# On the Existence of Conic Sasaki-Einstein Metrics on Log Fano Sasakian
Manifolds of Dimension Five

∗Shu-Cheng Chang  ∗Mathematical Science Research Center, Chongqing University
of Technology, 400054, Chongqing, P.R. China<EMAIL_ADDRESS>,
†Fengjiang Li  †Mathematical Science Research Center, Chongqing University of
Technology, 400054, Chongqing, P.R. China<EMAIL_ADDRESS>,  ††Chien Lin
††Mathematical Science Research Center, Chongqing University of Technology,
400054, Chongqing, P.R. China<EMAIL_ADDRESS>and  ∗∗Chin-Tung Wu
∗∗Department of Applied Mathematics, National Pingtung University, Pingtung
90003, Taiwan<EMAIL_ADDRESS>
###### Abstract.

In this paper, we derive the uniform $L^{4}$-bound of the transverse conic
Ricci curvature along the conic Sasaki-Ricci flow on a compact transverse log
Fano Sasakian manifold $M$ of dimension five and the space of leaves of the
characteristic foliation is not well-formed. Then we first show that any
solution of the conic Sasaki-Ricci flow converges in the Cheeger-Gromov sense
to the unique singular orbifold conic Sasaki-Ricci soliton on
$M_{\mathrm{\infty}\text{ }}$which is a $\mathbb{S}^{1}$-orbibundle over the
unique singular conic Kähler-Ricci soliton on a log del Pezzo orbifold
surface. As a consequence, there exists a Kähler-Ricci soliton orbifold metric
on its leave space which is a log del Pezzo orbifold surface. Second, we show
that the conic Sasaki-Ricci soliton is the conic Sasaki-Einstein if $M$ is
transverse log $K$-polystable. In summary, we have the existence theorems of
orbifold Sasaki-Ricci solitons and Sasaki-Einstein metrics on a compact quasi-
regular Sasakian manifold of dimension five.

###### Key words and phrases:

Conic Sasaki-Ricci flow, Conic Sasaki-Ricci soliton, Log Fano Sasakian
manifold, Log del Pezzo orbifold surface, Log Sasaki-Mabuchi $K$-energy, Log
Sasaki-Donaldson-Futaki invariant, Transverse log $K$-polystable.

###### 1991 Mathematics Subject Classification:

Primary 53E50, 53C25; Secondary 53C12, 14E30.

∗Shu-Cheng Chang is partially supported by Funds of MSRC, CQUT 0625199005\.
∗∗Chin-Tung Wu is partially supported by NSTC 112-2115-M-153-001. †Fengjiang
Li is partially supported by NSFC 12301062, CQUT 2021ZDZ023 and CQEC
KJZD-K202201102. ††Chien Lin is partially supported by KJQN202201165 and
QN2022035003L

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
    1. 2.1 Sasakian Geometry
    2. 2.2 Log Pair with the KLT Foliation Singularities
    3. 2.3 The Twisted and Conic Sasaki-Ricci Flow
  3. 3 Twisted Sasaki-Ricci Flow on Transverse Log Fano Sasakian Manifolds
    1. 3.1 $L^{4}$-Bound of the Transverse Conic Ricci Curvature
    2. 3.2 Longtime Behavior under the Twisted Sasaki-Ricci Flow
    3. 3.3 Regularity of the Limit Space and Its Smooth Convergence
  4. 4 The Transverse Log K-Polystable
    1. 4.1 Log Sasaki–Donaldson-Futaki Invariant
    2. 4.2 Log Sasaki-Mabuchi $K$-energy
    3. 4.3 Transverse Log K-Polystable

##  1\. Introduction

Let $(M,\eta,\xi,\Phi,g)$ be a compact Sasakian manifold of dimension $2n+1.$
If the orbits of the Reeb vector field $\xi$ are all closed and hence circles,
then integrates to an isometric $U(1)$ action on $(M,g)$. Since it is nowhere
zero this action is locally free, that is, the isotropy group of every point
in $S$ is finite. If the $U(1$) action is in fact free then the Sasakian
structure is said to be regular. Otherwise, it is said to be quasi-regular. If
the orbits of $\xi$ are not all closed, the Sasakian structure is said to be
irregular. However, by the second structure theorem ([Ru], [BG]), any Sasakian
structure $(\xi,\eta,\Phi)$ on $(M,g)$ is either quasi-regular or there is a
sequence of quasi-regular Sasakian structures
$(M,\xi_{i},\eta_{i},\Phi_{i},g_{i})$ converging in the compact-open
$C^{\infty}$-topology to $(\xi,\eta,\Phi,g).$ It means that there always
exists a quasi-regular Sasakian structure $(\xi,\eta,\Phi)$ on $(M,g)$.

A Sasakian $(2n+1)$-manifold is served as the odd-dimensional analogous of
Kähler manifolds. For instance, the Kähler cone of a Sasaki-Einstein
$5$-manifold is a Calabi-Yau 3-fold. It provides interesting examples of the
AdS/CFT correspondence. On the other hand, the class of simply connected,
closed, oriented, smooth $5$-manifolds is classifiable under diffeomorphism
due to Smale-Barden [S], [B].

In a compact quasi-regular Sasakian manifold, the unit Reeb vector field
induces a $\mathbb{S}^{1}$-action which generates the finite isotropy groups.
It is the regular free action if the isotropy subgroup of every point is
trivial. In general, as in [CLW], the space of leaves has at least the
codimension two fixed point set of every non-trivial isotropy subgroup or the
codimension one fixed point set of some non-trivial isotropy subgroup.

In a series of previous articles, it is our goal to address the related issues
on the geometrization and classification problems of Sasakian manifolds of
dimension five of Type I foliation singularities ([CHLW], [CLW], [CCHLW]) in
the sense that the space of leaves has at least codimension two fixed point
set of every non-trivial isotropy subgroup.

In this article, we deal with the old dimensional counterpart of Yau-Tian-
Donaldson conjecture ([CDS1], [CDS2], [CDS3], [T5]) for conic Sasaki-Einstein
metrics on a compact Sasakian manifolds of dimension five of Type II foliation
singularities in the sense that the space of leaves admits the codimension one
fixed point set of some non-trivial isotropy subgroup. It can be viewed as a
Sasaki analogue of Liu-Zhang [LZ] for the conic Kähler-Ricci flow (1.1).

