lemma 3.1.

Let $\omega(r):[0,1]\rightarrow\mathbb{R}^{+}$ be a continuous and strictly
increasing function such that $\omega(0)=0$. Define $r_{q}$ as given by (3.1).
Let $\chi>1$, then there exists $C_{3.3}$, which depends only on $\chi$, such
that

$\sum_{k=k_{0}}^{\infty}\frac{1}{\chi^{k}}\log(r_{\chi^{k}+\frac{1}{2}}^{-1})\leq
C_{3.3}\int_{0}^{1}\frac{\omega(r)^{2}}{r}dr.$

Here $k_{0}$ is the minimal integer for which $\chi^{k_{0}}>9+\frac{1}{2}$.

###### Proof.

First we observe that $\int_{0}^{1}\frac{\omega^{2}(r)}{r}dr<\infty$ implies
that

(3.8)
$\int_{0}^{\omega(1)}s\log\big{(}\frac{1}{\omega^{-1}(s)}\big{)}ds<\infty.$

Here $\omega^{-1}(s)$ is the inverse function of $\omega(r)$ on the interval
$r\in[0,1]$.

To see (3.8) holds, we can calculate:

$\begin{split}\int_{0}^{\omega(1)}&s\log\big{(}\frac{1}{\omega^{-1}(s)}\big{)}ds=\int_{0}^{1}\log(r^{-1})\omega(r)\omega^{\prime}(r)dr=\frac{\omega(r)^{2}}{2}\log(r^{-1})|_{r=1}-\frac{\omega(r)^{2}}{2}\log(r^{-1})|_{r=0^{+}}\\\
&+\int_{0}^{1}\frac{\omega(r)^{2}}{2r}dr\leq\int_{0}^{1}\frac{\omega(r)^{2}}{2r}dr.\end{split}$

It only remains to show that (3.8) implies (3.7). First we observe that from
the definition of $r_{q}$, we have
$r_{q}=\omega^{-1}(c_{4}^{-1}(8(q-1))^{-\frac{1}{2}})$, hence

(3.9)
$r_{\chi^{k}+\frac{1}{2}}=\omega^{-1}\big{(}\frac{1}{c_{4}\sqrt{8(\chi^{k}-\frac{1}{2})}}\big{)}\geq\omega^{-1}\big{(}\frac{1}{4c_{4}\chi^{\frac{k}{2}}}\big{)}.$

There is no loss of generality to assume $\omega(r)\geq r$(by changing
$\omega(r)$ to $\omega(r)+r$), then we have $\omega(1)\geq 1$ and

$\begin{split}\int_{0}^{1}&s\log\big{(}\frac{1}{\omega^{-1}(s)}\big{)}ds\geq\sum_{k=k_{0}+1}^{\infty}\int_{\frac{1}{4c_{4}}\chi^{-\frac{k+1}{2}}}^{\frac{1}{4c_{4}}\chi^{\frac{-k}{2}}}s\log\big{(}\frac{1}{\omega^{-1}(s)}\big{)}ds\\\
&\geq\sum_{k=k_{0}}^{\infty}\big{(}\frac{1}{4c_{4}}\chi^{\frac{-k}{2}}-\frac{1}{4c_{4}}\chi^{-\frac{k+1}{2}}\big{)}(4c_{4}\chi^{-\frac{k+1}{2}})\log\big{(}\frac{1}{\omega^{-1}(\frac{1}{4c_{4}}\chi^{-\frac{k}{2}})}\big{)}\\\
&\geq\chi^{-\frac{1}{2}}(1-\chi^{-\frac{1}{2}})\sum_{k=k_{0}}^{\infty}\chi^{-k}\log(r_{\chi^{k}+\frac{1}{2}}^{-1}).\end{split}$

The last inequality used (3.9). ∎

##  4\. Integrability estimate for $e^{-\varphi}$

In this section, we will show the following result:

######  Proposition 4.1.

Let $\varphi$ solves
$(\omega_{0}+\sqrt{-1}\partial\bar{\partial}\varphi)^{n}=e^{F}\omega_{0}^{n}$,
with normalization $\sup_{M}\varphi=0$. Suppose that there exists a function
$\Phi:[0,\infty)\rightarrow\mathbb{R}_{+}$, with $\Phi(t)\rightarrow+\infty$
as $t\rightarrow+\infty$ and $e^{F}\Phi(e^{F})\in L^{1}(\omega_{0}^{n})$, then
for any $p<\infty$, we have $\int_{M}e^{-p\varphi}\omega_{0}^{n}\leq C$. Here
the constant $C$ depends on $p$, the background metric $\omega_{0}$, the
function $\Phi$ and the integral bound
$\int_{M}e^{F}\Phi(e^{F})\omega_{0}^{n}$.

This result actually follows from the following theorem of Guedj and Zeriahi
in [9]:

######  Theorem 4.1.

For any positive measure $\mu$ which does not charge pluripolar sets, there
exists a unique function $\varphi\in\mathcal{E}(M,\omega_{0})$, such that
$\mu=(\omega_{0}+\sqrt{-1}\partial\bar{\partial}\varphi)^{n}$.

We refer the readers to [9] for more details about the definition of the space
$\mathcal{E}$. Among other properties, the authors also showed that any
$\varphi\in\mathcal{E}$ has zero Lelong number everywhere. Hence Proposition
4.1 follows from the Skoda’s integrability theorem.

Here we will give a more elementary and direct proof of Proposition 4.1 which
avoids the use of pluripotential machinery. Moreover, it has the merit of
explicit estimates. The idea is similar to Theorem 5.2 of [4].

To start, denote $A=\int_{M}e^{F}\Phi(e^{F})\omega_{0}^{n}$ and let $\psi\in
psh(\omega_{0})$ be the solution of the equation:

$\begin{split}&(\omega_{0}+\sqrt{-1}\partial\bar{\partial}\psi)^{n}=\frac{e^{F}\Phi(e^{F})}{A}\omega_{0}^{n},\\\
&\sup_{M}\psi=0.\end{split}$

Proposition 4.1 will follow from the following lemma:

######  Lemma 4.2.

For any $0<\varepsilon<\frac{1}{2}$, there exists a constant
$C_{\varepsilon}$, which has the same dependence as described in Proposition
4.1 but additionally on $\varepsilon$, such that $\varphi-\varepsilon\psi\geq-
C_{\varepsilon}$.

