# Approximation of weak geodesics and subharmonicity of Mabuchi energy, II:
$\varepsilon$-geodesics

Long Li  Mathematics Institute of ShanghaiTech University, 393 Middle Huaxia
Road, Pudong 201210, Shanghai, China<EMAIL_ADDRESS>Science
Institute, University of Iceland, Reykjavik, Iceland<EMAIL_ADDRESS>
###### Abstract.

The purpose of this article is to study the strict convexity of the Mabuchi
functional along a ${\mathcal{C}}^{1,\bar{1}}$-geodesic, with the aid of the
$\varepsilon$-geodesics. We proved the $L^{2}$-convergence of the fiberwise
volume element of the $\varepsilon$-geodesic. Moreover, the geodesic is proved
to be uniformly fiberwise non-degenerate if the Mabuchi functional is
$\varepsilon$-affine.

##  1\. Introduction

In order to study the uniqueness and existence problems of the Kähler-Einstein
metrics on a Fano manifold $X$, Mabuchi ([Ma], [BM]) introduced a useful
energy functional ${\mathcal{M}}$, called the $K$-energy or Mabuchi
functional, on the space ${\mathcal{H}}$ of all the smooth Kähler potentials
in a given cohomology class $[\omega]$. In other words, we can write the space
as

${\mathcal{H}}:=\\{\varphi\in C^{\infty}(X);\ \ \omega+dd^{c}\varphi>0\\}.$

It is observed that this functional ${\mathcal{M}}$ is convex along a smooth
geodesic ${\mathcal{G}}$, connecting with arbitrary two points in
${\mathcal{H}}$. Moreover, if the Mabuchi functional is affine along a smooth
geodesic ${\mathcal{G}}$, then ${\mathcal{G}}$ must be generated by a
holomorphic vector field. In this case, we say that the Mabuchi functional is
strictly convex along the geodesic ${\mathcal{G}}$.

It turns out that the Mabuchi functional ${\mathcal{M}}$ has also played an
important role in the study of the constant scalar curvature Kähler(cscK)
metrics. In particular, the convexity of ${\mathcal{M}}$ is crucial in the
proof of the uniqueness of the cscK metrics ([BB], [CLP], [CPZ]). Moreover,
the asymptotic behavior of this convex function ${\mathcal{M}}(\varphi_{t})$
along a geodesic ray is an invariant on a Kähler manifold $X$, and it is
proved ([CC1], [CC2], [CC3]) that the manifold $X$ admits a cscK metric if and
only if this invariant is non-negative.

However, one can not expect that there always exists a smooth geodesic
connecting two points in ${\mathcal{H}}$, due to the example in Darvas-Lempert
([LD]). What we can rely on in realty is the so called
${\mathcal{C}}^{1,\bar{1}}$-geodesic (as the solution of certain homogenous
complex Monge-Ampère equation), whose existence was guaranteed in the work of
Chen ([C00]). The difficulty to deal with a
${\mathcal{C}}^{1,\bar{1}}$-geodesic ${\mathcal{G}}$ is the lack of the
regularities and the possible degeneracy on the geodesic ${\mathcal{G}}$.

Nevertheless, the Mabuchi functional ${\mathcal{M}}$ was proved to be convex
and continuous along a ${\mathcal{C}}^{1,\bar{1}}$-geodesic, by the work of
Berman-Berndtsson ([BB]) and also Chen-Li-Păun ([CLP]). Then people try to ask
the question if the Mabuchi functional is also strictly convex along such a
geodesic.

If we allow that the boundary of ${\mathcal{G}}$ has merely
${\mathcal{C}}^{1,\bar{1}}$-regularities, then the answer is negative due to
the example of Berman ([Ber]). On the other hand, the answer is affirmative
([LL]), when ${\mathcal{G}}$ is connecting with two non-degenerate energy
minimizers of ${\mathcal{M}}$. In fact, the boundary of ${\mathcal{G}}$ must
be smooth in the later case by the work of He-Zeng ([HZ]). Therefore, a
geodesic ${\mathcal{G}}$ will always be assumed to have
${\mathcal{C}}^{1,\bar{1}}$-regularities and its boundary belongs to the space
${\mathcal{H}}$ through out this paper.

In order to circumvent the difficulties arising from a
${\mathcal{C}}^{1,\bar{1}}$-geodesic ${\mathcal{G}}$, we are appealing to
using the so called $\varepsilon$-geodesic ${\mathcal{G}}_{\varepsilon}$
instead ([C00]). The $\varepsilon$-geodesic ${\mathcal{G}}_{\varepsilon}$ is a
sequence of smooth approximation of ${\mathcal{G}}$, satisfying some non-
homogenous complex Monge-Ampère equation. This is easier to handle in the
practical computation, and we have proved in ([CLP]) that the Mabuchi
functional is _almost convex_ along the $\varepsilon$-geodesic.

However, the new difficulty is that the convergence
${\mathcal{G}}_{\varepsilon}\rightarrow{\mathcal{G}}$ is merely weakly $L^{p}$
for all $p>1$. In other words, it is even not clear to us if the Mabuchi
functional ${\mathcal{M}}(\varphi_{\varepsilon})$ along
${\mathcal{G}}_{\varepsilon}$ converges to the Mabuchi functional
${\mathcal{M}}(\varphi)$ along ${\mathcal{G}}$. Therefore, we can not conclude
the convexity of ${\mathcal{M}}$ by using the $\varepsilon$-geodesic
${\mathcal{G}}_{\varepsilon}$ in [CLP].

The new observation is that the convergence
${\mathcal{M}}(\varphi_{\varepsilon})\rightarrow{\mathcal{M}}(\varphi)$ is
true, provided that the Mabuchi function ${\mathcal{M}}$ is affine along
${\mathcal{G}}$. This first leads us to the following $L^{2}$-convergence of
the fiberwise volume element of ${\mathcal{G}}_{\varepsilon}$. Write
$\omega_{\varepsilon}:={\mathcal{G}}_{\varepsilon}|_{X_{t}}$ and
$\omega_{\varphi}:={\mathcal{G}}|_{X_{t}}$ on a fiber $X_{t}:=\\{t\\}\times X$
for any $t\in[0,1]$.

######  Theorem 1.1 (Theorem (4.1)).

Suppose the Mabuchi functional ${\mathcal{M}}$ is affine along a
${\mathcal{C}}^{1,\bar{1}}$-geodesic ${\mathcal{G}}$. Then the fiber-wise
volume element of the $\varepsilon$-geodesic converges to the volume element
of the geodesic in the strong $L^{2}$ sense. In other words, we have on each
fiber $X_{t}$

(1.1)
$\frac{\omega_{\varepsilon}^{n}}{\omega^{n}}\rightarrow\frac{\omega_{\varphi}^{n}}{\omega^{n}},\
\ \ \varepsilon\rightarrow 0,$

under the $L^{2}$-norm, possibly after passing to a subsequence.

We emphasis that the $L^{2}$-convergence of the volume element (equation
(1.1)) may not be true in general. Next, a slightly stronger condition than
the affine Mabuchi functional will be introduced, aiming to resolve the
possible degeneracy on the geodesic ${\mathcal{G}}$. Write the restriction of
the Mabuchi functional along a $\varepsilon$-geodesic
${\mathcal{G}}_{\varepsilon}$ as

${\mathcal{K}}_{\varepsilon}(t):={\mathcal{M}}(\varphi_{\varepsilon}(t))$

for all $t\in[0,1]$. Then we say that the Mabuchi functional is _$\varepsilon$
-affine_ along the geodesic ${\mathcal{G}}$ if it satisfies

(1.2)
$\frac{d{\mathcal{K}}_{\varepsilon}}{dt}|_{t=1}-\frac{d{\mathcal{K}}_{\varepsilon}}{dt}|_{t=0}=O(\varepsilon),$

for all $\varepsilon>0$ small. We note that ${\mathcal{G}}$ and
${\mathcal{G}}_{\varepsilon}$ are uniquely determined if the boundary of
${\mathcal{G}}$ is given. Thanks to a result in ([BB]), the Mabuchi functional
must be affine along the ${\mathcal{C}}^{1,\bar{1}}$-geodesic ${\mathcal{G}}$
if it is $\varepsilon$-affine. Moreover, we proved the following result.

######  Theorem 1.2 (Theorem 5.4).

Suppose the Mabuchi functional ${\mathcal{M}}$ is $\varepsilon$-affine along a
${\mathcal{C}}^{1,\bar{1}}$-geodesic ${\mathcal{G}}$. Then ${\mathcal{G}}$ is
uniformly fiberwise non-degenerate, namely, there exists a uniform constant
${\kappa}_{1}>0$ such that

${\mathcal{G}}|_{X_{t}}>\kappa_{1}\omega,$

for almost everywhere $t\in[0,1]$.

There are three main steps of the proof for the above Theorem. Step (1) is to
figure out the so called _gap phenomenon_ of the geodesic ${\mathcal{G}}$,
which is first observed in our previous work ([LL]). Step (2) is to establish
a kind of $W^{1,2}$-estimate for the volume element
$\omega_{\varepsilon}^{n}$, which is provided from the $\varepsilon$-affine
condition. Step (3) (see Proposition (5.7)) is to prove that a non-negative
function must have a positive lower bound if it has the gap phenomenon and
satisfies a certain partial $W^{1,2}$-estimate.

As an application of our Theorem (1.1) and (1.2), we can prove (see Theorem
(6.1)) that ${\mathcal{M}}(\varphi_{\varepsilon})$ is not only converging to
${\mathcal{M}}(\varphi)$ in the pointwise sense, but also in its complex
Hessian, if the later is $\varepsilon$-affine. That is to say, we actually
have

(1.3) $\frac{d^{2}{\mathcal{K}}_{\varepsilon}}{dt^{2}}|_{t}\rightarrow 0,$

for almost everywhere $t\in[0,1]$.

Another application is that we can recover the strict convexity result in
([LL]), as stated before. Moreover, if the manifold $X$ satisfies $c_{1}(X)=0$
or $c_{1}(X)<0$, then we can utilize Chen’s argument ([C00]) to conclude the
strict convexity of the Mabuchi functional, provided the $\varepsilon$-affine
condition (see Theorem (6.3)).

Therefore, we conjecture that the Mabuchi functional is strictly convex along
a ${\mathcal{C}}^{1,\bar{1}}$-geodesic, if it is $\varepsilon$-affine. In
fact, the convergence (equation (1.3)) further implies an $L^{2}$-estimate on
$\bar{\partial}v_{\varepsilon}$ for a sequence of smooth vector fields
$v_{\varepsilon}$ (see Theorem (6.1)). Unfortunately, there is still some
difficulties to conclude the holomorphicity of $v_{\infty}$ as the limit of
this sequence $v_{\varepsilon}$.

Finally, it might be worthy to point out that a geodesic ${\mathcal{G}}$
possibly possess more regularities than ${\mathcal{C}}^{1,\bar{1}}$, when the
Mabuchi functional is $\varepsilon$-affine along it. Hopefully, we will see
more examples about this fact, and the regularity problem will be considered
in our following projects.

Acknowledgment:  The author is very grateful to Prof. Chen and Prof. Păun who
introduced this problem, and have given continuous encouragement. He also
wants to thank Prof. Chengjian Yao, Dr. Jingchen Hu and Prof. Wei Sun for lots
of useful discussion.

##  2\. Preliminary

Suppose $\Sigma$ is an annulus in ${\mathbb{C}}$ with boundary, and $\pi$ is
the holomorphic projection from the product space $Y:=X\times\Sigma$ to $X$.
Therefore, $Y$ is a compact complex Kähler manifold with boundary. Let $\Phi$
be a quasi-plurisubharmonic function on $Y$ continuous up to the boundary.
Denote ${\mathcal{G}}$ by the closed positive $(1,1)$ current

$\pi^{*}\omega+dd^{c}\Phi$

on $Y$. We say that ${\mathcal{G}}$ is a geodesic in the space of Kähler
potential, if it is $S^{1}$-invariant in the argument direction of $\Sigma$,
and satisfies the following _Homogeneous complex Monge-Ampère_ (HCMA) equation

(2.1) ${\mathcal{G}}^{n+1}=(\pi^{*}\omega+dd^{c}\Phi)^{n+1}=0,$

in a suitable sense on $Y$.

The boundary value of $\Phi$ is required to be in the space ${\mathcal{H}}$ of
the smooth Kähler potentials. We say that ${\mathcal{G}}$ is a geodesic
connecting two points $\varphi_{0},\varphi_{1}\in{\mathcal{H}}$ if

$\Phi|_{X\times\\{0\\}}=\varphi_{0};\ \ \ \Phi|_{X\times\\{1\\}}=\varphi_{1},$

where we identify the annulus $\Sigma$ by a cylinder $[0,1]\times S^{1}$ via
the standard diffeomorphism.

It is proved by Chen ([C00]) that such a geodesic is unique with fixed
boundary value, and has the so called
${\mathcal{C}}^{1,\bar{1}}$-regularities, namely, writing ${\mathcal{G}}$
locally as

$g_{{\tau}\bar{\tau}}d{\tau}\wedge
d\bar{\tau}+\sum_{{\alpha},\beta=1}^{n}(g_{{\tau}\bar{\beta}}d{\tau}\wedge
d\bar{z}^{\beta}+g_{{\alpha}\bar{\tau}}dz^{{\alpha}}\wedge
d\bar{\tau}+g_{{\alpha}\bar{\beta}}dz^{{\alpha}}\wedge d\bar{z}^{\beta}),$

we have

$||g_{{\tau}\bar{\tau}}||_{L^{\infty}}+\sum_{{\alpha},\beta=1}^{n}(||g_{{\tau}\bar{\beta}}||_{L^{\infty}}+||g_{{\alpha}\bar{\tau}}||_{L^{\infty}}+||g_{{\alpha}\bar{\beta}}||_{L^{\infty}})<+\infty.$

In other words, there exist a uniform constant $C>0$ such that we have

$0\leq{\mathcal{G}}\leq C(\pi^{*}\omega+id{\tau}\wedge d\bar{\tau})$

on $Y$. Therefore the quasi-plurisubharmonic function $\Phi$ is of class
$C^{1,{\alpha}}$ for any ${\alpha}\in(0,1)$, and the wedge product
${\mathcal{G}}^{n+1}$ can be interpreted in the sense of Bedford and Talyor
([BT]).

There is another way to describe the domain of ${\mathcal{G}}$, which will be
useful in our later consideration on energy functionals. Let

$\Gamma:=\\{z\in{\mathbb{C}};\ \ 0\leq Rez\leq 1\\}$

be a strip domain in ${\mathbb{C}}$. Then there is a holomorphic map from
$\Gamma$ to $\Sigma$ as

${\tau}(z):=e^{z}.$

This is a branched cover of $\Sigma$, and the inverse map is multi-valued in
general.

Fortunately, ${\mathcal{G}}$ is $S^{1}$-invariant in the argument direction of
$\Sigma$. Therefore, we can select one of the branches as the inverse map of
$\tau$. Then the pull back of ${\mathcal{G}}$ under $\tau$ is the unique
solution of the HCMA equation on $\Gamma\times X$, which is independent of the
imaginary part of $z$. For this reason, the $z$-variable on $\Gamma$ can be
taken as the complex coordinate of the cylinder

$\mathcal{R}:=[0,1]\times S^{1},$

and then we can view that the solution ${\mathcal{G}}$ is actually defined on
$\mathcal{R}\times X$. Furthermore, we can identify the complex variable
$z:=t+is$ with its real part $t$ for the same reason.

