#  On the residual Monge-Ampère mass of plurisubharmonic functions with
symmetry in ${\mathbb{C}}^{2}$

Long Li  IMS of ShanghaiTech University, 393 Middle Huaxia Road, Pudong
201210, Shanghai, China<EMAIL_ADDRESS>
###### Abstract.

The aim of this paper is to study the residual Monge-Ampère mass of a
plurisubharmonic function with isolated singularity at the origin in
${\mathbb{C}}^{2}$. We prove that the residual mass is zero if its Lelong
number is zero at the origin, provided that it is $S^{1}$-invariant. This
result answers the zero mass conjecture raised by Guedj and Rashkovskii in
this special case. More generally, we obtain an estimate on the residual mass
by the maximal directional Lelong number and Lelong number at the origin.

##  1\. Introduction

Let $D$ be a bounded domain in ${\mathbb{C}}^{n}$, and $u$ a
$C^{2}$-continuous plurisubharmonic function on $D$. Then the Monge-Ampère
operator operates on $u$ and equals the following positive measure as

(1.1) $\mbox{MA}(u):=(dd^{c}u)^{n}\geq 0,$

where $d:=\partial+\bar{\partial}$ and
$d^{c}:=\frac{i}{2}(\bar{\partial}-\partial)$. This operator has great
importance in pluripotential theory. However, it is fully non-linear and can
not be defined for all plurisubharmonic functions on $D$, cf. [Ceg86], [Kis84]
and [Siu75].

On the other hand, there are several ways to define the Monge-Ampère measure
for a plurisubharmonic function $u$, if extra conditions have been assumed.
For instance, Bedford and Talyor [BT] have shown that $(dd^{c}u)^{n}$ is well
defined, if $u$ is further in $L^{\infty}_{loc}(D)$. Later Demailly [Dem93]
extended this definition to all plurisubharmonic functions whose unbounded
locus are relatively compact in $D$. In particular, the operator
$(dd^{c})^{n}$ acts well on plurisubharmonic functions with isolated
singularity.

For simplicity, we take $D$ as the unit ball $B_{1}$ in ${\mathbb{C}}^{n}$.
Let $u$ be a plurisubharmonic function on $B_{1}$ that is locally bounded
outside the origin. Then Guedj and Rashkovskii (Question 7, [GZ15]) raised the
following question:

######  Conjecture 1.1.

Assume that $(dd^{c}u)^{n}$ has a Dirac mass at the origin. Does it imply that
$u$ has a positive Lelong number at the origin  _?_

The atomic mass of $(dd^{c}u)^{n}$ at the origin is called the _residual
Monge-Ampère mass_ of $u$, and we can write it as

(1.2) $\tau_{u}(0):=\frac{1}{\pi^{n}}\mbox{MA}(u)(\\{0\\}).$

Here the normalization is chosen in such a way that we have
$\tau_{\log|z|}(0)=1$. Denote the Lelong number of $u$ at the origin by
$\nu_{u}(0)$. Then the above Conjecture (1.1) can be rephrased as follows.

######  Conjecture 1.2.

$\nu_{u}(0)=0\Rightarrow\tau_{u}(0)=0\ ?$

For this reason, this problem is also called the _zero mass conjecture_ for
plurisubharmonic functions. In history, there have been many works that
contribute to this problem, cf. [Ceg02], [Ra01], [Ra13], [KR21], [BFJ07] and
[G10]. In particular, Rashkovskii [Ra01] confirmed this conjecture, provided
with toric symmetry on $u$. In this paper, we will study a more general
symmetry called circular symmetry for plurisubharmonic functions.

Let $(z_{1},\cdots,z_{n})$ be the complex Euclidean coordinate on
${\mathbb{C}}^{n}$, and then there is a natural $S^{1}$-action on it as

(1.3) $z\rightarrow e^{i\theta}z:=(e^{i\theta}z_{1},\cdots,e^{i\theta}z_{n}),$

for all $\theta\in{\mathbb{R}}$. A domain is balanced if it is invariant under
this $S^{1}$-action. We say that a function $u$ on a balanced domain $D$ is
circular symmetric, or $S^{1}$-invariant if

$u(e^{i\theta}z)=u(z),$

for all $z\in D$. Then it is apparent that any toric symmetric function is
also $S^{1}$-invariant.

In fact, there is a deep connection between $S^{1}$-invariant plurisubharmonic
functions and _the Schwarz symmetrization_ technique in classical analysis.
Berman and Berndtsson [BB] proved that the Schwarz symmetrization of any
$S^{1}$-invariant plurisubharmonic function is also plurisubharmonic.
Moreover, the Lelong number at the origin is always increasing under this
symmetrization, cf. [Li19].

On the other hand, this $S^{1}$-action is highly related to _the Hopf-
fiberation_ of the unit sphere $S^{2n-1}$ in
${\mathbb{R}}^{2n}\cong{\mathbb{C}}^{n}$, cf. [GWZ86]. As the first attempt to
utilize this geometric picture, we will restrict to ${\mathbb{C}}^{2}$ in this
paper, where the structure of the Hopf-fiberation $S^{1}\hookrightarrow
S^{3}\xrightarrow{p}S^{2}$ is fully understood.

For this reason, the domain $D$ is assumed to be the unit ball
$B_{1}\subset{\mathbb{C}}^{2}$ from now on. Then we introduce the family
${\mathcal{F}}(B_{1})$ (Definition (2.1)) as a collection of all
$S^{1}$-invariant plurisubharmonic functions on $B_{1}$ that is
$L^{\infty}_{loc}$ outside the origin. In order to perform calculus, we
further introduce the family ${\mathcal{F}}^{\infty}(B_{1})$ (Definition
(2.2)) as a sub-collection of ${\mathcal{F}}(B_{1})$ that is
$C^{2}$-continuous outside the origin, and then we first confirm the zero mass
conjecture for this family.

######  Theorem 1.3 (Theorem (5.5)).

For any function $u\in{\mathcal{F}}^{\infty}(B_{1})$, we have

$\nu_{u}(0)=0\Rightarrow\tau_{u}(0)=0.$

The key observation is a decomposition formula (Theorem (4.4)) for the complex
Monge-Ampère mass. It decomposes the measure $(dd^{c}u)^{2}$ on the ball
$B_{R}$ into two integrals on the boundary $S_{R}:=\partial B_{R}$. The first
integral corresponds to the so called _pluricomplex energy_ on
${\mathbb{C}}{\mathbb{P}}^{1}$ (Section (7.2)), and the second integral is a
kind of $L^{2}$-Lelong number (Section (5.1)).

In fact, this decomposition formula implies a stronger result. For a general
function $u\in{\mathcal{F}}^{\infty}(B_{1})$ without zero Lelong number
condition, we obtain the following estimate on the residual Monge-Ampère mass.

######  Theorem 1.4 (Theorem (5.9)).

For any function $u\in{\mathcal{F}}^{\infty}(B_{1})$, we have

$[\nu_{u}(0)]^{2}\leq\tau_{u}(0)\leq
2\lambda_{u}(0)\cdot\nu_{u}(0)+[\nu_{u}(0)]^{2}.$

The one side inequality $[\nu_{u}(0)]^{2}\leq\tau_{u}(0)$ was indicated by
Cegrell, cf. [Ceg04], and it holds for a general function
$u\in{\mathcal{F}}(B_{1})$. The other side follows from our Lemma (5.6). Here
$\lambda_{u}(0)$ is the so called _maximal directional Lelong number_ of $u$
at the origin, cf. Definition (5.8). It is actually the decreasing limit of
$M_{A}(u)$ as $A\rightarrow+\infty$, where the constant $M_{A}(u)$ is the
_maximal directional Lelong number_ of $u$ at the distance $A$, cf. Definition
(5.7). For a function $u\in{\mathcal{F}}^{\infty}(B_{1})$, it is apparent that
they are all non-negative real numbers. However, there is no a priori reason
that they are finite for a general plurisubharmonic function.

Surprisingly, the finiteness of $\lambda_{u}(0)$ and $M_{A}(u)$ are always
true for a function $u\in{\mathcal{F}}(B_{1})$, cf. Lemma (6.1). This is due
to the following two facts: first, the function $u$ is bounded from above on
the ball $B_{1}$ and from below on each boundary sphere $S_{R}$ with
$R\in(0,1)$; second, the restriction of $u$ to each complex line through the
origin is actually a convex function in the variable $t:=\log
r\in(-\infty,0)$. Therefore, the slope of the tangent line of this convex
function can not grow arbitrarily large, when the angle of the line varies on
${\mathbb{C}}{\mathbb{P}}^{1}$.

Then the idea is to utilize a regularization sequence
$u_{\varepsilon}\in{\mathcal{F}}^{\infty}(B_{1})$ to approximate the function
$u\in{\mathcal{F}}(B_{1})$. The key step is to obtain a uniform estimate on
the maximal directional Lelong number $M_{B}(u_{\varepsilon})$ by $M_{A}(u)$
for any constant $B>A>1$, cf. Lemma (6.6). These facts enable us to generalize
Theorem (1.3) and (1.4) as follows.

######  Theorem 1.5 (Theorem (6.8)).

For any function $u\in{\mathcal{F}}(B_{1})$, we have

$\nu_{u}(0)=0\Rightarrow\tau_{u}(0)=0.$

######  Theorem 1.6 (Theorem (6.9)).

For any function $u\in{\mathcal{F}}(B_{1})$, we have

$[\nu_{u}(0)]^{2}\leq\tau_{u}(0)\leq
2\lambda_{u}(0)\cdot\nu_{u}(0)+[\nu_{u}(0)]^{2}.$

In order to illustrate the inequality in the above Theorem, we compute several
cases, including Demailly’s example [Dem93] and Chi Li’s construction [Chi21],
at the end of Section (6.5). Moreover, we provide an counter-example of this
inequality, when the function $u$ is no longer $S^{1}$-invariant.

There is another point of view to look at a function
$u\in{\mathcal{F}}(B_{1})$ via a variational approach. First we recall a few
basic facts in Kähler geometry, cf. [Sem74], [Don97], [C00] and [Li22].
Consider a sub-geodesic ray in the space of Kähler potentials on
${\mathbb{C}}{\mathbb{P}}^{1}$. It is actually a local plurisubharmonic
function $u$ on the product space
${\mathbb{D}}^{*}\times{\mathbb{C}}{\mathbb{P}}^{1}$ that is $S^{1}$-invariant
in the argument direction of ${\mathbb{D}}^{*}$. Moreover, it is a geodesic
ray if the following _homogeneous complex Monge-Ampère equation_ holds

(1.4) $(dd^{c}u)^{2}=0,$

on the product ${\mathbb{D}}^{*}\times{\mathbb{C}}{\mathbb{P}}^{1}$.

On the other hand, we note that the punctured disk ${\mathbb{D}}^{*}$ acts on
$B_{1}^{*}\subset{\mathbb{C}}^{2}$ in a natural way. Then the punctured ball
$B_{1}^{*}$ can be thought of as a non-trivial ${\mathbb{D}}^{*}$-fiberation
over ${\mathbb{C}}{\mathbb{P}}^{1}$, i.e. we have the following fiber bundle
structure

(1.5) ${\mathbb{D}}^{*}\hookrightarrow
B_{1}^{*}\xrightarrow{p}{\mathbb{C}}{\mathbb{P}}^{1}.$

Comparing with the manifold
${\mathbb{D}}^{*}\times{\mathbb{C}}{\mathbb{P}}^{1}$, we have a simpler total
space since the Euclidean metric on $B_{1}^{*}$ is flat. However, the
fiberation structure corresponds to the Hopf-fiberation that is more
complicated. In particular, the usual complex structure on
$B_{1}^{*}\subset{\mathbb{C}}^{2}$ is no longer a product of the complex
structures on ${\mathbb{D}}^{*}$ and ${\mathbb{C}}{\mathbb{P}}^{1}$.

In this way, a function $u\in{\mathcal{F}}(B_{1})$ can be viewed as a sub-
geodesic ray on this non-trivial ${\mathbb{D}}^{*}$-bundle (Definition (7.1)),
and it is a geodesic ray on this bundle if equation (1.4) holds on
$B_{1}^{*}$.

This observation leads us to a new understanding about the decomposition
formula and the zero mass conjecture. In fact, the decomposition formula can
be fit into an energy picture as shown in Theorem (7.4). There a sub-geodesic
ray corresponds to a convex energy functional on $(-\infty,0)$, and geodesic
rays are exactly the affine ones. The zero mass conjecture has also been
rephrased under this picture, and it describes simultaneous zero asymptotic
behaviors of two energy functionals as in Theorem (7.5).

Finally, we would like to point out that the decomposition formula (Theorem
(4.4)) is very likely to be generalized to all dimensions. Then the zero mass
conjecture can be proved in ${\mathbb{C}}^{n}$ for all $S^{1}$-invariant
plurisubharmonic functions in a similar manner. Moreover, an estimate on the
residual mass as in Theorem (1.6) may also be obtained in higher dimensions.

Acknowledgment:  The author is very grateful to Prof. Xiuxiong Chen and Prof.
Mihai Păun for their continuous support and encouragement in mathematics. This
problem has been raised to the author when he was studying in Fourier
Institute, Grenoble. It is also a great pleasure to thank Chengjian Yao,
Xiaojun Wu, Jian Wang, Wei Sun for lots of useful discussions.

Finally, the author wishes to thank Prof. Berndtsson, Prof. Rashkovskii, Prof.
Xiangyu Zhou and Prof. Chi Li for their valuable suggestions on the first
version of this paper.

##  2\.  Plurisubharmonic functions with isolated singularity

Denote $z:=(z_{1},z_{2})$ by the complex Euclidean coordinate on
${\mathbb{C}}^{2}$. There is a natural $S^{1}$-action on it as

$z\rightarrow e^{i\theta}z,$

where $e^{i\theta}z:=(e^{i\theta}z_{1},e^{i\theta}z_{2})$, and $\theta$ is an
arbitrary real number. We say that a domain $D$ is balanced if it is invariant
under this action. Moreover, a function $u$ on a balanced domain is said to be
$S^{1}$-invariant if for all $z\in D$

$u(e^{i\theta}z)=u(z).$

Assume that the origin $0\in{\mathbb{C}}^{2}$ is contained in a balanced
domain $D$, and we denote $D^{*}$ by the set $D-\\{0\\}$. Consider a
plurisubharmonic function $u$ on $D$, and we adapt to the following
definition.

######  Definition 2.1.

A plurisubharmonic function $u$ on $D$ belongs to the family
${\mathcal{F}}(D)$, if it is $S^{1}$-invariant and $L_{loc}^{\infty}$ on
$D^{*}$.

We say that $u$ has an isolated singularity at the origin, if
$u\in{\mathcal{F}}(D)$ and $u(0)=-\infty$. In order to preform calculus, we
also introduce the following collection of functions with better regularities.

######  Definition 2.2.

A plurisubharmonic function $u$ on $D$ belongs to the family
${\mathcal{F}}^{\infty}(D)$, if it is $S^{1}$-invariant and $C^{2}$-continuous
on $D^{*}$.

We note that a function $u\in{\mathcal{F}}^{\infty}(D)$ also belongs to the
family ${\mathcal{F}}(D)$. By shrinking $D$ to a smaller balanced domain if
necessary, a function $u\in{\mathcal{F}}(D)$, or ${\mathcal{F}}^{\infty}(D)$
always has an upper bound. After adjusting a constant, we can further assume
the following normalization condition

$\sup_{D}u\leq-1,$

for all $u\in{\mathcal{F}}(D)$, or ${\mathcal{F}}^{\infty}(D)$.

###  2.1. The residual mass

In the following, we will focus on the local behavior of a plurisubharmonic
function $u$ near the origin. To this purpose, it is enough to consider the
balanced domain $D$ as a small ball centered at the origin.

Let $B_{R}\subset{\mathbb{C}}^{2}$ be the open ball with radius $R$ centered
at the origin, and $B^{*}_{R}:=B_{R}-\\{0\\}$ be the corresponding punctured
ball. Denote its boundary as $S_{R}:=\partial B_{R}$, and then $S_{R}$ is
actually a $3$-sphere in ${\mathbb{R}}^{4}$.

Obviously, these two domains $B_{R}$, $B^{*}_{R}$ are balanced for each $R>0$,
and then we can consider plurisubharmonic functions in the family
${\mathcal{F}}(B_{R})$ and ${\mathcal{F}}^{\infty}(B_{R})$.

Thanks to Demailly’s work [Dem93], the complex Monge-Ampère measure of
$u\in{\mathcal{F}}(B_{1})$ or ${\mathcal{F}}^{\infty}(B_{1})$ is well defined,
namely, the following wedge product is a bidegree-$(2,2)$ closed positive
current

$\text{MA}(u):=dd^{c}u\wedge dd^{c}u,$

and then it is also a positive Borel measure on $B_{1}$. (Here we have used
$dd^{c}=i\partial\bar{\partial}$). Fixing an $R\in(0,1)$, we take this measure
on the ball as

(2.1)
$\text{MA}(u)(B_{R}):=\int_{B_{R}}(dd^{c}u)^{2}=\int\chi_{B_{R}}(dd^{c}u)^{2},$

where $\chi_{B_{R}}$ is the characteristic function of the ball $B_{R}$. Then
it builds a decreasing sequence of non-negative real numbers as $R\rightarrow
0$. Thanks to the dominated convergence theorem, this limit is exactly the
residual Monge-Ampère mass of $u$ at the origin, i.e. we have

(2.2) $\tau_{u}(0)=\frac{1}{\pi^{2}}\lim_{R\rightarrow 0}\text{MA}(u)(B_{R}).$

In order to calculate the measure in equation (2.1), we first observe the
following analogue of the Portemanteau theorem.

######  Lemma 2.3.

Suppose $u_{j}$ is a sequence of smooth plurisubharmonic functions on $B_{1}$,
decreasing to $u\in{\mathcal{F}}^{\infty}(B_{1})$. Then we have

(2.3) $\emph{MA}(u)(B_{R})=\lim_{j\rightarrow+\infty}\emph{MA}(u_{j})(B_{R}),$

for all $R\in(0,1)$.

###### Proof.

It is enough to prove the following two inequalities. First, we claim

(2.4)
$\text{MA}(u)(\overline{B}_{R})\geq\limsup_{j\rightarrow+\infty}\text{MA}(u_{j})(\overline{B}_{R}),$

on any closed ball $\overline{B}_{R}$ in $B_{1}$. Second, we claim

(2.5)
$\text{MA}(u)(B_{R})\leq\liminf_{j\rightarrow+\infty}\text{MA}(u_{j})(B_{R}),$

on any open ball $B_{R}\subsetneq B_{1}$. Then we have

(2.6) $\text{MA}(u)(\overline{B}_{R})=\text{MA}(u)(B_{R}),$

since $u$ is $C^{2}$-continuous near the boundary $S_{R}$. Hence our result
follows from equation (2.4) and (2.5).

To prove the first claim, we observe that there exists a sequence of smooth
cut off functions $\chi_{k}$, such that $\chi_{k}=1$ on $\overline{B}_{R}$ and
$\chi_{k}=0$ outside of $B_{R+\frac{1}{k}}$. Therefore, we have for a fixed
$k$

(2.7)
$\displaystyle\limsup_{j\rightarrow+\infty}\text{MA}(u_{j})(\overline{B}_{R})$
$\displaystyle=$
$\displaystyle\limsup_{j\rightarrow+\infty}\int\chi_{\overline{B}_{R}}(dd^{c}u_{j})^{2}$
$\displaystyle\leq$
$\displaystyle\limsup_{j\rightarrow+\infty}\int\chi_{k}(dd^{c}u_{j})^{2}$
$\displaystyle=$
$\displaystyle\int\chi_{k}(dd^{c}u)^{2}\leq\text{MA}(u)(B_{R+\frac{1}{k}}).$

The equality on the last line of the above equation follows from the
convergence $(dd^{c}u_{j})^{2}\rightarrow(dd^{c}u)^{2}$ in the sense of
currents, cf. [Dem93].