We recall the starting issue from our previous paper. We assume that $M$ is a
compact quasi-regular transverse Fano $5$-Sasakian manifold and the space $Z$
of leaves is a del Pezzo orbifold surface of cyclic quotient singularities
(well-formed) which means its orbifold singular locus and algebro-geometric
singular locus coincide, equivalently $Z$ has no branch divisors. Then we have

######  Proposition 1.1.

([CCHLW]) Let $(M,\xi,\eta_{0},g_{0})$ be a compact quasi-regular transverse
Fano Sasakian manifold of dimension five and
$(Z_{0}=M/\mathcal{F}_{\xi},h_{0},\omega_{h_{0}})$ denote the space of leaves
of the characteristic foliation which is a del Pezzo orbifold surface of
cyclic quotient klt singularities with codimension two orbifold singularities
$\Sigma_{0}$. Then, by a $D$-homothetic deformation, under the Sasaki-Ricci
flow

$\left\\{\begin{array}[c]{lcl}\frac{\partial}{\partial
t}\omega(t)&=&-Ric^{T}(\omega(t))+\omega(t),\\\
\omega(0)&=&\omega_{0}.\end{array}\right.$

$(M(t),\xi,\eta(t),g(t))$ converges to a compact quasi-regular transverse Fano
Sasaki-Ricci soliton orbifold $(M_{\infty},\xi,\eta_{\infty},g_{\infty})$ with
the leave space of a del Pezzo orbifold surface
$(Z_{\infty}=M_{\infty}/\mathcal{F}_{\xi},h_{\infty})$ which can have at worst
codimension two klt orbifold singularities $\Sigma_{\infty}.$ Furthermore,
$g^{T}(t_{i})$ converges to a Sasaki-Ricci soliton orbifold metric
$g_{\infty}^{T}$ on $M_{\infty}$ with $g_{\infty}^{T}=\pi^{\ast}(h_{\infty})$
such that $h_{\infty}$ is the smooth gradient Kähler-Ricci soliton metric in
the Cheeger-Gromov topology on $Z_{\infty}\backslash\Sigma_{\infty}.$ In
additional, if $M$ is transverse $K$-stable, then $M(t)$ converges to a
compact transverse Fano Sasakian manifold $M_{\infty}$ which is isomorphic to
$M$ endowed with a smooth Sasaki-Einstein metric.

Now we focus on a compact quasi-regular Sasakian $5$-manifold of the Type II
foliation singularities. It means that its leave space $Z$ has the orbifold
structure $\mathcal{Z=}(Z,\Delta)$ with the codimension one fixed point set of
some non-trivial isotropy subgroup with branch divisors $\Delta$. Then, Up to
conjugation, a representation $\mu_{p}$ of $\mathbb{Z}_{r}$ on
$\mathbb{C}^{2}$ is conjugate to the $\mathbb{S}^{1}$-action

$\begin{array}[c]{c}\mu_{\mathbb{Z}_{r}}:(z_{1},z_{2})\rightarrow(e^{2\pi\sqrt{-1}\frac{a_{1}}{r_{1}}}z_{1},e^{2\pi\sqrt{-1}\frac{a_{2}}{r_{2}}}z_{2})\end{array}$

for some positive integers $r_{1,}$ $r_{2}$ whose least common multiplier is
$r$, and $a_{i},$ $i=1,$ $2$ are integers coprime to $r_{i},$ $i=1,$ $2$. By
the first structure theory as in Proposition 2.1, the foliation singular set
contains some $3$-dimensional Sasakian submanifolds of $M$, the corresponding
foliation singularities $\Delta^{T}$ in $(M,\Delta^{T})$ is the Hopf
$\mathbb{S}^{1}$-orbibundle over a Riemann surface $\Sigma_{h}.$ We refer to
section $2$ in some detail ([CLW]).

More precisely, suppose that $(M,\xi,\eta,g)$ is a compact quasi-regular
Sasakian $5$-manifold. Then, by the first structure theorem, its leave space
of the characteristic foliation is a log del Pezzo orbifold surface
$(Z,\Delta)$ with klt singularities and

$\begin{array}[c]{c}\Delta=\sum_{\alpha=1}^{k}(1-\frac{1}{m_{\alpha}})\Delta_{\alpha},\end{array}$

where $m_{\alpha}$ is called the ramification index of $\Delta_{\alpha}$.
Moreover, for its submersion

$\pi:(M,\Delta^{T},g,\omega)\rightarrow(Z,\Delta,h,\omega_{h})$

with the transverse Kähler structure
$\frac{1}{2}d\eta=\pi^{\ast}(\omega_{h})=\omega$ and the transverse divisor
([CHLW]) $\Delta_{\alpha}^{T}$ such that $\Delta^{T}=\pi_{\ast}^{-1}(\Delta)$
with

$\begin{array}[c]{c}\Delta^{T}=\sum_{\alpha=1}^{k}(1-\frac{1}{m_{\alpha}})\Delta_{\alpha}^{T}\end{array}$

and the canonical (determinant) bundle $K_{M}^{T}$ of $T^{1,0}(M)$ which is a
basic CR holomorphic line bundle ([CLW])

$K_{M}^{T}=\pi^{\ast}(K_{Z}^{orb})=\pi^{\ast}(K_{Z}+\Delta).$

However, it follows from Remark 2.1 that if there is a Sasaki-Einstein metric
$\omega$ on a log Fano quasi-regular Sasakian manifold $(M,\Delta^{T})$ with
klt foliation singularities, then $\widetilde{\phi}^{\ast}\omega$ will be a
so-called conic Sasaki-Einstein metric on a log Fano regular Sasakian manifold
$\widetilde{M}$ with the transverse cone angle $2\pi(1-\beta)$ along $D^{T}$
for a log resolution
$\widetilde{\phi}:(\widetilde{M},D^{T})\rightarrow(M,\Delta^{T}).$ Therefore
we can turn around to use this to construct a Sasaki-Einstein metric on a log
Fano quasi-regular Sasakian manifold $(M,\Delta^{T})$.

In view of the observation above, instead we shall work on the conic Sasaki-
Ricci flow

(1.1) $\left\\{\begin{array}[c]{lcl}\frac{\partial}{\partial
t}\omega(t)&=&-Ric^{T}(\omega(t))+\beta\omega(t)+(1-\beta)[D^{T}]\\\
\omega(0)&=&\omega_{\ast}\end{array}\right.$

in a compact regular transverse Fano Sasakian manifold
$(M,\xi,\eta_{0},\Phi_{0},g_{0},\omega_{0})$ of dimension five (subsection
$2.1$ and $2.2$). More precisely, it follows from Theorem 3.2, Theorem 3.3,
and Corollary 3.3 that we first derive the existence theorem of the gradient
conic Sasaki-Ricci soliton orbifold metric:

######  Theorem 1.1.

Let $(M,\xi,\eta_{0},g_{0},(1-\beta)D^{T})$ be a compact transverse log Fano
regular Sasakian manifold of dimension five and
$(Z_{0}=M/\mathcal{F}_{\xi},h_{0},\omega_{h_{0}},(1-\beta)D)$ be the space of
leaves of the characteristic foliation which is a log del Pezzo surface with
the klt singularities. Then the solution
$(M(t),\xi,\eta(t),g(t),(1-\beta)D^{T})$ of the conic Sasaki-Ricci flow
converges smoothly to a compact transverse log Fano quasi-regular Sasakian
orbifold manifold $(M_{\infty},\xi,\eta_{\infty},g_{\infty},(1-\beta)D^{T})$
with the leave space of log del Pezzo orbifold surface
$(Z_{\infty}=M_{\infty}/\mathcal{F}_{\xi},h_{\infty},(1-\beta)D)$ outside
$\Sigma_{\infty}\cup(1-\beta)D$, where the singularities $\Sigma_{\infty}$ is
at worst of the codimension two orbifold singularities $\Sigma_{\infty}$ in
$Z_{\infty}.$ Furthermore, $g^{T}(t_{i})$ converges smoothly to a gradient
conic Sasaki-Ricci soliton orbifold metric $g_{\infty}^{T}$ on $M_{\infty}$
with $g_{\infty}^{T}=\pi^{\ast}(h_{\infty})$ such that $h_{\infty}$ is the
smooth conic Kähler-Ricci soliton metric in the Cheeger-Gromov topology on
$Z_{\infty}\backslash(\Sigma_{\infty}\cup(1-\beta)D).$

All these results confirm the Hamilton-Tian conjecture that $(M,g(t))$
converges (along a subsequence) to a shrinking Sasaki-Ricci soliton with mild
singularities on a compact transverse Fano Sasakian manifold of dimension
five.

Second, we show that it is indeed a conic Sasaki-Einstein metric if
$(M,(1-\beta)D^{T})$ is transverse log $K$-polystable (Definition 4.5). This
is an old dimensional counterpart of Yau-Tian-Donaldson conjecture on a
compact log $K$-polystable Kähler manifold ([CDS1], [CDS2], [CDS3], [T5]). It
can be viewed as a parabolic Sasaki analogue of Tian-Wang [TW1] and a
generalized theorem of Chen-Sun-Wang [CSW], Liu-Zhang [LZ] and Collins-Jacob
[CJ].

######  Theorem 1.2.

Let $(M,\xi,\eta_{0},g_{0},(1-\beta)D^{T})$ be a compact transverse log Fano
regular Sasakian manifold of dimension five and
$(Z_{0}=M/\mathcal{F}_{\xi},h_{0},\omega_{h_{0}},(1-\beta)D)$ be the space of
leaves of the characteristic foliation which is a log del Pezzo surface. If
$(M,(1-\beta)D^{T})$ is transverse log $K$-polystable, then there exists a
conic Sasaki-Einstein metric of $(M,(1-\beta)D^{T})$.

It follows from Remark 2.1 and Theorem 1.2 that we have the existence theorem
of Sasaki-Ricci soliton and Sasaki-Einstein metric on a compact transverse log
Fano quasi-regular Sasakian manifold $(M,\Delta^{T})$ of dimension five.

######  Corollary 1.1.

Let $(M,\Delta^{T})$ be a compact quasi-regular transverse log Fano Sasakian
manifold of dimension five and $(Z,\Delta)$ be the space of leaves of the
characteristic foliation which is not well-formed and the orbifold structure
has the codimension one fixed point set of some non-trivial isotropy subgroup.
Then

  1. (1)

There always exists a Sasaki-Ricci soliton orbifold metric on
$(M,\Delta^{T})$.

  2. (2)

In additional, suppose that $(\widetilde{M},(1-\beta)\widetilde{D}^{T})$ is
transverse log $K$-polystable for a log resolution
$\widetilde{\phi}:(\widetilde{M},D^{T})\rightarrow(M,\Delta^{T})$. Then there
exists a Sasaki-Einstein metric on $(M,\Delta^{T}).$

Finally, it is proved by Wang-Zhu [WZ] that all smooth del Pezzo surfaces
admit a Kähler-Ricci soliton metric and Shi-Zhu [SZ] extended to tori Fano
orbifolds. However, as a consequence of Proposition 1.1, Corollary 1.1 and
Guo-Phong-Song-Sturm [GPSS], we have

######  Corollary 1.2.

Let $(M,\Delta^{T})$ be a compact quasi-regular transverse log Fano Sasakian
manifold of dimension five and $(Z,\Delta)$ be the space of leaves of the
characteristic foliation which is a log del Pezzo orbifold surface. Then there
exists a Kähler-Ricci soliton orbifold metric on $(Z,\Delta)$.

It follow from Proposition 1.1 and Corollary 1.2 that

######  Corollary 1.3.

There exists a Sasaki-Ricci soliton orbifold metric on any compact quasi-
regular transverse Fano or log Fano Sasakian manifold of dimension five.

In our upcoming project ([CHT]) for any compact irregular Fano Sasakian
manifold of dimension five, by the second structure theorem ([Ru], [BG]), we
will focus on the compactness properties of a sequence of orbifold Sasaki-
Ricci solitons on a compact quasi-regular transverse Fano Sasakian
$5$-manifold and its leave space is normal projective Fano variety with
Kawamata log terminal (klt) singularities. More precisely, we are expected to
derive the Sasaki analogue of compactness theorems of compact Kähler-Ricci
solitons ([GPSS]) by applying algebraic and analytic Invariants such as the
$\alpha$-invariant due to Tian [T3], the log canonical threshold due to
Cheltsov-Shramov and Demailly [CheSh] and a uniform positive lower bound of
the log canonical threshold due to Birkar [Bir].

######  Remark 1.1.

  1. (1)

It follows from Proposition 1.1 and Corollary 1.1 that we have the existence
theorem of Sasaki-Einstein metrics on a compact transverse Fano quasi-regular
Sasakian manifold of dimension five. Moreover, there are quasi-regular Sasaki-
Einstein metrics on connected sums of$\ \mathbb{S}^{2}\times\mathbb{S}^{3},$
rational homology $5$-spheres and connected sums of these. We refer to [BG],
[K2], [K3], [Sp] and references therein.

  2. (2)

In fact, most $5$-manifolds do not admit a positive Sasakian structure and
there is a classification of (quasi-regular) Sasaki-Einstein structures on a
rational homology $5$-sphere which are quasi-regular only. We refer to [BG],
[K3], [Su], [PW] and references therein.

  3. (3)

The key ingredients for our proofs in this article remain hold for a quasi-
regular Sasakian manifold of dimension up to seven with its orbifold leave
space due to the $L^{4}$-Bound of the transverse conic Ricci curvature.

  4. (4)

The same results hold for the conic Kähler-Ricci flow on normal log pair of
Kähler orbifolds of dimension up to three.