###  2.1. The Mabuchi functional

It was in introduced in ([BM]) by Mabuchi the following functional on the
space ${\mathcal{H}}$

${\mathcal{M}}:=\underline{R}{\mathcal{E}}-{\mathcal{E}}^{Ric\omega}+H,$

where the constant $\underline{R}$ is the average of the scalar curvature

$\underline{R}=\frac{nc_{1}(X)\cdot[\omega]^{n-1}}{[\omega]^{n}}.$

The energy functional ${\mathcal{E}}$ is defined for any
$\varphi\in{\mathcal{H}}$ as

${\mathcal{E}}(\varphi):=\frac{1}{n+1}\sum_{i=0}^{n}\int_{X}\varphi\omega^{i}\wedge\omega_{\varphi}^{n-i}.$

The twisted energy functional ${\mathcal{E}}^{{\alpha}}$ (by a closed smooth
$(1,1)$ form ${\alpha}$) is defined as

${\mathcal{E}}^{{\alpha}}(\varphi):=\sum_{i=0}^{n-1}\int_{X}\varphi\omega^{i}\wedge\omega_{\varphi}^{n-i-1}\wedge{\alpha}.$

Finally, the entropy functional is

$H(\varphi):=\int_{X}\left(\log\frac{\omega_{\varphi}^{n}}{\omega^{n}}\right)\omega_{\varphi}^{n}.$

Suppose $\Phi$ is a $\pi^{*}\omega$-plurisubharmonic function on $Y$, which
corresponds to a geodesic ${\mathcal{G}}$. Then its restriction
$\varphi_{{\tau}}:=\Phi|_{X_{\tau}}$ on a fiber is actually a
$\omega$-plurisubharmonic function on $X_{{\tau}}$ and has the
${\mathcal{C}}^{1,\bar{1}}$-regularities. It is observed that ${\mathcal{M}}$
can be defined on such functions. Therefore, we can write the Mabuchi
functional along the geodesic ${\mathcal{G}}$ as

${\mathcal{K}}({\tau}):={\mathcal{M}}(\varphi_{\tau}),\ \ {\tau}\in\Sigma.$

Next, we introduce the following modified versions of the Mabuchi functional.
Suppose $\Psi(\tau,\cdot)=\psi_{{\tau}}(\cdot)$ is a locally bounded singular
metric on the relative canonical bundle $K_{Y/\Sigma}$, and then
$-\psi_{{\tau}}$ is a metric on the anti-canonical line bundle
$-K_{X_{\tau}}:=\bigwedge^{n}TX_{{\tau}}$. Therefore, the following is a
measure on $X$

$\mu:=e^{\psi_{{\tau}}}.$

We note that $\mu$ is absolutely continuous with respect to the Lebesgue
measure. Now define the following energy function on $\Sigma$ as

${\mathcal{K}}^{\Psi}({\tau}):=\underline{R}{\mathcal{E}}(\varphi_{{\tau}})-{\mathcal{E}}^{Ric\omega}(\varphi_{{\tau}})+\int_{X}\log\left(\frac{e^{\psi_{{\tau}}}}{\omega^{n}}\right)\omega_{\varphi_{\tau}}^{n}.$

This energy function will equal to ${\mathcal{K}}({\tau})$, if $\Psi$ is the
(unbounded) metric defined by $\omega_{\varphi_{{\tau}}}^{n}$. For any large
constant $A$, Berman-Berndtsson ([BB]) introduced the following _A-truncated
Mabuchi functional_ along the geodesic ${\mathcal{G}}$.

${\mathcal{K}}^{\Psi_{A}}({\tau}):=\underline{R}{\mathcal{E}}(\varphi_{\tau})-{\mathcal{E}}^{Ric\omega}(\varphi_{\tau})+\int_{X}\log\left(\max\left\\{\frac{\omega^{n}_{\varphi_{\tau}}}{\omega^{n}},\frac{h_{A}}{\omega^{n}}\right\\}\right)\omega_{\varphi_{\tau}}^{n},$

where $h_{A}:=e^{\chi-A}$, and $\chi$ is a fixed continuous metric on
$K_{Y/\Sigma}$ satisfying

$dd^{c}\chi\geq k_{0}(\pi^{*}\omega+dd^{c}\Phi),$

for some positive integer $k_{0}$.

The values of these energy functions
${\mathcal{K}},{\mathcal{K}}^{\Psi},{\mathcal{K}}^{\Psi_{A}}$ do not depend on
the argument part of $\Sigma$, and hence we can view that they are actually
defined on the cylinder ${\mathcal{R}}=[0,1]\times S^{1}$ by our previous
discussion. Moreover, it is proved ([BB]) that ${\mathcal{K}}^{\Psi_{A}}$ is a
convex function, and it converges to ${\mathcal{K}}$ as $A\rightarrow\infty$
by the dominated convergence theorem. Eventually, we conclude the following
convexity result ([BB], [CLP]).

######  Theorem 2.1.

The Mabuchi functional ${\mathcal{M}}$ is convex and continuous along a
${\mathcal{C}}^{1,\bar{1}}$-geodesic ${\mathcal{G}}$. That is to say,
${\mathcal{K}}$ is an $S^{1}$-invariant, convex and continuous function on
${\mathcal{R}}$.

##  3\. Energies on the $\varepsilon$-geodesics

Suppose we have two points $\varphi_{0},\varphi_{1}\in{\mathcal{H}}$. For each
$\varepsilon>0$ small enough, there exists a smooth strictly
$\pi^{*}\omega$-plurisubharmonic function $\Phi_{\varepsilon}$ on
$\Gamma\times X$ satisfying

(3.1)
${\mathcal{G}}_{\varepsilon}^{n+1}=(\pi^{*}\omega+dd^{c}\Phi_{\varepsilon})^{n+1}=\varepsilon\sqrt{-1}dt\wedge
d\bar{t}\wedge\omega^{n},$

with boundary conditions

$\Phi_{\varepsilon}(0,\cdot)=\varphi_{0}(\cdot);\ \ \
\Phi_{\varepsilon}(1,\cdot)=\varphi_{1}(\cdot).$

Then we say that
${\mathcal{G}}_{\varepsilon}:=\pi^{*}\omega+dd^{c}\Phi_{\varepsilon}$ is the
$\varepsilon$-geodesic connecting with $\varphi_{0}$ and $\varphi_{1}$. It is
proved ([C00]) that the $\varepsilon$-geodesic ${\mathcal{G}}_{\varepsilon}$
is uniformly bounded in the ${\mathcal{C}}^{1,\bar{1}}$-norm. Moreover, we
have known that $\Phi_{\varepsilon}\rightarrow\Phi$ in $C^{1,{\alpha}}$-norm
for any ${\alpha}\in(0,1)$, and also in weakly $W^{2,p}$-norm for all
$1<p<\infty$.

In general, we write the restriction of the geodesic potential on each fiber
$X_{t}:=\\{t\\}\times X$ as

$\varphi(t,\cdot):=\Phi|_{X_{t}};\ \ \
\omega_{\varphi}:={\mathcal{G}}|_{X_{t}}=\omega+dd^{c}\varphi.$

Similarly, we have for the $\varepsilon$-geodesic potential

$\varphi_{\varepsilon}(t,\cdot):=\Phi_{\varepsilon}|_{X_{t}};\ \ \
\omega_{\varepsilon}:={\mathcal{G}}_{\varepsilon}|_{X_{t}}=\omega+dd_{X}^{c}\varphi_{\varepsilon}.$

Then a standard computation shows the following equation

(3.2)
$(\pi^{*}\omega+dd^{c}\Phi_{\varepsilon})^{n+1}=\rho_{\varepsilon}\sqrt{-1}dt\wedge
d\bar{t}\wedge\omega_{\varepsilon}^{n},$

where
$\rho_{\varepsilon}:=\rho_{\varepsilon}(\varphi_{\varepsilon})=g_{t\bar{t}}-g^{\bar{\beta}{\alpha}}g_{{\alpha}\bar{t}}g_{t\bar{\beta}}$,
and hence the $\varepsilon$-geodesic equation (3.1) can be re-written as

(3.3)
$\rho_{\varepsilon}(\varphi_{\varepsilon})=\varepsilon\frac{\omega^{n}}{\omega_{\varepsilon}^{n}}.$

###  3.1. Energy and entropy

Denote ${\mathcal{K}}_{\varepsilon}:\Gamma\rightarrow\mathbb{R}$ by the
restriction of the Mabuchi functional ${\mathcal{M}}$ to the
$\varepsilon$-geodesic as

${\mathcal{K}}_{\varepsilon}(t):=\underline{R}{\mathcal{E}}(\varphi_{\varepsilon})-{\mathcal{E}}^{Ric\omega}(\varphi_{\varepsilon})+\int_{X}\log\frac{\omega^{n}_{\varepsilon}}{\omega^{n}}\omega^{n}_{\varepsilon}.$

Then the $A$-truncated Mabuchi functional along the $\varepsilon$-geodesic can
also be introduced as

${\mathcal{K}}_{\varepsilon,A}(t):=\underline{R}{\mathcal{E}}(\varphi_{\varepsilon})-{\mathcal{E}}^{Ric\omega}(\varphi_{\varepsilon})+\int_{X}\log\max\left\\{\frac{\omega_{\varepsilon}^{n}}{\omega^{n}},\frac{h_{A}}{\omega^{n}}\right\\}\omega_{\varepsilon}^{n},$

where $h_{A}:=h_{\varepsilon,A}=e^{\chi_{\varepsilon}-A}$ is a smooth volume
element whose associated curvature is greater than
$-C{\mathcal{G}}_{\varepsilon}$ for some fixed positive constant $C$. More
precisely, we construct this auxiliary element as follows.

Let $\chi_{0}$ be a smooth metric on the line bundle $K_{X}$, and $k_{0}$ be a
positive number such that $dd^{c}\chi_{0}+k_{0}\omega>0$. Then we set
$\chi_{\varepsilon}:=\pi^{*}\chi_{0}-k_{0}\Phi_{\varepsilon}$, and hence

$dd^{c}\chi_{\varepsilon}=\pi^{*}\chi_{0}+k_{0}\pi^{*}\omega-
k_{0}(\pi^{*}\omega+dd^{c}\Phi_{\varepsilon})\geq-
k_{0}{\mathcal{G}}_{\varepsilon}.$

We emphasis that the sub-index $\varepsilon$ in the notation $h_{A}$ is
omitted, since $\chi_{\varepsilon}$ is uniformly bounded in its
$C^{1,\bar{1}}$-norm, and converges uniformly to
$\chi:=\pi^{*}\chi_{0}-k_{0}\Phi$ in $C^{1,{\alpha}}$-norm for any
${\alpha}\in(0,1)$. Therefore, this omission will be harmless for our later
argument.

We note that ${\mathcal{K}}_{\varepsilon,A}$ is continuous in $t$ by the
construction. Moreover, its complex Hessian can be computed in sense of local
currents. Suppose $v$ is a locally compact supported smooth test function on
$\Gamma$

(3.4) $\displaystyle\langle dd^{c}{\mathcal{K}}_{\varepsilon,A},v\rangle$
$\displaystyle=$ $\displaystyle\underline{R}\int_{\Gamma\times
X}v(\pi^{*}\omega+dd^{c}\Phi_{\varepsilon})^{n+1}$ $\displaystyle-$
$\displaystyle\int_{\Gamma\times
X}v(\pi^{*}\omega+dd^{c}\Phi_{\varepsilon})^{n}\wedge\pi^{*}Ric(\omega)$
$\displaystyle+$ $\displaystyle\int_{\Gamma\times
X}vdd^{c}\left(\max\left\\{\log\frac{\omega_{\varepsilon}^{n}}{\omega^{n}},\log\frac{h_{A}}{\omega^{n}}\right\\}\right)\wedge(\pi^{*}\omega+dd^{c}\Phi_{\varepsilon})^{n}.$

In other words, if we take the fiberwise integral $\int_{X_{t}}$ as the push
forward operator acting on the currents from $\Gamma\times X$ to $\Gamma$, we
obtain the following equation.

(3.5) $\displaystyle dd^{c}{\mathcal{K}}_{\varepsilon,A}(t)$ $\displaystyle=$
$\displaystyle\frac{\underline{R}}{n+1}\int_{X_{t}}{\mathcal{G}}_{\varepsilon}^{n+1}-\int_{X_{t}}Ric(\omega)\wedge{\mathcal{G}}_{\varepsilon}^{n}$
$\displaystyle+$
$\displaystyle\int_{X_{t}}dd^{c}\left(\max\left\\{\log\frac{\omega_{\varepsilon}^{n}}{\omega^{n}},\log\frac{h_{A}}{\omega^{n}}\right\\}\right)\wedge{\mathcal{G}}_{\varepsilon}^{n}.$

On the one hand, we have seen locally

(3.6)
$\frac{1}{n+1}\left(\Delta_{{\mathcal{G}}_{\varepsilon}}\log{h_{A}}\right){\mathcal{G}}_{\varepsilon}^{n+1}=dd^{c}\log{h_{A}}\wedge{\mathcal{G}}_{\varepsilon}^{n}\geq-
k_{0}{\mathcal{G}}_{\varepsilon}^{n+1}.$

Next, we denote the function $f_{\varepsilon}:\Gamma\times
X\rightarrow\mathbb{R}$ by the equality

$\left.\frac{\omega^{n}_{\varepsilon}}{\omega^{n}}\right|_{X_{t}}=e^{f_{\varepsilon}(t,\cdot)},$

and introduce the following sub-level set

$P_{\varepsilon,A}:=\left\\{(t,z)\in\Gamma\times X;\ \ \
f_{\varepsilon}(t,z)>\log\frac{h_{A}}{\omega^{n}}(t,z)\right\\}.$

It is proved in ([CLP]) that there exists a positive number $c_{A}$ such that
the following inequality is satisfied in $P_{\varepsilon,A}$ (see more details
in the following sections).

(3.7)
$\left(dd^{c}f_{\varepsilon}-Ric(\omega)\right)\wedge{\mathcal{G}}_{\varepsilon}^{n}\geq-
c_{A}{\mathcal{G}}_{\varepsilon}^{n+1}.$

We emphasis that $c_{A}$ only depends on the constant $A$, the background
metric $\omega$ and the uniform upper bound of $\omega_{\varepsilon}$, and it
can be assumed to be increasing in $A$. Hence we have the following estimate
locally in $P_{\varepsilon,A+1}$.

(3.8)
$\left(\Delta_{{\mathcal{G}}_{\varepsilon}}\log\omega_{\varepsilon}^{n}\right){\mathcal{G}}_{\varepsilon}^{n+1}\geq-
c_{A+1}{\mathcal{G}}_{\varepsilon}^{n+1}.$

By a Theorem proved by Greene-Wu (see Lemma (5.2), [CLP]), we can establish
the following inequality in an open neighbourhood of each point on
$\Gamma\times X$.

(3.9)
$\left(\Delta_{{\mathcal{G}}_{\varepsilon}}\max\\{\log\omega_{\varepsilon}^{n},\log
h_{A}\\}\right){\mathcal{G}}_{\varepsilon}^{n+1}\geq-c^{\prime}_{A}{\mathcal{G}}_{\varepsilon}^{n+1},$

where $c^{\prime}_{A}:=\max\\{c_{A+1},(n+1)k_{0}\\}$. Thus we infer the
inequality

(3.10)
$\left(dd^{c}\max\left\\{\log\frac{\omega_{\varepsilon}^{n}}{\omega^{n}},\log\frac{h_{A}}{\omega^{n}}\right\\}-Ric\omega\right)\wedge{\mathcal{G}}_{\varepsilon}^{n}\geq-c^{\prime}_{A}{\mathcal{G}}_{\varepsilon}^{n+1},$

globally on $\Gamma\times X$. Combing this estimate with equation (3.5), we
have the following result.

######  Theorem 3.1 (Chen-Li-Păun).

For each positive number $A$, there is a uniform constant $C_{A}>0$ such that
the function

(3.11)
$\widetilde{{\mathcal{K}}}_{\varepsilon,A}:={\mathcal{K}}_{\varepsilon,A}(t)-\varepsilon
C_{A}t(1-t)$

is convex and continuous on $[0,1]$ for each $\varepsilon>0$ small.

One may expect that ${\mathcal{K}}_{\varepsilon,A}$ converges to the energy
${\mathcal{K}}$. If so, then we can directly conclude the convexity of
${\mathcal{M}}$ by Theorem (3.1).