To see that Proposition 4.1 follows from this lemma, we recall the following
lemma of Tian, which is based on the $L^{2}$ estimate of Hörmander:

######  Lemma 4.3.

Let $(M,\omega_{0})$ be a compact Kähler manifold. Then there exists a
constant $\alpha_{0}>0$, $C_{0}>0$, such that for all $\varphi\in
psh(\omega_{0})$, we have
$\int_{M}e^{-\alpha_{0}(\varphi-\sup_{M}\varphi)}\omega_{0}^{n}\leq C_{0}$.

For any $p>0$, we see from Lemma 4.2 that

$-p\varphi\leq pC_{\varepsilon}-p\varepsilon\psi.$

Hence if we take $\varepsilon$ such that $p\varepsilon=\alpha_{0}$ and apply
Lemma 4.3, we see that

$\int_{M}e^{-p\varphi}\omega_{0}^{n}\leq
e^{pC_{\varepsilon}}\int_{M}e^{-p\varepsilon\psi}\omega_{0}^{n}\leq
e^{pC_{\varepsilon}}C_{0}.$

Now it only remains to prove Lemma 4.2.

###### Proof.

First we can compute

$\begin{split}\Delta_{\varphi}&\psi=g_{\varphi}^{i\bar{j}}\partial_{i\bar{j}}\psi=g_{\varphi}^{i\bar{j}}(g_{i\bar{j}}+\psi_{i\bar{j}})-tr_{\varphi}g\geq
n\big{(}\det
g_{\varphi}^{i\bar{j}}\det(g_{i\bar{j}}+\psi_{i\bar{j}})\big{)}^{\frac{1}{n}}-tr_{\varphi}g\\\
&\geq
n\big{(}e^{-F}e^{F}\Phi(e^{F})A^{-1}\big{)}^{\frac{1}{n}}-tr_{\varphi}g=n\Phi^{\frac{1}{n}}(e^{F})A^{-\frac{1}{n}}-tr_{\varphi}g.\end{split}$

Therefore

$\Delta_{\varphi}(\varphi-\varepsilon\psi)=n-tr_{\varphi}g-\varepsilon\Delta_{\varphi}\psi\leq
n-\varepsilon
n\Phi^{\frac{1}{n}}(e^{F})A^{-\frac{1}{n}}-(1-\varepsilon)\varepsilon
tr_{\varphi}g.$

Suppose that the function $\varphi-\varepsilon\psi$ achieves minimum at
$p_{0}$. We may choose a ball $B_{d_{0}}(p_{0})$ centered at $p_{0}$, such
that under normal coordinates at $p_{0}$, we have $\frac{1}{2}\leq\det
g_{i\bar{j}}\leq 2$. Let $\eta:M\rightarrow\mathbb{R}^{+}$ be a cut-off
function on $M$, such that $\eta(p_{0})=1$, $\eta\equiv 1-\theta$ outside the
ball $B_{d_{0}}(p_{0})$, with $|\nabla\eta|\leq\frac{\theta}{d_{0}}$,
$|D^{2}\eta|\leq\frac{\theta}{d_{0}^{2}}$. Here $\theta>0$ is a parameter to
be chosen(small).

Now we choose $\delta=\frac{\alpha_{0}}{2n}$, where $\alpha_{0}>0$ is the
$\alpha$-invariant given by Lemma 4.3. We can compute

$\begin{split}&\Delta_{\varphi}\big{(}e^{-\delta(\varphi-\varepsilon\psi)}\big{)}=e^{-\delta(\varphi-\varepsilon\psi)}\delta^{2}|\nabla_{\varphi}(\varphi-\varepsilon\psi)|^{2}_{\varphi}-\delta
e^{-\delta(\varphi-\varepsilon\psi)}\Delta_{\varphi}(\varphi-\varepsilon\psi)\\\
&\geq\delta^{2}e^{-\delta(\varphi-\varepsilon\psi)}|\nabla_{\varphi}(\varphi-\varepsilon\psi)|^{2}_{\varphi}+e^{-\delta(\varphi-\varepsilon\psi)}\delta(\varepsilon
n\Phi^{\frac{1}{n}}(e^{F})A^{-\frac{1}{n}}+(1-\varepsilon)tr_{\varphi}g-n).\end{split}$

Hence

(4.1)
$\begin{split}\Delta_{\varphi}&\big{(}e^{-\delta(\varphi-\varepsilon\psi)}\eta\big{)}=e^{-\delta(\varphi-\varepsilon\psi)}\delta^{2}|\nabla_{\varphi}(\varphi-\varepsilon\psi)|^{2}_{\varphi}\eta\\\
&+e^{-\delta(\varphi-\varepsilon\psi)}\delta\big{(}\varepsilon
n\Phi^{\frac{1}{n}}(e^{F})A^{-\frac{1}{n}}+(1-\varepsilon)tr_{\varphi}g-n\big{)}\eta+e^{-\delta(\varphi-\varepsilon\psi)}\Delta_{\varphi}\eta\\\
&+e^{-\delta(\varphi-\varepsilon\psi)}(-\delta)\nabla_{\varphi}(\varphi-\varepsilon\psi)\cdot_{\varphi}\nabla_{\varphi}\eta\\\
&\geq e^{-\delta(\varphi-\varepsilon\psi)}\delta\big{(}\varepsilon
n\Phi^{\frac{1}{n}}(e^{F})A^{-\frac{1}{n}}+(1-\varepsilon)tr_{\varphi}g-\frac{|D^{2}\eta|}{\eta}tr_{\varphi}g-\frac{|\nabla\eta|^{2}}{\eta^{2}}tr_{\varphi}g-n\big{)}.\end{split}$

We then choose the parameter $\theta$ sufficiently small so that

$(1-\varepsilon)-\frac{\theta}{d_{0}^{2}(1-\theta)}-\frac{\theta^{2}}{d_{0}^{2}(1-\theta)^{2}}>0.$

With this choice, we conclude from (4.1) that

(4.2)
$\Delta_{\varphi}\big{(}e^{-\delta(\varphi-\varepsilon\psi)}\eta\big{)}\geq
e^{-\delta(\varphi-\varepsilon\psi)}\delta\big{(}\varepsilon
n\Phi^{\frac{1}{n}}(F)A^{-\frac{1}{n}}-n\big{)}.$