  5. (5)

In general, there is a orbifold stability in a compact quasi-regular Sasakian
manifold in the sense of Ross-Thomas [RT]. Instead, Collins and Székelyhidi
showed that the Sasakian manifold admits a Sasaki-Einstein metric if and only
if its Kähler cone is $K$-stable by the continuity method ([CSz2]) which
corresponds to the Type I foliation singularities ([CCHLW]). Along this
direction, Boyer and van Coevering [BvC] deal with the Sasakian extremal
metrics.

The crucial steps are followings. First, we are able to refine the Sasaki
analogue of a log pair quasi-regular Sasakian manifold $(M,\Delta^{T})$ with
klt foliation singularities. Then, for a Sasaki resolution
$\widetilde{\phi}:(\widetilde{M},D^{T})\rightarrow(M,\Delta^{T}),$ we can
construct a Sasaki-Einstein metric on such a log Fano quasi-regular Sasakian
manifold $M$ with the foliation singularities $\Delta^{T}$ by finding the
conic Sasaki-Einstein metric on a log Fano regular Sasakian manifold
$\widetilde{M}$ with the transverse cone angle $2\pi(1-\beta)$ along $D^{T}$.
We refer to Remark 2.1 in some detail.

Second, we study the structure and regularity theory of desired limit spaces
by applying $L^{p}$ version of the Cheeger-Colding-Tian structure theory
([CCT], [DS], [TW1], [CHT]), Then, based on Perelman’s uniform noncollapsing
condition and pseudolocality theorem of the Ricci flow, the limit solution
must be a gradient conic Sasaki-Ricci soliton orbifold metric $g_{\infty}^{T}$
on $M_{\infty}$. This is a Sasaki analogue of the partial $C^{0}$-estimate
([TZ, Theorem 5.1]) due to the first structure theorem on quasi-regular
Sasakian manifolds.

Finally, we are able to adapt the notions as in [D2] and [Li] in our Sasakian
setting for the log Sasaki-Donaldson-Futaki invariant and log Sasaki-Mabuchi
$K$-energy. Then we can have the Sasaki analogue of log $K$-polystable on
Sasakian manifolds which is served as the different view point from Collins-
Székelyhidi [CSz2] and Ross-Thomas [RT].

In section $2,$ we recall some preliminaries for Sasakian manifolds and
Sasakian structures. Then define a log pair $(M,\Delta^{T})$ with klt
foliation singularities. Based on this, we introduce the twisted and conic
Sasaki-Ricci flow. In section $3,$ we first prove the $L^{4}$-boundedness of
the transverse Ricci curvature along the twisted Sasaki-Ricci flows. Then, by
$L^{p}$ version of the Cheeger-Colding-Tian structure theory, to study the
structure of desired limit spaces. Finally, by a partial $C^{0}$-estimate, one
can refine the regularity of the desired limit space due to Donaldson [DS] and
Tian-Wang [TW1]. In the final section, we prove the main theorems. More
precisely, we derive all Sasaki analogues of log Donaldson-Futaki invariant,
log Mabuchi $K$-energy and Perelman’s $\mathcal{W}$-functional. We are able to
obtain a shrinking Sasaki-Ricci soliton with mild singularities and show that
it is indeed a conic Sasaki-Einstein metric if it is transverse log
$K$-polystable.

Acknowledgements. Part of the project was done during the first named author
visiting to Department of Mathematics under the supported by iCAG program of
Higher Education Sprout Project of National Taiwan Normal University and the
Ministry of Education (MOE) in Taiwan. He would like to express his gratitude
for the warm hospitality there.

##  2\. Preliminaries

In this section, we will recall some preliminaries for Sasakian manifolds and
Sasakian structures. Then define a log pair $(M,\Delta^{T})$ with klt
foliation singularities and introduce the twisted and conic Sasaki-Ricci flow.
We refer to [BG], [FOW], [Sp], [CLW], and references therein in some detail.

###  2.1. Sasakian Geometry

We review some basic facts from Sasakian geometry ([BG], [FOW], [CHLW]). Let
$(M,g)$ be a Riemannian $(2n+1)$-manifold. $(M,g)$ is called Sasaki if the
cone

$(C(M),\overline{g},J):=(\mathbb{R}^{+}\times M\mathbf{,}dr^{2}+r^{2}g)$

is Kähler. In that case $\\{r=1\\}=\\{1\\}\times M\subset C(M)$. Define the
Reeb vector field

$\begin{array}[c]{c}\xi=J(\frac{\partial}{\partial r})\end{array}$

and the contact $1$-form

$\eta(Y)=g(\xi,Y).$

Then $\eta(\xi)=1$ and $d\eta(\xi,X)=0.$ $\xi$ is a killing vector field with
unit length.

There are equivalent statements that there exists a Killing vector field $\xi$
of unit length on $M$ so that the tensor field of type $(1,1)$, defined by

$\Phi(Y)=\nabla_{Y}\xi,$

satisfies the condition

$(\nabla_{X}\Phi)(Y)=g(\xi,Y)X-g(X,Y)\xi$

for any pair of vector fields $X$ and $Y$ on $M$. Furthermore, the Riemann
curvature of $(M,g)$ satisfies the condition

(2.1) $R(X,\xi)Y=g(\xi,Y)X-g(X,Y)\xi.$

Now there is a natural splitting

$\begin{array}[c]{c}TC(M)=L_{r\frac{\partial}{\partial r}}\oplus L_{\xi}\oplus
H\end{array}$

and the relation

$\begin{array}[c]{c}JY=\Phi(Y)-\eta(Y)r\frac{\partial}{\partial
r}.\end{array}$

Then

$\Phi^{2}=-I+\eta\otimes\xi$

and

$g(\Phi X,\Phi Y)=g(X,Y)-\eta(X)\eta(Y)\text{ {which is}
}g=g^{T}+\eta\otimes\eta.$

Then $(M,\xi,\eta,g,\Phi)$ is said to be a Sasakian manifold $(M,g)$ with the
Sasakian structure $(\xi,\eta,g,\Phi)$. Let
$\\{U_{\alpha}\\}_{\alpha\in\Lambda}$ be an open covering of the Sasakian
manifold $(M,\xi,\eta,g,\Phi)$ and

$\pi_{\alpha}:U_{\alpha}\rightarrow V_{\alpha}\subset\mathbb{C}^{n}$

submersion such that
$\pi_{\alpha}\circ\pi_{\beta}^{-1}:\pi_{\beta}(U_{\alpha}\cap
U_{\beta})\rightarrow\pi_{\alpha}(U_{\alpha}\cap U_{\beta})$ is biholomorphic.
On each $V_{\alpha},$ there is a canonical isomorphism

$d\pi_{\alpha}:D_{p}\rightarrow T_{\pi_{\alpha}(p)}V_{\alpha}$

for any $p\in U_{\alpha},$ where the contact subbundle $D=\ker\eta\subset TM.$
$d\eta$ is the Levi form on $D.$ It is a global two form on $M$.