The convergence of the energy parts of ${\mathcal{M}}$, i.e.
${\mathcal{E}}(\varphi_{\varepsilon})\rightarrow{\mathcal{E}}(\varphi)$ and
${\mathcal{E}}^{{\alpha}}(\varphi_{\varepsilon})\rightarrow{\mathcal{E}}^{{\alpha}}(\varphi)$
as $\varepsilon\rightarrow 0$, is indeed true ([BB], [LL]). However, there is
a rather severe difficulty: the convergence of the fiber-wise volume element

$\frac{\omega_{\varepsilon}^{n}}{\omega^{n}}\rightharpoonup\frac{\omega_{\varphi}^{n}}{\omega^{n}}$

is only known to be in the weakly $L^{p}$ sense for any $1<p<\infty$.
Unfortunately, this is not enough to conclude the convergence of the entropy
functional $H(\varphi_{\varepsilon})$.

Nevertheless, we have the following estimate on the entropy functionals.
Recall that the entropy functional along a geodesic ${\mathcal{G}}$ is defined
as

$H(\varphi):=\int_{X}{\mathfrak{f}}(\varphi)\log{\mathfrak{f}}(\varphi)\cdot\omega^{n},$

where ${\mathfrak{f}}(\varphi):=\omega_{\varphi}^{n}/\omega^{n}$, and its
truncated version is

$H_{A}(\varphi):=\int_{X}{\mathfrak{f}}(\varphi)\log{\mathfrak{f}}_{A}(\varphi)\cdot\omega^{n},$

where

${\mathfrak{f}}_{A}(\varphi):=\max\left\\{\frac{\omega_{\varphi}^{n}}{\omega^{n}},\frac{h_{A}}{\omega^{n}}\right\\}.$

As before, we omit the sub-index $\varepsilon$ in the definition of $f_{A}$ as
in $h_{A}$’s, and hope that this will be clear from the context. Then the
energy function ${\mathcal{K}}_{\varepsilon,A}$ can be re-written as

(3.12)
${\mathcal{K}}_{\varepsilon,A}(t)=\underline{R}{\mathcal{E}}(\varphi_{\varepsilon})-{\mathcal{E}}^{Ric\omega}(\varphi_{\varepsilon})+H_{A}(\varphi_{\varepsilon}).$

Finally we state the following lower semi-continuity type property (see Lemma
(4.8), [CLP]) for the truncated entropy functionals.

######  Lemma 3.2.

There exists a function $\eta:\mathbb{R}^{+}\rightarrow\mathbb{R}^{+}$ with
$\eta(x)\rightarrow 0$ as $x\rightarrow+\infty$ such that we have

(3.13) $\liminf_{\varepsilon\rightarrow 0}H_{A}(\varphi_{\varepsilon})\geq
H_{A}(\varphi)-\eta(A),$

for all $A$ large enough.

We emphasis that $\eta(A)$ is independent of $\varepsilon$ and $t$.

###  3.2. The affine energy

From now on, we assume that the Mabuchi functional ${\mathcal{M}}$ is affine
along ${\mathcal{G}}$, namely, we have ${\mathcal{K}}(t)$ is a linear function
on $[0,1]$. Denote ${\mathcal{K}}_{A}$ by the following limit for each
$t\in[0,1]$ and $A$ large

${\mathcal{K}}_{A}(t):=\limsup_{\varepsilon\rightarrow
0}{\mathcal{K}}_{\varepsilon,A}(t)=\limsup_{\varepsilon\rightarrow
0}\widetilde{{\mathcal{K}}}_{\varepsilon,A}(t),$

where $\widetilde{{\mathcal{K}}}_{\varepsilon,A}$ is defined in equation
(3.11). Then ${\mathcal{K}}_{A}$ is a convex function on the unit interval
$[0,1]$, since the $\limsup$ of a sequence of convex functions which are
locally bounded above is still convex.

Eventually, we will see that this convex function ${\mathcal{K}}_{A}$ obtained
from taking the limit of ${\mathcal{K}}_{\varepsilon,A}$ is exactly equal to
${\mathcal{K}}^{\Psi_{A}}$ (see Corollary (3.4)). First, we note that
${\mathcal{K}}_{A}$ is actually a decreasing sequence in $A$. This is because
${\mathcal{K}}_{\varepsilon,A}$ is a decreasing sequence in $A$ for each
$\varepsilon$ fixed. In fact, we have

$H_{A^{\prime}}(\varphi_{\varepsilon})\leq H_{A}(\varphi_{\varepsilon}),$

for each $t\in[0,1]$ and any $A^{\prime}\geq A$, since
$f_{A^{\prime}}(\varphi_{\varepsilon})\leq f_{A}(\varphi_{\varepsilon})$ and
$f_{\varphi}\geq 0$ in this case. Bearing this in mind, we conclude the
following result.

######  Theorem 3.3.

Suppose the Mabuchi functional ${\mathcal{M}}$ is affine along a geodesic
${\mathcal{G}}$. Then there is a positive number $A_{0}$ such that we have

${\mathcal{K}}_{A}(t)={\mathcal{K}}(t),$

for all $t\in[0,1]$ and $A\geq A_{0}$.

###### Proof.

Up to a linear function on $[0,1]$, we can assume that the Mabuchi functional
${\mathcal{M}}$ is identically zero along ${\mathcal{G}}$, namely we have on
$[0,1]$

${\mathcal{K}}(t)\equiv 0.$

The first observation is that we have

(3.14) ${\mathcal{K}}_{A}(0)=0;\ \ \ {\mathcal{K}}_{A}(1)=0,$

since the $\varepsilon$-geodesic potential $\Phi_{\varepsilon}$ coincides with
the geodesic potential $\Phi$ for each $\varepsilon$ on the boundary of
${\mathcal{R}}\times X$. Then it is easy to see that
${\mathcal{K}}_{\varepsilon,A}(0)={\mathcal{K}}_{\varepsilon,A}(1)=0$ for all
$A$ large enough.

As a convex function on $[0,1]$, ${\mathcal{K}}_{A}$ is upper semi-continuous
near the boundaries, and then we have

$\limsup_{t\rightarrow 0,1}{\mathcal{K}}_{A}(t)\leq 0.$

Thanks to the convexity again, ${\mathcal{K}}_{A}$ must be below the line
segment joining its two boundaries. Therefore, it is non-positive under our
assumption and we have

(3.15) ${\mathcal{K}}_{A}(t)\leq 0,$

for all $t\in[0,1]$. On the other hand , we have for each $t\in(0,1)$

(3.16) ${\mathcal{K}}_{A}(t)\geq{\mathcal{K}}(t)-\eta(A),$

This directly follows from Lemma (3.2), where we have proved

$\limsup_{\varepsilon\rightarrow 0}H_{A}(\varphi_{\varepsilon})\geq
H_{A}(\varphi)-\eta(A).$

Next define a new function on $[0,1]$ as

$\widetilde{{\mathcal{K}}}(t):=\limsup_{A\rightarrow+\infty}{\mathcal{K}}_{A}(t).$

For the same reason, $\widetilde{{\mathcal{K}}}$ is a convex function which
verifies the boundary condition
$\widetilde{{\mathcal{K}}}(0)=\widetilde{{\mathcal{K}}}(1)=0$ by equation
(3.14). Moreover, inequality (3.16) implies that
$\widetilde{{\mathcal{K}}}(t)\geq 0$ for each $t\in(0,1)$, since
$\eta(A)\rightarrow 0$ as $A\rightarrow+\infty$ by Lemma (3.2). Therefore, we
conclude that $\widetilde{{\mathcal{K}}}$ must be identically equal to zero
for all $t\in[0,1]$.

However, as mentioned before, ${\mathcal{K}}_{A}$ is actually a decreasing
sequence in $A$, namely, we have
${\mathcal{K}}_{A}(t)\searrow\widetilde{{\mathcal{K}}}(t)$ for each
$t\in[0,1]$. Therefore, the following inequality follows

${\mathcal{K}}_{A}(t)\geq 0,$

for each $t\in[0,1]$. Combining this with equation (3.15), we have for all $A$
large enough

${\mathcal{K}}_{A}(t)\equiv 0$

on $[0,1]$, and our result follows.

∎

In other words, the limit of the $A$-truncated Mabuchi functional on the
$\varepsilon$-geodesics will coincide with ${\mathcal{M}}$ for all $A$ large
enough, provided the linearity of the Mabuchi functional along
${\mathcal{G}}$.

###  3.3. Gap phenomenon

When the Mabuchi functional is affine along ${\mathcal{G}}$, we have seen from
([LL]) that the $A$-truncated Mabuchi functional ${\mathcal{K}}^{\Psi_{A}}$
will also coincide with ${\mathcal{K}}$ for all $A$ large. This implies

(3.17) $H_{A}(\varphi)=H(\varphi),$

for all $A$ large enough, and then we have the following corollary.

######  Corollary 3.4.

Suppose the Mabuchi functional ${\mathcal{M}}$ is affine along a geodesic
${\mathcal{G}}$. Then for all $A$ large enough, the following limit exists for
each $t\in[0,1]$ and satisfies

(3.18) $\lim_{\varepsilon\rightarrow
0}{\mathcal{K}}_{\varepsilon,A}(t)={\mathcal{K}}^{\Psi_{A}}(t)={\mathcal{K}}(t).$

###### Proof.

For each $t\in[0,1]$ fixed, we define

$\widetilde{H}_{A}(t):=\liminf_{\varepsilon\rightarrow
0}H_{A}(\varphi_{\varepsilon}).$

Thanks to Lemma (3.2), we have

(3.19) $\lim_{A\rightarrow+\infty}\widetilde{H}_{A}(t)\geq H(\varphi),$

by equation (3.17). However, $\widetilde{H}_{A}$ is actually a decreasing
sequence in $A$. This follows from its construction and the fact that
$H_{A}(\varphi_{\varepsilon})$ is decreasing for each $\varepsilon$ fixed.
Hence, we conclude

(3.20) $\widetilde{H}_{A}(t)\geq H(\varphi).$

On the other hand, Theorem (3.3) implies for all $A$ large

(3.21) $\limsup_{\varepsilon\rightarrow
0}H_{A}(\varphi_{\varepsilon})=H(\varphi)=H_{A}(\varphi).$

Combined the two inequalities above together, we have

(3.22) $H(\varphi)=\limsup_{\varepsilon\rightarrow
0}H_{A}(\varphi_{\varepsilon})\geq\liminf_{\varepsilon\rightarrow
0}H_{A}(\varphi_{\varepsilon})\geq H(\varphi),$

at each $t\in[0,1]$, and then our result follows.

∎

Moreover, equation (3.17) implies the so called “gap phenomenon” ([LL]) for
the fiber-wise volume element of the geodesic ${\mathcal{G}}$. Denote $P$ by
the following measurable subset

$P:=\left\\{(t,z)\in\Gamma\times X;\ \ \ {\mathfrak{f}}(\varphi)>0\right\\},$

and $P_{A}$ by the following sub-level set

$P_{A}:=\left\\{(t,z)\in\Gamma\times X;\ \ \
{\mathfrak{f}}(\varphi)>\frac{e^{\chi-A}}{\omega^{n}}\right\\}.$

For each $t\in[0,1]$, we introduce their fiber-wise restrictions as

$P_{t}:=P\bigcap X_{t};\ \ \ P_{A,t}:=P_{A}\bigcap X_{t}.$

Then there is a large constant $A_{0}$ such that for all $A\geq A_{0}$, and
each $t\in[0,1]$, the following equality holds

(3.23) $X_{t}=P_{A,t}\bigcup P_{t}^{c}$

up to a set of measure zero. In other words, there exists a uniform constant
${\kappa}_{0}>0$ such that either we have on each fiber $X_{t}$

(3.24) ${\mathfrak{f}}(\varphi)>{\kappa}_{0}\ \ \emph{or}\ \ \ \
{\mathfrak{f}}(\varphi)=0$

almost everywhere. We emphasis that ${\kappa}_{0}$ does not depend on $t$ or
$x$. This is called the gap phenomenon for the fiberwise volume element of the
geodesic ${\mathcal{G}}$.

##  4\. Convergence of the volume element

Recall that the $\varepsilon$-geodesic potential $\Phi_{\varepsilon}$
converges to the geodesic potential $\Phi$ in the weak $C^{1,\bar{1}}$-norm.
Fixing a fiber $X_{\tau}$, we can pick up a convergent subsequence of the
volume elements on this fiber

$f_{\ell}(x):={\mathfrak{f}}(\varphi_{\varepsilon_{{\ell}}})(x);\ \ \
f(x):={\mathfrak{f}}(\varphi)(x),$

satisfying

$f_{\ell}\rightarrow f$

weakly in $L^{p}$ for all $p\geq 1$ on $X_{\tau}$. It is interesting to know
whether we have strong convergence in $L^{p}$ for this sequence or not, and it
turns out that this is indeed the case if the Mabuchi functional is affine.

######  Theorem 4.1.

Suppose the Mabuchi functional ${\mathcal{M}}$ is affine along a geodesic
${\mathcal{G}}$. Then the fiber-wise volume element of the
$\varepsilon$-geodesic converges to the volume element of the geodesic in the
strong $L^{2}$ sense as $\varepsilon\rightarrow 0$. More precisely, we have

$\lim_{{\ell}\rightarrow+\infty}||f_{\ell}-f||_{L^{2}}=0,$

on each fiber $X_{\tau}$.

In the following, we will provide two different proves of Theorem (4.1). The
first one is more complicated, but it will give an accurate estimate for the
$L^{2}$-norm of the difference $f_{\ell}-f$ directly from the convergence of
the truncated entropies. We expect that this estimate will be useful for some
independent interests.

For the beginning, denote ${\kappa}$ by the auxiliary function on the fiber
$X_{\tau}$

(4.1) ${\kappa}(x,A):=\frac{e^{\chi-A}}{\omega^{n}}(x).$

As before, we omit the sub-index $\varepsilon_{{\ell}}$ in $\chi$ due to the
uniform control on their ${\mathcal{C}}^{1,\bar{1}}$-norms. Then the following
result holds.

######  Lemma 4.2.

Suppose we have

$\lim_{{\ell}\rightarrow\infty}\int_{X}{\mathfrak{f}}(\varphi_{\varepsilon_{\ell}})\log{\mathfrak{f}}_{A}(\varphi_{\varepsilon_{\ell}})=\int_{X}{\mathfrak{f}}(\varphi)\log{\mathfrak{f}}_{A}(\varphi).$

Then there exists a uniform constant $C$, only depending on the upper bound of
$f_{\ell}$ and $f$, satisfying

(4.2)
$\lim_{{\ell}\rightarrow\infty}\int_{X}(f_{\ell}-f)^{2}\omega^{n}\leq\left(2+\frac{8C}{{\kappa}_{0}}\right)\max_{X_{{\tau}}}{\kappa},$

for all ${\kappa}$ small enough (or $A$ large enough).

Here ${\kappa}_{0}$ is the gap of the volume element ${\mathfrak{f}}(\varphi)$
(equation (3.24)), which is a fixed constant. Therefore, Theorem (4.1)
directly follows from Lemma (4.2) if we take $A\rightarrow\infty$.