Apply the Alexandrov estimate in the ball $B_{d_{0}}(p_{0})$, we obtain that

(4.3)
$\begin{split}\sup_{B_{d_{0}}(p_{0})}&e^{-\delta(\varphi-\varepsilon\psi)}\eta\leq\sup_{\partial
B_{d_{0}}(p_{0})}e^{-\delta(\varphi-\varepsilon\psi)}\eta\\\
&+C_{n}d_{0}\bigg{(}\int_{B_{d_{0}}(p_{0})}e^{-2n\delta(\varphi-\varepsilon\psi)}\delta^{2n}\big{(}(\varepsilon
n\Phi^{\frac{1}{n}}(F)A^{-\frac{1}{n}}-n)^{-}\big{)}^{2n}e^{2F}\omega_{0}^{n}\bigg{)}^{\frac{1}{2n}}.\end{split}$

In order for $(\varepsilon n\Phi^{\frac{1}{n}}(F)A^{-\frac{1}{n}}-n)^{-}$ to
be nonzero, we must have that

$\varepsilon n\Phi^{\frac{1}{n}}(F)A^{-\frac{1}{n}}-n<0.$

This gives an upper bound for $F$, say $F\leq C_{1}$, where $C_{1}$ has the
said dependence as in the lemma. Hence

(4.4) $\begin{split}\textrm{The integral in
(\ref{4.3})}&\leq\int_{B_{d_{0}}(p_{0})\cap\\{F\leq
C_{1}\\}}e^{-2n\delta(\varphi-\varepsilon\psi)}\delta^{2n}n^{2n}e^{2C_{1}}\omega_{0}^{n}\\\
&\leq\delta^{2n}n^{2n}e^{2C_{1}}\int_{M}e^{-\alpha_{0}\varphi}\omega_{0}^{n}\leq\delta^{2n}n^{2n}e^{2C_{1}}C_{0}.\end{split}$

On the other hand, note that $\eta\leq 1-\theta$ on $\partial
B_{d_{0}}(p_{0})$, and $e^{-\delta(\varphi-\varepsilon\psi)}$ achieves maximum
over $M$ at $p_{0}$. Then (4.3) implies:

(4.5)
$\sup_{M}e^{-\delta(\varphi-\varepsilon\psi)}\leq\frac{1}{\theta}C_{n}d_{0}\delta^{2n}n^{2n}e^{2C_{1}}C_{0}.$

This finishes the proof. ∎

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# Mott-driven BEC-BCS crossover in a doped spin liquid candidate,
$\kappa$-(BEDT-TTF)4Hg2.89Br8

Y. Suzuki1,† K. Wakamatsu1,† J. Ibuka1 H. Oike1 T. Fujii2 K. Miyagawa1 H.
Taniguchi3 K. Kanoda1,∗ 1Department of Applied Physics, University of Tokyo,
Bunkyo-ku, Tokyo, 113-8656, Japan
2Cryogenic Research Center, University of Tokyo, Bunkyo-ku, Tokyo, 113-0032,
Japan
3Graduate School of Science and Engineering, Saitama University, Saitama
338-8570, Japan

###### Abstract

The pairing of interacting fermions leading to superﬂuidity has two limiting
regimes: the Bardeen-Cooper-Schrieﬀer (BCS) scheme for weakly interacting
degenerate fermions and the Bose-Einstein condensation (BEC) of bosonic pairs
of strongly interacting fermions. While the superconductivity that emerges in
most metallic systems is the BCS-like electron pairing, strongly correlated
electrons with poor Fermi liquidity can condense into the unconventional BEC-
like pairs. Quantum spin liquids harbor extraordinary spin correlation free
from order and the superconductivity that possibly emerges by carrier doping
of the spin liquids is expected to have a peculiar pairing nature. The present
study experimentally explores the nature of the pairing condensate in a doped
spin-liquid candidate material and under varying pressure, which changes the
electron-electron Coulombic interactions across the Mott critical value in the
system. The transport measurements reveal that the superconductivity at low
pressures is a BEC-like condensate from a non-Fermi liquid and crosses over to
a BCS-like condensate from a Fermi liquid at high pressures. The Nernst-eﬀect
measurements distinctively illustrate the two regimes of the pairing in terms
of its robustness to the magnetic ﬁeld. The present Mott tuning of the BEC-BCS
crossover can be compared to the Feshbach tuning of the BEC-BCS crossover of
fermionic cold atoms.

##  I Introduction

Strong interactions between itinerant electrons in solids cause peculiar
correlations or organizations among the electrons beyond the Fermi-liquid
framework [1]. Superconductivity emerging in these circumstances can be
outside the celebrated Bardeen-Cooper-Schrieffer (BCS) framework, which
assumes an instability of Fermi surfaces formed by degenerate fermions with
well-deﬁned momenta [2], in that the strong correlation of electrons makes
their momenta ill-deﬁned and the electron pairing in momentum space, the
central idea of BCS theory, is less pertinent. In this case, superconductivity
is of the Bose-Einstein condensation (BEC) type, as the size of Cooper pairs
is as small as their mean distance, or equivalently, the interaction energy is
comparable to the Fermi energy. Several unconventional superconductors have
been discussed in the framework of a BEC scenario [3, 4, 5, 6, 7]. More
broadly, the superfluidity of cold fermionic atoms is known to be controllable
between the BEC and BCS regimes by the magnetic-field tuning of the
interatomic interaction through the Feshbach resonance [8, 9]. The relation
between the pairing nature and the interparticle correlation is a seminal
issue in the instability of interacting fermions at large [10, 11]. Thus, it
is expected to identify interacting systems having novel types of correlation
and explore the nature of the pairing condensate therein under variation in
the interaction in a controllable manner.

Quantum spin liquids emerging on geometrically frustrated lattices are argued
to possess peculiar spin correlations possibly linked to fractionalization,
entanglement and/or chirality in magnetic excitations [12, 13, 14]. The
superconductivity possibly brought about by doping against such anomalous
magnetic state [18, 23, 16, 15, 19, 20, 21, 22, 17] is expected to have
unconventional pairing nature out of the BCS framework; however, it has yet to
be explored experimentally. In the present study, we address this issue by
working with a doped spin liquid candidate under pressure variation, and
reveal that superconductivity in this system exhibits a novel type Mottness-
controlled BEC-BCS crossover.