Finally, the inequality (equation (2.4)) follows by taking
$k\rightarrow+\infty$ in equation (2.7). The second claim (equation (2.5)) can
also be proved in a similar way.

∎

######  Remark 2.4.

For a general $u\in{\mathcal{F}}(B_{1})$, Lemma (2.3) may fail to be true for
all $R\in(0,1)$. However, if the Monge-Ampère measure of $u$ has no mass on
the boundary of a ball, namely, we assume

$\int_{S_{R}}(dd^{c}u)^{2}=0,$

for a fixed $R$, then equation (2.6) still holds, and the convergence
(equation (2.3)) follows from the same argument.

Another advantage to deal with the family ${\mathcal{F}}^{\infty}(B_{R})$ is
that we can perform integration by parts on the current $(dd^{c}u)^{2}$ as
follows.

######  Proposition 2.5.

For a plurisubharmonic function $u\in{\mathcal{F}}^{\infty}(B_{1})$, we have

(2.8) $\int_{B_{R}}(dd^{c}u)^{2}=\int_{S_{R}}d^{c}u\wedge dd^{c}u,$

for all $R\in(0,1)$.

###### Proof.

Let $\rho(z):=\rho(|z|)\in C^{\infty}_{0}({\mathbb{C}}^{2})$ be a non-negative
function, satisfying $\rho(r)=0$ for $r\geq 1$, and
$\int_{{\mathbb{C}}^{2}}\rho\ d\lambda=1$. For each $\varepsilon>0$ small, we
rescale it as

$\rho_{\varepsilon}(z):=\varepsilon^{-4}\rho(z/\varepsilon).$

Then we have the following standard regularization of
$u\in{\mathcal{F}}^{\infty}(B_{1})$ by convolution, namely, we have on any
closed ball contained in $B_{1}$

(2.9) $\displaystyle u_{\varepsilon}(z):$ $\displaystyle=$
$\displaystyle(u\ast\rho_{\varepsilon})(z)$ $\displaystyle=$
$\displaystyle\int_{|z-y|\leq\varepsilon}\rho_{\varepsilon}(z-y)u(y)d\lambda(y)$
$\displaystyle=$ $\displaystyle\int_{|w|\leq 1}u(z-\varepsilon
w)\rho(w)d\lambda(w).$

Hence $u_{\varepsilon}(z)$ is a sequence of smooth plurisubharmonic functions,
decreasing to $u(z)$ as $\varepsilon\rightarrow 0$. Moreover, we have
$u_{\varepsilon}\rightarrow u$ uniformly in $C^{2}$-norm on any compact subset
in $B^{*}_{1}$.

Due to Stokes’ theorem, we compute

(2.10)
$\int_{B_{R}}(dd^{c}u_{\varepsilon})^{2}=\int_{S_{R}}d^{c}u_{\varepsilon}\wedge
dd^{c}u_{\varepsilon},$

on any ball $B_{R}\subsetneq B_{1}$. Thanks to Lemma (2.3), The L.H.S. of
equation (2.10) converges to the measure $\text{MA}(u)(B_{R})$. Moreover, the
R.H.S. of this equation converges to the desired integral

$\int_{S_{R}}d^{c}u\wedge dd^{c}u,$

since $u_{\varepsilon}\rightarrow u$ uniformly in $C^{2}$-norm near the sphere
$S_{R}$. Then our result follows. ∎

Take an $S^{1}$-action on the regularization $u_{\varepsilon}(z)$, we note
that it is also invariant under this action. In fact, if put
$w^{\prime}=e^{-i\theta}w$, then we have

(2.11) $\displaystyle u_{\varepsilon}(e^{i\theta}z)$ $\displaystyle=$
$\displaystyle\int u(e^{i\theta}z-\varepsilon w)\rho(w)d\lambda(w)$
$\displaystyle=$ $\displaystyle\int u\\{e^{i\theta}(z-\varepsilon
w^{\prime})\\}\rho(w^{\prime})d\lambda(w^{\prime})$ $\displaystyle=$
$\displaystyle\int u(z-\varepsilon
w^{\prime})\rho(w^{\prime})d\lambda(w^{\prime})=u_{\varepsilon}(z).$

Therefore, the following result holds.

######  Corollary 2.6.

For any function $u\in{\mathcal{F}}(B_{1})$, there exists a sequence of
$S^{1}$-invariant smooth plurisubharmonic function $u_{j}$ decreasing
pointwise to $u$, possible on a slightly smaller ball.

In order to illustrate the use of Proposition (2.5), let us consider a simpler
case. Suppose $u$ is a subharmonic function on the unit disk
${\mathbb{D}}\subset{\mathbb{C}}$, which is also $C^{2}$-continuous on
${\mathbb{D}}^{*}$. Then equation (2.8) reduces to

(2.12) $\int_{|z|<R}dd^{c}u=\int_{|z|=R}d^{c}u,$

for all $R\in(0,1)$.

Utilizing the polar coordinate $z=re^{i\theta}$ on ${\mathbb{C}}^{*}$, we have
the computation

(2.13) $d^{c}u=-\mathop{\rm
Im}\nolimits(\bar{\partial}u)=\frac{1}{2}\left\\{(r\partial_{r}u)d\theta-(r^{-1}\partial_{\theta}u)dr\right\\}.$

Denote the circular average of $u$ by

$\hat{u}(r):=\frac{1}{2\pi}\int_{0}^{2\pi}u(re^{i\theta})d\theta,$

Then it follows

$\frac{1}{\pi}\int_{|z|<R}dd^{c}u=r\partial_{r}\hat{u}|_{r=R}\rightarrow\nu_{u}(0),$

as $R\rightarrow 0$. Therefore, we prove that the residual mass of $u$ at the
origin is zero if its Lelong number is in ${\mathbb{C}}$.

##  3\. The Hopf-coordinates

In this section, we are going to compute the integral on the R.H.S. of
equation (2.8). It boils down to calculate the following $3$-form on the
$3$-sphere

(3.1) $d^{c}u\wedge dd^{c}u|_{S_{R}},$

for $u\in{\mathcal{F}}^{\infty}(B_{1})$, and $R\in(0,1)$. First, we note that
the $S^{1}$-action also performs on this $3$-sphere, and it induces the Hopf
fiberation. The Hopf fiberation $S^{1}\hookrightarrow
S^{3}\xrightarrow{p}S^{2}$ is an example of non-trivial $S^{1}$-fiber bundle
over $S^{2}$. It can be illustrated via the following real coordinates.

Write the unit $3$-sphere as

$S^{3}:=\\{x^{2}_{1}+y_{1}^{2}+x_{2}^{2}+y_{2}^{2}=1\\},$

where $(x_{1},y_{1},x_{2},y_{2})\in{\mathbb{R}}^{4}$ is the Euclidean
coordinate. Let $\theta\in[0,\pi],\varphi\in[0,2\pi]$ be the spherical
coordinate of the unit $2$-sphere $S^{2}\subset{\mathbb{R}}^{3}$. Define the
following coordinate for $\eta\in[0,4\pi]$

$x_{1}=\cos\left(\frac{\eta+\varphi}{2}\right)\sin\left(\frac{\theta}{2}\right),\
\ \ \
y_{1}=\sin\left(\frac{\eta+\varphi}{2}\right)\sin\left(\frac{\theta}{2}\right),$
$x_{2}=\cos\left(\frac{\eta-\varphi}{2}\right)\cos\left(\frac{\theta}{2}\right),\
\ \ \
y_{2}=\sin\left(\frac{\eta-\varphi}{2}\right)\cos\left(\frac{\theta}{2}\right).$

It is clear that $\eta$ is the direction under the $S^{1}$-action, and the
Hopf fiberation $p:S^{3}\rightarrow S^{2}$ is the submersion

(3.2) $\left(2(x_{1}x_{2}+y_{1}y_{2}),\ 2(x_{2}y_{1}-x_{1}y_{2}),\
x_{2}^{2}+y_{2}^{2}-x_{1}^{2}-y_{1}^{2}\right).$

This leads us to introduce the following _real Hopf-coordinate_

$(r,\eta,\theta,\varphi)$

for all $r\in{\mathbb{R}}_{+}$, $\theta\in[0,\pi]$ and
$\eta,\varphi\in{\mathbb{R}}$ to represent a point in
${\mathbb{R}}^{4}-\\{0\\}$. Then the change of variables is

$x_{1}=r\cos\left(\frac{\eta+\varphi}{2}\right)\sin\left(\frac{\theta}{2}\right),\
\ \ \
y_{1}=r\sin\left(\frac{\eta+\varphi}{2}\right)\sin\left(\frac{\theta}{2}\right),$
$x_{2}=r\cos\left(\frac{\eta-\varphi}{2}\right)\cos\left(\frac{\theta}{2}\right),\
\ \ \
y_{2}=r\sin\left(\frac{\eta-\varphi}{2}\right)\cos\left(\frac{\theta}{2}\right).$

It follows that a $3$-sphere $S_{R}$ as the boundary of the ball $B_{R}$ can
be written as

$S_{R}:=\\{x^{2}_{1}+y_{1}^{2}+x_{2}^{2}+y_{2}^{2}=R^{2}\\}\subset{\mathbb{R}}^{4},$

for a fixed $R>0$. Moreover, if the angle $\varphi$ varies in $[0,2\pi)$ and
$\eta$ in $[0,4\pi)$, then this coordinate runs over all the points on the
$3$-sphere $S_{R}$ exactly once.

###  3.1. Complex version

There is another way to view the Hopf fiberation through the complex
coordinates. Let $z:=(z_{1},z_{2})$ be the complex Euclidean coordinate on
${\mathbb{C}}^{2}$, and then the unit $3$-sphere $S^{3}$ can be characterized
as

$S^{3}:=\\{|z_{1}|^{2}+|z_{2}|^{2}=1\\}.$

Topologically, the $2$-sphere $S^{2}$ can be identified with the extended
complex plane ${\mathbb{C}}_{\infty}:={\mathbb{C}}\bigcup\\{\infty\\}$ via the
stereographic projection. Moreover, the complex projective line
${\mathbb{C}}{\mathbb{P}}^{1}$ can also be identified with
${\mathbb{C}}_{\infty}$ via the continuous map $f:S^{3}\rightarrow S^{2}$ as

$f:(z_{1},z_{2})\rightarrow\frac{z_{1}}{z_{2}}\in{\mathbb{C}}_{\infty}.$

Then we can define the fiber map $p$ in the fiber bundle $S^{1}\hookrightarrow
S^{3}\xrightarrow{p}S^{2}$ as

$p:(z_{1},z_{2})\rightarrow[z_{1}:z_{2}]\in{\mathbb{C}}{\mathbb{P}}^{1},$

and the pre-image of each point in ${\mathbb{C}}{\mathbb{P}}^{1}$ is a great
circle in $S^{3}$.

In order to illustrate the idea, we first introduce the following easier
version. It is well known that ${\mathbb{C}}{\mathbb{P}}^{1}$ can be covered
by two holomorphic coordinate charts, consisting of

$U_{1}:={\mathbb{C}}{\mathbb{P}}^{1}-[1:0],\ \ \
U_{2}:={\mathbb{C}}{\mathbb{P}}^{1}-[0:1],$

where $U_{1}$ is identified with ${\mathbb{C}}_{\infty}-\\{\infty\\}$ and
$U_{2}$ with ${\mathbb{C}}_{\infty}-\\{0\\}$. Write their corresponding
holomorphic coordinates as ${\zeta}:=z_{1}/z_{2}$ and $\xi={\zeta}^{-1}$. Then
we give the following two homeomorphisms as the local trivializations of the
fiber map $p$: define $\psi_{1}:{\mathbb{C}}\times S^{1}\rightarrow
p^{-1}({\mathbb{C}})$ and $\psi_{2}:({\mathbb{C}}_{\infty}-\\{0\\})\times
S^{1}\rightarrow p^{-1}({\mathbb{C}}_{\infty}-\\{0\\})$ as

(3.3)
$\psi_{1}({\zeta},\eta^{\prime})=\left(\frac{{\zeta}e^{i\eta^{\prime}}}{(1+|{\zeta}|^{2})^{1/2}},\
\frac{e^{i\eta^{\prime}}}{(1+|{\zeta}|^{2})^{1/2}}\right),$

and

(3.4)
$\psi_{2}(\xi,\eta^{\prime})=\left(\frac{e^{i\eta^{\prime}}}{(1+|\xi|^{2})^{1/2}},\
\frac{\xi e^{i\eta^{\prime}}}{(1+|\xi|^{2})^{1/2}}\right),$

for all ${\zeta},\xi\in{\mathbb{C}}$ and $\eta^{\prime}\in{\mathbb{R}}$. It is
apparent that we have $p\circ\psi_{1}({\zeta},\eta^{\prime})=[{\zeta}:1]$ on
$U_{1}$ and $p\circ\psi_{2}(\xi,\eta^{\prime})=[1:\xi]$ on $U_{2}$. Then these
two local trivializations describe the Hopf fiberation in one way.

###  3.2. Another version

Next, we can write the complex variables of ${\mathbb{C}}^{2}$ in terms of the
real Hopf-coordinate as

(3.5) $z_{1}=(r\sin(\theta/2))e^{\frac{i}{2}(\eta+\varphi)},\ \ \
z_{2}=(r\cos(\theta/2))e^{\frac{i}{2}(\eta-\varphi)},$

for all $r>0$, $\theta\in[0,\pi]$, $\varphi\in[0,2\pi]$ and $\eta\in[0,4\pi]$.
Moreover, the complex variable on ${\mathbb{C}}_{\infty}$ is

(3.6) ${\zeta}:=\frac{z_{1}}{z_{2}}=\tan(\theta/2)e^{i\varphi}.$

and we obtain the following change of variables

(3.7)
$z_{1}=re^{\frac{i}{2}\eta}\frac{({\zeta}\cdot|{\zeta}|)^{1/2}}{(1+|{\zeta}|^{2})^{1/2}},\
\ \
z_{2}=re^{\frac{i}{2}\eta}\frac{(\bar{\zeta}/|{\zeta}|)^{1/2}}{(1+|{\zeta}|^{2})^{1/2}}.$

In fact, they can be viewed as different local trivializations of the Hopf-
fiberation with the fiber map $p$. Denote ${\ell}_{+},{\ell}_{-}$ by the
following half circles on $S^{2}\cong{\mathbb{C}}{\mathbb{P}}^{1}$

${\ell}_{+}:=\left\\{[t:1]\in{\mathbb{C}}{\mathbb{P}}^{1};\ \
t\in[0,+\infty]\right\\};$
${\ell}_{-}:=\left\\{[t:1]\in{\mathbb{C}}{\mathbb{P}}^{1};\ \
t\in[-\infty,0]\right\\},$

and define two holomorphic coordinate charts as

$V_{1}:=S^{2}-{\ell}_{+};\ \ \ V_{2}:=S^{2}-{\ell}_{-}.$

Similarly, we have another two half circles as

$\jmath_{+}:=\left\\{[e^{i\theta^{\prime}}:1]\in{\mathbb{C}}{\mathbb{P}}^{1};\
\ \theta^{\prime}\in[0,\pi]\right\\};$
$\jmath_{-}:=\left\\{[e^{i\theta^{\prime}}:1]\in{\mathbb{C}}{\mathbb{P}}^{1};\
\ \theta^{\prime}\in[\pi,2\pi]\right\\},$

and another two charts are defined as

$V_{3}:=S^{2}-\jmath_{+};\ \ \ V_{4}:=S^{2}-\jmath_{-}.$

It is apparent that each of the charts $V_{i},i=1,2,3,4$ can be identified to
the slit plane ${\mathbb{C}}_{+}$ (or ${\mathbb{C}}_{-}$) via the
stereographic projections, and they together cover the whole sphere $S^{2}$.
Then we can introduce homeomorphisms $\psi^{\prime}_{i}$ between $V_{i}\times
S^{1}$ and $p^{-1}(V_{i})$ for all $i=1,2,3,4$, and they will build the local
trivializations for the fiber map $p$ in a different way.

In particular, equation (3.7) gives the map $\psi^{\prime}_{1}:V_{1}\times
S^{1}\rightarrow p^{-1}(V_{1})$ as

(3.8)
$\psi^{\prime}_{1}({\zeta},\eta):=\left(e^{\frac{i}{2}\eta}\frac{({\zeta}\cdot|{\zeta}|)^{1/2}}{(1+|{\zeta}|^{2})^{1/2}},\
\ \
e^{\frac{i}{2}\eta}\frac{(|{\zeta}|/{\zeta})^{1/2}}{(1+|{\zeta}|^{2})^{1/2}}\right).$

Therefore, we introduce the following coordinate

$(r,\eta,{\zeta},\bar{\zeta})$

for all $r>0$, $\eta\in{\mathbb{R}}$ and ${\zeta}\in{\mathbb{C}}_{\infty}$ to
represent a point in ${\mathbb{C}}^{2}-\\{0\\}$, and refer it as the _complex
Hopf-coordinate_.

We note that the coordinate ${\zeta}$ (under the trivialization
$\psi^{\prime}_{1}$) is no longer continuous across the line ${\ell}_{+}$,
since its fractional power ${\zeta}^{1/2}$ is multi-valued. However, we do not
worry about this problem if the function $u$ is $S^{1}$-invariant, and the
reason is as follows.

In fact, the trivialization $\psi^{\prime}_{2}$ on $V_{2}\times S^{1}$ will
take another analytic branch of the two-valued holomorphic function
${\zeta}^{1/2}$, and then we can write

(3.9)
$\psi^{\prime}_{2}({\zeta},\eta):=\left(e^{\frac{i}{2}\eta}\frac{e^{i\pi}({\zeta}\cdot|{\zeta}|)^{1/2}}{(1+|{\zeta}|^{2})^{1/2}},\
\ \
e^{\frac{i}{2}\eta}\frac{e^{-i\pi}(|{\zeta}|/{\zeta})^{1/2}}{(1+|{\zeta}|^{2})^{1/2}}\right).$

Hence it follows $u\circ\psi_{1}=u\circ\psi_{2}$ for all $({\zeta},\eta)$ in
the overlapping area. In other words, the function $u$ is periodic in the
angle $\varphi$ direction with period $2\pi$(instead of $4\pi$!).

######  Remark 3.1.

In fact, there is no difference between the two trivializations $\psi_{1}$ and
$\psi^{\prime}_{1}$ for any $S^{1}$-invariant function, since we can rewrite
$\psi^{\prime}_{1}$ as follows from equation (3.6)

(3.10)
$\psi^{\prime}_{1}({\zeta},\eta):=\left(e^{-\frac{i}{2}\varphi}\frac{{\zeta}e^{\frac{i}{2}\eta}}{(1+|{\zeta}|^{2})^{1/2}},\
\ \
e^{-\frac{i}{2}\varphi}\frac{e^{\frac{i}{2}\eta}}{(1+|{\zeta}|^{2})^{1/2}}\right),$

and then it follows $u\circ\psi_{1}=u\circ\psi^{\prime}_{1}$ for all
$({\zeta},\eta)$ in the overlapping area.

##  4\. The decomposition formula

Now we are going to perform local computations near a point $b\in S_{R}$,
under the complex Hopf-coordinate. It is noted that we will directly calculate
on $\bar{z}^{2}_{1}$ and $\bar{z}^{2}_{2}$ in the following, and the
fractional power ${\zeta}^{1/2}$ will not be used essentially.