###  4.1. The maximum function

In order to prove this lemma, the first step is to investigate the following
maximum function. Define a function $F:[0,+\infty)\rightarrow\mathbb{R}$ as

$F(x):=x\log x,$

and $F(0)=0$. Then $F$ is a convex continuous function on its domain. In fact,
it is smooth in $\mathbb{R}^{+}$, and its first and second derivatives are

$F^{\prime}(x)=\log x+1;\ \ \ F^{\prime\prime}(x)=x^{-1}$

Moreover, we can truncate $F$ by a small number ${\kappa}>0$ and introduce the
following maximum function

$h_{{\kappa}}(x):=\max\\{x\log x,x\log{\kappa}\\}.$

This is also a convex and continuous function on $[0,+\infty)$, and it is
piecewise smooth in this domain. Its first derivative exists everywhere on
$\mathbb{R}^{+}$ except at the point $x={\kappa}$, and we have

(4.3)
$h^{\prime}_{{\kappa}}(x)=\left\\{\begin{array}[]{rcl}\log{\kappa},&\mbox{for}&x<{\kappa}\\\
1+\log x,&\mbox{for}&x>{\kappa}.\\\ \end{array}\right.$

Furthermore, we can also compute its second derivative on $\mathbb{R}^{+}$ as

(4.4)
$h^{\prime\prime}_{{\kappa}}(x)=\left\\{\begin{array}[]{rcl}0,&\mbox{for}&x<{\kappa}\\\
\delta({\kappa}),&\mbox{for}&x={\kappa}\\\
\frac{1}{x},&\mbox{for}&x>{\kappa},\\\ \end{array}\right.$

where $\delta({\kappa})$ is the _Dirac-delta_ function at the point
$x={\kappa}$. We note that the Fundamental Theorem of Calculus is still
satisfied for $h_{{\kappa}},h^{\prime}_{{\kappa}},h^{\prime\prime}_{{\kappa}}$
on the interval $[0,1]$, since we can take the differentiation in the sense of
the generalized derivatives.

Fixing a point $x$ on the fiber, we introduce another variable $t\in[0,1]$ and
take

$u_{t}:=tf_{\ell}+(1-t)f=at+b,$

where $a:=(f_{\ell}-f)(x)$ and $b:=f(x)$. We note that $u_{t}$ is strictly
positive and has a uniform upper bound for all $t$, ${\ell}$ and $x$. Define a
new composition function

$F_{{\ell},{\kappa}}(t):=h_{{\kappa}}(u_{t}),$

and then its derivatives can be written as

(4.5)
$F^{\prime}_{{\ell},{\kappa}}(t)=\left\\{\begin{array}[]{rcl}a\log{\kappa},&\mbox{for}&at+b<{\kappa}\\\
a\log(at+b)+a,&\mbox{for}&at+b>{\kappa};\\\ \end{array}\right.$

(4.6)
$F^{\prime\prime}_{{\ell},{\kappa}}(t)=\left\\{\begin{array}[]{rcl}0,&\mbox{for}&at+b<{\kappa}\\\
a\delta(t_{0}),&\mbox{for}&at+b={\kappa}\\\
\frac{a^{2}}{at+b},&\mbox{for}&at+b>{\kappa},\\\ \end{array}\right.$

where $t_{0}$ is determined by the equation

$at_{0}+b={\kappa}.$

In particular, we have $F^{\prime}_{{\ell},{\kappa}}(0)=a\log{\kappa}$ if
$f(x)<{\kappa}$, and $F^{\prime}_{{\ell},{\kappa}}(0)=a\log b+a$ if
$f(x)>{\kappa}$. Then the following convergence holds.

######  Lemma 4.3.

For all $A$ large enough, we have

$\lim_{{\ell}\rightarrow+\infty}\int_{X}F^{\prime}_{{\ell},{\kappa}}(0)\omega^{n}=0.$

###### Proof.

When the constant $A$ is large enough, we can assume ${\kappa}<{\kappa}_{0}/2$
on the fiber $X_{{\tau}}$, where ${\kappa}_{0}$ is the gap of the fiber-wise
volume element of ${\mathcal{G}}$ defined in equation (3.24). Then the fiber
can be completely decomposed into two parts as in equation (3.23)

$X_{{\tau}}=P_{A,{\tau}}\bigcup P_{{\tau}}^{c},$

up to a set of measure zero. Recall that the two sets can be re-written as
follows

$P_{A,{\tau}}=\\{x\in X_{{\tau}};\ \ \ f(x)>{\kappa}(x)\\};\ \ \
P^{c}_{{\tau}}:=\\{x\in X_{\tau};\ \ \ f(x)=0\\}.$

Then we have

(4.7) $\displaystyle\int_{X}F^{\prime}_{{\ell},{\kappa}}(0)$ $\displaystyle=$
$\displaystyle\int_{P_{A,{\tau}}}F^{\prime}_{{\ell},{\kappa}}(0)+\int_{P_{{\tau}}^{c}}F^{\prime}_{{\ell},{\kappa}}(0)$
$\displaystyle=$ $\displaystyle\int_{P_{A,{\tau}}}(f_{{\ell}}-f)\log
f+\int_{P_{A,{\tau}}}(f_{{\ell}}-f)+\int_{P_{\tau}^{c}}f_{{\ell}}\log{\kappa}$
$\displaystyle=$ $\displaystyle\int_{P_{A,{\tau}}}(f_{{\ell}}-f)\log
f+\int_{P_{{\tau}}^{c}}(f_{\ell}-f)\log{\kappa}+\int_{P_{{\tau}}^{c}}f\log{\kappa}$
$\displaystyle+$
$\displaystyle\int_{X}(f_{\ell}-f)-\int_{P^{c}_{\tau}}(f_{\ell}-f)$
$\displaystyle=$
$\displaystyle\int_{X}(f_{\ell}-f)\log\max\\{f,{\kappa}\\}+\int_{X}(f_{\ell}-f)-\int_{P^{c}_{{\tau}}}f_{\ell}.$

The three terms on the RHS of equation (4.7) will all converge to zero as
${\ell}\rightarrow+\infty$, since
$||f||_{L^{\infty}},||f_{\ell}||_{L^{\infty}}$ are uniformly bounded and
$f_{\ell}\rightarrow f$ weakly in $L^{p}$ for any $p\geq 1$, and then our
result follows.

∎

###  4.2. The four cases

Next we will apply the Fundamental Theorem of Calculus on the function
$F_{{\ell},{\kappa}}(t)$ and its first derivative, namely, we have

(4.8) $\displaystyle F_{{\ell},{\kappa}}(1)-F_{{\ell},{\kappa}}(0)$
$\displaystyle=$ $\displaystyle h_{{\kappa}}(f_{\ell})-h_{{\kappa}}(f)$
$\displaystyle=$
$\displaystyle\int_{0}^{t_{0}}F^{\prime}_{{\ell},{\kappa}}(t)dt+\int_{t_{0}}^{1}F^{\prime}_{{\ell},{\kappa}}(t)dt.$

A first observation is that the point $t_{0}$ may not be in the integration
domain above. Suppose the point $x$ is in the subset $P_{{\tau}}^{c}$, and
then we have $b=f(x)=0$, $a=f_{{\ell}}(x)>0$. Therefore, the point $t_{0}$
belongs to the interval $(0,1)$ if and only if $0<{\kappa}<a$.

Otherwise, we have $t_{0}\geq 1$ if $f_{\ell}(x)\leq{\kappa}$, but $t_{0}\leq
0$ is not possible in this case since it means ${\kappa}\leq 0$.

On the other hand, suppose the point $x$ is in the subset $P_{A,{\tau}}$. Then
we have $b=f(x)>{\kappa}$ and $a=f_{\ell}(x)-f(x)$. Hence $t_{0}\in(0,1)$ if
and only if $a<0$ and $b>{\kappa}>a+b=f_{{\ell}}(x)$.

Otherwise, when $a\geq 0$, we have $f_{l}(x)\geq f(x)>{\kappa}$ for $t_{0}\leq
0$; or when $a<0$, we have ${\kappa}\leq f_{\ell}(x)$ for $t_{0}\geq 1$.

In conclusion, we distinguish all situations into the following four cases:

  1. (i)

$x\in P_{A,{\tau}}$, $f_{\ell}(x)\geq f(x)>{\kappa}$ or
$f(x)>f_{\ell}(x)\geq{\kappa}$;

  2. (ii)

$x\in P^{c}_{{\tau}}$, $f(x)=0$ and $f_{\ell}(x)\leq{\kappa}$;

  3. (iii)

$x\in P^{c}_{{\tau}}$, $f(x)=0$ and $f_{\ell}(x)>{\kappa}$;

  4. (iv)

$x\in P_{A,{\tau}}$, $f_{\ell}(x)<{\kappa}<f(x)$.

We note that these four cases are disjoint from each other. Then we will
discuss case by case. For Case (i), we note that $u_{t}>{\kappa}$ and then
$h_{{\kappa}}(u_{t})=u_{t}\log u_{t}$ for all $t\in[0,1]$. Therefore, we can
further compute as follows.

(4.9) $\displaystyle F_{{\ell},{\kappa}}(1)-F_{{\ell},{\kappa}}(0)$
$\displaystyle=$ $\displaystyle
F^{\prime}_{{\ell},{\kappa}}(0)+\int_{0}^{1}\int_{0}^{t}F_{{\ell},{\kappa}}^{\prime\prime}(s)dsdt$
$\displaystyle=$ $\displaystyle
F^{\prime}_{{\ell},{\kappa}}(0)+\int_{0}^{1}\int_{0}^{t}\frac{a^{2}ds}{as+b}dt$
$\displaystyle\geq$ $\displaystyle
F^{\prime}_{{\ell},{\kappa}}(0)+\frac{a^{2}}{2C},$

where the constant $C$ is the uniform upper bound of $u_{t}$.

For $\mathbf{Case(ii)}$, we note $u_{t}\leq{\kappa}$ and then
$h_{{\kappa}}(u_{t})=u_{t}\log{\kappa}$. Hence we have

(4.10) $\displaystyle F_{{\ell},{\kappa}}(1)-F_{{\ell},{\kappa}}(0)$
$\displaystyle=$ $\displaystyle
F^{\prime}_{{\ell},{\kappa}}(0)+\int_{0}^{1}\int_{0}^{t}F_{{\ell},{\kappa}}^{\prime\prime}(s)dsdt$
$\displaystyle=$ $\displaystyle F^{\prime}_{{\ell},{\kappa}}(0).$

The two cases above are the easy ones. For the remaining cases, we will
utilise equations (4.5) and (4.6) in the computation. In $\mathbf{Case(iii)}$,
we note that $h_{{\kappa}}(u_{t})=u_{t}\log{\kappa}$ for $t\leq t_{0}$ and
$h_{{\kappa}}(u_{t})=u_{t}\log u_{t}$ for $t>t_{0}$. Recall that
$t_{0}={\kappa}/a$ in this case, and hence the computation follows.

(4.11) $\displaystyle F_{{\ell},{\kappa}}(1)-F_{{\ell},{\kappa}}(0)$
$\displaystyle=$
$\displaystyle\int_{0}^{{\kappa}/a}F^{\prime}_{{\ell},{\kappa}}(t)dt+\int_{{\kappa}/a}^{1}F^{\prime}_{{\ell},{\kappa}}(t)dt$
$\displaystyle=$
$\displaystyle\frac{{\kappa}}{a}F^{\prime}_{{\ell},{\kappa}}(0)+\int_{0}^{{\kappa}/a}\int_{0}^{t}F^{\prime\prime}_{{\ell},{\tau}}(s)dsdt+\int_{{\kappa}/a}^{1}F_{{\ell},{\kappa}}^{\prime}(t)dt$
$\displaystyle=$ $\displaystyle
F^{\prime}_{{\ell},{\kappa}}(0)+\int_{{\kappa}/a}^{1}\left\\{a+\int_{{\kappa}/a}^{t}F^{\prime\prime}_{{\ell},{\kappa}}(s)ds\right\\}dt$
$\displaystyle=$ $\displaystyle
F^{\prime}_{{\ell},{\kappa}}(0)+(a-{\kappa})+\int_{{\kappa}/a}^{1}\int_{{\kappa}/a}^{t}\frac{a^{2}ds}{as+b}dt$
$\displaystyle\geq$ $\displaystyle
F^{\prime}_{{\ell},{\kappa}}(0)+(a-{\kappa})+\frac{a^{2}}{2C}(1-{\kappa}/a)^{2}$
$\displaystyle\geq$ $\displaystyle
F^{\prime}_{{\ell},{\kappa}}(0)+\frac{a^{2}}{2C}+a(1-{\kappa}/C)-{\kappa}.$

Recall that $a>0$ in this case, and then we have

(4.12) $F_{{\ell},{\kappa}}(1)-F_{{\ell},{\kappa}}(0)\geq
F^{\prime}_{{\ell},{\kappa}}(0)+\frac{a^{2}}{2C}-{\kappa},$

for all ${\kappa}$ small enough. Finally, the most difficult one is
$\mathbf{Case(iv)}$. As before, we first note that
$h_{{\kappa}}(u_{t})=u_{t}\log u_{t}$ for $t\leq t_{0}$ and
$h_{{\kappa}}(u_{t})$ and $h_{{\kappa}}(u_{t})=u_{t}\log{\kappa}$ for
$t>t_{0}$. Then we compute in a similar way.

(4.13) $\displaystyle F_{{\ell},{\kappa}}(1)-F_{{\ell},{\kappa}}(0)$
$\displaystyle=$
$\displaystyle\int_{0}^{\frac{{\kappa}-b}{a}}F^{\prime}_{{\ell},{\kappa}}(t)dt+\int_{\frac{{\kappa}-b}{a}}^{1}F^{\prime}_{{\ell},{\kappa}}(t)dt$
$\displaystyle=$
$\displaystyle\frac{{\kappa}-b}{a}F^{\prime}_{{\ell},{\kappa}}(0)+\int_{0}^{\frac{{\kappa}-b}{a}}\int_{0}^{t}F^{\prime\prime}_{{\ell},{\tau}}(s)dsdt+\int_{\frac{{\kappa}-b}{a}}^{1}F_{{\ell},{\kappa}}^{\prime}(t)dt$
$\displaystyle=$ $\displaystyle
F^{\prime}_{{\ell},{\kappa}}(0)+\int_{0}^{\frac{{\kappa}-b}{a}}\int_{0}^{t}\frac{a^{2}ds}{as+b}dt+\int^{1}_{\frac{{\kappa}-b}{a}}\left\\{\int_{0}^{\frac{{\kappa}-b}{a}}F^{\prime\prime}_{{\ell},{\kappa}}(s)ds+a\right\\}$
$\displaystyle\geq$ $\displaystyle
F^{\prime}_{{\ell},{\kappa}}(0)+a+b+\frac{({\kappa}-b)^{2}}{2C}+\frac{({\kappa}-b)(a+b-{\kappa})}{C}-{\kappa}.$

Recall that we have $a+b=f_{\ell}(x)>0$, ${\kappa}-b={\kappa}-f(x)<0$ and
$a+b-{\kappa}=f_{\ell}(x)-{\kappa}<0$. Hence the following estimate holds.

(4.14) $\displaystyle F_{{\ell},{\kappa}}(1)-F_{{\ell},{\kappa}}(0)$
$\displaystyle\geq$ $\displaystyle
F^{\prime}_{{\ell},{\kappa}}(0)+\frac{({\kappa}-b)^{2}}{2C}-{\kappa}$
$\displaystyle\geq$ $\displaystyle
F^{\prime}_{{\ell},{\kappa}}(0)+\frac{{\kappa}_{0}^{2}}{8C}-{\kappa}.$

The last inequality in equation (4.14) is because that we have picked up
${\kappa}<{\kappa}_{0}/2$, and $P_{A,{\tau}}$ is actually the set where
$f(x)>0$ on the fiber which is equal to

$\\{x\in X_{{\tau}};\ \ \ f(x)>{\kappa}_{0}\\}$

up to a set of measure zero by the gap phenomenon.

Combining with equations (4.9) - (4.14) above, we conclude the following
inequality after taking the integral on $X_{{\tau}}$.

(4.15)
$\displaystyle\int_{X}(h_{{\kappa}}(f_{\ell})-h_{{\kappa}}(f))\omega^{n}$
$\displaystyle\geq$
$\displaystyle\int_{X}F^{\prime}_{{\ell},{\kappa}}(0)\omega^{n}-\max_{X_{{\tau}}}{\kappa}$
$\displaystyle+$
$\displaystyle\frac{1}{2C}\int_{\left(P^{c}_{{\tau}}\bigcap\\{f_{\ell}\leq{\kappa}\\}\right)^{c}\bigcap\left(P_{A,{\tau}}\bigcap\\{f_{l}<{\kappa}<f\\}\right)^{c}}(f_{\ell}-f)^{2}\omega^{n}$
$\displaystyle+$
$\displaystyle\frac{1}{8C}{\kappa}_{0}^{2}\mu\left(P_{A,{\tau}}\bigcap\\{f_{l}<{\kappa}<f\\}\right).$

Then we are ready to prove the main theorem in this section.