Since $\xi$ generates isometries, the restriction of the Sasakian metric $g$
to $D$ gives a well-defined Hermitian metric $g_{\alpha}^{T}$ on $V_{\alpha}$.
In fact it is Kähler. The Kähler $2$-form $\omega_{\alpha}^{T}$ of the
Hermitian metric $g_{\alpha}^{T}$ on $V_{\alpha}$ is the same as the
restriction of the Levi form $\frac{1}{2}d\eta$ to
$\widetilde{D_{\alpha}^{n}}$, the slice $\\{x=$ constant$\\}$ in $U_{\alpha}.$
The collection of Kähler metrics $\\{g_{\alpha}^{T}\\}$ on $\\{V_{\alpha}\\}$
is so-called a transverse Kähler metric. We often refer to $\frac{1}{2}d\eta$
as the Kähler form of the transverse Kähler metric $g^{T}$ in the leaf space
$\widetilde{D^{n}}.$

We denote $\nabla^{T},$ $\mathrm{Rm}^{T},$ $\mathrm{Ric}^{T}$ and
$\mathrm{R}^{T}$ for its Levi-Civita connection, the curvature, the Ricci
tensor and the scalar curvature with respect to $g^{T}$. For $\widetilde{X},$
$\widetilde{Y},$ $\widetilde{W},$ $\widetilde{Z}\in\Gamma(TD)$ and the
$d\pi_{\alpha}$-corresponding $X,$ $Y,$ $W,$ $Z\in\Gamma(TV_{\alpha}),$ we
have the following:

$\begin{array}[c]{c}\nabla_{X}^{T}Y=d\pi_{\alpha}(\nabla_{\widetilde{X}}\widetilde{Y}),\text{
}\widetilde{\nabla_{X}^{T}Y}=\nabla_{\widetilde{X}}\widetilde{Y}+g(JX,Y)\xi,\end{array}$
(2.2)
$\begin{array}[c]{ccl}\mathrm{Rm}^{T}(X,Y,Z,W)&=&\mathrm{Rm}(\widetilde{X},\widetilde{Y},\widetilde{Z},\widetilde{W})+2g(J\widetilde{X},\widetilde{Y})g(J\widetilde{Z},\widetilde{W})\\\
&&+g(J\widetilde{X},\widetilde{Z})g(J\widetilde{Y},\widetilde{W})-g(J\widetilde{X},\widetilde{W})g(J\widetilde{Y},\widetilde{Z})\end{array}$

and

(2.3)
$\mathrm{Ric}^{T}(X,Z)=\mathrm{Ric}(\widetilde{X},\widetilde{Z})+2g(\widetilde{X},\widetilde{Z}).$

Furthermore, the Sasaki metric can be written as

$g=g^{T}\oplus\eta\otimes\eta$

and transversal Ricci curvature $\mathrm{Ric}^{T}$ is

$\mathrm{Ric}^{T}=\mathrm{Ric}+2g^{T}.$

In terms of the foliation normal coordinate which are simultaneously foliated
and Riemann normal coordinates ([GKN], [CHLW]) along the leaves, we have a
coordinate $\\{x,z^{1},z^{2},\cdots,z^{n}\\}$ on a neighborhood $U$ of $p$,
such that $x$ is the coordinate along the leaves with
$\xi=\frac{\partial}{\partial x}$ on $U$. Moreover,
$\\{z^{1},z^{2},\cdots,z^{n}\\}$ is the local holomorphic coordinates on
$V_{\alpha}$. We pull back these to $U_{\alpha}$ and still write the same.
Then we have the foliation local coordinate $\\{x,z^{1},z^{2},\cdots,z^{n}\\}$
on $U_{\alpha}\ $and $(D\otimes\mathbb{C})$ is spanned by the fields
$Z_{j}=\frac{\partial}{\partial z^{j}}+\sqrt{-1}h_{j}\frac{\partial}{\partial
x},\ j\in\left\\{1,2,\cdots,n\right\\}$ with

$\begin{array}[c]{c}\eta=dx-\sqrt{-1}h_{j}dz^{j}+\sqrt{-1}h_{\overline{j}}d\overline{z}^{j}\end{array}$

and its dual frame

$\\{\eta,dz^{j},\ j=1,2,\cdots,n\\}.$

Here $h$ is a basic function such that $\frac{\partial h}{\partial x}=0$ and
$h_{j}=\frac{\partial h}{\partial z^{j}},$
$h_{j\overline{l}}=\frac{\partial^{2}h}{\partial
z^{j}\partial\overline{z}^{l}}$ with the foliation normal coordinate

(2.4) $h_{j}(p)=0,\text{ }h_{j\overline{l}}(p)=\delta_{j}^{l},\text{
}dh_{j\overline{l}}(p)=0.$

Now in terms of the normal coordinate, we have

$\begin{array}[c]{c}g^{T}=g_{i\overline{j}}^{T}dz^{i}d\overline{z}^{j}\end{array}$

and

$\begin{array}[c]{c}\omega=2\sqrt{-1}h_{i\overline{j}}dz^{i}\wedge
d\overline{z}^{j}.\end{array}$

Here $g_{i\overline{j}}^{T}=g^{T}(\frac{\partial}{\partial
z^{i}},\frac{\partial}{\partial\overline{z}^{j}})=d\eta(\frac{\partial}{\partial
z^{i}},\Phi\frac{\partial}{\partial\overline{z}^{j}})=2h_{i\overline{j}}.$
Furthermore,

(2.5)
$\begin{array}[c]{c}\mathrm{R}_{i\overline{j}}^{T}=-\frac{\partial^{2}}{\partial
z^{i}\partial\overline{z}^{j}}\log\det(g_{\alpha\overline{\beta}}^{T})\end{array}$

and the transversal Ricci form $\rho^{T}$

$\begin{array}[c]{c}\rho^{T}=\mathrm{Ric}^{T}(J\cdot,\cdot)=-\sqrt{-1}\mathrm{R}_{i\overline{j}}^{T}dz^{i}\wedge
d\overline{z}^{j}.\end{array}$

Finally, we define the basic form $\gamma$ in Sasakian manifold so that

$i(\xi)\gamma=0\text{\ {and} }\mathcal{L}_{\xi}\gamma=0.$

In this case the closure of the $1$-parameter subgroup of the isometry group
of $(M,g)$ is isomorphic to a torus $\mathbb{T}^{k}$, for some positive
integer $k$ called the rank of the Sasakian structure. In particular,
irregular Sasakian manifolds have at least a $\mathbb{T}^{2}$ isometry.