###  4.1. The $1$-form

Recall that we have

(4.1)
$z_{1}=re^{\frac{i}{2}\eta}\frac{({\zeta}\cdot|{\zeta}|)^{1/2}}{(1+|{\zeta}|^{2})^{1/2}},\
\ \
z_{2}=re^{\frac{i}{2}\eta}\frac{(\bar{\zeta}/|{\zeta}|)^{1/2}}{(1+|{\zeta}|^{2})^{1/2}},$

and then the following relations exist:

(4.2) $z_{1}\cdot\bar{z}_{2}=r^{2}\frac{{\zeta}}{1+|{\zeta}|^{2}},\ \ \
z_{1}\cdot z_{2}=r^{2}\frac{e^{i\eta}|{\zeta}|}{1+|{\zeta}|^{2}},$

and

(4.3) $|z_{1}|^{2}+|z_{2}|^{2}=r^{2},$

together with

(4.4)
$|z_{2}|^{2}-|z_{1}|^{2}=r^{2}\frac{1-|{\zeta}|^{2}}{1+|{\zeta}|^{2}}=r^{2}\cos\theta.$

The first goal is to calculate the following $1$-form

(4.5) $\displaystyle d^{c}u$ $\displaystyle=$
$\displaystyle\frac{i}{2}(\bar{\partial}u-\partial u)=-\mathop{\rm
Im}\nolimits(\bar{\partial}u)$ $\displaystyle=$ $\displaystyle-\mathop{\rm
Im}\nolimits\left(\frac{\partial
u}{\partial\bar{z}_{1}}d\bar{z}_{1}\right)-\mathop{\rm
Im}\nolimits\left(\frac{\partial u}{\partial\bar{z}_{2}}d\bar{z}_{2}\right).$

The first term on the R.H.S. of the above equation can be computed as

(4.6)
$\bar{z}_{1}=re^{-\frac{i}{2}\eta}\frac{(\bar{\zeta}\cdot|{\zeta}|)^{1/2}}{(1+|{\zeta}|^{2})^{1/2}},\
\ \
\bar{z}^{2}_{1}=r^{2}e^{-i\eta}\frac{\bar{\zeta}\cdot|{\zeta}|}{(1+|{\zeta}|^{2})},$

and it follows

(4.7) $\displaystyle 2\bar{z}_{1}d\bar{z}_{1}$ $\displaystyle=$ $\displaystyle
2re^{-i\eta}\frac{\bar{\zeta}\cdot|{\zeta}|}{(1+|{\zeta}|^{2})}dr-
ir^{2}e^{-i\eta}\frac{\bar{\zeta}\cdot|{\zeta}|}{(1+|{\zeta}|^{2})}d\eta$
$\displaystyle+$ $\displaystyle
r^{2}e^{-i\eta}\left\\{\partial_{\zeta}\left(\frac{\bar{\zeta}\cdot|{\zeta}|}{1+|{\zeta}|^{2}}\right)d{\zeta}+\partial_{\bar{\zeta}}\left(\frac{\bar{\zeta}\cdot|{\zeta}|}{1+|{\zeta}|^{2}}\right)d\bar{\zeta}\right\\},$

and then we have

(4.8)
$\partial_{\zeta}\left(\frac{\bar{\zeta}\cdot|{\zeta}|}{1+|{\zeta}|^{2}}\right)=\frac{\bar{\zeta}^{2}}{2|{\zeta}|}\cdot\frac{1-|{\zeta}|^{2}}{(1+|{\zeta}|^{2})^{2}},$

and

(4.9)
$\partial_{\bar{\zeta}}\left(\frac{\bar{\zeta}\cdot|{\zeta}|}{1+|{\zeta}|^{2}}\right)=\frac{|{\zeta}|}{2}\cdot\frac{3+|{\zeta}|^{2}}{(1+|{\zeta}|^{2})^{2}}.$

Combing equations (4.8), (4.9) with (4.7), we obtain

(4.10) $\displaystyle 2\bar{z}_{1}d\bar{z}_{1}$ $\displaystyle=$
$\displaystyle
2re^{-i\eta}\frac{\bar{\zeta}\cdot|{\zeta}|}{(1+|{\zeta}|^{2})}dr-
ir^{2}e^{-i\eta}\frac{\bar{\zeta}\cdot|{\zeta}|}{(1+|{\zeta}|^{2})}d\eta$
$\displaystyle+$ $\displaystyle
r^{2}e^{-i\eta}\frac{\bar{\zeta}^{2}}{2|{\zeta}|}\cdot\frac{1-|{\zeta}|^{2}}{(1+|{\zeta}|^{2})^{2}}d{\zeta}$
$\displaystyle+$ $\displaystyle
r^{2}e^{-i\eta}\frac{|{\zeta}|}{2}\cdot\frac{3+|{\zeta}|^{2}}{(1+|{\zeta}|^{2})^{2}}d\bar{\zeta},$

and it follows

(4.11) $\displaystyle 4\ d\bar{z}_{1}$ $\displaystyle=$ $\displaystyle
4e^{-\frac{i}{2}\eta}\frac{(\bar{\zeta}\cdot|{\zeta}|)^{1/2}}{(1+|{\zeta}|^{2})^{1/2}}dr-2ire^{-\frac{i}{2}\eta}\frac{(\bar{\zeta}\cdot|{\zeta}|)^{1/2}}{(1+|{\zeta}|^{2})^{1/2}}d\eta$
$\displaystyle+$ $\displaystyle
re^{-\frac{i}{2}\eta}\left(\frac{\bar{\zeta}}{|{\zeta}|}\right)^{\frac{3}{2}}\cdot\frac{1-|{\zeta}|^{2}}{(1+|{\zeta}|^{2})^{3/2}}d{\zeta}$
$\displaystyle+$ $\displaystyle
re^{-\frac{i}{2}\eta}\left(\frac{{\zeta}}{|{\zeta}|}\right)^{\frac{1}{2}}\cdot\frac{3+|{\zeta}|^{2}}{(1+|{\zeta}|^{2})^{3/2}}d\bar{\zeta}.$

In fact, we can divide $\bar{z}^{2}_{1}$ in equation (4.10), and obtain

$\displaystyle 4\
d\bar{z}_{1}=\bar{z}_{1}\left\\{4r^{-1}dr-2id\eta+\frac{1}{{\zeta}}\cdot\frac{1-|{\zeta}|^{2}}{(1+|{\zeta}|^{2})}d{\zeta}+\frac{1}{\bar{\zeta}}\cdot\frac{3+|{\zeta}|^{2}}{(1+|{\zeta}|^{2})}d\bar{\zeta}\right\\}.$

Similarly, we have

(4.13)
$\bar{z}_{2}=re^{-\frac{i}{2}\eta}\frac{({\zeta}/|{\zeta}|)^{1/2}}{(1+|{\zeta}|^{2})^{1/2}},\
\ \
\bar{z}^{2}_{2}=r^{2}e^{-i\eta}\frac{{\zeta}/|{\zeta}|}{(1+|{\zeta}|^{2})},$

and then it gives

(4.14) $\displaystyle 2\bar{z}_{2}d\bar{z}_{2}$ $\displaystyle=$
$\displaystyle 2re^{-i\eta}\frac{{\zeta}/|{\zeta}|}{(1+|{\zeta}|^{2})}dr-
ir^{2}e^{-i\eta}\frac{{\zeta}/|{\zeta}|}{(1+|{\zeta}|^{2})}d\eta$
$\displaystyle+$ $\displaystyle
r^{2}e^{-i\eta}\left\\{\partial_{\zeta}\left(\frac{{\zeta}/|{\zeta}|}{1+|{\zeta}|^{2}}\right)d{\zeta}+\partial_{\bar{\zeta}}\left(\frac{{\zeta}/|{\zeta}|}{1+|{\zeta}|^{2}}\right)d\bar{\zeta}\right\\}.$

Moreover, we have

(4.15)
$\partial_{\zeta}\left(\frac{{\zeta}/|{\zeta}|}{1+|{\zeta}|^{2}}\right)=\frac{1}{2|{\zeta}|}\cdot\frac{1-|{\zeta}|^{2}}{(1+|{\zeta}|^{2})^{2}},$

and

(4.16)
$\partial_{\bar{\zeta}}\left(\frac{{\zeta}/|{\zeta}|}{1+|{\zeta}|^{2}}\right)=-\frac{{\zeta}^{2}}{2|{\zeta}|^{3}}\cdot\frac{1+3|{\zeta}|^{2}}{(1+|{\zeta}|^{2})^{2}}.$

It follows

(4.17) $\displaystyle 2\bar{z}_{2}d\bar{z}_{2}$ $\displaystyle=$
$\displaystyle 2re^{-i\eta}\frac{{\zeta}/|{\zeta}|}{(1+|{\zeta}|^{2})}dr-
ir^{2}e^{-i\eta}\frac{{\zeta}/|{\zeta}|}{(1+|{\zeta}|^{2})}d\eta$
$\displaystyle+$ $\displaystyle
r^{2}e^{-i\eta}\frac{1}{2|{\zeta}|}\cdot\frac{1-|{\zeta}|^{2}}{(1+|{\zeta}|^{2})^{2}}d{\zeta}$
$\displaystyle-$ $\displaystyle
r^{2}e^{-i\eta}\frac{{\zeta}^{2}}{2|{\zeta}|^{3}}\cdot\frac{1+3|{\zeta}|^{2}}{(1+|{\zeta}|^{2})^{2}}d\bar{\zeta},$

and we further simplify as

$\displaystyle 4\
d\bar{z}_{2}=\bar{z}_{2}\left\\{4r^{-1}dr-2id\eta+\frac{1}{{\zeta}}\cdot\frac{1-|{\zeta}|^{2}}{(1+|{\zeta}|^{2})}d{\zeta}-\frac{1}{\bar{\zeta}}\cdot\frac{1+3|{\zeta}|^{2}}{(1+|{\zeta}|^{2})}d\bar{\zeta}\right\\}.$

Next we are going to use the chain rule as follows

$\frac{\partial u}{\partial\bar{z}_{1}}=\frac{\partial u}{\partial
r}\frac{\partial r}{\partial\bar{z}_{1}}+\frac{\partial
u}{\partial{\zeta}}\frac{\partial{\zeta}}{\partial\bar{z}_{1}}+\frac{\partial
u}{\partial\bar{\zeta}}\frac{\partial\bar{\zeta}}{\partial\bar{z}_{1}},$

and

$\frac{\partial u}{\partial\bar{z}_{2}}=\frac{\partial u}{\partial
r}\frac{\partial r}{\partial\bar{z}_{2}}+\frac{\partial
u}{\partial{\zeta}}\frac{\partial{\zeta}}{\partial\bar{z}_{2}}+\frac{\partial
u}{\partial\bar{\zeta}}\frac{\partial\bar{\zeta}}{\partial\bar{z}_{2}}.$

Since ${\zeta}$ is a holomorphic function of $z$, it is clear that
$\partial{\zeta}/\partial\bar{z}_{1}=0$, and
$\partial{\zeta}/\partial\bar{z}_{2}=0$. Moreover, it follows from equation
(4.3) that $\partial r/\partial\bar{z}_{1}=z_{1}/2r$ and $\partial
r/\partial\bar{z}_{2}=z_{2}/2r$. Hence we have

$\displaystyle 4\mathop{\rm Im}\nolimits\left(\frac{\partial u}{\partial
r}\frac{\partial r}{\partial\bar{z}_{1}}d\bar{z}_{1}\right)$ $\displaystyle=$
$\displaystyle(r^{-1}\partial_{r}u)|z_{1}|^{2}\left\\{-d\eta+\frac{1-|{\zeta}|^{2}}{2(1+|{\zeta}|^{2})}\mathop{\rm
Im}\nolimits\left(\frac{d{\zeta}}{{\zeta}}\right)+\frac{3+|{\zeta}|^{2}}{2(1+|{\zeta}|^{2})}\mathop{\rm
Im}\nolimits\left(\frac{d\bar{\zeta}}{\bar{\zeta}}\right)\right\\}$
$\displaystyle=$
$\displaystyle-(r^{-1}\partial_{r}u)|z_{1}|^{2}\left\\{d\eta+\mathop{\rm
Im}\nolimits\left(\frac{d{\zeta}}{{\zeta}}\right)\right\\},$

where we used the equality $\mathop{\rm
Im}\nolimits({\zeta}^{-1}d{\zeta})=-\mathop{\rm
Im}\nolimits(\bar{\zeta}^{-1}d\bar{\zeta})$. Similarly, it follows

$\displaystyle 4\mathop{\rm Im}\nolimits\left(\frac{\partial u}{\partial
r}\frac{\partial r}{\partial\bar{z}_{2}}d\bar{z}_{2}\right)$ $\displaystyle=$
$\displaystyle(r^{-1}\partial_{r}u)|z_{2}|^{2}\left\\{-d\eta+\frac{1-|{\zeta}|^{2}}{2(1+|{\zeta}|^{2})}\mathop{\rm
Im}\nolimits\left(\frac{d{\zeta}}{{\zeta}}\right)-\frac{1+3|{\zeta}|^{2}}{2(1+|{\zeta}|^{2})}\mathop{\rm
Im}\nolimits\left(\frac{d\bar{\zeta}}{\bar{\zeta}}\right)\right\\}$
$\displaystyle=$
$\displaystyle-(r^{-1}\partial_{r}u)|z_{2}|^{2}\left\\{d\eta-\mathop{\rm
Im}\nolimits\left(\frac{d{\zeta}}{{\zeta}}\right)\right\\}.$

Combing with equation (4.1) and (4.1), we further have

(4.21) $\displaystyle-4\mathop{\rm Im}\nolimits\left(\frac{\partial
u}{\partial r}\frac{\partial
r}{\partial\bar{z}_{1}}d\bar{z}_{1}\right)-4\mathop{\rm
Im}\nolimits\left(\frac{\partial u}{\partial r}\frac{\partial
r}{\partial\bar{z}_{2}}d\bar{z}_{2}\right)$ $\displaystyle=$
$\displaystyle(r\partial_{r}u)\left\\{d\eta-\frac{1-|{\zeta}|^{2}}{1+|{\zeta}|^{2}}\mathop{\rm
Im}\nolimits\left(\frac{d{\zeta}}{{\zeta}}\right)\right\\},$

and note that the R.H.S. of equation (4.21) equals to

(4.22) $(r\partial_{r}u)(d\eta-\cos\theta d\varphi)$

in the real Hopf-coordiante.

Furthermore, it is straightforward to have

$\frac{\partial\bar{\zeta}}{\partial\bar{z}_{1}}=\frac{1}{\bar{z}_{2}},\ \ \
\frac{\partial\bar{\zeta}}{\partial\bar{z}_{2}}=-\frac{\bar{z}_{1}}{(\bar{z}_{2})^{2}}.$

Then we obtain

$\displaystyle 4\frac{\partial
u}{\partial\bar{\zeta}}\frac{\partial\bar{\zeta}}{\partial\bar{z}_{1}}d\bar{z}_{1}$
$\displaystyle=$
$\displaystyle(\bar{\zeta}\partial_{\bar{\zeta}}u)\left\\{4r^{-1}dr-2id\eta+\frac{1-|{\zeta}|^{2}}{(1+|{\zeta}|^{2})}\left(\frac{d{\zeta}}{{\zeta}}\right)+\frac{3+|{\zeta}|^{2}}{(1+|{\zeta}|^{2})}\left(\frac{d\bar{\zeta}}{\bar{\zeta}}\right)\right\\},$

and

$\displaystyle 4\frac{\partial
u}{\partial\bar{\zeta}}\frac{\partial\bar{\zeta}}{\partial\bar{z}_{2}}d\bar{z}_{2}$
$\displaystyle=$
$\displaystyle-(\bar{\zeta}\partial_{\bar{\zeta}}u)\left\\{4r^{-1}dr-2id\eta+\frac{1-|{\zeta}|^{2}}{(1+|{\zeta}|^{2})}\left(\frac{d{\zeta}}{{\zeta}}\right)-\frac{1+3|{\zeta}|^{2}}{(1+|{\zeta}|^{2})}\left(\frac{d\bar{\zeta}}{\bar{\zeta}}\right)\right\\}.$

Moreover, we note the first three terms on the R.H.S. of equation (4.1) and
(4.1) are only differ by a minus sign. Hence it follows

(4.25) $\displaystyle-\mathop{\rm Im}\nolimits\left(\frac{\partial
u}{\partial\bar{\zeta}}\frac{\partial\bar{\zeta}}{\partial\bar{z}_{1}}d\bar{z}_{1}+\frac{\partial
u}{\partial\bar{\zeta}}\frac{\partial\bar{\zeta}}{\partial\bar{z}_{2}}d\bar{z}_{2}\right)$
$\displaystyle=$ $\displaystyle-\mathop{\rm
Im}\nolimits\left(\partial_{\bar{\zeta}}u\cdot d\bar{\zeta}\right)=\mathop{\rm
Im}\nolimits\left(\partial_{{\zeta}}u\cdot d{\zeta}\right).$

In conclusion, we obtain the following formula.

######  Lemma 4.1.

For any $u\in{\mathcal{F}}^{\infty}(B_{1})$, we have

(4.26) $\displaystyle 4\ d^{c}u$ $\displaystyle=$
$\displaystyle(r\partial_{r}u)\left\\{d\eta-\cos\theta\cdot\mathop{\rm
Im}\nolimits\left(\frac{d{\zeta}}{{\zeta}}\right)\right\\}+4\mathop{\rm
Im}\nolimits(\partial_{{\zeta}}u\cdot d{\zeta})$ $\displaystyle=$
$\displaystyle(ru_{r})d\eta+(2\sin\theta\cdot u_{\theta}-\cos\theta\cdot
ru_{r})d\varphi-\frac{2u_{\varphi}}{\sin\theta}d\theta.$

where $\cos\theta=(1-|{\zeta}|^{2})(1+|{\zeta}|^{2})^{-1}$.