###### Proof of Lemma (4.2).

By our choice on the function ${\kappa}$ and the maximum function
$h_{{\kappa}}$, it follows

(4.16)
$H_{A}(\varphi_{\varepsilon_{\ell}})-H_{A}(\varphi)=\int_{X}(h_{{\kappa}}(f_{\ell})-h_{{\kappa}}(f))\omega^{n}.$

Thanks to Corollary (3.4), the LHS of equation (4.15) converges to zero as
${\ell}\rightarrow+\infty$. Meanwhile, our Lemma (4.3) implies the first term
on the RHS of equation (4.15) also converges to zero. Therefore, it implies

(4.17)
$\frac{8C}{{\kappa}_{0}^{2}}(\max_{X_{\tau}}{\kappa})\geq\lim_{{\ell}\rightarrow+\infty}\mu\left(P_{A,{\tau}}\bigcap\\{f_{l}<{\kappa}<f\\}\right).$

Furthermore, we note that the two subsets
$P_{A,{\tau}}\bigcap\\{f_{l}<{\kappa}<f\\}$ and
$P^{c}_{{\tau}}\bigcap\\{f_{\ell}\leq{\kappa}\\}$ are mutually disjoint. Then
the third term on the RHS of equation (4.15) can be decomposed into the
following three parts.

$\int_{X}(f_{\ell}-f)^{2}-\int_{P^{c}_{{\tau}}\bigcap\\{f_{\ell}\leq{\kappa}\\}}(f_{\ell}-f)^{2}-\int_{P_{A,{\tau}}\bigcap\\{f_{l}<{\kappa}<f\\}}(f_{\ell}-f)^{2}.$

The first negative term in the equation above can be estimated as

(4.18)
$\int_{P^{c}_{{\tau}}\bigcap\\{f_{\ell}\leq{\kappa}\\}}(f_{\ell}-f)^{2}\leq\max_{X_{\tau}}{\kappa}^{2},$

and the second negative term can be estimated by equation (4.17) as

(4.19) $\int_{P_{A,{\tau}}\bigcap\\{f_{l}<{\kappa}<f\\}}(f_{\ell}-f)^{2}\leq
C^{2}\mu\left(P_{A,{\tau}}\bigcap\\{f_{l}<{\kappa}<f\\}\right).$

Combing with equations (4.16) - (4.20) and take the limit in ${\ell}$, we
eventually conclude the following estimate

(4.20)
$\left(2+\frac{8C^{3}}{{\kappa}_{0}}\right)\max_{X_{\tau}}{\kappa}\geq\lim_{{\ell}\rightarrow+\infty}\int_{X}(f_{\ell}-f)^{2}\omega^{n},$

and then our result follows.

∎

###  4.3. Yet another proof

From the previous argument, we can see that the $L^{2}$-convergence of the
difference $(f_{\ell}-f)$ actually follows from the convergence of the entropy
as

(4.21)
$\lim_{{\ell}\rightarrow\infty}\int_{X}{\mathfrak{f}}(\varphi_{\varepsilon_{\ell}})\log{\mathfrak{f}}(\varphi_{\varepsilon_{\ell}})=\int_{X}{\mathfrak{f}}(\varphi)\log{\mathfrak{f}}(\varphi),$

but this is a direct consequence of our Corollary (3.4) as follows.

######  Corollary 4.4.

Suppose the Mabuchi functional is affine along a geodesic ${\mathcal{G}}$.
Then on each fiber $X_{{\tau}},{\tau}\in[0,1]$, we have

$\lim_{\varepsilon\rightarrow 0}H(\varphi_{\varepsilon})=H(\varphi).$

###### Proof.

During the proof of Corollary (3.4), we have seen that
$H_{A}(\varphi_{\varepsilon})$ is decreasing in $A$ for each $\varepsilon$
fixed. Moreover, Theorem (3.3) and the gap phenomenon imply that

$\limsup_{\varepsilon\rightarrow
0}H_{A}(\varphi_{\varepsilon})=H_{A}(\varphi)=H(\varphi),$

for all $A$ large enough. Therefore, we conclude the following inequality

(4.22) $\limsup_{\varepsilon\rightarrow
0}H(\varphi_{\varepsilon})\leq\limsup_{\varepsilon\rightarrow
0}H_{A}(\varphi_{\varepsilon})=H(\varphi).$

On the other hand, by the lower semi-continuity property of the entropy
functional, we have

(4.23) $\liminf_{\varepsilon\rightarrow 0}H(\varphi_{\varepsilon})\geq
H(\varphi),$

and then our result follows.

∎

In particular, the above corollary implies the convergence of the Mabuchi
functional along the $\varepsilon$-geodesic, namely, we have

$\lim_{\varepsilon\rightarrow
0}{\mathcal{K}}_{\varepsilon}(t)={\mathcal{K}}(t),$

for each $t\in[0,1]$.

As a simple application of Theorem (4.1), the fiber-wise volume element
$f_{\ell}$ of the $\varepsilon$-geodesic converges to the volume element $f$
of the geodesic in measure. Moreover, thanks to the Riesz-Lebesgue Theorem, we
have

$f_{\ell}\rightarrow f$

almost everywhere on each fiber, possibly after passing to a subsequence.

##  5\. The $\varepsilon$-affine energy and non-degneracy

Suppose the Mabuchi functional ${\mathcal{M}}$ is affine along a geodesic
${\mathcal{G}}$. For the convergence
$\widetilde{{\mathcal{K}}}_{\varepsilon,A}\rightarrow{\mathcal{K}}_{A}$ of a
sequence of convex functions, the first derivative
$\widetilde{{\mathcal{K}}}^{\prime}_{\varepsilon,A}(t)$ also converges
uniformly to the slope $k:={\mathcal{K}}^{\prime}_{A}(t)$ on the closed
interval $[\delta,1-\delta]$ for any $\delta>0$ small. However, the boundary
behaviour of $\widetilde{{\mathcal{K}}}^{\prime}_{\varepsilon,A}(t)$ is
unclear in general. Therefore, we impose the following condition on the
boundary of ${\mathcal{G}}$.

######  Definition 5.1.

The Mabuchi functional ${\mathcal{M}}$ is essentially affine along a geodesic
${\mathcal{G}}$, if ${\mathcal{K}}(t)$ is a linear function on $[0,1]$ with
slope $k$ and we have

(5.1)
$-\int_{X}\dot{\varphi}_{t}(R_{\varphi_{t}}-\underline{R})\omega_{\varphi_{t}}^{n}=k,$

at $t=0,1$.

It is proved in Berman-Berndtsson ([BB]) that the one side inequality of
equation (5.1) always holds at the two boundaries.

$-\int_{X}\dot{\varphi}_{0}(R_{\varphi_{0}}-\underline{R})\omega_{\varphi_{0}}^{n}\leq
k,$

and

$-\int_{X}\dot{\varphi}_{1}(R_{\varphi_{1}}-\underline{R})\omega_{\varphi_{1}}^{n}\geq
k.$

As we have seen before, the potential $\varphi_{\varepsilon}(t,\cdot)$ of the
$\varepsilon$-geodesic converges to the potential $\varphi(t,\cdot)$ of the
geodesic ${\mathcal{G}}$ in $C^{1,{\alpha}}$-norm for each ${\alpha}\in(0,1)$
on $\Gamma\times X$. Therefore, we have

(5.2) $\lim_{\varepsilon\rightarrow
0}\int_{X}\dot{\varphi}_{\varepsilon}(R_{\varphi_{\varepsilon}}-\underline{R})\omega_{\varphi_{\varepsilon}}^{n}=\int_{X}\dot{\varphi}(R_{\varphi}-\underline{R})\omega_{\varphi}^{n},$

at the boundaries $t=0,1$, since $\omega_{\varepsilon}$ coincides with
$\omega_{\varphi}$ at these two boundaries. Moreover, we note that the volume
element $\omega_{\varepsilon}^{n}$ varies smoothly near the boundaries, and
then ${\mathfrak{f}}_{A}(\varphi_{\varepsilon})$ will keep to be the same as
${\mathfrak{f}}(\varphi_{\varepsilon})$ for all $A$ large in a small
neighbourhood near the boundaries.

Therefore, the $A$-truncated Mabuchi functional along
${\mathcal{G}}_{\varepsilon}$ is equal to the Mabuchi functional along
${\mathcal{G}}_{\varepsilon}$ in a small neighbourhood of the two boundaries.
That is to say, there exist a small number $\delta>0$, possibly depends on
$\varepsilon$ and $A$, such that we have

(5.3) ${\mathcal{K}}_{\varepsilon,A}(t)={\mathcal{K}}_{\varepsilon}(t),$

for all $t\in[0,\delta)\bigcup(1-\delta,1]$. Therefore, we have for all $t$ in
this small interval

(5.4)
${\mathcal{K}}^{\prime}_{\varepsilon,A}(t)={\mathcal{K}}^{\prime}_{\varepsilon}(t)=-\int_{X}\dot{\varphi}_{\varepsilon}(R_{\varphi_{\varepsilon}}-\underline{R})\omega_{\varphi_{\varepsilon}}^{n}.$

In particular, we conclude the following result.

######  Lemma 5.2.

Suppose the Mabuchi functional ${\mathcal{M}}$ is essentially affine along a
geodesic ${\mathcal{G}}$ with slope $k$. Then
${\mathcal{K}}^{\prime}_{\varepsilon,A}$ converges uniformly to $k$ on $[0,1]$
for all $A$ large enough.

###### Proof.

Combing with equation (5.2) and (5.4), we have for each $A$ large

(5.5) $\lim_{\varepsilon\rightarrow
0}\widetilde{\mathcal{K}}^{\prime}_{\varepsilon,A}(t)=\lim_{\varepsilon\rightarrow
0}{\mathcal{K}}^{\prime}_{\varepsilon,A}(t)=k,$

for $t=0,1$. Moreover, the function
$\widetilde{\mathcal{K}}_{\varepsilon,A}(t)$ is convex on the interval
$[0,1]$, and then we have the control

$\widetilde{\mathcal{K}}_{\varepsilon,A}^{\prime}(0)\leq\widetilde{\mathcal{K}}^{\prime}_{\varepsilon,A}(t)\leq\widetilde{\mathcal{K}}^{\prime}_{\varepsilon,A}(1),$

for all $t\in(0,1)$. Therefore,
$\widetilde{\mathcal{K}}^{\prime}_{\varepsilon,A}(t)\rightarrow k$ uniformly
as $\varepsilon\rightarrow 0$.

∎

However, this condition is still too difficult to handle in the application.
Therefore, we will introduce an even stronger one based on the
$\varepsilon$-geodesic as follows.

######  Definition 5.3.

The Mabuchi functional ${\mathcal{M}}$ is $\varepsilon$-affine along a
geodesic ${\mathcal{G}}$, if ${\mathcal{K}}(t)$ is a linear function on
$[0,1]$ and we have

${\mathcal{K}}_{\varepsilon}^{\prime}(1)-{\mathcal{K}}_{\varepsilon}^{\prime}(0)=O(\varepsilon),$

where ${\mathcal{K}}_{\varepsilon}$ is the Mabuchi functional along the
$\varepsilon$-geodesic ${\mathcal{G}}_{\varepsilon}$.

For the Mabuchi functional ${\mathcal{M}}$ along ${\mathcal{G}}$, it is easy
to see that the $\varepsilon$-affine is a stronger condition than the
essentially affine. Therefore, Lemma (5.2) implies that
${\mathcal{K}}^{\prime}_{\varepsilon,A}$ converges to the constant slope $k$
uniformly on $[0,1]$ if ${\mathcal{M}}$ is $\varepsilon$-affine. Moreover, we
have the following result under this assumption.

######  Theorem 5.4.

Suppose the Mabuchi functional ${\mathcal{M}}$ is $\varepsilon$-affine along a
geodesic ${\mathcal{G}}$. Then ${\mathcal{G}}$ is uniformly fiberwise non-
degenerate, namely, there exists a uniform constant ${\kappa}_{1}>0$ such that

${\mathcal{G}}|_{X_{t}}>{\kappa}_{1}\omega,$

for almost everywhere $t\in[0,1]$.

We emphasis that the constant ${\kappa}_{1}$ does not depend on $t$. Before
moving on, we need to recall and improve some computations in [CLP].

###  5.1. Computations

We will take a closer look at equation (3.7), and try to evaluate the lower
bound of the following $(n+1,n+1)$ form on $P_{\varepsilon,A}$

$(dd^{c}f_{\varepsilon}-Ric(\omega))\wedge{\mathcal{G}}_{\varepsilon}^{n}.$

Locally near a point $p\in\Gamma\times X$, we write the $\varepsilon$-geodesic
as follows

${\mathcal{G}}_{\varepsilon}:=g_{t\bar{t}}dt\wedge
d\bar{t}+\sum_{{\alpha},\beta=1}^{n}\left(g_{t\bar{\beta}}dt\wedge
d\bar{z}^{\beta}+g_{{\alpha}\bar{t}}dz^{{\alpha}}\wedge
d\bar{t}+g_{{\alpha}\bar{\beta}}dz^{{\alpha}}\wedge d\bar{z}^{\beta}\right).$

This is a Kähler metric on $\Gamma\times X$, and its restriction on the fiber
$X_{t}:=\\{t\\}\times X$ can be written as

$\omega_{\varepsilon}(t,\cdot):={\mathcal{G}}_{\varepsilon}|_{X_{t}}=\sum_{{\alpha},\beta=1}^{n}g_{{\alpha}\bar{\beta}}dz^{{\alpha}}\wedge
d\bar{z}^{\beta}.$

Up to a change of holomorphic coordinates on $X_{t}$, we can assume

(5.6) $g_{{\alpha}\bar{\beta}}=\delta_{{\alpha}\beta};\ \ \ dg=0$

at this particular point $p$. Near this point, the $\varepsilon$-geodesic
equation can be re-written as

(5.7)
$\rho_{\varepsilon}:=g_{t\bar{t}}-g^{\bar{\beta}{\alpha}}g_{{\alpha}\bar{t}}g_{t\bar{\beta}}=\varepsilon
e^{-f_{\varepsilon}}.$

Then we introduce another $(1,1)$ form $\chi_{\varepsilon}$ defined by the
following equation

$\chi_{\varepsilon}:={\mathcal{G}}_{\varepsilon}-\rho_{\varepsilon}\sqrt{-1}dt\wedge
d\bar{t}.$

This $(1,1)$-form may not be closed anymore, but it is still positive definite
on each fiber $X_{t}$, and satisfies

$\chi_{\varepsilon}^{n+1}=0.$

Hence we have

(5.8) $\displaystyle{\mathcal{G}}_{\varepsilon}^{n}$ $\displaystyle=$
$\displaystyle\chi_{\varepsilon}^{n}+n\rho_{\varepsilon}\sqrt{-1}dt\wedge
d\bar{t}\wedge\chi_{\varepsilon}^{n-1}$ $\displaystyle=$
$\displaystyle\chi_{\varepsilon}^{n}+n\rho_{\varepsilon}\sqrt{-1}dt\wedge
d\bar{t}\wedge{\mathcal{G}}_{\varepsilon}^{n-1}$

The first factor

$n\rho_{\varepsilon}(dd^{c}f_{\varepsilon}-Ric(\omega))\wedge\sqrt{-1}dt\wedge
d\bar{t}\wedge{\mathcal{G}}_{\varepsilon}^{n-1}$

can be computed as follows.