The material studied is a layered organic superconductor, $\kappa$-(BEDT-
TTF)4Hg2.89Br8 ($\kappa$-HgBr hereafter), in which dimers of BEDT-TTF
molecules constitute a nearly perfect isotropic triangular lattice, separated
by insulating Hg2.89Br8 layers (Figs. 1(a) and 1(b)) [24, 25]. The
nonstoichiometry of Hg arises from its sublattice incommensurate with the host
lattice formed by BEDT-TTF and Br, and the composition of Hg is reproducibly
given by the mismatch between the two lattices precisely determined by x-ray
diffraction [24, 25]. The triangular lattice structure is almost identical to
that of the Mott-insulating spin liquid candidate $\kappa$-(BEDT-TTF)2Cu2(CN)3
($\kappa$-Cu2(CN)3 hereafter), in which the spin-1/2 localized on each BEDT-
TTF dimer interacts with its neighbors through an antiferromagnetic exchange
energy $J$ of 250 K but does not order down to 30 mK [26, 27, 28]. A
distinction of $\kappa$-HgBr from $\kappa$-Cu2(CN)3 is that the
nonstoichiometric Hg ions give 11$\%$ hole doping to the Mott insulator,
resulting in a metallic state [25, 29, 30]. Surprisingly, the $J$-scaled spin
susceptibility, namely, $J\chi_{\textrm{spin}}$, versus
$T$/$J\chi_{\textrm{spin}}$, ($J$ = 140 and 250 K for $\kappa$-HgBr and
$\kappa$-Cu2(CN)3, respectively) nearly coincide between $\kappa$-HgBr and
$\kappa$-Cu2(CN)3, and are well reproduced by the triangular-lattice
antiferromagnetic Heisenberg model, suggesting that $\kappa$-HgBr hosts a QSL
in the spin sector while remaining metallic in the charge sector [31]. Thus,
$\kappa$-HgBr is a possible realization of the long sought-after doped QSL
[14]. According to the transport study [29, 32], the metallic state at low
pressures is a non-Fermi liquid (NFL), and the doped holes, not particles of
the entire band, carry mobile charges. Superconductivity in such a doped QSL
candidate [24, 29], possibly appearing in place of the low-temperature
instability of the QSL [33], is of profound interest [18, 23, 16, 15, 19, 20,
21, 22, 17].

Pressure is an effective control parameter of electron-electron interactions
in molecular materials, which are highly compressible in general [34]. The
transport measurements indicate that when $\kappa$-HgBr is pressurized in
excess of 5 kbar, all band particles become mobile [29], and concomitantly,
the NFL at low pressures changes to a Fermi liquid (FL). Comparative phase
diagrams (Fig. 1(c)) illustrate that the Mott-insulating QSL phase in
$\kappa$-Cu2(CN)3 is replaced by the NFL in $\kappa$-HgBr [29, 35, 36].
Remarkably, $T_{\textrm{c}}$ forms a dome around the NFL-FL crossover pressure
[29, 32]. In the present work, we investigate the pressure evolution of the
nature of electron pairing that occurs in a doped QSL candidate $\kappa$-HgBr
by electron transport and Nernst effect measurements. The results demonstrate
the pressure-tuned BEC-BCS crossover and clarify its relation to the Fermi
liquidity above $T_{\textrm{c}}$.

##  II RESULTS and DISCUSSION

###  II.1 Crossover from non-Fermi liquid to Fermi liquid

To precisely associate the nature of electron pairing with that of the normal-
state fluidity above $T_{\textrm{c}}$, we performed resistivity measurements
under finely incremented pressures [25]. To avoid resistivity jumps due to
microcracking of the crystal encountered in the ambient-pressure measurements,
we had to apply finite pressures to obtain jump-free resistivity data. As seen
in Fig. 2(a), the temperature dependence of the in-plane resistivity
$\rho$($T$) varies systematically with pressure; the linear temperature
dependence changes into superlinear dependences at elevated pressures. Fitting
the form of $\rho_{\textrm{n}}$($T$)=$\rho_{0}$+$AT^{\alpha}$ to the normal-
state resistivity$\rho$($T$) below 10 K measured under 9 T yields the value of
the residual resistivity $\rho_{0}$ for each pressure. The logarithmic plot of
($\rho$($T$) - $\rho_{0}$) versus $T$ (Fig. 2(b)) highlights the pressure
evolution of the temperature-dependent part of resistivity. The slopes of
these curves, given by $\alpha$($T$) $\equiv$ $d\log$($\rho$($T$) -
$\rho_{0}$)/$d\log$$T$, are defined as the exponent at temperature $T$. The
contour plot of the deduced $\alpha$($T$) values on the pressure-temperature
plane (Fig. 2(c)) shows that at low temperatures, $\alpha$($T$) is
approximately unity before the pressure reaches 0.4 GPa and sharply switches
to the value of 2 at higher pressures, demonstrating that an NFL crosses over
to an FL at approximately 0.35-0.40 GPa. This finding reproduces the previous
results [29] and additionally reveals that the crossover is so sharp, as
theoretically suggested [37], that it is reminiscent of a quantum phase
transition [38, 39, 40, 41]. This sharp crossover is regarded as a
correspondence to the Mott transition in $\kappa$-Cu2(CN)3, as implied by Fig.
1(c); at the crossover, the interaction strength crosses a Mott critical value
at which the double occupancies of holes on a site, namely, Mottness,
drastically changes [29], as theoretically suggested [30]. Furthermore, the
NFL state does not only occur at the quantum critical point but extends to
lower pressures suggesting a stable quantum critical phase [42]. Such an NFL
phase is also argued in a heavy electron system presumably hosting a QSL [43].
Charge carriers moving around in QSL phases may not achieve sufficient
coherence, losing Fermi degeneracy and resulting in NFL phases. Theoretically,
the temperature-linear resistivity observed in $\kappa$-HgBr at low pressures
is suggested for bosonic charge excitations in a spin-charge separated doped
spin liquid [17].