The first structure theory of Sasakian manifolds reads as

######  Proposition 2.1.

([Ru], [Sp], [BG]) Let $(M,\eta,\xi,\Phi,g)$ be a compact quasi-regular
Sasakian manifold of dimension $2n+1$ and $Z$ denote the space of leaves of
the characteristic foliation $\mathcal{F}_{\xi}$ (just as topological space).
Then

  1. (1)

$Z$ carries the structure of a Hodge orbifold $\mathcal{Z=}(Z,\Delta)$ with an
orbifold Kähler metric $h$ and Kähler form $\omega$ which defines an integral
class in $H_{orb}^{2}(Z,\mathbb{Z}\mathbf{)}$ in such a way that
$\pi:(M,g,\omega)\rightarrow(Z,h,\omega_{h})$ is an orbifold Riemannian
submersion, and a principal $\mathbb{S}^{1}$-orbibundle ($V$-bundle) over $Z.$
Furthermore,it satisfies $\frac{1}{2}d\eta=\pi^{\ast}(\omega_{h}).$ The fibers
of $\pi$ are geodesics.

  2. (2)

$Z$ is also a $\mathbb{Q}$-factorial, polarized, normal projective algebraic
variety.

  3. (3)

The orbifold $Z$ is Fano if and only if $\mathrm{Ric}_{g}>-2$. In this case
$Z$ as a topological space is simply connected; and as an algebraic variety is
uniruled with Kodaira dimension $-\infty$.

  4. (4)

$(M,\xi,g)$ is Sasaki-Einstein if and only if $(Z,h)$ is Kähler-Einstein with
scalar curvature $4n(n+1).$

  5. (5)

If $(M,\eta,\xi,\Phi,g)$ is regular then the orbifold structure is trivial and
$\pi$ is a principal circle bundle over a smooth projective algebraic variety.

  6. (6)

As real cohomology classes, there is a relation between the first basic Chern
class and the first orbifold Chern class

$c_{1}^{B}(M):=c_{1}(\emph{F}_{\xi})=\pi^{\ast}c_{1}^{orb}(Z).$

  7. (7)

Conversely, let $\pi:M\rightarrow Z$ be a $\mathbb{S}^{1}$-orbibundle over a
compact Hodge orbifold $(Z,h)$ whose first Chern class is an integral class
defined by $[\omega_{Z}]$, and $\eta$ be a $1$-form with
$\frac{1}{2}d\eta=\pi^{\ast}\omega_{Z}$. Then
$(M,\pi^{\ast}h+\eta\otimes\eta)$ is a Sasakian manifold if all the local
uniformizing groups inject into the structure group $U(1).$

###  2.2. Log Pair with the KLT Foliation Singularities

In this subsection, we follow the notations as in [K1], [TW1], [CLW] and
references therein. Let $(M,\xi,\alpha_{\xi})$ be a compact quasi-regular
Sasakian $5$-manifold and the leave space $Z:=M/\mathbb{S}^{1}$ be an orbifold
Kähler surface. Denote by the quotient map

$\pi:M\rightarrow Z$

as before, and call such a $5$-manifold $(M,\xi)$ an
$\mathbb{S}^{1}$-orbibundle. More precisely, $M$ admits a locally free,
effective $\mathbb{S}^{1}$-action

$\begin{array}[c]{c}\alpha_{\xi}:\mathbb{S}^{1}\times M\rightarrow
M\end{array}$

such that $\alpha_{\xi}(t)$ is orientation-preserving, for every
$t\in\mathbb{S}^{1}$. Since $M$ is compact, $\alpha_{\xi}$ is proper and the
isotropy group $\Gamma_{p}$ of every point $p\in M$ is finite.

######  Definition 2.1.

The principal orbit type $M_{\operatorname{reg}}$ corresponds to points in
$(M,\xi,\alpha_{\xi})$ with the trivial isotropy group and
$M_{\operatorname{reg}}\rightarrow M_{\operatorname{reg}}/\mathbb{S}^{1}$ is a
principle $\mathbb{S}^{1}$-bundle. Furthermore, the orbit $\mathbb{S}_{p}^{1}$
of a point $p\in M$ is called a regular fiber if $p\in
M_{\operatorname{reg}}$, and a singular fiber otherwise. In this case,
$M_{\mathrm{sing}}/\mathbb{S}^{1}\simeq\Sigma^{orb}(Z).$

######  Proposition 2.2.

([CLW]) Let $(M,\eta,\xi,\Phi,g)$ be a compact quasi-regular Sasakian
$5$-manifold and $Z$ be its leave space of the characteristic foliation. Then
$Z$ is a $\mathbb{Q}$-factorial normal projective algebraic orbifold surface
satisfying

  1. (1)

if its leave space $(Z,\emptyset)$ has at least codimension two fixed point
set of every non-trivial isotropy subgroup. That is to say $Z$ is well-formed,
then $Z$ has isolated singularities of a finite cyclic quotient of
$\mathbb{C}^{2}$ and the action is

$\begin{array}[c]{c}\mu_{\mathbb{Z}_{r}}:(z_{1},z_{2})\rightarrow(\zeta^{a}z_{1},\zeta^{b}z_{2}),\end{array}$

where $\zeta$ is a primitive $r$-th root of unity. We denote the cyclic
quotient singularity by $\frac{1}{r}(a,b)$ with $(a,r)=1=(b,r)$. In
particular, the action can be rescaled so that every cyclic quotient
singularity corresponds to a $\frac{1}{r}(1,a)$-type singularity with
$(r,a)=1,$ $\zeta=e^{2\pi\sqrt{-1}\frac{1}{r}}$. In particular, it is klt
(Kawamata log terminal) singularities. Moreover, the corresponding
singularities in $(M,\eta,\xi,\Phi,g)$ is called foliation cyclic quotient
singularities of type

$\begin{array}[c]{c}\frac{1}{r}(1,a)\end{array}$

at a singular fibre $\mathbb{S}_{p}^{1}$ in $M$.