(5.9) $n\rho_{\varepsilon}dd^{c}f_{\varepsilon}\wedge\sqrt{-1}dt\wedge
d\bar{t}\wedge{\mathcal{G}}_{\varepsilon}^{n-1}=\varepsilon(\Delta_{\varepsilon}f_{\varepsilon})\sqrt{-1}dt\wedge
d\bar{t}\wedge\omega^{n},$

and

(5.10) $-n\rho_{\varepsilon}Ric(\omega)\wedge\sqrt{-1}dt\wedge
d\bar{t}\wedge{\mathcal{G}}_{\varepsilon}^{n-1}=-\varepsilon(\mathop{\rm
tr}\nolimits_{\omega_{\varepsilon}}Ric\omega)\sqrt{-1}dt\wedge
d\bar{t}\wedge\omega^{n}.$

We note that there exist a constant $c_{A}$ such that we have

$\mathop{\rm tr}\nolimits_{\omega_{\varepsilon}}Ric\omega<c_{A},$

on $P_{\varepsilon,A}$, since the eigenvalues of $\omega_{\varepsilon}$ are
bounded from below (and above) by a uniform constant on this set. Therefore,
we conclude the inequality

(5.11) $\displaystyle
n\rho_{\varepsilon}(dd^{c}f_{\varepsilon}-Ric(\omega))\wedge\sqrt{-1}dt\wedge
d\bar{t}\wedge{\mathcal{G}}_{\varepsilon}^{n-1}$ $\displaystyle\geq$
$\displaystyle(\Delta_{\varepsilon}f_{\varepsilon}){\mathcal{G}}_{\varepsilon}^{n+1}-c_{A}{\mathcal{G}}_{\varepsilon}^{n+1}.$

Next, we compute the second factor as follows. Introduce the following vector
field on $\Gamma\times X$ as

$v:=\frac{\partial}{\partial
t}-g^{\bar{\beta}{\alpha}}g_{t\bar{\beta}}\frac{\partial}{\partial
z^{{\alpha}}}.$

Then one observes that this vector field $v$ generates the kernel of the
$(1,1)$-form $\chi_{\varepsilon}$. Hence we have

(5.12)
$(dd^{c}f_{\varepsilon}-Ric(\omega))\wedge\chi_{\varepsilon}^{n}=(dd^{c}f_{\varepsilon}-Ric(\omega))(v,\bar{v})\sqrt{-1}dt\wedge
d\bar{t}\wedge\chi_{\varepsilon}^{n}$

Then the goal is to compute the lower bound of the following term

$\partial\bar{\partial}\log\mathop{\rm
det}\nolimits(g_{{\alpha}\bar{\beta}})(v,\bar{v}).$

At this point $p$, a standard computation shows the following equation (here
we are using the Einstein summation convention)

(5.13) $\displaystyle\partial\bar{\partial}\log\mathop{\rm
det}\nolimits(g_{{\alpha}\bar{\beta}})(v,\bar{v})$ $\displaystyle=$
$\displaystyle
g_{t\bar{t},{\alpha}\bar{\alpha}}-\sum_{{\alpha},\beta}|g_{t\bar{\beta},{\alpha}}|^{2}$
$\displaystyle-$ $\displaystyle
g_{t\bar{\gamma}}g_{\gamma\bar{t},{\alpha}\bar{\alpha}}-g_{\gamma\bar{t}}g_{t\bar{\gamma},{\alpha}\bar{\alpha}}$
$\displaystyle+$ $\displaystyle
R_{{\alpha}\bar{\gamma}}g_{t\bar{\alpha}}g_{\gamma\bar{t}}$

Taking the Laplacian $\Delta_{\varepsilon}$ with respect to the metric
$\omega_{\varepsilon}$ on the both sides of equation (5.7), we have

(5.14) $\displaystyle
g_{t\bar{t},{\alpha}\bar{\alpha}}-\varepsilon\Delta_{\varepsilon}(e^{-f_{\varepsilon}})$
$\displaystyle=$
$\displaystyle\sum_{{\alpha},p}|g_{p\bar{t},\bar{\alpha}}|^{2}+\sum_{\beta,q}|g_{t\bar{q},\bar{\beta}}|^{2}$
$\displaystyle-$ $\displaystyle R_{q\bar{p}}g_{p\bar{t}}g_{t\bar{q}}$
$\displaystyle+$ $\displaystyle
g_{t\bar{p}}g_{p\bar{t},{\alpha}\bar{\alpha}}+g_{p\bar{t}}g_{t\bar{p},{\alpha}\bar{\alpha}}.$

Combing with the two equations above, it turns out that

(5.15) $\partial\bar{\partial}\log\mathop{\rm
det}\nolimits(g_{{\alpha}\bar{\beta}})(v,\bar{v})=||\bar{\partial}_{X}v||^{2}_{\varepsilon}+\varepsilon
e^{-f_{\varepsilon}}|\nabla_{\varepsilon}f_{\varepsilon}|^{2}-\varepsilon
e^{-f_{\varepsilon}}(\Delta_{\varepsilon}f_{\varepsilon}).$

Furthermore, we can improve the above equality as follows. First, we observe

$\bar{\partial}_{X}v=-g^{\bar{\beta}{\alpha}}g_{t\bar{\beta},\bar{\lambda}}d\bar{z}^{\lambda}\otimes\frac{\partial}{\partial
z^{{\alpha}}}.$

In other words, it can be written in tensors as

(5.16)
$\bar{\partial}_{\lambda}v^{{\alpha}}=-g_{t\bar{\alpha},\bar{\lambda}};\ \ \
\bar{\partial}_{\lambda}v^{t}=0,$

at the point $p$. Moreover, we have in the time direction

(5.17)
$\bar{\partial}_{t}v^{\gamma}=-g_{t\bar{t},\bar{\gamma}}+g_{t\bar{\beta}}g_{\beta\bar{\gamma},\bar{t}};\
\ \ \bar{\partial}_{t}v^{t}=0.$

By differentiating equation (5.7) once we get

(5.18) $\varepsilon
e^{-f_{\varepsilon}}(\partial_{{\alpha}}f_{\varepsilon})=g_{t\bar{t},{\alpha}}-g_{t\bar{p}}g_{p\bar{t},{\alpha}}-g_{p\bar{t}}g_{t\bar{p},{\alpha}},$

and similarly

(5.19) $\varepsilon
e^{-f_{\varepsilon}}(\partial_{\bar{\beta}}f_{\varepsilon})=g_{t\bar{t},\bar{\beta}}-g_{t\bar{q}}g_{q\bar{t},\bar{\beta}}-g_{q\bar{t}}g_{t\bar{q},\bar{\beta}}.$

Hence we have

(5.20) $\displaystyle(\varepsilon
e^{-f_{\varepsilon}})^{2}|\nabla_{\varepsilon}f_{\varepsilon}|^{2}$
$\displaystyle=$ $\displaystyle
g^{\bar{\beta}{\alpha}}\left\\{(g_{t\bar{t},{\alpha}}-g_{p\bar{t}}g_{{\alpha}\bar{p},t})-g_{t\bar{p}}g_{p\bar{t},{\alpha}}\right\\}$
$\displaystyle\cdot\left\\{(g_{t\bar{t},\bar{\beta}}-g_{t\bar{q}}g_{q\bar{\beta},\bar{t}})-g_{q\bar{t}}g_{t\bar{q},\bar{\beta}}\right\\}$
$\displaystyle=$ $\displaystyle
g^{\bar{\beta}{\alpha}}(\partial_{t}v^{\bar{\alpha}}-g_{t\bar{p}}\partial_{p}v^{\bar{\alpha}})(\bar{\partial}_{t}v^{\beta}-g_{q\bar{t}}\bar{\partial}_{q}v^{\beta})$
$\displaystyle=$
$\displaystyle\sum_{\beta}|\bar{\partial}_{t}v^{\beta}|^{2}+g_{t\bar{\mu}}g_{\lambda\bar{t}}\partial_{\mu}v^{\bar{\beta}}\bar{\partial}_{\lambda}v^{\beta}$
$\displaystyle-$ $\displaystyle
g_{\mu\bar{t}}\partial_{t}v^{\bar{\beta}}\bar{\partial}_{\mu}v^{\beta}-g_{t\bar{\lambda}}\bar{\partial}_{t}v^{\beta}\partial_{\lambda}v^{\bar{\beta}}.$

On the other side, we can compute the inverse matrix of
${\mathcal{G}}_{\varepsilon}$ at the point $p$. First notice that

$\mathop{\rm det}\nolimits{\mathcal{G}}_{\varepsilon}(p)=\varepsilon
e^{-f_{\varepsilon}(p)},$

since $g_{{\alpha}\bar{\beta}}=\delta_{{\alpha}\beta}$ at this point. Then a
standard calculation shows the following equations.

${\mathcal{G}}_{\varepsilon}^{\bar{t}t}=(\varepsilon
e^{-f_{\varepsilon}})^{-1};\ \ \
{\mathcal{G}}_{\varepsilon}^{\bar{t}p}=-(\varepsilon
e^{-f_{\varepsilon}})^{-1}g_{t\bar{p}},$

and also

(5.21) ${\mathcal{G}}_{\varepsilon}^{\bar{p}q}=\delta_{pq}+(\varepsilon
e^{-f_{\varepsilon}})^{-1}g_{p\bar{t}}g_{t\bar{q}}$

Therefore, the four terms on the RHS of equation (5.20) can be re-written as

(5.22) $(\varepsilon
e^{-f_{\varepsilon}})^{-1}\sum_{\beta}|\bar{\partial}_{t}v^{\beta}|^{2}={\mathcal{G}}_{\varepsilon}^{\bar{t}t}g_{{\alpha}\bar{\beta}}\bar{\partial}_{t}v^{{\alpha}}\partial_{t}v^{\bar{\beta}};$
(5.23) $-(\varepsilon
e^{-f_{\varepsilon}})^{-1}g_{\mu\bar{t}}\partial_{t}v^{\bar{\beta}}\bar{\partial}_{\mu}v^{\beta}={\mathcal{G}}_{\varepsilon}^{\bar{\mu}t}g_{{\alpha}\bar{\beta}}\bar{\partial}_{\mu}v^{{\alpha}}\partial_{t}v^{\bar{\beta}};$
(5.24) $(\varepsilon
e^{-f_{\varepsilon}})^{-1}g_{t\bar{\mu}}g_{\lambda\bar{t}}\partial_{\mu}v^{\bar{\beta}}\bar{\partial}_{\lambda}v^{\beta}={\mathcal{G}}_{\varepsilon}^{\bar{\lambda}\mu}g_{{\alpha}\bar{\beta}}\bar{\partial}_{\lambda}v^{{\alpha}}\partial_{\mu}v^{\bar{\beta}}-g_{{\alpha}\bar{\beta}}\bar{\partial}_{\lambda}v^{{\alpha}}\partial_{\lambda}v^{\bar{\beta}}.$

Combining with equations (5.15) - (5.24) above, we eventually derive the
following equality.

######  Lemma 5.5.

We have

(5.25) $\displaystyle\partial\bar{\partial}\log\mathop{\rm
det}\nolimits(g_{{\alpha}\bar{\beta}})(v,\bar{v})$ $\displaystyle=$
$\displaystyle||\bar{\partial}_{X}v||^{2}_{\omega_{\varepsilon}}+\varepsilon
e^{-f_{\varepsilon}}|\nabla_{\varepsilon}f_{\varepsilon}|^{2}-\varepsilon
e^{-f_{\varepsilon}}(\Delta_{\varepsilon}f_{\varepsilon})$ $\displaystyle=$
$\displaystyle||\bar{\partial}v||^{2}_{{\mathcal{G}}_{\varepsilon}}-\varepsilon
e^{-f_{\varepsilon}}(\Delta_{\varepsilon}f_{\varepsilon}).$

Combing with Lemma (5.5) and equations (5.11), (5.12), we infer as equation
(3.10) the following

(5.26)
$\displaystyle\left(dd^{c}\max\left\\{\log\frac{\omega_{\varepsilon}^{n}}{\omega^{n}},\log\frac{h_{A}}{\omega^{n}}\right\\}-Ric\omega\right)\wedge{\mathcal{G}}_{\varepsilon}^{n}+c^{\prime}_{A}{\mathcal{G}}_{\varepsilon}^{n+1}$
$\displaystyle\geq$
$\displaystyle\chi_{P_{\varepsilon,A}}\left(||\bar{\partial}_{X}v||^{2}_{\omega_{\varepsilon}}+\varepsilon
e^{-f_{\varepsilon}}|\nabla_{\varepsilon}f_{\varepsilon}|^{2}\right)idt\wedge
d\bar{t}\wedge\omega_{\varepsilon}^{n},$

where $\chi_{P_{\varepsilon,A}}$ is the characteristic function of the set
$P_{\varepsilon,A}$. Finally, by using the modified energy
$\widetilde{\mathcal{K}}_{\varepsilon,A}$ (defined in the Theorem (3.1)), we
conclude the following integral estimate.

(5.27)
$\displaystyle\widetilde{\mathcal{K}}^{\prime}_{\varepsilon,A}(1)-\widetilde{\mathcal{K}}^{\prime}_{\varepsilon,A}(0)$
$\displaystyle\geq$
$\displaystyle\int_{P_{\varepsilon,A}}||\bar{\partial}v||^{2}_{{\mathcal{G}}_{\varepsilon}}idt\wedge
d\bar{t}\wedge\omega_{\varepsilon}^{n}$ $\displaystyle\geq$
$\displaystyle\int_{P_{\varepsilon,A}}||\bar{\partial}_{X}v||^{2}_{\omega_{\varepsilon}}idt\wedge
d\bar{t}\wedge\omega_{\varepsilon}^{n}+\varepsilon\int_{P_{\varepsilon,A}}|\nabla_{\varepsilon}f_{\varepsilon}|^{2}idt\wedge
d\bar{t}\wedge\omega^{n}.$

If we assume that the Mabuchi functional is essentially affine along
${\mathcal{G}}$, then the RHS of equation (5.27) converges to zero as
$\varepsilon\rightarrow 0$. Moreover, suppose the Mabuchi functional is
$\varepsilon$-affine along the geodesic. Then we have

(5.28)
$\displaystyle\widetilde{\mathcal{K}}^{\prime}_{\varepsilon,A}(1)-\widetilde{\mathcal{K}}^{\prime}_{\varepsilon,A}(0)$
$\displaystyle=$
$\displaystyle{\mathcal{K}}^{\prime}_{\varepsilon}(1)-{\mathcal{K}}^{\prime}_{\varepsilon}(0)+2\varepsilon
C_{A}$ $\displaystyle\leq$ $\displaystyle\varepsilon C^{\prime}_{A},$

for some constant $C^{\prime}_{A}$. Therefore, we conclude the following
estimate from equations (5.27) and (5.28).

(5.29)
$\displaystyle\int_{P_{\varepsilon,A}}|\nabla{\mathfrak{f}}(\varphi_{\varepsilon})|^{2}idt\wedge
d\bar{t}\wedge\omega^{n}$ $\displaystyle\leq$ $\displaystyle
C^{2}\int_{P_{\varepsilon,A}}\frac{|\nabla{\mathfrak{f}}(\varphi_{\varepsilon})|^{2}}{{\mathfrak{f}}^{2}(\varphi_{\varepsilon})}idt\wedge
d\bar{t}\wedge\omega^{n}$ $\displaystyle\leq$ $\displaystyle
C^{2}\int_{P_{\varepsilon,A}}|\nabla\log{\mathfrak{f}}(\varphi_{\varepsilon})|^{2}idt\wedge
d\bar{t}\wedge\omega^{n}$ $\displaystyle\leq$ $\displaystyle
C^{\prime}\int_{P_{\varepsilon,A}}|\nabla_{\varepsilon}\log{\mathfrak{f}}(\varphi_{\varepsilon})|^{2}idt\wedge
d\bar{t}\wedge\omega^{n}\leq C^{\prime\prime}_{A},$

where we used the fact $\omega_{\varepsilon}\leq c\omega$ for a uniform
constant $c$. Moreover, if we take

$F_{\varepsilon,A}(t):=\int_{P_{\varepsilon,A}\bigcap
X_{t}}|\nabla{\mathfrak{f}}(\varphi_{\varepsilon})|^{2}\omega^{n},$

then Fatou’s lemma implies

(5.30)
$\int_{0}^{1}\liminf_{\varepsilon}F_{\varepsilon,A}(t)dt\leq\liminf_{\varepsilon}\int_{0}^{1}F_{\varepsilon,A}(t)dt\leq
C^{\prime\prime}_{A}.$

Therefore, for almost everywhere $t\in[0,1]$, there exist a constant $C_{t}$
and a subsequence $F_{\varepsilon_{\ell},A}$ such that

$\lim_{{\ell}\rightarrow+\infty}F_{\varepsilon_{\ell},A}(t)\leq C_{t}.$

Hence we have the following result.