###### Proof.

Combine equation (4.21) and (4.25), and then the equality follows in the
complex Hopf-coordinate. By utilizing the real Hopf-coordinate, we obtain the
second line on the R.H.S. of equation (4.26), and one can check the following
change of variables

$2\mathop{\rm Im}\nolimits(\partial_{{\zeta}}u\cdot
d{\zeta})=\sin\theta(\partial_{\theta}u)d\varphi-\frac{1}{\sin\theta}(\partial_{\varphi}u)d\theta,$

Then our result follows. ∎

###  4.2. The $3$-form

In this section, we continue our computation on the complex hessian $dd^{c}u$
for a $u\in{\mathcal{F}}^{\infty}(B_{1})$ in the complex Hopf-coordinate.
Again, we perform the calculation near a point on $S_{R}$ for an $R\in(0,1)$.
Moreover, we only need to know the formula for its restriction as

$dd^{c}u|_{S_{R}}=d(d^{c}u)|_{S_{R}}.$

For the first term in equation (4.26) we have

(4.27)
$d\left\\{(ru_{r})d\eta\right\\}|_{S_{R}}=(ru_{r})_{,{\zeta}}d{\zeta}\wedge
d\eta+(ru_{r})_{,\bar{\zeta}}d\bar{\zeta}\wedge d\eta,$

where we used the notation

$(ru_{r})_{,{\zeta}}=\frac{\partial(r\partial_{r}u)}{\partial{\zeta}},\ \
(ru_{r})_{,\bar{\zeta}}=\frac{\partial(r\partial_{r}u)}{\partial\bar{\zeta}}.$

For the second term, we can write it as

(4.28)
$d\left\\{(ru_{r})\cos\theta\frac{1}{2i}\left(\frac{d\bar{\zeta}}{\bar{\zeta}}-\frac{d{\zeta}}{{\zeta}}\right)\right\\}|_{S_{R}}.$

Then we have

(4.29)
$\partial_{{\zeta}}\cos\theta=\partial_{{\zeta}}\left(\frac{1-|{\zeta}|^{2}}{1+|{\zeta}|^{2}}\right)=\frac{-2\bar{\zeta}}{(1+|{\zeta}|^{2})^{2}},$

and the following is clear

(4.30) $d\left(\frac{d{\zeta}}{{\zeta}}\right)=0,\ \ \
d\left(\frac{d\bar{\zeta}}{\bar{\zeta}}\right)=0.$

Thus equation (4.28) is equal to

(4.31)
$\displaystyle\frac{1}{2i}\left\\{(ru_{r})_{,{\zeta}}\cos\theta\frac{{\zeta}}{|{\zeta}|^{2}}-(ru_{r})\frac{2}{(1+|{\zeta}|^{2})^{2}}\right\\}d{\zeta}\wedge
d\bar{\zeta}$ $\displaystyle+$
$\displaystyle\frac{1}{2i}\left\\{(ru_{r})_{,\bar{\zeta}}\cos\theta\frac{\bar{\zeta}}{|{\zeta}|^{2}}-(ru_{r})\frac{2}{(1+|{\zeta}|^{2})^{2}}\right\\}d{\zeta}\wedge
d\bar{\zeta}$ $\displaystyle=$ $\displaystyle(ru_{r})\frac{2id{\zeta}\wedge
d\bar{\zeta}}{(1+|{\zeta}|^{2})^{2}}-\mathop{\rm
Re}\nolimits({\zeta}\cdot(ru_{r})_{,{\zeta}})\frac{\cos\theta}{|{\zeta}|^{2}}id{\zeta}\wedge
d\bar{\zeta},$

and then we obtain the following equality.

######  Lemma 4.2.

For any $u\in{\mathcal{F}}^{\infty}(B_{1})$, we have

(4.32) $\displaystyle 4\ dd^{c}u|_{S_{R}}$ $\displaystyle=$
$\displaystyle(ru_{r})_{,{\zeta}}d{\zeta}\wedge
d\eta+(ru_{r})_{,\bar{\zeta}}d\bar{\zeta}\wedge d\eta$ $\displaystyle+$
$\displaystyle\left\\{4u_{,{\zeta}\bar{\zeta}}-\mathop{\rm
Re}\nolimits({\zeta}\cdot(ru_{r})_{,{\zeta}})\frac{\cos\theta}{|{\zeta}|^{2}}+\frac{2ru_{r}}{(1+|{\zeta}|^{2})^{2}}\right\\}id{\zeta}\wedge
d\bar{\zeta}.$

###### Proof.

It is left to compute the third term in equation (4.26), and we can perform
like follows

$\frac{1}{2i}d(\partial_{{\zeta}}u\cdot d{\zeta}-\partial_{\bar{\zeta}}u\cdot
d\bar{\zeta})|_{S_{R}}=\frac{\partial^{2}u}{\partial{\zeta}\partial\bar{\zeta}}\
id{\zeta}\wedge d\bar{\zeta},$

and then the equality follows. ∎

In terms of the real Hopf-coordinate, we observe that

$id{\zeta}\wedge
d\bar{\zeta}=\frac{\sin\theta}{2\cos^{4}(\theta/2)}d\theta\wedge d\varphi,$

and then we obtain another way to describe the above $2$-form as

(4.33) $\displaystyle 4\ dd^{c}u|_{S_{R}}$ $\displaystyle=$
$\displaystyle(ru_{,r\theta})d\theta\wedge
d\eta+(ru_{,r\varphi})d\varphi\wedge d\eta$ $\displaystyle+$
$\displaystyle\left\\{(ru_{r})\sin\theta-(ru_{,r\theta})\cos\theta\right\\}d\theta\wedge
d\varphi$ $\displaystyle+$ $\displaystyle 2\left\\{\sin\theta\cdot
u_{,\theta\theta}+\cos\theta\cdot
u_{\theta}+(\sin\theta)^{-1}u_{,\varphi\varphi}\right\\}d\theta\wedge
d\varphi.$

Next we are going to compute the following $3$-form. Combing Lemma (4.1) with
Lemma (4.2), it equals to

$16\ d^{c}u\wedge dd^{c}u|_{S_{R}}=\mbox{I}+\mbox{II},$

where

$\displaystyle\mbox{I}:$ $\displaystyle=$
$\displaystyle\left((r\partial_{r}u)\left\\{d\eta-\cos\theta\cdot\mathop{\rm
Im}\nolimits\left(\frac{d{\zeta}}{{\zeta}}\right)\right\\}+4\mathop{\rm
Im}\nolimits(\partial_{{\zeta}}u\cdot d{\zeta})\right)$ $\displaystyle\wedge$
$\displaystyle 2\mathop{\rm
Re}\nolimits\left\\{(ru_{r})_{,{\zeta}}d{\zeta}\wedge d\eta\right\\},$

and

$\displaystyle\mbox{II}:$ $\displaystyle=$
$\displaystyle\left((r\partial_{r}u)\left\\{d\eta-\cos\theta\cdot\mathop{\rm
Im}\nolimits\left(\frac{d{\zeta}}{{\zeta}}\right)\right\\}+4\mathop{\rm
Im}\nolimits(\partial_{{\zeta}}u\cdot d{\zeta})\right)$ $\displaystyle\wedge$
$\displaystyle\left\\{4u_{,{\zeta}\bar{\zeta}}-\mathop{\rm
Re}\nolimits({\zeta}\cdot(ru_{r})_{,{\zeta}})\frac{\cos\theta}{|{\zeta}|^{2}}+\frac{2ru_{r}}{(1+|{\zeta}|^{2})^{2}}\right\\}id{\zeta}\wedge
d\bar{\zeta}.$

Then the first term is

I $\displaystyle=$ $\displaystyle
d^{c}u\wedge\left\\{(ru_{r})_{,{\zeta}}d{\zeta}\wedge
d\eta+(ru_{r})_{,\bar{\zeta}}d\bar{\zeta}\wedge d\eta\right\\}$
$\displaystyle=$
$\displaystyle\frac{i}{2}(ru_{r})\frac{\cos\theta}{|{\zeta}|^{2}}\left\\{\bar{\zeta}\cdot(ru_{r})_{,\bar{\zeta}}+{\zeta}\cdot(ru_{r})_{,{\zeta}}\right\\}d{\zeta}\wedge
d\bar{\zeta}\wedge d\eta$ $\displaystyle-$ $\displaystyle
2i\left\\{(\partial_{\bar{\zeta}}u)(ru_{r})_{,{\zeta}}+(\partial_{{\zeta}}u)(ru_{r})_{,\bar{\zeta}}\right\\}d{\zeta}\wedge
d\bar{\zeta}\wedge d\eta$ $\displaystyle=$
$\displaystyle(ru_{r})\frac{\cos\theta}{|{\zeta}|^{2}}\mathop{\rm
Re}\nolimits\left\\{{\zeta}\cdot(ru_{r})_{,{\zeta}}\right\\}id{\zeta}\wedge
d\bar{\zeta}\wedge d\eta$ $\displaystyle-$ $\displaystyle 4\mathop{\rm
Re}\nolimits\left\\{(\partial_{\bar{\zeta}}u)(ru_{r})_{,{\zeta}}\right\\}id{\zeta}\wedge
d\bar{\zeta}\wedge d\eta,$

and the second term is

II $\displaystyle=$ $\displaystyle
d^{c}u\wedge\left\\{4u_{,{\zeta}\bar{\zeta}}-\mathop{\rm
Re}\nolimits({\zeta}\cdot(ru_{r})_{,{\zeta}})\frac{\cos\theta}{|{\zeta}|^{2}}+\frac{2ru_{r}}{(1+|{\zeta}|^{2})^{2}}\right\\}id{\zeta}\wedge
d\bar{\zeta}$ $\displaystyle=$
$\displaystyle-(ru_{r})\frac{\cos\theta}{|{\zeta}|^{2}}\mathop{\rm
Re}\nolimits\left\\{{\zeta}\cdot(ru_{r})_{,{\zeta}}\right\\}id{\zeta}\wedge
d\bar{\zeta}\wedge d\eta$ $\displaystyle+$
$\displaystyle\left\\{4(ru_{r})u_{,{\zeta}\bar{\zeta}}+\frac{2(ru_{r})^{2}}{(1+|{\zeta}|^{2})^{2}}\right\\}id{\zeta}\wedge
d\bar{\zeta}\wedge d\eta.$

Observe that the first line on the R.H.S. of equation (4.2) cancels with the
first line on the R.H.S. of equation (4.2), and then we eventually obtain the
following formula.

######  Proposition 4.3.

For any $u\in{\mathcal{F}}^{\infty}(B_{1})$, we have the following $3$-form on
the $3$-sphere $S_{R}$, in terms of the complex Hopf-coordinate as

(4.38) $\displaystyle 8\ d^{c}u\wedge dd^{c}u|_{S_{R}}$ $\displaystyle=$
$\displaystyle 2\left((ru_{r})u_{,{\zeta}\bar{\zeta}}-\mathop{\rm
Re}\nolimits\left\\{(\partial_{\bar{\zeta}}u)(ru_{r})_{,{\zeta}}\right\\}\right)id{\zeta}\wedge
d\bar{\zeta}\wedge d\eta$ $\displaystyle+$
$\displaystyle\frac{(ru_{r})^{2}}{(1+|{\zeta}|^{2})^{2}}id{\zeta}\wedge
d\bar{\zeta}\wedge d\eta.$

Moreover, we note that ${\zeta}$ is actually a coordinate on
${\mathbb{C}}_{\infty}\cong{\mathbb{C}}{\mathbb{P}}^{1}$, and this leads us to
consider the Fubini-Study metric on this Kähler manifold as

$\omega:=\frac{id{\zeta}\wedge d\bar{\zeta}}{2(1+|{\zeta}|^{2})^{2}},$

and it has volume $\int_{{\mathbb{C}}{\mathbb{P}}^{1}}\omega=\pi$. Then we can
rewrite the above $3$-form in the following coordinate-free way

(4.39) $\displaystyle 8\ d^{c}u\wedge dd^{c}u|_{S_{R}}$ $\displaystyle=$
$\displaystyle 2(ru_{r}\cdot\Delta_{\omega}u)\omega\wedge d\eta$
$\displaystyle-$ $\displaystyle\\{\langle\nabla
u,\nabla(ru_{r})\rangle_{\omega}+\langle\nabla(ru_{r}),\nabla
u\rangle_{\omega}\\}\omega\wedge d\eta$ $\displaystyle+$ $\displaystyle
2(ru_{r})^{2}\omega\wedge d\eta,$

where $\nabla$ is the complex gradient for ${\zeta}$, and the inner product is
taken on ${\mathbb{C}}{\mathbb{P}}^{1}$ as

$\langle\nabla v,\nabla
w\rangle_{\omega}=\mbox{tr}_{\omega}(\partial_{{\zeta}}v\wedge\bar{\partial}_{{\zeta}}w).$

Next, we are going to use another change of variables as $t:=\log
r\in(-\infty,0)$, and then $u$ can be rewritten as

(4.40) $u_{t}({\zeta}):=\hat{u}(t,{\zeta})=u(e^{t},{\zeta},\bar{\zeta}).$

Along each complex line through the origin of ${\mathbb{C}}^{2}$, we have

(4.41) $r\partial_{r}u=\partial_{t}\hat{u}=\dot{u}_{t},$

where $r=e^{t}$. Therefore, we obtain the following decomposition formula for
the complex Monge-Ampère mass.

######  Theorem 4.4.

For any $u\in{\mathcal{F}}^{\infty}(B_{1})$, we have

(4.42) $\frac{1}{\pi}\int_{S_{R}}d^{c}u\wedge
dd^{c}u=2\int_{{\mathbb{C}}{\mathbb{P}}^{1}}(\dot{u}_{t}\Delta_{\omega}u_{t})\omega+\int_{{\mathbb{C}}{\mathbb{P}}^{1}}(\dot{u}_{t})^{2}\omega,$

where $\dot{u}_{t}=\frac{du_{t}}{dt}|_{t=T}$ for $e^{T}=R$.

###### Proof.

The idea is to integrate both sides of equation (4.39) on the $3$-sphere
$S_{R}$. Due to equation (3.9) and Remark (3.1), it is legal to perform the
integration under the complex Hopf-coordinate
$({\zeta},\eta)\in{\mathbb{C}}_{\infty}\times S^{1}$. After applying Fubini’s
Theorem, it boils down to take the following integration by parts on
${\mathbb{C}}{\mathbb{P}}^{1}$.

(4.43) $\displaystyle-$
$\displaystyle\int_{{\mathbb{C}}{\mathbb{P}}^{1}}\partial_{{\zeta}}u\wedge\bar{\partial}_{{\zeta}}(ru_{r})-\int_{{\mathbb{C}}{\mathbb{P}}^{1}}\partial_{{\zeta}}(ru_{r})\wedge\bar{\partial}_{{\zeta}}u$
$\displaystyle=$
$\displaystyle\int_{{\mathbb{C}}{\mathbb{P}}^{1}}\partial\bar{\partial}_{{\zeta}}u\cdot(ru_{r})+\int_{{\mathbb{C}}{\mathbb{P}}^{1}}(ru_{r})\cdot\partial\bar{\partial}_{{\zeta}}u$
$\displaystyle=$ $\displaystyle
2\int_{{\mathbb{C}}{\mathbb{P}}^{1}}(ru_{r}\cdot\Delta_{\omega}u)\omega,$

where Stoke’s theorem is used in the first equality. Equipped with the other
two terms in the R.H.S. of equation (4.39), and then our result follows.

∎

In order to illustrate the above computation in a clear way, we will also
invoke the real Hopf-coordinate, and perform the integration under it. First
we note again that the function $u$ is periodic in the angle $\varphi$
direction with period $2\pi$, and this can be seen directly as follows.

(4.44) $\displaystyle
u\left(|z_{1}|e^{\frac{i}{2}(\eta+\varphi)+i\pi},|z_{2}|e^{\frac{i}{2}(\eta-\varphi)-i\pi}\right)$
$\displaystyle=$ $\displaystyle
u\left(|z_{1}|e^{\frac{i}{2}(\eta+\varphi)+i\pi},|z_{2}|e^{\frac{i}{2}(\eta-\varphi)+i\pi}\right)$
$\displaystyle=$ $\displaystyle
u\left(|z_{1}|e^{\frac{i}{2}(\eta+\varphi)},|z_{2}|e^{\frac{i}{2}(\eta-\varphi)}\right).$

Then in the real Hopf-coordinate, we take the wedge product of equation (4.26)
and (4.33), and obtain

(4.45) $\displaystyle 8\ d^{c}u\wedge dd^{c}u|_{S_{R}}$ $\displaystyle=$
$\displaystyle ru_{r}^{2}\left\\{\frac{\partial_{\varphi}(u_{\varphi}\cdot
u^{-1}_{r})}{\sin\theta}+\sin\theta\cdot\partial_{\theta}(u_{\theta}\cdot
u^{-1}_{r})\right\\}d\theta\wedge d\varphi\wedge d\eta$ $\displaystyle+$
$\displaystyle ru_{r}^{2}\left\\{\cos\theta\cdot(u_{\theta}\cdot
u^{-1}_{r})+\frac{r}{2}\sin\theta\right\\}d\theta\wedge d\varphi\wedge d\eta.$

Next we will integrate the two sides of this equation on the $3$-sphere
$S_{R}$. Thanks to Fubini’s theorem, we can perform the integration by parts
in the $\varphi$-direction as follows

(4.46)
$\displaystyle\int_{0}^{\pi}d\theta\int_{0}^{2\pi}\sin^{-1}\theta(ru_{r}^{2})\partial_{\varphi}(u_{\varphi}\cdot
u^{-1}_{r})d\varphi$ $\displaystyle=$
$\displaystyle\int_{0}^{\pi}d\theta\int_{0}^{2\pi}\sin^{-1}\theta\left\\{(ru_{r})u_{,\varphi\varphi}-ru_{\varphi}u_{,r\varphi}\right\\}d\varphi$
$\displaystyle=$ $\displaystyle
2\int_{0}^{2\pi}\int_{0}^{\pi}(ru_{r})\frac{u_{,\varphi\varphi}}{\sin\theta}d\theta
d\varphi.$

On the other hand, we can also perform the integration by parts in
$\theta$-direction as follows.

$\displaystyle\int_{0}^{2\pi}d\varphi\int_{0}^{\pi}\cos\theta(ru_{r}^{2})(u_{\theta}\cdot
u^{-1}_{r})d\theta$ $\displaystyle=$
$\displaystyle\int_{0}^{2\pi}d\varphi\left(\sin\theta(u_{\theta}\cdot
ru_{r})\right)\big{|}_{0}^{\pi}-\int_{0}^{2\pi}d\varphi\int_{0}^{\pi}\sin\theta\partial_{\theta}(u_{\theta}\cdot
ru_{r})d\theta$ $\displaystyle=$
$\displaystyle-\int_{0}^{2\pi}d\varphi\int_{0}^{\pi}\sin\theta\left\\{r\partial_{r}(u_{\theta})^{2}+ru_{r}^{2}\cdot\partial_{\theta}(u_{\theta}\cdot
u^{-1}_{r})\right\\}d\theta,$

and the first term on the R.H.S. of equation (4.2) reads as

$\displaystyle-\int_{0}^{2\pi}d\varphi\int_{0}^{\pi}\sin\theta\left\\{2r(u_{\theta}u_{,r\theta})\right\\}d\theta$
$\displaystyle=$
$\displaystyle-\int_{0}^{2\pi}d\varphi\left(2\sin\theta(ru_{r}\cdot
u_{\theta})\right)\big{|}_{0}^{\pi}+2\int_{0}^{2\pi}\int_{0}^{\pi}(ru_{r})\partial_{\theta}(\sin\theta
u_{\theta})d\theta d\varphi.$

Finally we combine equation (4.45), (4.2), (4.2) and (4.46) together to obtain

(4.49) $\displaystyle 8\int_{S_{R}}d^{c}u\wedge dd^{c}u$ $\displaystyle=$
$\displaystyle 8\pi\int_{0}^{2\pi}\int_{0}^{\pi}(ru_{r})\left\\{\
\frac{u_{,\varphi\varphi}}{\sin\theta}+\partial_{\theta}(\sin\theta
u_{\theta})\right\\}d\theta d\varphi$ $\displaystyle+$ $\displaystyle
2\pi\int_{0}^{2\pi}\int_{0}^{\pi}(ru_{r})^{2}\sin\theta d\theta d\varphi$
$\displaystyle=$ $\displaystyle
8\pi\int_{S^{2}}(ru_{r}\cdot\Delta_{\Theta}u)d\sigma_{2}+2\pi\int_{S^{2}}(ru_{r})^{2}d\sigma_{2},$

where $d\sigma_{2}=\sin\theta d\theta\wedge d\varphi$ is the area form of the
unit $2$-sphere $S^{2}$, and $\Delta_{\Theta}$ is the standard Laplacian on
$S^{2}$ w.r.t. the round metric, i.e. we have

(4.50)
$\displaystyle\Delta_{\Theta}u=\left\\{\frac{1}{\sin\theta}\frac{\partial}{\partial\theta}\left(\sin\theta\frac{\partial
u}{\partial\theta}\right)+\frac{1}{\sin^{2}\theta}\frac{\partial^{2}u}{\partial\varphi^{2}}\right\\}.$

Then it is clear that equation (4.49) is equivalent to (4.42) after the change
of variables of the real and complex Hopf-coordinates and $r=e^{t}$.