###  II.2 Upper critical field and superconducting coherence length : BEC-BCS
crossover

Next, we explore how the nature of the superconducting condensate changes upon
crossover from the anomalous metal to the ordinary metal by pressure. To
assess the BEC or BCS nature, we evaluate the in-plane Ginzburg-Landau
coherence length $\xi_{\parallel}$, which measures the Cooper-pair size,
albeit not strictly in the BEC regime [44], from the perpendicular upper
critical field, $H_{\textrm{c2}\perp}$. To determine $H_{\textrm{c2}\perp}$,
we traced the in-plane resistivity upon sweeping the magnetic field normal to
the conducting layers shown at fixed temperatures [25]. Figure 3(a) shows the
resistivity evolution upon the destruction of superconductivity by magnetic
field at 0.32 GPa as an example (see [25] for the resistivity data at other
pressures, 0.24, 0.27, 0.32, 0.39, 0.52, 0.58, 0.73, 0.83 and 1.0 GPa). As the
normal state shows no appreciable magnetoresistance as seen in the figure, the
field dependence of the resistivity stems from the mobile vortices and/or
superconducting fluctuations. The resistivity behavior in the magnetic field-
temperature ($H$-$T$) plane at each pressure is shown in the contour plot of
the reduced resistivity $\rho$($T$, $H$)/$\rho_{\textrm{n}}$($T$), where
$\rho_{\textrm{n}}$($T$) is the normal-state resistivity determined above, in
Figs. 3(b)-3(j). In the blue area, the resistivity vanishes or is small. As
the magnetic field increases, the system enters the mixed state with mobile
vortices, $\rho$($T$, $H$)/$\rho_{\textrm{n}}$($T$) takes finite values (light
blue-yellow area), and eventually, the normal-state resistivity is restored,
i.e., $\rho$($T$, $H$)/$\rho_{\textrm{n}}$($T$) $\approx$ 1 (red area). In the
low-pressure regime, there is a wide region with finite values in $\rho$($T$,
$H$)/$\rho_{\textrm{n}}$($T$), indicating that mobile vortices or
superconducting fluctuations are developed even at high fields. In contrast,
at high pressures, the superconductivity is fully suppressed by low magnetic
fields, and the transition to the normal state is sharp, as in conventional
superconductors. Overall, superconductivity is robust to magnetic fields at
low pressures.

The coherence length $\xi_{\parallel}$ is evaluated from the temperature
derivative of $H_{\textrm{c2}\perp}$ near $T_{\textrm{c}}$ through
$T_{\textrm{c}}$[$dH_{\textrm{c2}\perp}$($T$)/$dT$]${}_{T=T_{\textrm{c}}}$ =
$\phi_{0}$/2$\pi$$\xi_{\parallel}^{2}$, where $\phi_{0}$ is the flux quantum.
In highly two-dimensional layered superconductors, finite resistivity is
caused by vortex flow under perpendicular magnetic fields, as observed in this
system [45]; therefore, the definition of $H_{\textrm{c2}\perp}$ is not
straightforward. In such a case, $H_{\textrm{c2}\perp}$ is better
characterized by an initial drop in resistivity. Here, we adopt as
$H_{\textrm{c2}\perp}$ the 80$\%$ and 90$\%$ transition points ($\rho$($T$,
$H$)/$\rho_{\textrm{n}}$($T$) = 0.8 and 0.9), which are indicated by bold
contours in Fig. 3(b)-3(j). The values of
[$dH_{\textrm{c2}\perp}$($T$)/$dT$]${}_{T=T_{\textrm{c}}}$ for the the 80$\%$
and 90$\%$ transition lines are shown in Fig. 4. Using these values, we
determined the $\xi_{\parallel}$ values, whose pressure dependence is also
shown in Fig. 4. At pressures below 0.4 GPa, $\xi_{\parallel}$ $\sim$ 3 nm,
whereas, it increases steeply with pressure and reaches $\sim$ 20 nm at 1 GPa.
Assuming a cylindrical Fermi surface, we evaluated the Fermi wavenumber
$k_{\textrm{F}}$ from the Hall coefficient under pressure [25] and obtained
the $k_{\textrm{F}}$$\xi_{\parallel}$ value, an index of the BEC/BCS nature
or, roughly speaking, the degree of overlap of Cooper pairs. The pressure
dependence of $k_{\textrm{F}}$$\xi_{\parallel}$ is shown along with
$T_{\textrm{c}}$ in Fig. 5. At high pressures, e.g., 1 GPa,
$k_{\textrm{F}}$$\xi_{\parallel}$ reaches $\sim$ 50, pointing to highly
overlapping pairs as in the conventional BCS regime, whereas at low pressures,
$k_{\textrm{F}}$$\xi_{\parallel}$ is as small as $\sim$ 3; the size of the
Cooper pair may become even smaller than the evaluated $\xi_{\parallel}$ value
when approaching the BEC regime according to a theoretically suggestion [44].
The $k_{\textrm{F}}$$\xi_{\parallel}$ value of the order of unity signifies
that the pairing is BEC-like, while the value of several tens points to BCS-
like pairing, thus demonstrating that pressure induces a BEC-to-BCS crossover.
The field-robust vortex liquid state and fluctuations at low pressures are
consistent with the short coherence length and small density of the bosonic
pairs inherent in BEC-like pairing, as shown theoretically [46].