######  Lemma 5.6.

Suppose the Mabuchi functional is $\varepsilon$-affine along the geodesic.
Then there exists a constant $C$ (possibly depending on $t$ and $A$), and a
subsequence of volume elements ${\mathfrak{f}}(\varphi_{\varepsilon_{\ell}})$
satisfying

(5.31) $\int_{P_{\varepsilon_{\ell},A}\bigcap
X_{t}}|\nabla{\mathfrak{f}}(\varphi_{\varepsilon_{\ell}})|^{2}\omega^{n}\leq
C,$

for almost everywhere $t\in[0,1]$ and any ${\ell}$ large enough.

Thank to Theorem (4.1), we can further assume that
${\mathfrak{f}}(\varphi_{\varepsilon_{\ell}})\rightarrow{\mathfrak{f}}(\varphi)$
in $L^{2}$ on $X_{t}$, possibly after passing to a subsequence.

###  5.2.  Positive lower bound

To deal with the non-degeneracy of the fiber-wise volume element of
${\mathcal{G}}$, we would like to utilise the partial $W^{1,2}$-estimate
obtained in equation (5.31) and the $L^{2}$ convergence of the volume
elements. However, the difficulty is that the integral on the LHS of this
equation is not taken on the whole manifold $X$, and the integration domain
varies with respect to $\varepsilon$ and $A$. In order to overcome this
difficulty, we first investigate a local model as follows.

Suppose $f_{{\ell}}$ is a sequence of positive smooth functions on the domain
$D:=[0,1]^{m}\subset\mathbb{R}^{m}$ with uniformly bounded $L^{\infty}$-norm,
and $f$ is an $L^{\infty}$ non-negative function on $D$ such that $f_{\ell}$
converges to $f$ in $L^{2}$-norm. We further assume that the function $f$
satisfies the gap phenomenon, namely, there exists a constant ${\kappa}_{0}>0$
such that we have

$\\{f>{\kappa}_{0}\\}\bigcup\\{f=0\\}=D,$

up to a set of measure zero. In the following, we set

$P:=\\{f>{\kappa}_{0}\\};\ \ \ P^{c}:=\\{f=0\\},$

and assume $\mu(P)>0$. Let ${\kappa}$ denote a continuous function on $D$ and
we set

$P_{{\ell},{\kappa}}:=\\{x\in D;\ \ \ f_{\ell}(x)>{\kappa}\\}.$

Then the following result is crucial.

######  Proposition 5.7.

Assume $\max{\kappa}<{\kappa}_{0}/4$ on $D$. Suppose there exists a constant
$C>0$, such that the following estimate holds

(5.32) $\int_{P_{{\ell},{\kappa}}}|\nabla f_{\ell}|^{2}<C,$

for a fixed ${\kappa}$ and all ${\ell}$ large enough. Then $f>{\kappa}_{0}$
almost everywhere on $D$.

First we will prove that Proposition (5.7) holds in $\mathbb{R}$, namely, we
assume $D=[0,1]$, and then the following fact is clear by Hölder’s inequality.

######  Lemma 5.8.

Suppose $u$ is a smooth positive function on $[0,1]$, and we assume that
$u(a)>2k$ and $u(b)<k$ for some $0\leq a<b\leq 1$ and a constant $k>0$. Then
we have

$|b-a|>\frac{k^{2}}{\int_{a}^{b}(f^{\prime}(t))^{2}dt}.$

Suppose $E_{1},E_{2}$ are two subsets of $D$, and we denote $d(E_{1},E_{2})$
by the distance between them

$d(E_{1},E_{2}):=\inf_{x_{1}\in E_{1},x_{2}\in E_{2}}d(x_{1},x_{2}).$

Then the following observation is crucial.

######  Lemma 5.9.

Suppose $P,Q$ are two non-empty disjoint subsets of the interval $[0,1]$.
Assume that the union of $P,Q$ is the whole interval $[0,1]$ up to a set with
measure zero. Then we have $d(P,Q)=0$.

###### Proof.

We will prove by contradiction. Suppose the distance between the two sets is
positive as

$d(P,Q)>\delta>0.$

Take a large number $m\in\mathbb{Z}$ satisfying
$\frac{1}{m}<\frac{\delta}{100}$. For each point $p\in P$, we define the
following open interval as

$\mathcal{I}_{p,m}:=(p-\frac{1}{m},p+\frac{1}{m}),$

and similarly for each point $q\in Q$

$\mathcal{I}_{q,m}:=(q-\frac{1}{m},q+\frac{1}{m}).$

We note that $\mathcal{I}_{p,m}$ is disjoint from the set $Q$ for each $p\in
P$, and $\mathcal{I}_{q,m}$ is disjoint from the set $P$ for each $q\in Q$ by
our assumptions. Therefore, the following two unions

$\mathcal{U}:=\bigcup_{p\in P}\mathcal{I}_{p,m},\ \ \
\mathcal{V}:=\bigcup_{q\in Q}\mathcal{I}_{q,m}$

are mutually disjoint open subsets of the interval $[0,1]$.

Moreover, there is a subset $E$ of $[0,1]$ with measure zero satisfying

$P\bigcup Q\bigcup E=[0,1].$

We claim that $E$ is contained in the union $\mathcal{U}\bigcup\mathcal{V}$.
Otherwise, there is a point $a\in E$ such that we have

$d(\\{a\\},P)>\frac{1}{2m},\ \ \ d(\\{a\\},Q)>\frac{1}{2m}.$

Then the open interval $(a-\frac{1}{2m},a+\frac{1}{2m})$ must be contained in
$E$, but this is impossible since $E$ has measure zero.

Therefore, we have proved that the union $\mathcal{U}\bigcup\mathcal{V}$ is
exactly the interval $[0,1]$, which is impossible since $[0,1]$ is a connected
set.

∎

After passing to a subsequence, we can assume that $f_{\ell}$ converges to $f$
almost everywhere on $[0,1]$ due to the $L^{2}$ convergence. Therefore, there
is a subset $E\subset[0,1]$ with measure zero such that $f_{\ell}\rightarrow
f$ in the poinwise sense outside of $E$. Then we are ready to prove the
$1$-dimensional case as follows.

######  Lemma 5.10.

Proposition (5.7) holds in one dimension.

###### Proof.

Thanks to Lemma (5.9), everything boils down to prove

$d(P-E,P^{c}-E)>0,$

if $P^{c}$ is non-empty.

We will prove by contradiction again. If not, then there exists a sequence of
pairs $(a_{j},b_{j})\in(P-E)\times(P^{c}-E)$ such that
$|a_{j}-b_{j}|\rightarrow 0$ as $j\rightarrow\infty$. Without loss of
generality, we assume $a_{j}<b_{j}$ in the following.

Fix one interval $[a_{j},b_{j}]$. For all ${\ell}$ large enough, we have

$f_{\ell}(a_{j})>\frac{3{\kappa}_{0}}{4};\ \ \ \
f_{{\ell}}(b_{j})<\frac{{\kappa}_{0}}{4}.$

Therefore, there exists a point $c_{j}\in[a_{j},b_{j}]$ such that
$f_{\ell}(c_{j})=\frac{{\kappa}_{0}}{2}$ by the continuity of $f_{{\ell}}$.
Moreover, the point $c_{j}$ can be chosen close enough to $a_{j}$ such that we
have

(5.33) $f_{\ell}|_{[a_{j},c_{j}]}>\frac{{\kappa}_{0}}{4},$

since the first derivative $f^{\prime}_{\ell}$ is bounded. Hence we have
$[a_{j},c_{j}]\subset P_{{\ell},{\kappa}}$ as $\max{\kappa}<{\kappa}_{0}/4$.
Then the following inequality holds by Lemma (5.8) and equation (5.32)

(5.34) $|b_{j}-a_{j}|>|c_{j}-a_{j}|\geq\frac{{\kappa}^{2}_{0}}{16C}.$

However, this contradicts to the fact that $d(a_{j},b_{j})\rightarrow 0$ as
$j\rightarrow\infty$, and then our result follows.

∎

###### Proof of Proposition (5.7).

We will use induction on the dimension $m$. Lemma (5.10) implies that the
result is true for $m=1$, and we assume that Proposition (5.7) holds on
$\mathbb{R}^{m-1}$ for some integer $m\geq 2$.

Write the coordinate $x\in[0,1]^{m}$ as
$(x_{1},\cdots,x_{m-1},x_{m})=(x^{\prime},y)$, where
$x^{\prime}=(x_{1},\cdots,x_{m-1})$ and $y=x_{m}$. Denote $X_{y}$ the
$(m-1)$-dimensional slice by

$X_{y}:=\\{x\in[0,1]^{m};\ \ \ x=(x^{\prime},y)\\}.$

Consider the $L^{2}$ difference of $f_{\ell}$ and $f$ on each slice as

$F_{{\ell}}(y):=\int_{X_{y}}|f_{\ell}-f|^{2}dx^{\prime}$

Then the $L^{2}$-convergence of $f_{\ell}-f$ on $[0,1]^{m}$ implies that
$F_{{\ell}}$ converges to zero as $L^{1}$ functions on $[0,1]$, and then it
converges in measure. By the Riesz-Lebesgue Theorem, it follows that there
exists a subsequence $F_{{\ell}_{k}}$ such that

$\lim_{{\ell}_{k}\rightarrow\infty}F_{{\ell}_{k}}(y)\rightarrow 0,$

for almost everywhere $y\in[0,1]$. We emphasis that this subsequence does not
depend on $y$. Therefore, after re-writing the sub-index, we can assume

(5.35) $||f_{\ell}-f||_{L^{2}(X_{y})}\rightarrow 0,\ \ \ \
{\ell}\rightarrow\infty$

for almost everywhere $y\in[0,1]$. Moreover, equation (5.32) can be written as

(5.36) $\int_{0}^{1}\left(\int_{X_{y}\bigcap P_{{\ell},{\kappa}}}|\nabla
f_{\ell}|^{2}dx^{\prime}\right)dy<C$

Then by Fatou’s lemma (as we argued in equation (5.30)), there exists a
constant $C_{y}>0$ and a subsequence $f_{\ell}$, possibly depending on $y$,
satisfying

(5.37) $\int_{X_{y}\bigcap
P_{{\ell},{\kappa}}}|\nabla_{x^{\prime}}f_{\ell}|^{2}\leq\int_{X_{y}\bigcap
P_{{\ell},{\kappa}}}|\nabla f_{\ell}|^{2}dx^{\prime}<C_{y},$

for almost everywhere $y\in[0,1]$ and all ${\ell}$ large.

Furthermore, the gap phenomenon (either we have $f>{\kappa}_{0}$ or $f=0$
almost everywhere ) must be satisfied on $X_{y}$ for almost everywhere
$y\in[0,1]$.

Up to this point, all conditions in Proposition (5.7), possibly except
$\mu(P\bigcap X_{y})>0$, are satisfied on such a slice $X_{y}$ and a
subsequence $f_{\ell}$. Then for almost everywhere $y\in[0,1]$, from our
induction hypothesis, we conclude that either one of the following two cases
happens:

  1. (1)

$f>{\kappa}_{0}$ almost everywhere on $X_{y}$;

  2. (2)

$f=0$ almost everywhere on $X_{y}$.

In fact, we claim that the $\mathbf{Case\ (1)}$ occurs for almost everywhere
$y\in[0,1]$, and then our result follows. Otherwise, the following set

$S:=\\{y\in[0,1];\ \ \ f=0\ \ \emph{a.e. on $X_{y}$}\\},$

will have positive measure on $[0,1]$. Therefore, the set

$\mathcal{S}:=\bigcup_{y\in S}X_{y}$

has positive measure on $[0,1]^{m}$.

On the other hand, switch the direction and take another slicing as

$x:=(x_{1},x_{2},\cdots,x_{m})=(y^{\prime},x^{\prime\prime}),$

where $x^{\prime\prime}=(x_{2},\cdots,x_{m})$. Repeating our previous argument
on this new slicing, we can also conclude that for almost everywhere
$y^{\prime}\in[0,1]$, either $f>{\kappa}_{0}$ or $f=0$ as an $L^{\infty}$
function on the slice $X_{y^{\prime}}$.

However, the measure of the following set on each slice
$X_{y^{\prime}}\approx[0,1]^{m-1}$

$\mathcal{S}_{y^{\prime}}:=\bigcup_{y\in S}\left(X_{y^{\prime}}\bigcap
X_{y}\right)$

is the same for different $y^{\prime}$, and then
$\mu(\mathcal{S}_{y^{\prime}})$ must be positive by Fubini’s Theorem. In other
words, the set $P^{c}\bigcap X_{y^{\prime}}$ has positive measure for each
$y^{\prime}\in[0,1]$. Therefore, we conclude that $f=0$ on $X_{y^{\prime}}$
for almost everywhere $y^{\prime}\in[0,1]$. This contradicts to the fact that
$\mu(P)>0$, and our claim follows.

∎

Now we are going to prove the main theorem in this section.

###### Proof of Theorem (5.4).

For almost everywhere $t\in[0,1]$, we fix a fiber $X_{t}$ such that the
estimate in Lemma (5.6) holds for a sequence
${\mathfrak{f}}(\varphi_{\varepsilon_{\ell}})$. Then we will prove that the
volume element ${\mathfrak{f}}(\varphi)=\omega_{\varphi}^{n}/\omega^{n}$ is
bounded below by the gap ${\kappa}_{0}$ on $X_{t}$ as an $L^{\infty}$
function. Hence the restriction of the geodesic ${\mathcal{G}}|_{X_{t}}$, as a
metric on the fiber, must have a lower bound determined by ${\kappa}_{0}$ and
its uniform upper bound.

We will prove by contradiction. Suppose the set
$P^{c}=\\{{\mathfrak{f}}(\varphi)=0\\}$ has a positive measure. As before, we
denote $P$ by the set

$P=\\{{\mathfrak{f}}(\varphi)>{\kappa}_{0}\\},$

and it also has a positive measure. Recall that $P_{\varepsilon,A}$ is the
subset of $\Gamma\times X$ where $f_{\varepsilon}$ is larger than the
auxiliary function $\log\frac{e^{\chi-A}}{\omega^{n}}$, or equivalently,

$P_{\varepsilon,A}\bigcap X_{t}=\left\\{x\in X_{t};\ \ \
{\mathfrak{f}}(\varphi_{\varepsilon}(t,x))>{\kappa}\right\\},$

where ${\kappa}=e^{\chi-A}/\omega^{n}$ is the auxiliary function (equation
(4.1)). When the constant $A$ is large enough, we can assume
$\max_{X_{t}}{\kappa}<{\kappa}_{0}/4$ for all $t\in[0,1]$.