##  5\. The residual Monge-Ampère mass

For the next step, we are going to estimate the two integrals on the R.H.S. of
equation (4.42) as $t\rightarrow-\infty$. First, we note that the last
integral has a deep connection with the Lelong number of $u$.

###  5.1. The $L^{2}$-Lelong number

On the one hand, it is well known that the following limit is exactly the
Lelong number at the origin of a plurisubharmonic function $u$ in
${\mathbb{C}}^{2}$.

$\nu_{u}(0)=\lim_{r\rightarrow 0^{+}}\nu_{u}(0,r),$

if we take

(5.1)
$\nu_{u}(0,r):=r\partial^{-}_{r}\left(\frac{1}{2\pi^{2}}\int_{|\xi|=1}u(r\xi)\
d\sigma_{3}(\xi)\right),$

where $d\sigma_{3}$ is the area form of the unit $3$-sphere $S^{3}$. In fact,
the integral on the R.H.S. of equation (5.1) is $\log$-convex and non-
decreasing in $r$. Therefore, the number $\nu_{u}(0,r)$ is non-negative and
non-decreasing in the radial direction.

In fact, it is a standard fact that the Lelong number of a plurisubharmonic
function $u$ is invariant under restriction to almost all complex directions
in ${\mathbb{C}}^{2}$. From now on, we assume that the plurisubharmonic
function $u$ is in the space ${\mathcal{F}}(B_{1})$ with the normalization
$\sup_{B_{1}}u=-1$.

Fixing a ${\zeta}\in{\mathbb{C}}_{\infty}$, the complex line through the
origin of ${\mathbb{C}}^{2}$ in the direction ${\zeta}$ can be written as
${\ell}_{{\zeta}}:=(\lambda{\zeta},\lambda)$ for all $\lambda\in{\mathbb{C}}$
and ${\zeta}\neq\infty$. If ${\zeta}=\infty$, then we have
${\ell}_{\infty}:=(\lambda,0)$. Then the following restriction

$u|_{{\ell}_{{\zeta}}}=u(\lambda{\zeta},\lambda)=u(|\lambda|{\zeta},|\lambda|)$

is actually a non-decreasing convex function of $\log|\lambda|$ (see [BB]),
and hence is also a non-decreasing convex function of

$t=\log r=\log|\lambda|+\frac{1}{2}\log(1+|{\zeta}|^{2}),$

for all $t\in(-\infty,0)$. In other words, the negative function
$u_{t}({\zeta})=\hat{u}(t,{\zeta})$ (defined in equation (4.40)) is also non-
decreasing and convex in $t$. Therefore, we have for each ${\zeta}$ fixed

(5.2) $\dot{u}_{t}\geq 0,\ \ and\ \ \ddot{u}_{t}\geq 0,$

for almost all $t\in(-\infty,0)$. Moreover, we infer from equation (4.41) and
(5.5) that the Lelong number at zero of $u|_{{\ell}_{{\zeta}}}$ is equal to

(5.3)
$\nu_{u|_{{\ell}_{{\zeta}}}}(0)=\lim_{t\rightarrow-\infty}\dot{u}_{t}({\zeta}).$

Then the invariance of the Lelong number under restrictions reads as follows.

######  Lemma 5.1 (Remark (2.38), [GZ]).

For a plurisubharmonic function $u$, its Lelong number at the origin
$\nu_{u}(0)$ is equal to $\nu_{u|_{{\ell}_{{\zeta}}}}(0)$ for almost
everywhere ${\zeta}\in{\mathbb{C}}_{\infty}$. Moreover, for such a ${\zeta}$,
it is the decreasing limit of $\dot{u}_{t}({\zeta})$ as $t\rightarrow-\infty$.

Suppose that $u$ is further in ${\mathcal{F}}^{\infty}(B_{1})$. Recall that
the total area of a unit $3$-sphere is $2\pi^{2}$. Then we can rewrite the
area form in the real Hopf-coordinate as

(5.4) $8\ d\sigma_{3}=\sin\theta d\theta\wedge d\varphi\wedge d\eta,$

and then equation (5.1) can be re-written as

(5.5) $\nu_{u}(0,r)=\frac{1}{4\pi}\int_{S^{2}}(r\partial_{r}u)\
d\sigma_{2}=\frac{1}{\pi}\int_{{\mathbb{C}}{\mathbb{P}}^{1}}\dot{u}_{t}\omega.$

Then we can obtain an estimate on the last term of equation (4.42), and it
will reveal that this term actually behaves like a kind of _$L^{2}$ -Lelong
number_.

######  Lemma 5.2.

For any $u\in{\mathcal{F}}^{\infty}(B_{1})$, we have

(5.6)
$[\nu_{u}(0)]^{2}=\lim_{t\rightarrow-\infty}\frac{1}{\pi}\int_{{\mathbb{C}}{\mathbb{P}}^{1}}(\dot{u}_{t})^{2}\omega.$

###### Proof.

For each fixed ${\zeta}$, we denote $v_{{\zeta}}$ by the limit of the non-
decreasing sequence $\dot{u}_{t}({\zeta})$ as

$v_{{\zeta}}=\lim_{t\rightarrow-\infty}\dot{u}_{t}({\zeta})\geq 0.$

Thanks to Lemma (5.1), it coincides with the Lelong number $\nu_{u}(0)$ for
almost everywhere ${\zeta}\in{\mathbb{C}}{\mathbb{P}}^{1}$. Take

(5.7)
$M_{A}:=\max_{{\zeta}\in{\mathbb{C}}{\mathbb{P}}^{1}}\dot{u}_{-A}({\zeta}),$

for some constant $A>0$. Then we have

$\dot{u}_{t}({\zeta})\leq M_{A},$

for all $t<-A$ and ${\zeta}\in{\mathbb{C}}{\mathbb{P}}^{1}$, and then our
result follows from the dominated convergence theorem as

(5.8)
$\lim_{t\rightarrow-\infty}\frac{1}{\pi}\int_{{\mathbb{C}}{\mathbb{P}}^{1}}(\dot{u}_{t})^{2}\omega=\frac{1}{\pi}\int_{{\mathbb{C}}{\mathbb{P}}^{1}}v^{2}_{{\zeta}}\omega=\nu^{2}_{u}(0).$

∎

###  5.2. Laplacian estimates

The next goal is to estimate the first term on the R.H.S. of equation (4.42).
Before moving on, we need to take a closer look at the Laplacian
$\Delta_{\omega}$ as follows.

First, it is a standard fact that the Laplacian $\Delta_{e}$ with respect to
the Euclidean coordinate in ${\mathbb{R}}^{4}$ has the following decomposition
in the hyper-spherical coordinate

(5.9) $\displaystyle\Delta_{e}=\Delta_{r}+r^{-2}\Delta_{\Xi},$

where $\Delta_{r}=r^{-3}\partial_{r}(r^{3}\partial_{r}\cdot)$ is the radial
part, and $\Delta_{\Xi}$ is the standard Laplacian on the unit $3$-sphere.
That is to say, this operator $\Delta_{\Xi}$ only consists of derivatives in
the directions perpendicular to the radial direction. In the real Hopf-
coordinate, it is standard to compute (cf. [LUW])

$\displaystyle\frac{1}{4}\
\Delta_{\Xi}=\frac{1}{\sin\theta}\frac{\partial}{\partial\theta}\left(\sin\theta\frac{\partial}{\partial\theta}\right)+\frac{1}{\sin^{2}\theta}\left(\frac{\partial^{2}}{\partial\varphi^{2}}+\frac{\partial^{2}}{\partial\eta^{2}}\right)+\frac{2\cos\theta}{\sin^{2}\theta}\frac{\partial^{2}}{\partial\varphi\partial\eta}.$

If the function $u$ is also $S^{1}$-invariant, then we can compare equation
(5.2) with (4.50), and obtain

(5.11) $\displaystyle\Delta_{\Xi}u=4\ \Delta_{\Theta}u=2\ \Delta_{\omega}u.$

Therefore, we have the following Laplacian decomposition formula for any
function $u\in{\mathcal{F}}^{\infty}(B_{1})$

(5.12) $\displaystyle\Delta_{e}u=\Delta_{r}u+2r^{-2}\Delta_{\omega}u.$

######  Lemma 5.3.

For any $u\in{\mathcal{F}}^{\infty}(B_{1})$, there exists a positive constant
$M_{A}$ such that we have

(5.13) $\displaystyle
2\int_{{\mathbb{C}}{\mathbb{P}}^{1}}\dot{u}_{t}(\Delta_{\omega}u_{t})\omega\leq
M_{A}\int_{{\mathbb{C}}{\mathbb{P}}^{1}}(\ddot{u}_{t}+2\dot{u}_{t})\omega,$

for all $t\leq-A$.

###### Proof.

Thanks to the Laplacian decomposition formula (equation (5.12)), the first
integral on the R.H.S. of equation (4.42) can be rewritten as

(5.14)
$2\int_{{\mathbb{C}}{\mathbb{P}}^{1}}\dot{u}_{t}(\Delta_{\omega}u_{t})\omega=\int_{{\mathbb{C}}{\mathbb{P}}^{1}}\dot{u}_{t}(r^{2}\Delta_{e}u)\omega-\int_{{\mathbb{C}}{\mathbb{P}}^{1}}\dot{u}_{t}(r^{2}\Delta_{r}u)\omega.$

Moreover, the second term on the R.H.S. of equation (5.14) is

(5.15)
$\displaystyle\int_{{\mathbb{C}}{\mathbb{P}}^{1}}\dot{u}_{t}(r^{2}\Delta_{r}u)\omega$
$\displaystyle=$
$\displaystyle\int_{{\mathbb{C}}{\mathbb{P}}^{1}}(ru_{r})\left\\{(r^{2}u_{,rr}+ru_{r})+2ru_{r}\right\\}\omega$
$\displaystyle=$
$\displaystyle\int_{{\mathbb{C}}{\mathbb{P}}^{1}}\dot{u}_{t}(\ddot{u}_{t}+2\dot{u}_{t})\omega.$

Thanks to equation (5.2), the integral in equation (5.15) is actually non-
negative, and we obtain

(5.16)
$2\int_{{\mathbb{C}}{\mathbb{P}}^{1}}\dot{u}_{t}(\Delta_{\omega}u_{t})\omega\leq\int_{{\mathbb{C}}{\mathbb{P}}^{1}}\dot{u}_{t}(r^{2}\Delta_{e}u)\omega.$

Furthermore, we have $\Delta_{e}u\geq 0$ since $u$ is plurisubharmonic on
${\mathbb{C}}^{2}$. Take $M_{A}$ as the maximum of $\dot{u}_{-A}$ on
${\mathbb{C}}{\mathbb{P}}^{1}$ (equation (5.7)), and it follows for all $t<-A$

(5.17) $\displaystyle
2\int_{{\mathbb{C}}{\mathbb{P}}^{1}}\dot{u}_{t}(\Delta_{\omega}u_{t})\omega$
$\displaystyle\leq$
$\displaystyle\int_{{\mathbb{C}}{\mathbb{P}}^{1}}\dot{u}_{t}(r^{2}\Delta_{e}u)\omega$
$\displaystyle\leq$ $\displaystyle
M_{A}\int_{{\mathbb{C}}{\mathbb{P}}^{1}}(r^{2}\Delta_{r}u+2\Delta_{\omega}u)\omega$
$\displaystyle=$ $\displaystyle
M_{A}\int_{{\mathbb{C}}{\mathbb{P}}^{1}}(\ddot{u}_{t}+2\dot{u}_{t})\omega.$

Here we have used

$\int_{{\mathbb{C}}{\mathbb{P}}^{1}}(\Delta_{\omega}u)\omega=0$

in the last equality in equation (5.17). Then our result follows.

∎

Now everything boils down to study the asymptotic behavior of the following
integral

(5.18)
$\int_{{\mathbb{C}}{\mathbb{P}}^{1}}\ddot{u}_{t}\omega=\frac{d}{dt}\left(\int_{{\mathbb{C}}{\mathbb{P}}^{1}}\dot{u}_{t}\omega\right),$

as $t\rightarrow-\infty$. To this purpose, we introduce the following two non-
negative functionals

(5.19) $I_{u}(t):=\int_{{\mathbb{C}}{\mathbb{P}}^{1}}\dot{u}_{t}\omega;\ \ \
J_{u}(t):=\int_{{\mathbb{C}}{\mathbb{P}}^{1}}(\dot{u}_{t})^{2}\omega,$

and the negative functional as a primitive of $I_{u}$

(5.20) ${\mathcal{I}}(u_{t}):=\int_{{\mathbb{C}}{\mathbb{P}}^{1}}u_{t}\omega,$

for all $t\in(-\infty,0)$. Then the following observation is crucial.

######  Proposition 5.4.

Suppose $u\in{\mathcal{F}}^{\infty}(B_{1})$ has the Lelong number
$\nu_{u}(0)\geq 0$ at the origin. Then there exists a sequence
$t_{i}\in(-\infty,0)$ converging to $-\infty$ such that

$\lim_{i\rightarrow+\infty}\frac{dI_{u}}{dt}(t_{i})=0.$

###### Proof.

First we note that ${\mathcal{I}}(u_{t})$ is a non-decreasing convex function
along $t\in(-\infty,0)$. As its first derivative, $I_{u}$ is a non-negative,
non-decreasing function in $t$. Moreover, we have

(5.21)
$I^{\prime}_{u}(t):=\frac{dI_{u}}{dt}(t)=\int_{{\mathbb{C}}{\mathbb{P}}^{1}}\ddot{u}_{t}\omega\geq
0.$

Thanks to equation (5.5), we have its limit as a decreasing sequence in $t$

(5.22) $\lim_{t\rightarrow-\infty}I_{u}(t)=\pi\nu_{u}(0)\geq 0.$

Hence $I_{u}$ is a non-negative $C^{1}$-continuous function decreasing to
$\pi\nu_{u}(0)$ as $t\rightarrow-\infty$. Then there exists a $T_{0}<0$ such
that we have

$I_{u}(t)\leq\pi\nu_{u}(0)+1,$

for all $t<T_{0}$. Now it is sufficient to prove

(5.23) $\liminf_{t\rightarrow-\infty}I^{\prime}_{u}(t)=0.$

Suppose not, and then there exists an $\varepsilon>0$ such that

(5.24) $\liminf_{t\rightarrow-\infty}I^{\prime}_{u}(t)>\varepsilon,$

and then there exists a $T_{1}<0$ such that for all $t<T_{1}$

(5.25) $I^{\prime}_{u}(t)>\varepsilon/2.$

Take $T_{2}:=\min\\{T_{0},T_{1}\\}-1$, and then the graph of $I_{u}$ will be
under the following straight line for all $t<T_{2}$

$y(x)=\frac{\varepsilon}{2}x+I_{u}(T_{2})-\frac{\varepsilon
T_{2}}{2}\leq\frac{\varepsilon}{2}x+\pi\nu_{u}(0)+1-\frac{\varepsilon
T_{2}}{2},$

but this implies $I_{u}(t)<0$ for all $t$ negative enough, which is a
contradiction.

∎

Then we are ready to prove the first main theorem.

######  Theorem 5.5.

For any $u\in{\mathcal{F}}^{\infty}(B_{1})$, its residual Monge-Ampère mass
$\tau_{u}(0)$ is zero, if its Lelong number $\nu_{u}(0)$ is zero at the
origin.

###### Proof.

Thanks to Proposition (2.5) and the decomposition formula (equation (4.42)),
we have

(5.26)
$\pi^{-1}\mbox{MA}(u)(B_{r})=2\int_{{\mathbb{C}}{\mathbb{P}}^{1}}(\dot{u}_{t}\Delta_{\omega}u_{t})\omega+\int_{{\mathbb{C}}{\mathbb{P}}^{1}}(\dot{u}_{t})^{2}\omega,$

for $r=e^{t}$. Thanks to Lemma (5.9), we can further estimate

(5.27) $\pi^{-1}\mbox{MA}(u)(B_{r})\leq
M_{A}\int_{{\mathbb{C}}{\mathbb{P}}^{1}}(\ddot{u}_{t}+2\dot{u}_{t})\omega+\int_{{\mathbb{C}}{\mathbb{P}}^{1}}(\dot{u}_{t})^{2}\omega,$

for a uniform constant $M_{A}\geq 0$ and all $t\leq-A$. In fact, we can re-
write the above estimate (equation (5.27) as

(5.28) $\pi^{-1}\mbox{MA}(u)(B_{r})\leq
M_{A}\left\\{I^{\prime}_{u}(t)+2I_{u}(t)\right\\}+J_{u}(t),$

for $r=e^{t}\leq e^{-A}$. Due to Lemma (5.2) and equation (5.5), the zero
Lelong number $\nu_{u}(0)=0$ implies

$I_{u}(t)\rightarrow 0,\ \ \ \mbox{and}\ \ \ J_{u}(t)\rightarrow 0,$

as $t\rightarrow-\infty$. Moreover, we infer from Proposition (5.4) that there
exists a sequence $t_{i}=\log r_{i}$ converging to $-\infty$ satisfying

(5.29) $\pi^{-1}\mbox{MA}(u)(B_{r_{i}})\leq
M_{A}\left\\{I^{\prime}_{u}(t_{i})+2I_{u}(t_{i})\right\\}+J_{u}(t_{i}),$

and the R.H.S. of equation (5.29) converges to zero as $r_{i}\rightarrow 0$.
However, the Monge-Ampère mass of $u$ on the $r$-ball $\mbox{MA}(u)(B_{r})$ is
non-decreasing in $r$. Then it follows

(5.30) $\tau_{u}(0)=\frac{1}{\pi^{2}}\lim_{r\rightarrow
0^{+}}\mbox{MA}(u)(B_{r})=0.$

∎

###  5.3. Maximal directional Lelong numbers

Recall that the residual Monge-Ampère mass at the origin of a plurisubharmonic
function $u$ is equal to the following decreasing limit

${\tau}_{u}(0):=\frac{1}{\pi^{2}}\lim_{r\rightarrow
0^{+}}\mbox{MA}(u)(B_{r}).$

As we have seen in the proof of the above Theorem (5.5), the inequality in
equation (5.28) has played a major role. In fact, this inequality implies a
stronger result on the estimate of the residual mass. First we will re-state
it as follows.

######  Lemma 5.6.

For a function $u\in{\mathcal{F}}^{\infty}(B_{1})$, we have

(5.31) $\pi^{-1}\emph{MA}(u)(B_{r})\leq
M_{A}\left\\{I^{\prime}_{u}(t)+2I_{u}(t)\right\\}+J_{u}(t),$

for all $t=\log r\in(-\infty,-A]$.

Here the uniform constant $M_{A}$ is defined as

$M_{A}:==\sup_{{\zeta}\in{\mathbb{C}}{\mathbb{P}}^{1}}\dot{u}_{-A}({\zeta})\geq
0,$

for any $A>0$. Furthermore, it can be generalized to to the family
${\mathcal{F}}(B_{1})$.