###  II.3 Nernst effect and superconducting fluctuations

We further verified the preformed nature and magnetic field robustness of the
pairing by examining the Nernst effect, in which a thermal flow generated by a
temperature gradient $\nabla_{x}T$ under a perpendicular magnetic field
induces a transverse electric field $E_{y}$ (see the inset of Fig.6). This
effect, characterized by $e_{N}$ $\equiv$ $\pm$$E_{y}$/$\nabla_{x}T$ (with the
sign depending on the field direction), is known to be enhanced by phase
fluctuations arising from vortex liquid or preformed pairs in the
superconducting state under magnetic fields, as observed in real materials
such as cuprate and organic superconductors [10, 47, 48, 49, 50, 51]. Figure 6
shows the temperature variation of $e_{N}$ at several fixed magnetic fields
under a pressure of 0.3 GPa. The $e_{N}$ value, which is small in the normal
state, starts to increase at temperatures well above $T_{\textrm{c}}$ upon
cooling, signaling superconducting preformation; it is remarkable that $e_{N}$
shows a field-insensitive universal temperature variation at high magnetic
fields. Figures 7(a)-7(d) display the magnetic-field variation of $e_{N}$ at
several fixed temperatures under the pressures of 0.3, 0.5, 0.65 and 0.9 GPa.
It is evident that the magnetic field profile of $e_{N}$ is strongly pressure-
dependent; the $e_{N}$ value keeps large even at the maximum field, 9T, under
0.3 GPa whereas under 0.9 GPa it rapidly drops above 1 T. To visualize the
pressure evolution of the Nernst effect, we displays the contour plots of
$e_{N}$ in the temperature-magnetic field plane for each pressure in Figs.
7(e)-7(h). At 0.3 GPa, the ridge fields at which $e_{N}$ takes large values
are extended vertically and even tend to go beyond the experimental maximum
field, 9 T. In general, quasi-two-dimensional superconductivity is easily
suppressed by the perpendicular magnetic field far less than the Pauli
limiting field, which is $\sim$ 10 T at 0.3 GPa, due to the orbital depairing.
No such symptom up to 9 T and higher evidences unusual robustness of the
pairing to the magnetic field, in line with the view of the BEC condensate. At
higher pressures, the ridge fields become suppressed along with
$T_{\textrm{c}}$ and inclined, as in conventional superconductors. To see the
pressure-temperature profile of $e_{N}$, we make a contour plot of the
$T_{\textrm{c}}$ values at the maximum experimental field, 9 T (Fig. 8), which
illustrates that the field-robust Nernst signal and its persistence at high
temperatures far beyond $T_{\textrm{c}}$ is specific to the low-pressure
phase. This is the Nernst behavior typical of BEC-like condensates; the effect
of preformed pairs is expected to set in at higher temperatures in the Nernst
signal than in the resistivity [10], consistent with the present observations.
The systematic change of the Nernst profile with pressure lends support to the
BEC-BCS crossover of the superconducting condensate.

It is reported that the 13C NMR relaxation rate divided by temperature,
1/$T_{1}T$, shows a decrease below 7-9 K [52] despite of $T_{\textrm{c}}$
$\sim$ 4.2 K at ambient pressure and the decrease becomes less prominent at
higher pressures. This behavior is a likely manifestation of the preformed
pairs in the BEC-like regime at ambient pressure and its crossover to the BCS
regime at high pressures. We also note that the recent torque measurements
suggest the fluctuating superconductivity that sets in at around 7 K at
ambient pressure, implying the preformation of the Cooper pairs [53].

##  III CONCLUDING REMARKS

The electron transport and Nernst effect in the doped triangular system,
$\kappa$-HgBr, revealed that the nature of the superconductivity crosses over
from BEC-like to BCS pairing with increasing pressure. The present BEC-BCS
crossover is associated with an NFL-FL crossover, at which the Coulombic
repulsive energy relative to the band width exceeds a critical value for a
Mott transition that would occur unless doped. To explore the nature of
fermionic pairing that encompasses the BEC and BCS regimes, it is crucial to
vary the control parameter linked to the interparticle interaction across some
critical value at which the nature of the interaction dramatically changes. In
cold fermionic atoms, the controlling parameter is the magnetic field, which
varies interatomic interactions across a critical value through the Feshbach
resonance. In the present case of interacting electrons, pressure is a
parameter that varies the Coulombic interactions across the Mott critical
value at which the double occupancies of carriers are critically allowed or
forbidden [30]. This Mott-driven BEC-BCS crossover is a novel addition to the
physics of the pairing instability of interacting fermions.

###### Acknowledgements.

We thank M. Ogata for his fruitful discussions. This work was supported by
Japan Society for the Promotion of Science (grant Nos. 18H05225, 19H01846,
20K20894 and 20KK0060). Several experiments were performed using facilities of
the Cryogenic Research Center, the University of Tokyo.

*

##  Appendix A METHODS

###  A.1 Measurements of resistivity

The in-plane resistivity was measured by the conventional four-probe method;
four gold wires were attached on the planar surface of a crystal with carbon
paste. The Quantum Design Physical Property Measurement System (PPMS) was used
for the resistivity measurements. To investigate the perpendicular upper
critical field, $H_{c2}$⟂, we traced the resistivity with the magnetic field
applied perpendicular to the conducting layers up to 9 T. In organic
conductors, the rapid cooling often causes unwanted crystal cracking and/or
conformational disorder of terminal ethylene groups in BEDT-TTF. To avoid
suffering from these, the cooling rate was kept below 0.5 K/min.

###  A.2 Measurements of the Nernst effect

The Nernst voltage, $e_{N}$, was generated in a direction perpendicular to
both the directions of an externally created temperature gradient and an
applied magnetic field (see the inset of Fig. 6). The magnetic field was
applied perpendicular to the conducting layers. The temperature difference
across the crystal, $\Delta$$T$, along the two-dimensional plane was kept less
than 0.5 K in the field-sweep measurements and less than $T$/10 in the
temperature-sweep measurements. The temperatures were measured with the Cernox
sensors (Lake Shore Cryotronics, Inc.) attached on the cold and hot side of
the crystal. The Nernst-effect measurements were performed under several
pressures and the Cernox sensors were calibrated every time pressure is
changed, using a reference thermometer. To determine the Nernst voltage,
$e_{N}$, the transverse voltage $V$ was measured under both positive and
negative fields (+$H$ and -$H$), and then the $e_{N}$ was obtained by
antisymmetrizing the two voltages by $e_{N}$($H$) = ($V$(+$H$)-$V$(-$H)$)/2.
Through this measurement, PPMS was used as a controller of magnetic field and
temperature; the temperature gradient in the sample was generated by a heater
attached on one side of the crystal and the sample temperature was defined by
the mean value of the temperatures of the hot and cold sides. As in the
resistivity measurements, the cooling rate was kept below 0.5 K/min. The
thermopower was also measured along with the Nernst voltage for the identical
sample in order to know the $T_{\textrm{c}}$ of the sample used.

###  A.3 Pressurization

To apply hydrostatic pressures to the sample, a dual structured clamp-type
pressure cell made from BeCu and NiCrAl cylinders was used with the Daphne
7373 oil as a pressure-transmitting medium. The pressure values quoted in this
article are the internal pressures at low temperatures determined with the
manganin pressure gauge or the tin manometer. According to Ref. [54], the
clumped pressure gradually decreases by 1.5-2.0 kbar on cooling from 300 K to
50 K and is nearly constant below that.