As a compact connected Kähler manifold, the fiber $X_{t}$ has an open covering
by holomorphic coordinate charts, namely, we have

$X_{t}=\bigcup_{j=1}^{N}U_{j},$

and the local trivialisation map $\pi_{j}:B_{j}\rightarrow U_{j}$, where
$B_{j}\subset\mathbb{C}^{n}$ is an open ball. Without loss of generality, we
assume that each ball $B_{j}$ is centred at the origin of $\mathbb{C}^{n}$ and
has radius larger than $2$. In fact, we can further assume that the manifold
$X_{t}$ is covered by the union of $V_{j}\subset U_{j}$, where $V_{j}$ denotes
the open set

$V_{j}:=\pi_{j}(B),$

for the unit ball $B\subset\mathbb{C}^{n}$. Therefore, there exists at least
one $j$, such that we have on the coordinate $U_{j}$

(5.38) $\mu\left(\\{{\mathfrak{f}}(\varphi)=0\\}\bigcap V_{j}\right)>0.$

Then we claim that among all these $j$’s, there exists at least one
$j^{\prime}$ such that the set $P^{c}$ does not cover the whole unit ball,
namely, we have on $U_{j^{\prime}}$

(5.39) $a_{2n}>\mu\left(\\{{\mathfrak{f}}(\varphi)=0\\}\bigcap
V_{j^{\prime}}\right)>0.$

If not, then we can find a non-empty subset
$\Lambda\subsetneqq\\{1,\cdots,N\\}$ such that we have ${\mathfrak{f}}=0$
pointwise a.e. on $V_{j}$ for each $j\in\Lambda$ and
${\mathfrak{f}}>{\kappa}_{0}$ pointwise a.e. on $V_{k}$ for each
$k\in\Lambda^{c}$. Take two open sets as

$\mathcal{U}:=\bigcup_{j\in\Lambda}V_{j};\ \ \ \
\mathcal{V}:=\bigcup_{k\in\Lambda^{c}}V_{k}.$

Then we note that ${\mathfrak{f}}=0$ pointwise a.e. on ${\mathcal{U}}$ and
${\mathfrak{f}}>{\kappa}_{0}$ pointwise a.e. on ${\mathcal{V}}$. This implies
that the intersection of the two open sets ${\mathcal{U}}\bigcap{\mathcal{V}}$
is empty. Otherwise, the intersection is an open set with positive measure,
and we both have ${\mathfrak{f}}=0$ and ${\mathfrak{f}}>{\kappa}_{0}$
pointwise a.e. on ${\mathcal{U}}\bigcap{\mathcal{V}}$, which is not possible.
Therefore, we have

(5.40) ${\mathcal{U}}\bigcap{\mathcal{V}}=\emptyset;\ \ \
{\mathcal{U}}\bigcup{\mathcal{V}}=X_{t},$

but this is also impossible, since $X_{t}$ is a compact connected manifold and
${\mathcal{U}},{\mathcal{V}}$ are both non-empty finite union of coordinate
charts. Hence our claim follows.

Pick up such a $j^{\prime}$ that equation (5.39) holds, and then we take

$f_{\ell}:={\mathfrak{f}}(\varphi_{\varepsilon_{\ell}})|_{U_{j^{\prime}}},\ \
\ f:={\mathfrak{f}}(\varphi)|_{U_{j^{\prime}},}$

and consider the $m^{th}$-cube $D:=[-1,1]^{m}$ in $U_{j^{\prime}}$. Combined
with equation (5.31) and (5.39), all the conditions in Proposition (5.7) are
satisfied on this domain $D$ for the sequence $f_{\ell}$. Therefore, we
conclude that $f>{\kappa}_{0}$ almost everywhere on $D$. However, this
contradicts to the fact that $\\{f=0\\}$ has positive measure on
$V_{j^{\prime}}\subset D$, and our main result follows.

∎

##  6\. Applications

As one application of the main result (Theorem (5.4)), we will prove that the
Mabuchi functional
${\mathcal{K}}_{\varepsilon}:={\mathcal{M}}(\varphi_{\varepsilon})$ along the
$\varepsilon$-geodesic converges to the Mabuchi functional
${\mathcal{M}}(\varphi)$ along the geodesic, not only pointwise on each fiber
(Corollary (3.4)), but also in its complex Hessian.

For the first step, we will take a closer look at equations (5.7) as follows.
Recall that the $\varepsilon$-geodesic ${\mathcal{G}}_{\varepsilon}$ is
uniformly ${\mathcal{C}}^{1,\bar{1}}$ on $\Gamma\times X$. Therefore, we have

(6.1)
$C_{2}>g_{t\bar{t}}-g^{\bar{\beta}{\alpha}}g_{{\alpha}\bar{t}}g_{t\bar{\beta}}=\frac{\varepsilon\omega^{n}}{\omega_{\varepsilon}^{n}},$

for some uniform constant $C_{2}$. For the same reason, the eigenvalues of
$\omega_{\varepsilon}$ are bounded from above by a uniform constant $C_{0}>0$.
Moreover, at a particular point $p\in\Gamma\times X$, we can write

$\omega=\sum_{j=1}^{n}dz^{j}\wedge d\bar{z}^{j};\ \ \
\omega_{\varepsilon}=\sum_{j=1}^{n}\lambda_{j}dz^{j}\wedge d\bar{z}^{j}.$

where $0<\lambda_{1}\leq\cdots\leq\lambda_{n}$ are the $n$ eigenvalues of the
metric $\omega_{\varepsilon}$ at this point. Hence we have the following
inequality.

(6.2) $\displaystyle\mathop{\rm tr}\nolimits_{\omega_{\varepsilon}}\omega$
$\displaystyle=$ $\displaystyle\sum_{k=1}^{n}\frac{1}{\lambda_{i}}$
$\displaystyle\leq$
$\displaystyle\frac{n\lambda_{2}\cdots\lambda_{n}}{\lambda_{1}\cdots\lambda_{n}}$
$\displaystyle\leq$ $\displaystyle\frac{nC_{0}^{n-1}}{\mathop{\rm
det}\nolimits(g_{{\alpha}\bar{\beta}})}\leq\frac{nC_{2}C_{0}^{n-1}}{\varepsilon}.$

Equipped equation (6.2) into the RHS of equation (5.10), we have

(6.3) $\displaystyle-\varepsilon\int_{X_{t}}(\mathop{\rm
tr}\nolimits_{\omega_{\varepsilon}}Ric\omega)\omega^{n}$ $\displaystyle\geq$
$\displaystyle-\varepsilon C\int_{X_{t}}(\mathop{\rm
tr}\nolimits_{\omega_{\varepsilon}}\omega)\omega^{n}$ $\displaystyle\geq$
$\displaystyle-\varepsilon C\int_{P_{\varepsilon,A_{1}}\bigcap
X_{t}}(\mathop{\rm
tr}\nolimits_{\omega_{\varepsilon}}\omega)\omega^{n}-C_{4}\mu\left(\\{{\mathfrak{f}}(\varphi_{\varepsilon})\leq{\kappa}_{1}\\}\right)$
$\displaystyle\geq$ $\displaystyle-\varepsilon
C_{A_{1}}-C_{4}\mu\left(\\{{\mathfrak{f}}(\varphi_{\varepsilon})\leq{\kappa}_{1}\\}\right),$

where we fix a constant $A_{1}$ large enough such that

$\max_{X_{t}}{\kappa}_{1}<{\kappa}_{0}/2.$

Then the first factor on the RHS of equation (6.3) converges to zero.
Moreover, Theorem (5.4) implies that the measure
$\mu(\\{{\mathfrak{f}}(\varphi_{\varepsilon})\leq{\kappa}_{1}\\})$ converges
to zero as $\varepsilon\rightarrow 0$, possibly after passing to a
subsequence. Therefore, we conclude the following result.

######  Theorem 6.1.

Suppose the Mabuchi functional is $\varepsilon$-affine along a geodesic. Then
for almost everywhere $t\in[0,1]$, we have on the fiber $X_{t}$

$dd^{c}{\mathcal{K}}_{\varepsilon}|_{t}\rightarrow 0,$

as $\varepsilon\rightarrow 0$, possible after passing to a subsequence. In
particular, we have the following convergence of the $L^{2}$-norms

(6.4)
$\int_{X_{t}}||\bar{\partial}_{X}v||_{\omega_{\varepsilon}}^{2}\omega_{\varepsilon}^{n},\int_{X_{t}}||\bar{\partial}v||_{{\mathcal{G}}_{\varepsilon}}^{2}\omega_{\varepsilon}^{n}\rightarrow
0,$

as $\varepsilon\rightarrow 0$ for this subsequence.

###### Proof.

The complex Hessian of ${\mathcal{K}}_{\varepsilon}$ can be computed as

(6.5)
$dd^{c}{\mathcal{K}}_{\varepsilon}(t)=\frac{\underline{R}}{n+1}\int_{X_{t}}{\mathcal{G}}_{\varepsilon}^{n+1}+\int_{X_{t}}(dd^{c}f_{\varepsilon}-Ric\omega)\wedge{\mathcal{G}}_{\varepsilon}^{n}.$

Repeat the previous computation on the factor

$(dd^{c}f_{\varepsilon}-Ric\omega)\wedge{\mathcal{G}}_{\varepsilon}^{n}.$

Equipped the estimate (6.3) into (5.10), and then we have .

(6.6)
$\displaystyle\left(dd^{c}\log\frac{\omega_{\varepsilon}^{n}}{\omega^{n}}-Ric\omega\right)\wedge{\mathcal{G}}_{\varepsilon}^{n}+C^{\prime}_{A_{1}}{\mathcal{G}}_{\varepsilon}^{n+1}$
$\displaystyle\geq$
$\displaystyle\left(||\bar{\partial}_{X}v||^{2}_{\omega_{\varepsilon}}+\varepsilon
e^{-f_{\varepsilon}}|\nabla_{\varepsilon}f_{\varepsilon}|^{2}\right)idt\wedge
d\bar{t}\wedge\omega_{\varepsilon}^{n},$

where

$C^{\prime}_{A_{1}}:=C_{A_{1}}+\chi_{\\{{\mathfrak{f}}(\varphi_{\varepsilon})\leq{\kappa}_{1}\\}}\varepsilon^{-1}C_{4}$

is a function on $\Gamma\times X$. Then our previous argument shows that the
following energy

${\mathcal{K}}_{\varepsilon}(t)-\varepsilon\left(\int_{X_{t}}C^{\prime}_{A_{1}}\right)t(1-t)$

is convex on the unit interval, and it converges to an affine function point-
wise as $\varepsilon\rightarrow 0$. Therefore, its second derivative converges
to zero for almost everywhere $t\in[0,1]$. Moreover, we have seen

$\mu\\{{\mathfrak{f}}(\varphi_{\varepsilon})\leq{\kappa}_{1}\\}\rightarrow 0,\
\ \ \varepsilon\rightarrow 0,$

possibly after passing to a subsequence on the fiber $X_{t}$, and then our
result follows.

∎

By utilizing the estimate in the above Theorem, i.e.
$\bar{\partial}v_{\varepsilon}\rightarrow 0$ in the $L^{2}$ sense, one may
expect that $v_{\varepsilon}$ would converge to a vector field $v_{\infty}$,
and this limit $v_{\infty}$ is holomorphic on $\Gamma\times X$. However, this
is still unclear to us since the fiberwise volume element
$\omega_{\varepsilon}^{n}$ may not converge uniformly to the volume element
$\omega^{n}_{\varphi}$ of the geodesic ${\mathcal{G}}$. Up to this stage, we
can only conclude the holomorphicity of $v_{\infty}$ under some special cases.

###  6.1. Special cases

Suppose the two boundaries $\varphi_{0},\varphi_{1}$ of ${\mathcal{G}}$ are
both non-degenerate energy minimizers of ${\mathcal{M}}$. Then
${\mathcal{M}}(\varphi_{t})$ keeps to be a constant along this geodesic, and
it satisfies the $\varepsilon$-affine condition automatically. Therefore, our
Theorem (5.4) implies that the geodesic ${\mathcal{G}}$ is fiberwise uniformly
non-degenerate, and then the regularities of ${\mathcal{G}}$ can be improved
by the work of He-Zeng([HZ]). Then we recover one of our result in [LL].

######  Theorem 6.2 (L.).

Suppose the two boundaries of a $C^{1,\bar{1}}$-geodesic ${\mathcal{G}}$ are
both non-degenerate energy minimizers of ${\mathcal{M}}$. Then the geodesic
${\mathcal{G}}$ is generated by a holomorphic vector field.

On the other hand, we can assume that the Kähler manifold $X$ satisfies
$c_{1}(X)<0$ or $c_{1}(X)=0$. Then our computation in Section (4) would
recover Chen’s estimates in [C00], under the $\varepsilon$-affine condition.
In conclusion, we can infer the following result by a similar argument as in
Section (6), [C00].

######  Theorem 6.3.

Suppose the manifold $X$ satisfies $c_{1}(X)<0$ or $c_{1}(X)=0$. Assume that
the Mabuchi functional ${\mathcal{M}}$ is $\varepsilon$-affine along a
$C^{1,\bar{1}}$-geodesic ${\mathcal{G}}$. Then ${\mathcal{G}}$ is generated by
a holomorphic vector field.

Finally, we would like to emphasis that the boundaries of the geodesic
${\mathcal{G}}$ are assumed to be smooth and non-degenerate in our set up.
Therefore, one possible way to utilize the $L^{2}$-convergence of
$\bar{\partial}v_{\varepsilon}$ is to consider their behavior close enough to
the boundary. Hence we will end up with the following observation, which may
be useful in our later consideration.

######  Proposition 6.4.

Suppose ${\mathcal{G}}$ is a ${\mathcal{C}}^{1,\bar{1}}$-geodesic connecting
two Kähler potentials $\varphi_{0},\varphi_{1}\in{\mathcal{H}}$. Then its
fiberwise volume element has the following convergence near the boundaries

${\mathfrak{f}}(\varphi_{t})\rightarrow{\mathfrak{f}}(\varphi_{0});\ \ \
{\mathfrak{f}}(\varphi_{s})\rightarrow{\mathfrak{f}}(\varphi_{1}),$

as $t\rightarrow 0$ and $s\rightarrow 1$ in the $L^{2}$-sense.

###### Proof.

By a Theorem proved in Chen-Tian (Theorem 7.1.1, [CT]), it is enough to show

$\lim_{t\rightarrow
0}\int_{X}{\mathfrak{f}}(\varphi_{t})\log{\mathfrak{f}}(\varphi_{t})\rightarrow\int_{X}{\mathfrak{f}}(\varphi_{0})\log{\mathfrak{f}}(\varphi_{0}),$

and

$\lim_{s\rightarrow
1}\int_{X}{\mathfrak{f}}(\varphi_{s})\log{\mathfrak{f}}(\varphi_{s})\rightarrow\int_{X}{\mathfrak{f}}(\varphi_{1})\log{\mathfrak{f}}(\varphi_{1}).$

In other words, the entropy $H(\varphi_{t})(H(\varphi_{s}))$ converges to
$H(\varphi_{0})(H(\varphi_{1}))$ as $t\rightarrow 0$ and $s\rightarrow 1$.
This is true because the Mabuchi functional ${\mathcal{M}}$ is convex and
continuous up to the boundaries of ${\mathcal{G}}$. ∎

## References


which there can be at most $O(w)$ introduce nodes, so there can be at most
$O(w^{2}n)$ introduce nodes total.

Because $G$ has carving width $w$, its treewidth and maximum degree are both
$O(w)$, so by Section 6.2 the number of states at each node of this
decomposition is exponential in $O(w^{4})$. By the lemmas above, we can
compute the set of valid states at each node by an algorithm that examines
each valid state at the child node (when there is one child) or each pair of
valid states at the two children (when there are two children), in an amount
of time per state or pair of states that is polynomial in $w$. Therefore the
total time per node is bounded by this polynomial multiplied by the square of
the number of states, which remains exponential in $O(w^{4})$. ∎

##  7 Conclusions and open problems

We have investigated the graphs of stably matchable pairs of stable matching
instances, with results including the characterization of special classes of
these graphs, characterization of the lattices of stable matchings
corresponding to these classes of graphs, the NP-completeness of recognizing
graphs of stably matchable pairs in general, and exact algorithms for
recognizing these graphs that run in singly exponential or fixed-parameter
tractable time.

A natural open problem that remains concerns whether our algorithm for carving
width can be extended to treewidth for graphs of unbounded degree. As a first
step towards this, what is the complexity of recognizing series-parallel
graphs of stably matchable pairs? Another problem concerns the analysis of
numbers of combinations of matchings and subsets of edges in Section 6.1,
which we used to analyze our exponential-time recognition algorithm. This
analysis seems unlike to be tight; can it be improved?

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