######  Definition 5.7.

For a function $u\in{\mathcal{F}}(B_{1})$, the maximal directional Lelong
number of $u$ at a distance $A\in(0,\infty)$ to the origin is defined as

$M_{A}(u):=\sup_{{\zeta}\in{\mathbb{C}}{\mathbb{P}}^{1}}\partial_{t}^{+}u_{t}({\zeta})|_{t=-A}\in[0,+\infty].$

As we have discussed in Section (5.1), the restriction $u|_{{\ell}_{{\zeta}}}$
is a non-decreasing convex function along $t=\log r$. Hence the right
derivative

(5.32)
$\partial^{+}_{t}u_{t}({\zeta})|_{t=t_{0}}=\partial^{+}_{t}\hat{u}(t,{\zeta})|_{t=t_{0}}=\frac{d}{dt}|_{t=t_{0}^{+}}(u|_{{\ell}_{{\zeta}}})$

is well defined everywhere on $B^{*}_{1}$. Moreover, it is also non-negative
and non-decreasing in $t$. Then it follows that this number $M_{A}(u)$ is non-
negative, and non-increasing in $A$. Therefore, we can take its decreasing
limit as $A\rightarrow+\infty$.

######  Definition 5.8.

For a function $u\in{\mathcal{F}}(B_{1})$, the maximal directional Lelong
number of $u$ at the origin is defined as

$\lambda_{u}(0):=\lim_{A\rightarrow+\infty}M_{A}(u).$

There is no a priori reason that this number is finite, but it is for any $u$
in the sub-collection ${\mathcal{F}}^{\infty}(B_{1})$. Moreover, the constant
$M_{A}(u)$ coincides with the uniform constant $M_{A}$ for any $A>0$ in
equation (5.31) in this case, and then we can obtain the following estimate.

######  Theorem 5.9.

For a function $u\in{\mathcal{F}}^{\infty}(B_{1})$, we have

(5.33) ${\tau}_{u}(0)\leq 2\lambda_{u}(0)\cdot\nu_{u}(0)+[\nu_{u}(0)]^{2}.$

###### Proof.

It is enough to prove the following inequality for all $A$ large

(5.34) ${\tau}_{u}(0)\leq 2M_{A}(u)\cdot\nu_{u}(0)+[\nu_{u}(0)]^{2}.$

Thanks to Lemma (5.2), we have

$\pi\nu_{u}(0)=\lim_{t\rightarrow-\infty}I_{u}(t);\ \ \
\pi[\nu_{u}(0)]^{2}=\lim_{t\rightarrow-\infty}J_{u}(t).$

Moreover, we can extract a sequence $t_{i}\rightarrow-\infty$ such that
$I^{\prime}_{u}(t_{i})\rightarrow 0$ due to Proposition (5.4). Then the result
follows by taking limits on both sides of equation (5.31), where we have used
the fact that the Monge-Ampère mass $\mbox{MA}(u)(B_{r})$ is non-decreasing in
$r$.

∎

It is apparent that the estimate in Theorem (5.9) implies the zero mass
conjecture for a function $u\in{\mathcal{F}}^{\infty}(B_{1})$.

##  6\. The general case

For a general plurisubharmonic function $u$ on a domain $D$, it is a standard
fact that $u$ is in the Sobolev space $W_{loc}^{1,p}(D)$ for any $1\leq p<2$.
If $D$ is in ${\mathbb{C}}^{2}$ and $u$ is in the Cegrell class, cf. [Ceg04],
then we can even take $p=2$ due to a result by Blocki, cf. [Blo04]. In
particular, for a function $u\in{\mathcal{F}}(B_{1})$, the weak derivative
$\nabla u$ exists almost everywhere as an $L^{2}_{loc}$-function on $B_{1}$.

###  6.1. The radial bound

Writing $t=\log r$, the first partial $t$-derivative of $u$ also exists as an
$L^{2}_{loc}$ function on $B_{1}$, since we have

(6.1) $\partial_{t}u(z)=r\partial_{r}u(z)=r\langle\nabla
u(z),\hat{r}_{\vec{z}}\rangle,$

where $\vec{z}$ is the point vector of $z$ in ${\mathbb{R}}^{4}$, and
$\hat{r}_{\vec{z}}$ is the unit normal vector in the direction of $\vec{z}$.
According to our definitions in Section (5.1), it is also equal to the
$t$-derivative of $u$ under the restriction to the complex line
${\ell}_{{\zeta}}$ as

(6.2)
$\partial_{t}u(z)=\partial_{t}u(e^{t},{\zeta},\bar{\zeta})=\dot{u}_{t}({\zeta})=\frac{d}{dt}u|_{{\ell}_{{\zeta}}},$

for almost all $t\in(-\infty,0)$ and ${\zeta}\in{\mathbb{C}}{\mathbb{P}}^{1}$,
where $(r,\eta,{\zeta},\bar{\zeta})$ is the complex Hopf-coordinate of $z\in
B_{1}^{*}$.

Next we claim that this partial derivative $\partial_{t}u$ is not merely
$L^{2}$, but actually bounded on $B^{*}_{R}$ for any $R\in(0,1)$, and this
will be implied by the following crucial observation.

######  Lemma 6.1.

For a function $u\in{\mathcal{F}}(B_{1})$, its maximal directional Lelong
number $M_{A}(u)$ at distance $A$ is finite for any $A>0$.

###### Proof.

For each boundary sphere $S_{R}=\partial B_{R}$ with $R\in(0,1)$, we claim
that there exists a constant $C_{R}>0$ such that it satisfies

(6.3) $u|_{S_{R}}\geq-C_{R}.$

This is because $u$ is an $L^{\infty}_{loc}$-function in $B_{1}^{*}$, and
hence it is in the space $L^{\infty}(S_{R})$ for each $R\in(0,1)$. Then our
claim follows since $u$ is actually everywhere defined as a locally bounded
plurisubharmonic function on $B^{*}_{1}$.

Suppose on the contrary, we have $M_{A}(u)=+\infty$ for some $A>0$. Then there
exists a sequence of points ${\zeta}_{j}\in{\mathbb{C}}{\mathbb{P}}^{1}$ such
that we have

(6.4)
$\partial_{t}^{+}u_{t}({\zeta}_{j})|_{t=-A}=\frac{d}{dt}|_{t=(-A)^{+}}\left(u|_{{\ell}_{{\zeta}_{j}}}\right)$

diverges to $+\infty$ as $j\rightarrow+\infty$. In particular, we can pick a
point $\xi\in{\mathbb{C}}{\mathbb{P}}^{1}$ among this sequence such that we
have

(6.5) $\partial_{t}^{+}u_{t}(\xi)|_{t=-A}>\frac{2C_{R}}{A},$

for $R=e^{-A}$. However, as a convex function of $t$, the graph of
$u_{t}(\xi)$ is above the following straight line

$y(x)=\frac{2C_{R}}{A}(x+A)-C_{R},$

for all $t\in[-A,-A/2]$, and then we have

$u_{-A/2}(\xi)\geq y(-A/2)\geq 0,$

but this contradicts to the normalization $\sup_{B_{1}}u\leq-1$. Therefore,
our result follows.

∎

One simple fact of the number $M_{A}(u)$ is that it is always positive if we
have a singularity of $u$ at the origin.

######  Lemma 6.2.

Suppose a function $u\in{\mathcal{F}}(B_{1})$ is equal to $-\infty$ at the
origin. Then we have $M_{A}(u)>0$ for all $A>0$.

###### Proof.

Suppose on the contrary, we have $M_{A}(u)=0$ for some $A>0$. Then for each
${\zeta}\in{\mathbb{C}}{\mathbb{P}}^{1}$ fixed, it follows from the convexity
of $u_{t}$ that we have $u_{t}({\zeta})=u_{-A}({\zeta})$ for all $t\leq-A$.
However, we have the convergence

$\limsup_{z\rightarrow 0}u(z)=-\infty,$

as a plurisubharmonic function. This forces that there exists a sequence of
points ${\zeta}_{j}\in{\mathbb{C}}{\mathbb{P}}^{1}$ such that
$u_{-A}({\zeta}_{j})\rightarrow-\infty$ as $j\rightarrow+\infty$. Hence it
contradicts to the fact that there is a uniform lower bound of $u$ on the
boundary sphere $S_{R}$ with $R=e^{-A}$.

∎

As a direct consequence of Lemma (6.1), the maximal directional Lelong number
$\lambda_{u}(0)$ of a function $u\in{\mathcal{F}}(B_{1})$ at the origin is
always finite, since it is the decreasing limit of $M_{A}(u)$ as
$A\rightarrow+\infty$.

######  Corollary 6.3.

For a function $u\in{\mathcal{F}}(B_{1})$, we have

$0\leq\lambda_{u}(0)<+\infty.$

Another upshot of Lemma (6.1) is that $\partial_{t}u$ is in the space
$L^{\infty}(B^{*}_{R})$ for all $R\in(0,1)$, since we have

$\dot{u}_{t}({\zeta})\leq\partial_{t}^{+}u_{t}({\zeta}),$

whenever the L.H.S. exists. Then we are ready to introduce the previous
functionals on the boundary sphere $S_{R}$ for almost all $R$. To this
purpose, a few facts of slicing theory ([Dem2], [Fe], [Siu74]) will be
invoked.

###  6.2. Slicing theory

The radius function $r:B_{1}^{*}\rightarrow(0,1)$ can be viewed as a proper
smooth submersion of differentiable manifolds. In fact, if we identify
$B_{1}^{*}$ with $S^{3}\times(0,1)$, then $r$ is locally the projection map

$S^{3}\times(0,1)\rightarrow(0,1)\ \ \ (r,\eta,{\zeta},\bar{\zeta})\rightarrow
r.$

Here we have used the complex Hopf-coordinate on $B_{1}^{*}$.

A function $U\in L^{1}_{loc}(B_{1}^{*})$ can be viewed as a order zero locally
flat current, and then we can define its slicing for $r^{\prime}\in(0,1)$ as

$U_{r^{\prime}}:=U|_{S_{r^{\prime}}}.$

Thanks to Fubini’s Theorem, this restriction of $U$ to the fibers exists for
almost everywhere $r^{\prime}\in(0,1)$, and we have $U_{r^{\prime}}\in
L^{1}(S_{r^{\prime}})$. Moreover, its regularization
$U_{\varepsilon}:=U*\rho_{\varepsilon}$ (equation (2.9)) has the property that
$U_{\varepsilon,r^{\prime}}\rightarrow U_{r^{\prime}}$ in
$L^{1}(S_{r^{\prime}})$ for almost all $r^{\prime}\in(0,1)$. Furthermore, we
have the following basic slicing formula

(6.6) $\int_{B_{1}^{*}}U{\alpha}\wedge
r^{*}\beta=\int_{r^{\prime}\in(0,1)}\left(\int_{S_{r^{\prime}}}U_{r^{\prime}}({\zeta},\eta)\alpha|_{S_{r^{\prime}}}\right)\beta(r^{\prime}),$

for every smooth $3$-form $\alpha$ compactly supported on $B_{1}^{*}$, and
smooth $1$-form $\beta$ on $(0,1)$.

Take a function $u$ in ${\mathcal{F}}(B_{1})$, and then it is in the space
$L^{p}_{loc}(B_{1})$ for any $p\geq 1$. Therefore, its slicing $u|_{S_{R}}$
exists as an $L^{p}$-function on $S_{R}$ for almost everywhere $R\in(0,1)$.
Moreover, its $t$-derivative $\partial_{t}u$ is in
$L^{\infty}(B^{*}_{1-\delta})$ for a small $\delta>0$. Hence its slicing

(6.7)
$(\partial_{t}u)|_{S_{R}}=(r\partial_{r}u)|_{S_{R}}=R(\partial_{r}u)|_{S_{R}}$

exists as an $L^{\infty}$-function on $S_{R}$ for almost all
$R\in(0,1-\delta)$. Thanks to the circular symmetry of $u$, the slices can
also be viewed as $L^{p}$($L^{\infty}$)-functions on
${\mathbb{C}}{\mathbb{P}}^{1}$ via the Hopf-fiberation. Then we can rewrite
for almost all $R=e^{T}$

(6.8) $u_{T}:=u|_{S_{R}};\ \ \ \dot{u}_{T}:=(\partial_{t}u)|_{S_{R}}.$

Therefore, it is legal to introduce the following functionals

${\mathcal{I}}(u_{t}):=\int_{{\mathbb{C}}{\mathbb{P}}^{1}}u_{t}\omega;\ \ \
I_{u}(t):=\int_{{\mathbb{C}}{\mathbb{P}}^{1}}\dot{u}_{t}\omega;\ \ \
J_{u}(t):=\int_{{\mathbb{C}}{\mathbb{P}}^{1}}(\dot{u}_{t})^{2}\omega,$

for almost all $t\in(-\infty,-1)$. In the following, we summarize a few basic
properties of these functionals:

  1. (a)

the functional ${\mathcal{I}}(u_{t})$ is a negative, non-decreasing convex
function along $t\in(-\infty,-1)$;

  2. (b)

the functionals $I_{u}$ and $J_{u}$ are both non-negative and non-decreasing
$L^{\infty}$-functions along $t\in(-\infty,-1)$;

  3. (c)

the two functionals converge as $t\rightarrow-\infty$,

$I_{u}(t)\rightarrow\pi\nu_{u}(0);\ \ \
J_{u}(t)\rightarrow\pi[\nu_{u}(0)]^{2}.$

Property (a) follows from a basic fact of plurisubharmonic functions, and part
(b) follows since $\dot{u}_{t}({\zeta})$ is a non-negative and non-decreasing
function in $t$ for any fixed ${\zeta}\in{\mathbb{C}}{\mathbb{P}}^{1}$.
Finally, property (c) follows from Lemma (5.1), (6.1) and the dominated
convergence theorem, exactly as we have used in Lemma (5.2).

###  6.3. Regularization

Thanks to Corollary (2.6), the standard regularization
$u_{\varepsilon}:=u*\rho_{\varepsilon}$ builds a sequence of functions in
${\mathcal{F}}^{\infty}(B_{1})$ decreasing to $u$. Therefore, its first
derivative in the $t$-direction

(6.9)
$\dot{u}_{\varepsilon,t}:=\partial_{t}u_{\varepsilon}=r\partial_{r}(u*\rho_{\varepsilon})$

is a non-negative, non-decreasing smooth function along $t\in(-\infty,0)$.
Moreover, we will obtain a uniform control on $\dot{u}_{t,\varepsilon}$ as
follows.

It follows from Lemma (6.1) that the radial derivative $\partial_{r}u$ is an
$L^{\infty}_{loc}$-function on $B^{*}_{1-\delta}$ for any small $\delta>0$.
Then the standard convolution
$(\partial_{r}u)_{\varepsilon}:=(\partial_{r}u)*\rho_{\varepsilon}$ converges
to it strongly in $L^{p}$ for any $p\geq 1$ on a compact subset $K\subset
B^{*}_{1-\delta}$, and hence its slicing
$(\partial_{r}u)_{\varepsilon}|_{S_{R}}$ converges to
$(\partial_{r}u)|_{S_{R}}$ strongly in $L^{p}$ on $S_{R}$ for almost all
$R\in(0,1-\delta)$.

However, a subtle issue left to us is to compare the convolution
$(\partial_{r}u)_{\varepsilon}$ with $\partial_{r}(u*\rho_{\varepsilon})$.
Then we need the following version of Friedrichs’ lemma.

######  Lemma 6.4.

For a function $u\in{\mathcal{F}}(B_{1})$ and a point $z\in
B^{*}_{1-2\delta}$, we have

(6.10)
$\left|r\partial_{r}(u*\rho_{\varepsilon})(z)-r(\partial_{r}u*\rho_{\varepsilon})(z)\right|\leq
2\varepsilon||\nabla u||_{L^{1}(B_{1-\delta})},$

for all $\varepsilon<\min\\{|z|,\delta\\}$.

###### Proof.

First we evaluate

(6.11) $\displaystyle\partial_{r}(u*\rho_{\varepsilon})(z)$ $\displaystyle=$
$\displaystyle\left\langle\
\nabla_{\vec{z}}(u*\rho_{\varepsilon})(z),\hat{r}_{\vec{z}}\ \right\rangle$
$\displaystyle=$
$\displaystyle\left\langle\varepsilon^{-4}\int_{|z-y|<\varepsilon}\nabla_{\vec{z}}\rho\left(\frac{z-y}{\varepsilon}\right)u(y)d\lambda(y),\hat{r}_{\vec{z}}\right\rangle$
$\displaystyle=$
$\displaystyle\left\langle-\varepsilon^{-4}\int_{|z-y|<\varepsilon}\nabla_{\vec{y}}\rho\left(\frac{z-y}{\varepsilon}\right)u(y)d\lambda(y),\hat{r}_{\vec{z}}\right\rangle$
$\displaystyle=$
$\displaystyle\left\langle\varepsilon^{-4}\int_{|z-y|<\varepsilon}\rho\left(\frac{z-y}{\varepsilon}\right)\nabla
u(y)d\lambda(y),\hat{r}_{\vec{z}}\right\rangle$ $\displaystyle=$
$\displaystyle\int_{|w|<1}\left\langle\nabla u(z-\varepsilon
w),\hat{r}_{\vec{z}}\right\rangle\rho\left(w\right)d\lambda(w),$

where we used the change of variables $y:=z-\varepsilon w$. On the other hand,
we have

(6.12) $\displaystyle(\partial_{r}u*\rho_{\varepsilon})(z)$ $\displaystyle=$
$\displaystyle\varepsilon^{-4}\int_{|z-y|<\varepsilon}\left\langle\nabla
u(y),\hat{r}_{\vec{y}}\right\rangle\rho\left(\frac{z-y}{\varepsilon}\right)d\lambda(y)$
$\displaystyle=$ $\displaystyle\int_{|w|<1}\left\langle\nabla u(z-\varepsilon
w),\hat{r}_{(\vec{z}-\varepsilon\vec{w})}\right\rangle\rho(w)d\lambda(w).$

Take the difference of equation (6.11) and (6.12), and we obtain

(6.13)
$\displaystyle|(\partial_{r}u*\rho_{\varepsilon})(z)-\partial_{r}(u*\rho_{\varepsilon})(z)|$
$\displaystyle\leq$ $\displaystyle\int_{|w|<1}\left|\left\langle\nabla
u(z-\varepsilon
w),\hat{r}_{(\vec{z}-\varepsilon\vec{w})}-\hat{r}_{\vec{z}}\right\rangle\right|\rho(w)d\lambda(w)$
$\displaystyle\leq$ $\displaystyle C(\varepsilon,z)\int_{|w|<1}\left|\nabla
u(z-\varepsilon w)\right|\rho(w)d\lambda(w)$ $\displaystyle\leq$
$\displaystyle C(\varepsilon,z)||\nabla u||_{L^{1}(B_{1-\delta})}.$

Here we have used the constant

$C(\varepsilon,z):=\sup_{|w|<1}|\hat{r}_{(\vec{z}-\varepsilon\vec{w})}-\hat{r}_{\vec{z}}|,$

and it can be controlled as

(6.14)
$\displaystyle|\hat{r}_{(\vec{z}-\varepsilon\vec{w})}-\hat{r}_{\vec{z}}|=\left|\frac{z}{|z|}-\frac{z-\varepsilon
w}{|z-\varepsilon w|}\right|$ $\displaystyle\leq$
$\displaystyle\frac{\varepsilon}{|z|}+\left|(z-\varepsilon
w)\left(\frac{1}{|z|}-\frac{1}{|z-\varepsilon w|}\right)\right|$
$\displaystyle\leq$
$\displaystyle\frac{\varepsilon}{|z|}+\frac{\left||z|-|z-\varepsilon
w|\right|}{|z|}\leq\frac{2\varepsilon}{|z|},$

Then our result follows.