∗Corresponding author, e-mail<EMAIL_ADDRESS>
†These authors contributed equally to this work

Supplemental Material for the Papaer

”Mott-driven BEC-BCS crossover in a doped spin liquid

candidate, $\kappa$-(BEDT-TTF)4Hg2.89Br8”

Crystal structure and band filling in $\kappa$-(BEDT-TTF)4Hg2.89Br8

The family of organic charge-transfer salts, $\kappa$-(BEDT-TTF)${}_{2}X$,
where BEDT-TTF and $X$ stand for bis(ethylenedithio)tetrathiafulvalene and
anions, respectively, have layered structures composed of BEDT-TTF conducting
layers and insulating anion layers, both of which have monoclinic sublattices
with the space group $C2$/$c$ and $C2$/$m$, respectively [55]. In the
conducting layers, the BEDT-TTF molecules form dimers, which constitute a
quasi-triangular lattice characterized by two kinds of transfer integrals, $t$
and $t^{\prime}$, between the adjacent dimer molecular orbitals (Fig. 1(b)).
According to the molecular orbital calculations, the ratio, $t^{\prime}$/$t$,
in $\kappa$-(BEDT-TTF)4Hg2.89Br8 is 1.02 [56], which is compared to the value
in the Mott-insulating spin liquid candidate, $\kappa$-(BEDT-TTF)2Cu2(CN)3,
1.06 [57] (0.83 [58] and 0.83-0.99 [59] according to the first-principles
calculations).

Most of $\kappa$-(BEDT-TTF)${}_{2}X$ compounds have half-filled bands, or
equivalently, one hole carrier per one dimer because the valence of the anion
$X$ is -1. However, the band filling in $\kappa$-(BEDT-TTF)4Hg2.89Br8 deviates
from a half by 0.055 assuming Hg+2 and Br-1, because of the nonstoichiometry
of Hg, whose sublattice is incommensurate along the c-axis against those of
BEDT-TTF and Br; if the composition of Hg were 3.0, the band filling would be
a half. The compositional ratio of 4.0 : 2.89 : 8.0 cannot be varied because
the incommensurability precisely determined by x-ray diffraction [55, 24]
results from chemistry between BEDT-TTF, Hg and Br. Such non-stoichiometry
occurs in other Hg-containing compounds as well [61]. The band filling
slightly deviated from a half is evidenced by the valence of BEDT-TTF
determined by the Raman spectroscopy [62]. This band-filling deviation from a
half corresponds to a 11$\%$ hole doping to the half-filled band.

Determination of the upper critical field

The upper critical field is usually defined as the magnetic field at which the
vanishingly small resistivity transitions to the normal-state one. In highly
two-dimensional layered superconductors like the present material, however,
resistivity is induced by mobile vortices or, broadly speaking,
superconducting phase fluctuations under perpendicular field even when the
amplitude of the superconducting order parameter is developed. Figure 9 shows
the magnetic field dependence of the in-plane resistivities at each pressure,
where the magnetic field was applied normal to the conducting plane. As
expected, the resistive transition against field variation is gradual, in
particular, for low pressures. Therefore, we defined the perpendicular upper
critical filed $H_{\textrm{c2}}$⟂ by the field at which the resistivity drops
by 10$\%$ and 20$\%$ from the normal-state value, $\rho_{\textrm{n}}(T)$, of
the form, $\rho_{\textrm{n}}(T)$ = $\rho_{0}$ \+ $AT^{\alpha}$, fitting the
normal state resistivity; namely, $H_{\textrm{c2}}$⟂ is defined in two ways as
the fields of $\rho(T,H)$/$\rho_{\textrm{n}}(T)$ = 0.8 and 0.9, where
$\rho(T,H)$ is the resistivity at temperature $T$ and magnetic field $H$. The
contour curves of $\rho(T,H)$/$\rho_{\textrm{n}}(T)$ = 0.8 and 0.9 are shown
by the bold lines in Fig. 3(b)-(j).

Evaluation of $k_{\textrm{F}}\xi_{\parallel}$

The in-plane coherence length $\xi_{\parallel}$ was calculated from the slope
of $H_{\textrm{c2}}$⟂ near $T_{\textrm{c}}$, $dH_{\textrm{c2}}$⟂ /$dT$,
through the formula, $\xi_{\parallel}$ =
$\sqrt{\frac{\phi_{0}}{2\pi{T}_{\textrm{c}}|dH_{\textrm{c2}\perp}/dT|}}$ The
experimental $dH_{\textrm{c2}}$⟂ /$dT$ and $T_{\textrm{c}}$ values are shown
in Fig. 4 and Fig. 5 in main text. The Fermi wavenumber, $k_{\textrm{F}}$, was
estimated from the Hall coefficient data [29] using the relation
$R_{\textrm{H}}$ = 1/$ne$ and assuming the two-dimensional cylindrical Fermi
surface, where $n$ is the carrier density and $e$ is the elementary charge.
The pressure dependences of the $R_{\textrm{H}}$ value at 10 K [29] and the
calculated $k_{\textrm{F}}$ values are shown in Fig. 10: the $k_{\textrm{F}}$
values at pressures studied here were obtained by interpolating the points in
Fig. 10(b).

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Figure 1:  (Color online) Structure and phase diagram of $\kappa$-(BEDT-
TTF)4Hg2.89Br8. (a) Layered structure consisting of BEDT-TTF conducting layers
and Hg2.89Br8 insulating layers. BEDT-TTF denotes
bis(ethylenedithio)tetrathiafulvalene. The nonstoichiometry in the Hg
composition comes from the fixed incommensurability between the Hg and BEDT-
TTF lattices originating from material chemistry, not from extrinsic defects.
(b) In-plane structure of the conducting layers. BEDT-TTF molecules form
dimers, which constitute a nearly isotropic isosceles triangular lattice with
the anisotropy of transfer integrals, $t^{\prime}$/$t$, close to unity. (c)
Pressure-temperature phase diagrams of $\kappa$-(BEDT-TTF)4Hg2.89Br8 and a
quantum spin liquid candidate, $\kappa$-(BEDT-TTF)2Cu2(CN)3. The red area
indicates the superconducting phase. The blue area in $\kappa$-(BEDT-
TTF)4Hg2.89Br8 depicts the non-Fermi liquid region (corresponding to the
yellow region in Fig. 2(c)). The purple area in $\kappa$-(BEDT-TTF)2Cu2(CN)3
depicts the Mott insulating spin liquid phase [35, 36]. $T_{\textrm{c}}$ in
$\kappa$-(BEDT-TTF)2Cu2(CN)3 is taken from Ref. [35], and $T_{\textrm{c}}$ in
$\kappa$-(BEDT-TTF)4Hg2.89Br8 corresponds to the midpoint of the resistive
transition observed in this study. MI, QSL, NFL and SC represent the Mott
insulator, quantum spin liquid, non-Fermi liquid, and superconductivity,
respectively. The white broken line indicates the location of the so-called 6
K-anomaly observed in $\kappa$-(BEDT-TTF)2Cu2(CN)3 [12, 33, 35], a possible
indication of an instability of QSL.