  2. (2)

if its leave space $(Z,\Delta)$ has the codimension one fixed point set of
some non-trivial isotropy subgroup. Then the action is

$\begin{array}[c]{c}\mu_{\mathbb{Z}_{r}}:(z_{1},z_{2})\rightarrow(e^{2\pi\sqrt{-1}\frac{a_{1}}{r_{1}}}z_{1},e^{2\pi\sqrt{-1}\frac{a_{2}}{r_{2}}}z_{2}),\end{array}$

for some positive integers $r_{1,}$ $r_{2}$ whose least common multiplier is
$r$, and $a_{i},$ $i=1,$ $2$ are integers coprime to $r_{i},$ $i=1,$ $2$. Then
the foliation singular set contains some $3$-dimensional submanifolds of $M.$
More precisely, the corresponding singularities in $(M,\eta,\xi,\Phi,g)$ is
called the Hopf $\mathbb{S}^{1}$-orbibundle over a Riemann surface
$\Sigma_{h}.$

Next we recall that a log pair $(Z,\Delta)$ is consisting of a connected
compact projective normal variety $Z$ and an effective $\mathbb{Q}$-divisor
$\Delta$ such that $(K_{Z}+\Delta)$ is $\mathbb{Q}$-Cartier. Along this
direction, we define a log pair $(M,\Delta^{T})$ with klt foliation
singularities.

######  Definition 2.2.

Let $(Z,\Delta)$ be a log pair with klt singularities and a log minimal
resolution $\phi:(Y,D_{Y})\rightarrow(Z,\Delta)$.

  1. (1)

It means that there exists a unique $\mathbb{Q}$-divisor
$D_{Y}=\sum_{j=1}b_{j}D_{j}$ with $b_{j}>-1$ such that

$K_{Y}=\phi^{\ast}(K_{Z}+\Delta)+D_{Y},\text{ }D_{Y}=E+\Delta_{Y}^{\prime},$

where the union of the $\phi$-exceptional divisor $E$ and the strict transform
$\Delta_{Y}^{\prime}$ of $\Delta$ is a simple normal crossing divisor defined
by

$\begin{array}[c]{c}E:=\sum_{-1<b_{j}\leq
0}b_{j}D_{j}=\phi^{-1}(Z_{\text{{sing}}})\text{\ }\end{array}$

and

$\begin{array}[c]{c}\Delta_{Y}^{\prime}:=\phi_{\ast}^{-1}\Delta=\sum_{b_{j}>0}b_{j}D_{j}.\end{array}$

  2. (2)

A conic Kähler metric $\omega$ on a compact Kähler $n$-manifold $Z$ along the
irreducible divisor $D$ with cone angle $2\pi\beta,$ $0<\beta<1$ is
asymptotically equivalent to the model metric on $\mathbb{C}^{n}$

$\begin{array}[c]{c}\omega_{\beta}=\sqrt{-1}\frac{dz_{1}\wedge
d\overline{z}_{1}}{|z_{1}|^{2(1-\beta)}}+\sum_{j=2}^{n}dz_{j}\wedge
d\overline{z}_{j}.\end{array}$

It is called conic Kähler-Einstein

$\mathrm{Ric}(\omega):=\beta\omega(t)+(1-\beta)[D]$

in the sense of currents with $D\thicksim-|K_{M}|$. That is, locally near a
point $p\in Z,$ there is a holomorphic coordinate system
$(V,z_{1},\cdots,z_{n})$ such that $D\cap V=\\{z\in V;$ $z_{1}=0\\}.$

Now given a log minimal resolution $\phi:(Y,D_{Y})\rightarrow(Z,\Delta)$ as
above. Since $Y$ is nonsingular, by the first Sasakian structure theorem
again, there exists a submersion
$\widetilde{\pi}:(\widetilde{M},D^{T})\rightarrow(Y,D_{Y})$ with the regular
Sasakian manifold $(\widetilde{M},D^{T}).$ Then we define the so-called log
pair with the klt foliation singularities as follows:

######  Definition 2.3.

Let $(M,\xi,\eta,g)$ be a compact quasi-regular Sasakian manifold with the
submersion $\pi:(M,\Delta^{T})\rightarrow(Z,\Delta)$. Given $\phi,$ $\pi$ and
$\widetilde{\pi}$ as above, the transverse log resolution is defined by
$\widetilde{\phi}:(\widetilde{M},D^{T})\rightarrow(M,\Delta^{T})$ so that it
is basic and the following diagram

$\begin{array}[c]{ccc}(\widetilde{M},\widetilde{\overline{E}}^{T},(\Delta_{Y}^{\prime})^{T},D_{\widetilde{M}}^{T})&\overset{\widetilde{\phi}}{\longrightarrow}&(M,\Delta^{T})\\\
\downarrow\widetilde{\pi}&\circlearrowright&\downarrow\pi\\\
(Y,E,\Delta_{Y}^{\prime},D_{Y})&\overset{\phi}{\longrightarrow}&(Z,\Delta)\end{array}$

is commutative $\pi\circ\widetilde{\phi}=\phi\circ\widetilde{\pi}.$

  1. (1)

We call $(M,\Delta^{T})$ is a log pair with klt foliation singularities if
there exists a unique transverse $\mathbb{Q}$-divisor

$\begin{array}[c]{c}D_{\widetilde{M}}^{T}=\sum_{j=1}b_{j}D_{j}^{T}\end{array}$

with $b_{j}>-1$ such that

$\begin{array}[c]{c}K_{\widetilde{M}}^{T}=\widetilde{\phi}^{\ast}(K_{M}^{T})+D_{\widetilde{M}}^{T},\text{
}D_{\widetilde{M}}^{T}=\widetilde{\overline{E}}^{T}+(\Delta_{Y}^{\prime})^{T}.\end{array}$

That is

$\begin{array}[c]{c}K_{\widetilde{M}}^{T}=\widetilde{\phi}^{\ast}(K_{M}^{T})+\sum_{-1<b_{j}\leq
0}b_{j}D_{j}^{T}+(\Delta_{Y}^{\prime})^{T}.\end{array}$

Here the $\widetilde{\phi}$-exceptional divisor $\overline{E}^{T}$

$\begin{array}[c]{c}\widetilde{\phi}^{-1}(\pi^{-1}(Z_{\text{{sing}}}))=\widetilde{\overline{E}}^{T}=\sum_{-1<b_{j}\leq
0}b_{j}D_{j}^{T}\end{array}$

with $\pi(M_{\text{{sing}}})=M_{\text{{sing}}}/\mathbb{S}^{1}=\Sigma^{orb}(Z)$
and the strict transform $(\Delta_{Y}^{\prime})^{T}$ of the transverse branch
divisor $\Delta^{T}$

$\begin{array}[c]{c}\widetilde{\phi}_{\ast}^{-1}(\Delta^{T})=\Delta^{\prime
T}=\sum_{b_{j}>0}b_{j}D_{j}^{T}.\end{array}$

Set $a_{j}:=-b_{j},$ if $-1<b_{j}\leq 0.$ Then

$\begin{array}[c]{c}K_{Y}^{-1}=\phi^{\ast}(K_{Z}^{-1}-\Delta)+\sum_{0\leq
a_{j}<1}a_{j}D_{j}-\Delta^{\prime}\end{array}$

and

$\begin{array}[c]{c}(K_{\widetilde{M}}^{T})^{-1}-\sum_{0\leq
a_{j}<1}a_{j}D_{j}^{T}=\widetilde{\phi}^{\ast}((K_{M}^{T})^{-1}))-(\Delta_{Y}^{\prime})^{T}=\widetilde{\phi}^{\ast}((K_{M}^{T})^{-1}-\Delta^{T}).\end{array}$