∎

Since the gradient of a function $u\in{\mathcal{F}}(B_{1})$ is also in
$L^{2}_{loc}(B_{1})$, Lemma (6.4) implies that we can write

(6.15)
$r(\partial_{r}u)_{\varepsilon}(z)=\dot{u}_{\varepsilon,t}(z)+O(\varepsilon),$

for all $z$ in a punctured smaller ball in $B_{1}$. Hence we have proved the
convergence

(6.16) $\dot{u}_{\varepsilon,t}\rightarrow\partial_{t}u$

strongly in $L^{p}$ for any $p\geq 1$ on a relatively compact subset $K\subset
B^{*}_{1}$. Then we can infer the following convergence of the functionals.

######  Lemma 6.5.

For a function $u\in{\mathcal{F}}(B_{1})$, and any constants $B>A>2$, there
exists a subsequence $u_{\varepsilon_{j}}$ such that we have the convergence

$I_{u_{\varepsilon_{j}}}(t)\rightarrow I_{u}(t);\ \ \
J_{u_{\varepsilon_{j}}}(t)\rightarrow J_{u}(t),$

for almost all $t\in[-B,-A]$.

###### Proof.

Let $\\{\rho_{j}\\}_{j=1}^{k}$ be a partition of unity of
${\mathbb{C}}{\mathbb{P}}^{1}$, and $\chi(r)$ a smooth cut-off function on the
unit interval that $\chi(r)=1$ for all $r\in(e^{-B},e^{-A})$ and $\chi(r)=0$
outside of the interval $(e^{-B-1},e^{-A+1})$. Then we introduce the following
compactly supported $3$-forms on $B_{1}^{*}$ as

${\alpha}_{j}:=\chi(r)\rho_{j}({\zeta})d\sigma_{3},$

where $d\sigma_{3}$ is the area form of the unit $3$-sphere. Take the $1$-form
as

$\beta:=r^{-1}dr,$

and then the basic slicing formula (equation (6.6)) implies the following
equality for $r=e^{t}$

(6.17)
$\int_{B_{1}}|\dot{u}_{\varepsilon,t}-\partial_{t}u|^{2}{\alpha}_{j}\wedge\beta=\int_{0}^{1}\left(\int_{S_{r}}|\dot{u}_{\varepsilon,t}-\dot{u}_{t}|^{2}\rho_{j}({\zeta})d\sigma_{3}\right)\chi(r)r^{-1}dr.$

Summing up with $j$, we obtain

(6.18)
$\int_{B_{1}}|\dot{u}_{\varepsilon,t}-\partial_{t}u|^{2}\chi(r)d\lambda_{t}=2\pi\int_{-\infty}^{0}\left(\int_{{\mathbb{C}}{\mathbb{P}}^{1}}|\dot{u}_{\varepsilon,t}-\dot{u}_{t}|^{2}\omega\right)\tilde{\chi}(t)dt,$

where $d\lambda_{t}:=r^{-1}drd\sigma_{3}$ is a measure on
${\mathbb{C}}^{2}-\\{0\\}$, and $\tilde{\chi}(t):=\chi(e^{t})$ is a cut-off
function on $(-\infty,0)$ that supports on $(-B-1,-A+1)$.

From equation (6.16), we infer that the L.H.S. of equation (6.18) converges to
zero as $\varepsilon\rightarrow 0$. Then it follows that we have

$\dot{u}_{\varepsilon,t}\rightarrow\dot{u}_{t}$

strongly in $L^{2}$ on the fiber ${\mathbb{C}}{\mathbb{P}}^{1}\times\\{t\\}$
for almost all $t\in[-B,-A]$, possibly after passing to a subsequence, and
then our result follows.

∎

###  6.4. Apriori estimates

The following idea is to utilize the regularization
$u_{\varepsilon}\in{\mathcal{F}}^{\infty}(B_{1})$ to approximate a function
$u\in{\mathcal{F}}(B_{1})$, and then apply our decomposition formula (equation
(5.31)) to this sequence to derive a contradiction. To this purpose, we first
need a uniform estimate on the maximal directional Lelong numbers of the
regularization.

######  Lemma 6.6.

Fix any two constants $B>A>1$. Then there exists a uniform constant $C>0$ such
that we have

(6.19) $M_{B}(u_{\varepsilon})\leq 2M_{A}(u)+C\varepsilon,$

for all $\varepsilon<\varepsilon_{0}$, where

$\varepsilon_{0}:=\frac{1}{2}\min\left\\{(e^{-A}-e^{-B}),e^{-B}\right\\}.$

###### Proof.

The maximal directional Lelong number of the regularization can be written as

$M_{B}(u_{\varepsilon})=\sup_{{\zeta}\in{\mathbb{C}}{\mathbb{P}}^{1}}(\partial_{t}u_{\varepsilon,t})({\zeta})|_{t=-B}.$

Thanks to Lemma (6.4), it boils down to estimate the partial derivatives
$r(\partial_{r}u)_{\varepsilon}(z)$ on the boundary sphere $S_{R}$ with
$R=e^{-B}$.

Hence we take a point $z\in S_{R}$, and compute for all
$\varepsilon<\varepsilon_{0}$

(6.20) $\displaystyle r(\partial_{r}u)_{\varepsilon}(z):$ $\displaystyle=$
$\displaystyle r(\partial_{r}u\ast\rho_{\varepsilon})(z)$ $\displaystyle=$
$\displaystyle\int_{|w|\leq 1}\frac{|z|}{|z-\varepsilon
w|}(r\partial_{r})u(z-\varepsilon w)\rho(w)d\lambda(w)$ $\displaystyle=$
$\displaystyle\int_{|w|\leq 1}\frac{|z|}{|z-\varepsilon
w|}\dot{u}_{t}({\zeta})\rho(w)d\lambda(w),$

where $t=\log|z-\varepsilon w|<-A$ and
${\zeta}\in{\mathbb{C}}{\mathbb{P}}^{1}$ is the angle corresponding to the
point $z-\varepsilon w$. Then consider the restriction of $u$ to the complex
line ${\ell}_{{\zeta}}$ through the origin. By utilizing the monotonicity of
$\dot{u}_{t}({\zeta})$, we have

$0\leq\dot{u}_{t}({\zeta})\leq M_{A}(u),$

for all ${\zeta}\in{\mathbb{C}}{\mathbb{P}}^{1}$. Therefore, we further have

(6.21) $\displaystyle r(\partial_{r}u)_{\varepsilon}(z)$ $\displaystyle\leq$
$\displaystyle M_{A}(u)\int_{|w|\leq 1}\frac{|z|}{|z-\varepsilon
w|}\rho(w)d\lambda(w)$ $\displaystyle\leq$ $\displaystyle(1+2\varepsilon
e^{-B})M_{A}(u)\leq 2M_{A}(u),$

due to our choice of small $\varepsilon$. Finally, our result follows from
Lemma (6.4).

∎

######  Remark 6.7.

If we choose a smaller $\varepsilon_{0}$ in Lemma (6.6), then it is possible
to obtain a shaper estimate. For instance, take an arbitrary real number
$\beta\in(0,1)$ and

(6.22) $\varepsilon_{0}:=\min\left\\{\frac{1}{2}(e^{-A}-e^{-B}),\frac{\beta
e^{-B}}{1+\beta}\right\\}.$

Then we have the following estimate for all $\varepsilon<\varepsilon_{0}$

(6.23) $M_{B}(u_{\varepsilon})\leq(1+\beta)M_{A}(u)+C\varepsilon,$

with the same constant $C$.

We have seen a useful estimate (equation (5.31)), based on the decomposition
formula of the Monge-Ampère mass. For a function
$u\in{\mathcal{F}}^{\infty}(B_{1})$ and a constant $A>0$ with $M_{A}(u)>0$, we
can re-write it as follows.

(6.24)
$I^{\prime}_{u}(t)\geq\frac{1}{M_{A}(u)}\left\\{\pi^{-1}\mbox{MA}(u)(B_{r})-J_{u}(t)\right\\}-2I_{u}(t),$

for all $t=\log r\in(-\infty,-A]$. This new version of the estimate will be
substantial for the following argument on the zero mass conjecture in the
general case.

######  Theorem 6.8.

For any $u\in{\mathcal{F}}(B_{1})$, its residual Monge-Ampère mass
$\tau_{u}(0)$ is zero, if its Lelong number $\nu_{u}(0)$ is zero at the
origin.

As in Theorem (5.5), a contradiction argument will be applied. In this case,
we can assume $u(0)=-\infty$, and then Lemma (6.2) implies that we have
$M_{A}(u)>0$ for all $A>0$. Therefore, there exists a small number $\delta>0$
such that we can make the assumption

(6.25) $\pi^{-1}\mbox{MA}(u)(B_{r})\geq 20(2M_{2}(u)+1)\delta,$

for all $r\in(0,1)$.

The first step is to fix a large constant $A>4$ such that the following
condition holds:

  1. (i)

for almost all $t\leq-A$, we have $0\leq I_{u}(t)\leq\delta$ and $0\leq
J_{u}(t)\leq\delta$.

This follows from the fact $\nu_{u}(0)=0$ and the properties (b) and (c) of
the functionals. Moreover, this condition will be kept for any larger
constant.

The second step is to pick up a subsequence $u_{\varepsilon}$ of the
regularization and a small $\varepsilon_{1}>0$ such that the following
estimates hold for all $\varepsilon<\varepsilon_{1}$:

  1. (ii)

we have at $t=-A$

$0\leq I_{u_{\varepsilon}}(-A)\leq 2\delta,\ \ \ 0\leq
J_{u_{\varepsilon}}(-A)\leq 2\delta;$

  2. (iii)

we have at $R=e^{-2A}$

$\pi^{-1}\mbox{MA}(u_{\varepsilon})(B_{R})\geq 10(2M_{2}(u)+1)\delta;$

  3. (iv)

for any $B\in[A,2A]$, we have

$M_{B}(u_{\varepsilon})\leq 2M_{2}(u)+1.$

Condition-(ii) follows from Lemma (6.5) and condition-(i), since we have the
convergence

$I_{u_{\varepsilon}}(t)\rightarrow I_{u}(t),\ \ \
J_{u_{\varepsilon}}(t)\rightarrow J_{u}(t),$

for almost all $t\in[-2A,A]$, possibly after passing to a subsequence. Here
the constant $A$ may be perturbed to a slightly larger one, but this will not
affect our condition-(i).

Condition-(iii) is implied by equation (6.25) and our previous estimate
(equation (2.5)). Finally condition-(iv) follows from Lemma (6.6), by taking
$\varepsilon_{1}<\min\\{\varepsilon_{0},\frac{1}{C}\\}$, where $C$ is the
uniform constant appearing in the R.H.S. of equation (6.19), and

$\varepsilon_{0}:=\frac{1}{2}\min\left\\{e^{-2}-e^{-4},e^{-2A}\right\\}.$

Equipped with the above conditions, we continue the argument as follows.

###### Proof of Theorem(6.8).

Suppose on the contrary that $\tau_{u}(0)$ is positive. Then we can pick up a
real number $\delta>0$ small enough such that equation (6.25) holds for all
$r\in(0,1)$. Choose a large constant $A$, and take a subsequence
$u_{\varepsilon}$ of the regularization with $\varepsilon<\varepsilon_{1}$
such that condition (i)-(iv) are all satisfied.

Fix an $\varepsilon<\varepsilon_{1}$, we will focus on the
$I_{u_{\varepsilon}}$-functional in the closed interval $[-2A,A]$. First we
claim $M_{B}(u_{\varepsilon})>0$ for all $B\in[A,2A]$ under our choices, and
the reason is as follows.

It is deduced from conditions (ii)-(iii) that we have for any $t\in[-2A,-A]$

(6.26)
$\displaystyle\pi^{-1}\mbox{MA}(u_{\varepsilon})(B_{r})-J_{u_{\varepsilon}}(t)$
$\displaystyle\geq$
$\displaystyle\pi^{-1}\mbox{MA}(u_{\varepsilon})(B_{R})-J_{u_{\varepsilon}}(-A)$
$\displaystyle\geq$ $\displaystyle 8(2M_{2}(u)+1)\delta>0,$

where $r=e^{t}$ and $R=e^{-2A}$. However, this contradicts to the inequality
in equation (5.31), if we have $M_{B}(u)=0$ for any $B\in[A,-t]$. Then our
claim follows.

Thanks to the estimate in equation (6.24), the first derivative of the
$I_{u_{\varepsilon}}$-functional can be estimated from below as

(6.27) $\displaystyle I^{\prime}_{u_{\varepsilon}}(t)$ $\displaystyle\geq$
$\displaystyle\frac{1}{M_{A}(u_{\varepsilon})}\left\\{8(2M_{2}(u)+1)\delta\right\\}-2I_{u_{\varepsilon}}(-A)$
$\displaystyle\geq$
$\displaystyle\frac{1}{2M_{2}(u)+1}\left\\{8(2M_{2}(u)+1)\delta\right\\}-4\delta$
$\displaystyle\geq$ $\displaystyle 4\delta,$

for all $t\in[-2A,-A]$. Here the first line on the R.H.S. of equation (6.27)
follows from equation (6.26), and the second line follows from condition-(iv).
However, it implies

(6.28)
$I_{u_{\varepsilon}}(-A)\geq\int_{-2A}^{A}I^{\prime}_{u_{\varepsilon}}(t)dt\geq
16\delta,$

which contradicts to the assumption on the $I_{u_{\varepsilon}}$-functional in
condition-(ii). Then our result follows.

∎

###  6.5. Positive Lelong numbers

We note that the zero Lelong number condition does not really used in the
apriori estimates during the proof of the Theorem (6.8). In fact, we can
obtain the same estimate (equation (5.33)) in Theorem (5.9) for a function $u$
in ${\mathcal{F}}(B_{1})$. In other words, its residual Monge-Ampère mass
$\tau_{u}(0)$ can be controlled by the maximal directional Lelong number
$\lambda_{u}(0)$ and Lelong number $\nu_{u}(0)$ at the origin.

######  Theorem 6.9.

For a function $u\in{\mathcal{F}}(B_{1})$, we have

(6.29) ${\tau}_{u}(0)\leq 2\lambda_{u}(0)\cdot\nu_{u}(0)+[\nu_{u}(0)]^{2}.$

###### Proof.

First we note that it is enough to prove the inequality

(6.30) ${\tau}_{u}(0)\leq
2(1+\beta)M_{A_{0}}(u)\cdot\nu_{u}(0)+[\nu_{u}(0)]^{2},$

for all $A_{0}>2$ large and $\beta\in(0,1)$ small. Moreover, we can also
assume $u(0)=-\infty$, and then $M_{A}(u)$ is always positive for all $A>0$.

Suppose on the contrary that equation (6.30) fails. Then there exists a small
number $\delta>0$ such that we have

(6.31) $\displaystyle\pi^{-1}\mbox{MA}(u)(B_{r})$ $\displaystyle\geq$
$\displaystyle
2(1+\beta)M_{A_{0}}(u)\cdot\pi\nu_{u}(0)+\pi[\nu_{u}(0)]^{2}+20(2M_{A_{0}}(u)+1)\delta,$

for all $r\in(0,1)$. Next, we can pick up a large constant $A>2A_{0}$, and
withdraw a subsequence $u_{\varepsilon}$ with $\varepsilon<\varepsilon_{1}$
such that the following conditions are satisfied:

  1. (v)

for almost all $t\leq-A$, we have

$\pi\nu_{u}(0)\leq I_{u}(t)\leq\pi\nu_{u}(0)+\delta,\ \ \
\pi[\nu_{u}(0)]^{2}\leq J_{u}(t)\leq\pi[\nu_{u}(0)]^{2}+\delta;$

  2. (vi)

we have at $t=-A$ and $t=-2A$

$\pi\nu_{u}(0)-\delta\leq I_{u_{\varepsilon}}(-2A)\leq
I_{u_{\varepsilon}}(-A)\leq\pi\nu_{u}(0)+2\delta$
$\pi[\nu_{u}(0)]^{2}-\delta\leq
J_{u_{\varepsilon}}(-A)\leq\pi[\nu_{u}(0)]^{2}+2\delta;$

  3. (vii)

we have at $R=e^{-2A}$

(6.32) $\displaystyle\pi^{-1}\mbox{MA}(u_{\varepsilon})(B_{R})$
$\displaystyle\geq$ $\displaystyle
2(1+\beta)M_{A_{0}}(u)\cdot\pi\nu_{u}(0)+\pi[\nu_{u}(0)]^{2}+10(2M_{A_{0}}(u)+1)\delta;$

  4. (viii)

for any $B\in[A,2A]$, we have for a small number ${\kappa}>0$

$M_{B}(u_{\varepsilon})\leq(1+\beta)M_{A_{0}}(u)+{\kappa}.$

As before, condition (v)-(vii) are satisfied due to Lemma (6.5) and properties
of $I_{u}$ and $J_{u}$ functionals. The points $t=-A$ and $t=-2A$ may also be
slightly perturbed to satisfy condition-(vi). Moreover, condition-(viii)
follows from Lemma (6.6) and remark (6.7), since we can take
$\varepsilon_{1}<\varepsilon_{0}$ where

$\varepsilon_{0}:=\min\left\\{\frac{1}{2}(e^{-A_{0}}-e^{-2A_{0}}),\
\frac{\beta e^{-2A}}{1+\beta},\ {\kappa}C^{-1}\right\\}.$

Equipped with these conditions (v)-(viii), and then we have the following
estimate for all $t\in[-2A,-A]$

(6.33)
$\displaystyle\pi^{-1}\mbox{MA}(u_{\varepsilon})(B_{r})-J_{u_{\varepsilon}}(t)$
$\displaystyle\geq$
$\displaystyle\pi^{-1}\mbox{MA}(u_{\varepsilon})(B_{R})-J_{u_{\varepsilon}}(-A)$
$\displaystyle\geq$ $\displaystyle
2(1+\beta)M_{A_{0}}(u)\cdot\pi\nu_{u}(0)+8(2M_{A_{0}}(u)+1)\delta>0,$

where $r=e^{t}$ and $R=e^{-2A}$. Thanks to the estimate in equation (5.31)
again, we can infer $M_{A}(u_{\varepsilon})>0$. Then the lower bound of
$I^{\prime}_{u_{\varepsilon}}(t)$ can be estimated for all $t\in[-2A,A]$ as

(6.34) $\displaystyle I^{\prime}_{u_{\varepsilon}}(t)$ $\displaystyle\geq$
$\displaystyle\frac{2(1+\beta)M_{A_{0}}(u)\cdot\pi\nu_{u}(0)+8(2M_{A_{0}}(u)+1)\delta}{M_{A}(u_{\varepsilon})}-2I_{u_{\varepsilon}}(-A)$
$\displaystyle\geq$
$\displaystyle\frac{2(1+\beta)M_{A_{0}}(u)\cdot\pi\nu_{u}(0)+8(2M_{A_{0}}(u)+1)\delta}{(1+\beta)M_{A_{0}}(u)+{\kappa}}-2I_{u_{\varepsilon}}(-A)$
$\displaystyle\geq$ $\displaystyle
4\delta-\frac{2\pi\nu_{u}(0){\kappa}}{(1+\beta)M_{A_{0}}(u)+{\kappa}}.$

Finally, we take the integral

(6.35)
$I_{u_{\varepsilon}}(-A)=I_{u_{\varepsilon}}(-2A)+\int_{-2A}^{A}I^{\prime}_{u_{\varepsilon}}(t)dt\geq\pi\nu_{u}(0)+15\delta+O({\kappa}),$

but this contradicts to condition-(vi), since we have $M_{A_{0}}(u)>0$ and
${\kappa}$ can be taken arbitrarily small. Then our result follows.

∎

It is also apparent that the inequality (equation (6.29)) in the Theorem (6.9)
implies the Theorem (6.8). In other words, this estimate on the residual
Monge-Ampère mass is a more general result than the zero mass conjecture.