Figure 2:  (Color online) Temperature dependence of resistivity under
pressures. (a) Temperature variation of the in-plane resistivity $\rho$($T$)
at 0 T and a perpendicular field of 9 T. (b) Logarithmic plot of the
temperature-dependent part of resistivity, $\rho$($T$) - $\rho_{0}$, with
$\rho_{0}$ being the residual resistivity (see main text for its
determination). The solid and dashed lines indicate the resistivity at $H$ = 0
and 9 T, respectively. (c) Contour plot of the exponent $\alpha$ in fits of
the form $AT^{\alpha}$ to the temperature-dependent part of the experimental
in-plane resistivity $\rho$($T$) - $\rho_{0}$; $\alpha$ is determined by
$\alpha$($T$) $\equiv$ $d\log$($\rho$($T$) - $\rho_{0}$)/$d\log$$T$. The
yellow region ($\alpha$ $\approx$ 1) corresponds to a non-Fermi-liquid (NFL)
phase, and the blue region ($\alpha$ $\approx$ 2) corresponds to the Fermi-
liquid (FL) phase.

Figure 3:  (Color online) Resistivity under magnetic field. (a) In-plane
resistivity as a function of magnetic field, $H$, applied perpendicular to the
layers under the pressure of 0.32 GPa. The temperature was set to the
temperatures between 1.8 and 6.6 K at 0.3 K intervals and to 10.0 K. (b)-(j)
Contour plots of the normalized in-plane resistivity,
$\rho$($T,H$)/$\rho_{\textrm{n}}$($T$), in the temperature ($T$)-magnetic
field ($H$) plane for the pressure of 0.24, 0.27, 0.32, 0.39, 0.52, 0.58,
0.73, 0.83 and 1.0 GPa, where $\rho$($T,H$) is the in-plane resistivity at $T$
and $H$ and $\rho_{\textrm{n}}$($T$) is the normal resistivity with the form
of ($\rho_{0}$ \+ $AT^{\alpha}$) fitting the high-field data (see text). The
solid lines indicate the contours of $\rho$($T,H$)/$\rho_{\textrm{n}}$($T$) =
0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8 and 0.9.

Figure 4:  (Color online) Upper critical field and Superconducting coherence
length. The in-plane Ginzburg-Landau coherence length $\xi_{\parallel}$
calculated from the slope of the upper critical field,
$dH_{\textrm{c2}}$⟂/$dT$ near $T_{\textrm{c}}$, where the $H_{\textrm{c2}}$⟂
is determined from the line at $\rho$($T,H$)/$\rho_{\textrm{n}}$($T$) = 0.8,
0.9 (80$\%$ and 90$\%$ of the normal-state resistivity) in Fig. 3. The squares
and diamonds mean the values determined for
$\rho$($T,H$)/$\rho_{\textrm{n}}$($T$) = 0.8 and 0.9, respectively

Figure 5:  (Color online) Pressure dependence of
$k_{\textrm{F}}\xi_{\parallel}$ and BEC-BCS crossover. Superconducting
transition temperature, $T_{\textrm{c}}$, and $k_{\textrm{F}}\xi_{\parallel}$
as a function of temperature, where $k_{\textrm{F}}$ is the Fermi wavenumber
determined by the Hall coefficient [29]. The value of
$k_{\textrm{F}}\xi_{\parallel}$ is $\sim$3 at low pressures and increases up
to $\sim$50 at 1 GPa with pressure, pointing to a BEC-BCS crossover.
Schematics drawn underneath the panel depict the BEC- and BCS-like pairs on
the low- and high-pressure sides.

Figure 6:  (Color online) Temperature dependence of the Nernst signal $e_{N}$
under a pressure of 0.30 GPa. The measurements were made at the magnetic
fields of 0.2, 0.5, 1, 3, 5, 7 and 9 T applied perpendicular to the conducting
layers. The arrow indicates the zero-field $T_{\textrm{c}}$ determined by the
simultaneously measured thermopower. The geometry of the measurement setup is
depicted in the inset.

Figure 7:  (Color online) Magnetic field-temperature profile of the Nernst
signal $e_{N}$ under different pressures. (a)-(d) Nernst signal $e_{N}$ traced
upon sweeping the perpendicular magnetic field at several fixed temperatures
under the pressures studied, 0.3, 0.5 0.65 and 0.9 GPa. (e)-(h) Contour plots
of the Nernst signal eN in the temperature-magnetic field plane at the
pressures of 0.3, 0.5 0.65 and 0.9 GPa. The red areas indicate hot Nernst
regions. The arrows indicate the zero-field $T_{\textrm{c}}$.

Figure 8:  (Color online) Temperature-pressure profile of the Nernst signal
$e_{N}$. The contour plot of the Nernst signal $e_{N}$ at a field of 9T is
shown in the temperature-pressure plane. The white circles indicate the zero-
field $T_{\textrm{c}}$.

Figure 9: (Color online) Magnetic field dependence of the in-plane
resistivities under pressure. The magnetic field was applied perpendicular to
the conducting layer. Temperature was fixed at intervals of 0.3 K below
$T_{\textrm{c}}$. For reference data at a high temperature well above
$T_{\textrm{c}}$, the resistivity curves at 10 K were measured.

Figure 10:  (Color online) Hall coefficient of $\kappa$-HgBr at 10 K [29] and
the estimate of $k_{\textrm{F}}$. (a) Pressure dependence of the Hall
coefficient at 10 K. The values are taken from Ref. [29]. (b) $k_{\textrm{F}}$
values deduced from the Hall coefficient are shown in Fig. 10(a) on the
assumption of the relation, $R_{\textrm{H}}$ = 1/$ne$, and the two-dimensional
cylindrical Fermi surface.