In order to illustrate this inequality in a better way, we provide the
following typical examples.

######  Example 6.10.

Suppose a function $u\in{\mathcal{F}}(B_{1})$ is further radially symmetric,
cf. [Li19]. That is to say, we have $u(z)=u(|z|)$ for all $z\in B_{1}$. It is
a standard fact that we can compute

$\tau_{u}(0)=[\nu_{u}(0)]^{2},$

and then equation (6.29) holds in a trivial way.

######  Example 6.11.

Consider Demailly’s example as follows, cf. [Dem93]. For any $\varepsilon>0$,
we take

$u(z_{1},z_{2}):=\max\\{\varepsilon\log|z_{1}|,\varepsilon^{-1}\log|z_{2}|\\}.$

Then we have $\nu_{u}(0)=\varepsilon$ and $\tau_{u}(0)=1$. However, the
maximal directional Lelong number is determined on the complex line
$\\{z_{1}=0\\}$. Then it follows $M_{A}(u)=\varepsilon^{-1}$ for each $A>0$,
and equation (6.29) holds as

$1=\tau_{u}(0)\leq
2\lambda_{u}(0)\cdot\nu_{u}(0)+[\nu_{u}(0)]^{2}=2+\varepsilon^{2}.$

######  Example 6.12.

Consider Chi Li’s example as follows, cf. [Chi21]. Take the holomorphic
projection
$p:{\mathbb{C}}^{2}-\\{0\\}\rightarrow{\mathbb{C}}{\mathbb{P}}^{1}$, and
$\omega_{0}$ the standard Fubini-Study metric. Let $\varphi$ be the
$\omega_{0}$-plurisubharmonic function on ${\mathbb{C}}{\mathbb{P}}^{1}$ such
that its singular locus $\\{\varphi=-\infty\\}$ is a Cantor set $\mathcal{C}$,
and $\varphi$ is $\omega_{0}$-pluriharmonic outside $\mathcal{C}$. Then we can
construct

$u(z):=\max\\{\log|z|^{2}+p^{*}\varphi,2\log|z|^{2}\\}.$

Then compute and obtain $\tau_{u}(0)=8$ and $\nu_{u}(0)=2$. Here we note that
the function $p^{*}\varphi$ does not directly contribute to the variation in
the $t$-direction on each complex line. Therefore, we have $M_{A}(u)=4$ for
all $A>0$ large, and then it follows

$8=\tau_{u}(0)\leq 2\lambda_{u}(0)\cdot\nu_{u}(0)+[\nu_{u}(0)]^{2}=20.$

######  Example 6.13.

The following example is constructed by Coman and Guedj, cf. [CG09]. We note
that it is no longer $S^{1}$-invariant, and then equation (6.29) fails in this
case: Fix an integers $n>4$, and we write

$u(z):=\frac{1}{2n}\log\left(|z_{2}-z_{1}^{n}|^{2}+|z_{2}^{n}|^{2}\right).$

Then it follows $\nu_{u}(0)=\frac{1}{n}$ and $\tau_{u}(0)=1$. The maximal
direction of the Lelong numbers is along the line $\\{z_{2}=0\\}$, and then we
have $\lambda_{u}(0)=1$. Hence it follows

$1=\tau_{u}(0)>2\lambda_{u}(0)\cdot\nu_{u}(0)+[\nu_{u}(0)]^{2}=\frac{2}{n}+\frac{1}{n^{2}}.$

For $S^{1}$-invariant plurisubharmonic functions, we can consider the
following:

$u_{1}(z):=\frac{1}{2n}\log\left(|z_{2}-z_{1}|^{2}+|z_{2}^{n}|^{2}\right).$

Then we have $\nu_{u_{1}}(0)=\frac{1}{n}$, $\lambda_{u_{1}}(0)=1$ and
$\tau_{u_{1}}(0)=\frac{1}{n}$, and equation (6.29) holds as

$\frac{1}{n}=\tau_{u_{1}}(0)\leq
2\lambda_{u_{1}}(0)\cdot\nu_{u_{1}}(0)+[\nu_{u_{1}}(0)]^{2}=\frac{2}{n}+\frac{1}{n^{2}}.$

Another modification is

$u_{2}(z):=\frac{1}{2n}\log\left(|z_{2}^{n}-z_{1}^{n}|^{2}+|z_{2}^{n}|^{2}\right).$

Then we have $\nu_{u_{2}}(0)=1$, $\lambda_{u_{2}}(0)=1$ and
$\tau_{u_{2}}(0)=1$, and equation (6.29) holds as

$1=\tau_{u_{2}}(0)\leq
2\lambda_{u_{2}}(0)\cdot\nu_{u_{2}}(0)+[\nu_{u_{2}}(0)]^{2}=3.$

##  7\. Variational approach

In this section, we will introduce another point of view to look at the
decomposition formula (Theorem (4.4)) and also the zero mass conjecture. The
observation is that the function $u_{t}$ actually defines a curve in the space
of all quasi-plurisubharmonic functions on ${\mathbb{C}}{\mathbb{P}}^{1}$, and
it can be thought of as a subgeodesic with non-trivial $S^{1}$-fiberation in
the space of Kähler potentials. Then the first term on the R.H.S. of equation
(4.42) corresponds to the first variation of the so called _pluri-complex
energy_.

###  7.1. Subgeodesic on fiber bundles

We recall some basic facts in Kähler geometry. Let $X$ be a compact Riemann
surface without boundary, and $\omega_{0}$ a Kähler metric on this Riemann
surface. Denote $\mathbb{D}^{*}$ by the punctured unit disk in ${\mathbb{C}}$,
and it can be identified with the product $(0,1)\times S^{1}$ via the polar
coordinate $(r,s)$, namely, we can write

$z:=re^{is}$

as a complex variable in ${\mathbb{D}}^{*}$.

Consider the product space $Y:={\mathbb{D}}^{*}\times X$, and the projection
maps $\pi_{1}:Y\rightarrow{\mathbb{D}}^{*}$ and $\pi_{2}:Y\rightarrow X$. The
pull back $\pi_{2}^{*}\omega_{0}$ is a closed non-negative $(1,1)$-form on
$Y$. Then a _subgeodesic ray_ in the space of Kähler potentials is an
$S^{1}$-invariant $\pi^{*}_{2}\omega_{0}$-plurisubharmonic functions $v$ on
$Y$. That is to say, if we take any local potential $\Phi$ of $\omega_{0}$,
then the function

$V:=\pi_{2}^{*}\Phi+v$

is independent of the variable $s$, and plurisubharmonic in the product
manifold $(0,1)\times S^{1}\times X$. In other words, a subgeodesic $V$ is
locally an $S^{1}$-invariant plurisubharmonic function on the trivial
${\mathbb{D}}^{*}$-bundle of $X$, and hence the restriction
$v|_{X\times\\{t\\}}$ is a $\omega_{0}$-plurisubharmonic function on each
fiber $X\times\\{t\\}$.

Furthermore, a subgeodesic ray $v$ is a _geodesic ray_ , if it satisfies the
following _homogeneous complex Monge-Ampère equation_ on $Y$

(7.1) $(dd^{c}_{z,X}V)^{2}=(\pi_{2}^{*}\omega_{0}+dd^{c}_{z,X}v)^{2}=0.$

From now on, we put $X={\mathbb{C}}{\mathbb{P}}^{1}$ and $\omega_{0}$ the
Fubini-Study metric on it. In fact, the projective space can be viewed as the
moduli space of ${\mathbb{C}}^{2}-\\{0\\}$, under the natural
${\mathbb{C}}^{*}$-action. The punctured disk ${\mathbb{D}}^{*}$ acts in the
same way on the punctured ball $B_{1}^{*}$ in ${\mathbb{C}}^{2}$, and then
$B_{1}^{*}$ can be thought of as a non-trivial ${\mathbb{D}}^{*}$-bundle of
${\mathbb{C}}{\mathbb{P}}^{1}$ via the Hopf-fiberation, i.e. we can write the
bundle map as follows

$({\dagger})\ \ \ \ \ \ \ \ \ {\mathbb{D}}^{*}\hookrightarrow
B_{1}^{*}\xrightarrow{p}{\mathbb{C}}{\mathbb{P}}^{1}.$

If we take a function $u\in{\mathcal{F}}(B_{1})$, then it is naturally an
$S^{1}$-invariant plurisubharmonic function on this non-trivial
${\mathbb{D}}^{*}$-bundle. Writing $B_{1}^{*}$ as a product $(0,1)\times
S^{3}$, a fiber $\\{r\\}\times S^{3}$ is identified with the $3$-sphere
$S_{r}$ for all $r\in(0,1)$. Then the restriction $u|_{S_{r}}$ can be viewed
as a function on ${\mathbb{C}}{\mathbb{P}}^{1}$ via the Hopf-fiberation. In
fact, this restriction $u|_{S_{r}}$ is exactly the function $u_{t}$ defined in
equation (4.40), via the change of variables $r=e^{t}$.

If we further require $u\in{\mathcal{F}}^{\infty}(B_{1})$, then the Laplacian
decomposition formula (equation (5.12)) implies on each fiber
${\mathbb{C}}{\mathbb{P}}^{1}\times\\{t\\}$

(7.2)
$\frac{1}{2}\left(\ddot{u}_{t}+2\dot{u}_{t}\right)+\Delta_{\omega}u_{t}\geq
0.$

Therefore, the function $u_{t}$ is quasi-plurisubharmonic on each fiber
${\mathbb{C}}{\mathbb{P}}^{1}\times\\{t\\}$, but the lower bound of its
complex hessian varies with respect to $t$ and $u$ itself. For these reasons,
we introduce the following definition.

######  Definition 7.1.

For a function $u\in{\mathcal{F}}(B_{1})$, we say that $u_{t}$ is a bounded
subgeodesic ray on the fiber bundle (†). Moreover, it is a geodesic ray on
this fiber bundle if we have

$(dd^{c}u)^{2}=0,$

on $B_{1}^{*}$.

If $u$ is further in the family ${\mathcal{F}}^{\infty}(B_{1})$, then we say
that $u_{t}$ is a $C^{2}$-continuous subgeodesic ray on this fiber bundle.

###  7.2. Energy functionals

For a quasi-plurisubharmonic function on ${\mathbb{C}}{\mathbb{P}}^{1}$, the
pluri-complex energy ${\mathcal{E}}$ is defined as

${\mathcal{E}}(u_{t}):=\int_{{\mathbb{C}}{\mathbb{P}}^{1}}(-u_{t})dd^{c}_{{\zeta}}u_{t}=-\int_{{\mathbb{C}}{\mathbb{P}}^{1}}u_{t}(\Delta_{\omega}u_{t})\omega.$

It is well known that this energy is concave along a subgeodesic in the space
of Kähler potentials. In fact, the push-forward of the Monge-Ampère measure is
exactly the complex hessian of $-{\mathcal{E}}$ along a sub-geodesic ray, cf.
[BB22].

Next we will show that our decomposition formula is an analogue of this on a
non-trivial fiber bundle. Let $u_{t}$ be a $C^{2}$-continuous subgeodesic ray
on the fiber bundle $({\dagger})$. Then the first variation of this energy
with respect to $t$ can be computed as

(7.3)
$\frac{d}{dt}{\mathcal{E}}(u_{t})=-2\int_{{\mathbb{C}}{\mathbb{P}}^{1}}\dot{u}_{t}(\Delta_{\omega}u_{t})\omega,$

and this is exactly the negative of the first term on the R.H.S. of the
decomposition formula (equation (4.42)). Take the following non-negative
functional to represent the complex Monge-Ampère mass for $r=e^{t}$

(7.4) $K(u_{t}):=\pi^{-1}\cdot\mbox{MA}(u)(B_{r}).$

Then the decomposition formula can be rewritten as follows.

######  Lemma 7.2.

Suppose $u_{t}$ is a $C^{2}$-continuous subgeodesic ray on the fiber bundle
$({\dagger})$. Then we have

(7.5) $-\frac{d}{dt}{\mathcal{E}}(u_{t})+J_{u}(t)=K(u_{t}),$

for all $t\in(-\infty,0)$.

By invoking toric plurisubharmonic functions, we can take a closer look at
these energy functionals.

######  Example 7.3.

Suppose $u\in{\mathcal{F}}^{\infty}(B_{1})$ also has toric symmetry, i.e. we
have

$u(z_{1},z_{2})=u(e^{i\theta_{1}}z_{1},e^{i\theta_{2}}z_{2}),$

for arbitrary $\theta_{1},\theta_{2}\in{\mathbb{R}}$. Then in the real Hopf-
coordinate, we can write

$u(r,\eta,\theta,\varphi)=u(r,\theta).$

On the one hand, equation (4.49) boils down to the following

(7.6) $\displaystyle 8\int_{S_{R}}d^{c}u\wedge dd^{c}u$ $\displaystyle=$
$\displaystyle 16\pi^{2}\int_{0}^{\pi}(ru_{r})\partial_{\theta}(\sin\theta
u_{\theta})d\theta+4\pi^{2}\int_{0}^{\pi}(ru_{r})^{2}\sin\theta d\theta.$

Then the first variation of the energy ${\mathcal{E}}$ with respect to $t$ is

$-\frac{d}{dt}{\mathcal{E}}(u_{t})=2\pi\int_{0}^{\pi}(ru_{r})\partial_{\theta}(\sin\theta
u_{\theta})d\theta,$

and the $L^{2}$-Lelong number is

$J_{u}(t)=\frac{\pi}{2}\int_{0}^{\pi}(ru_{r})^{2}\sin\theta d\theta.$

On the other hand, we can also compute the complete complex Hessian of $u$ as
follows

(7.7) $\displaystyle 4\ dd^{c}u$ $\displaystyle=$
$\displaystyle\left\\{2\sin\theta
u_{,r\theta}-\partial_{r}(ru_{r})\cos\theta\right\\}dr\wedge
d\varphi+\partial_{r}(ru_{r})dr\wedge d\eta$ $\displaystyle+$
$\displaystyle\left\\{(ru_{r})\sin\theta-(ru_{,r\theta})\cos\theta\right\\}d\theta\wedge
d\varphi$ $\displaystyle+$ $\displaystyle 2\left\\{\sin\theta\cdot
u_{,\theta\theta}+\cos\theta\cdot u_{\theta}\right\\}d\theta\wedge
d\varphi+(ru_{,r\theta})d\theta\wedge d\eta.$

Therefore, we obtain a formula for the positive measure as

(7.8) $\displaystyle 8(dd^{c}u)^{2}$ $\displaystyle=$
$\displaystyle\left\\{2\partial_{r}(ru_{r})\partial_{\theta}(\sin\theta
u_{\theta})-2r\sin\theta(u_{,r\theta})^{2}+\partial_{r}(ru_{r})(ru_{r})\sin\theta\right\\}$
$\displaystyle dr\wedge d\theta\wedge d\varphi\wedge d\eta$

Then we continue to compute the second variation of the energy ${\mathcal{E}}$
as

(7.9) $\displaystyle-\frac{d^{2}{\mathcal{E}}(u_{t})}{dt^{2}}$
$\displaystyle=$ $\displaystyle 2\pi
r\int_{0}^{\pi}\left\\{\partial_{r}(ru_{r})\partial_{\theta}(\sin\theta
u_{\theta})+(ru_{r})\partial_{\theta}(\sin\theta
u_{,r\theta})\right\\}d\theta$ $\displaystyle=$ $\displaystyle 2\pi
r\int_{0}^{\pi}\left\\{\partial_{r}(ru_{r})\partial_{\theta}(\sin\theta
u_{\theta})-r\sin\theta(u_{,r\theta})^{2}\right\\}d\theta$ $\displaystyle\geq$
$\displaystyle-\pi r\int_{0}^{\pi}\dot{u}_{t}\ddot{u}_{t}\sin\theta d\theta.$

As we have expected, the last term on the R.H.S. of equation (7.9) is exactly
equal to $-dJ_{u}(t)/dt$. That is to say, the almost concavity of the
${\mathcal{E}}$ functional along the ray $u_{t}$ is due to the positivity of
$(dd^{c}u)^{2}$.

In fact, the functionals $J_{u}(t)$ and $K(u_{t})$ are both non-negative, non-
decreasing functions along $t\in(-\infty,0)$. Then denote $\mathcal{K}$,
$\mathcal{J}$ by their primitives along $t$, i.e. we have

$\frac{d}{dt}\mathcal{K}(u_{t})=K(u_{t}),\ \ \mbox{and }\ \ \
\frac{d}{dt}\mathcal{J}(u_{t})=J_{u}(t),$

for all $t\in(-\infty,0)$. Then both functionals are non-decreasing, convex
functions along $t$. Therefore, we can rephrase Lemma (7.2) as follows.

######  Theorem 7.4.

Suppose $u_{t}$ is a $C^{2}$-continuous subgeodesic ray on the fiber bundle
$({\dagger})$. Then we have the equality

(7.10) $\mathcal{K}(u_{t})=\mathcal{J}(u_{t})-\mathcal{E}(u_{t})+A,$

for a constant $A$ and all $t\in(-\infty,0)$. In particular, the difference
$\mathcal{J}-{\mathcal{E}}$ is non-decreasing and convex along the ray.
Moreover, it is affine if and only if $u_{t}$ is a geodesic ray on this fiber
bundle.

###### Proof.

It is left to prove the statement about the geodesic. For a function
$u\in{\mathcal{F}}^{\infty}(B_{1})$, it is clear that $K(u_{t})=0,$ for all
$t\in(-\infty,0)$ if and only if $(dd^{c}u)^{2}=0$ on $B_{1}^{*}$. Then our
result follows. ∎

Recall that a primitive of the $I_{u}$-functional is defined as

$\mathcal{I}(u_{t})=\int_{{\mathbb{C}}{\mathbb{P}}^{1}}u_{t}\omega.$

It is also non-decreasing and convex along $t$, and the asymptote of
$\pi^{-1}\cdot{\mathcal{I}}$ as $t\rightarrow-\infty$ is exactly the Lelong
number of $u$ at the origin. Then the zero mass conjecture can be rephrased in
the energy setting, and it reveals the relation between the asymptotes of
these two functionals along a subgeodesic ray.

######  Theorem 7.5.

Suppose $u_{t}$ is a bounded subgeodesic ray on the fiber bundle
$({\dagger})$. Assume that we have

$\lim_{t\rightarrow-\infty}\frac{d{\mathcal{I}}(u_{t})}{dt}=0.$

Then it follows

$\lim_{t\rightarrow-\infty}\frac{d{\mathcal{K}}(u_{t})}{dt}=0.$

Similarly, for a general function $u\in{\mathcal{F}}(B_{1})$, the estimate on
the residual Monge-Ampère mass in Theorem (6.9) can also be interpreted under
the energy picture. It says that the asymptote of the $\mathcal{K}$-functional
can be controlled by the maximal directional Lelong number and the asymptote
of the ${\mathcal{I}}$-functional along the ray.

######  Remark 7.6.

It is possible to prove equation (7.10) along a bounded subgeodesic ray
$u_{t}$. In fact, the pluri-complex energy ${\mathcal{E}}$ can be rewritten as

${\mathcal{E}}(u_{t})=\int_{{\mathbb{C}}{\mathbb{P}}^{1}}d_{{\zeta}}u_{t}\wedge
d^{c}_{{\zeta}}u_{t}.$

This is well defined for almost all $t$ since $u$ is in the Sobolev space
$W^{1,2}_{loc}(B_{1})$, cf. [Blo04]. Then one can utilize the regularization
sequence $u_{\varepsilon}$ to obtain the convergences on both sides of
equation (7.10) as before.

## References