  2. (2)

$(M,\Delta^{T})$ is a log Fano quasi-regular Sasakian manifold if
$(K_{M}^{T})^{-1}-\Delta^{T}$ is a basic ample $\mathbb{Q}$-line bundle. That
is

$c_{1}^{B}(M,\Delta^{T})=c_{1}^{B}(M)-c_{1}^{B}(\Delta^{T})>0.$

  3. (3)

$(\widetilde{M},D^{T})$ is a log Fano regular Sasakian manifold if $D^{T}$ is
a normal crossing smooth divisors such that $L^{T}$ is a basic ample
$\mathbb{Q}$-line bundle:

$\begin{array}[c]{c}L^{T}:=(K_{\widetilde{M}}^{T})^{-1}-\sum_{0\leq
a_{j}<1}a_{j}D_{j}^{T}=\widetilde{\phi}^{\ast}((K_{M}^{T})^{-1}-\Delta^{T}).\end{array}$

  4. (4)

Let $(\widetilde{M},D^{T})$ be a log Fano regular Sasakian manifold. A conic
Sasakian metric $\widetilde{\omega}$ is called a conic Sasaki-Einstein on
$(\widetilde{M},D^{T})$ with the transverse cone angle $2\pi(1-a_{j})$ along
$D_{j}^{T}$ with $\widetilde{\pi}_{\ast}^{-1}(D_{Y})=D_{j}^{T}$ if

$\begin{array}[c]{c}\widetilde{\mathrm{Ric}}^{T}(\widetilde{\omega})=\widetilde{\omega}+\sum_{0<a_{j}<1}a_{j}D_{j}^{T}\end{array}$

such that $2\pi(1-a_{j})$ is the cone angle along $D_{Y}$ with
$\widetilde{\pi}_{\ast}^{-1}(D_{Y})=D_{j}^{T}.$

######  Remark 2.1.

If there is a Sasaki-Einstein metric $\omega$ on a log Fano quasi-regular
Sasakian manifold $(M,\Delta^{T})$ with klt foliation singularities, then
$\widetilde{\phi}^{\ast}\omega$ will be a degenerate conic Sasaki-Einstein
metric on a log Fano regular Sasakian manifold $\widetilde{M}$ with the
transverse cone angle $2\pi(1-a_{j})$ along $D_{j}^{T}$:

$\begin{array}[c]{c}\mathrm{Ric}^{T}(\widetilde{\phi}^{\ast}\omega)=\widetilde{\phi}^{\ast}\omega+\sum_{0<a_{j}<1}a_{j}D_{j}^{T}.\end{array}$

Then, for a log Fano $(M,\Delta^{T})$ with such a resolution
$\widetilde{\phi}:(\widetilde{M},D^{T})\rightarrow(M,\Delta^{T})$, one might
expect that we can turn around to use this to construct a Sasaki-Einstein
metric on a log Fano quasi-regular Sasakian manifold $M$ with the foliation
singularities $\Delta^{T}$. Indeed, this is true due to Chang-Han-Tie [CHT]
and Tian-Wang [TW2] on the work of the compactness and existence of conic
Sasaki-Einstein metrics.

###  2.3. The Twisted and Conic Sasaki-Ricci Flow

Let $(M,D^{T})$ is a log Fano regular Sasakian manifold with a smooth basic
divisors $D^{T}$. We consider the transverse Kähler class which is not
proportional to the basic first Chern class such that

$c_{1}^{B}(M)-\beta[\omega_{0}]_{B}=[\alpha]_{B}\neq 0.$

Then for any fixed basic closed $(1,1)$-form $\theta\in[\alpha]_{B}$, it is
natural to ask if there exists a unique transverse Kähler form
$\omega\in[\omega_{0}]_{B}$ so that

$\mathrm{Ric}^{T}(\omega)=\beta\omega+\theta$

which is called a twisted Sasaki-Ricci metric. The corresponding twisted
Sasaki-Ricci flow reads as

(2.6) $\begin{array}[c]{l}\frac{\partial}{\partial
t}\omega(t)=-\mathrm{Ric}^{T}(\omega(t))+\beta\omega(t)+\theta\end{array}$

for $0<\beta<1$.

Let $\left(M,\eta,g_{0},\left(1-\beta\right)D^{T}\right)$ be a Sasakian
manifold with the transverse cone angle $2\pi\beta$ along $D^{T}$ and
$D^{T}\thicksim-K_{M}^{T}$ such that

$\begin{array}[c]{c}c_{1}^{B}(M,\left(1-\beta\right)D^{T})=c_{1}^{B}(M)-\left(1-\beta\right)c_{1}^{B}([D^{T}])=\beta[\omega_{0}]_{B}>0\end{array}$

where $\left[D^{T}\right]$ is the line bundle induced by the transverse
divisor $D^{T}$ and $\omega_{0}=d\eta$ denotes the transverse Kähler form of
$g_{0}.$ Now we consider the so-called conic Sasaki-Ricci flow

$\left\\{\begin{array}[c]{lcl}\frac{\partial}{\partial
t}\omega(t)&=&-\mathrm{Ric}^{T}(\omega(t))+\beta\omega(t)+(1-\beta)[D^{T}],\\\
\omega(0)&=&\omega_{\ast}.\end{array}\right.$

Here
$\omega_{\ast}=\omega_{0}+\delta\sqrt{-1}\partial_{B}\overline{\partial}_{B}||S^{T}||^{2\beta}$
and $S^{T}$ is the basic defined section of $D^{T}$ in transverse Fano log
pair $(M,D^{T}).$ $[D^{T}]$ is the current of integration along $D^{T}$.

##  3\. Twisted Sasaki-Ricci Flow on Transverse Log Fano Sasakian Manifolds

###  3.1. $L^{4}$-Bound of the Transverse Conic Ricci Curvature

In this subsection, we consider the $L^{4}$-boundedness of the transverse
Ricci curvature along the twisted Sasaki-Ricci flow (3.1) as follows.

Let $\left(M,\eta,g_{0},\left(1-\beta\right)D^{T}\right)$ be a Sasakian
manifold with the transverse cone angle $2\pi\beta$ along $D^{T}.$ In this
article, we use the following twisted Sasaki-Ricci flow to approach the conic
Sasaki-Ricci flow (1.1)

(3.1)
$\begin{array}[c]{c}\left\\{\begin{array}[c]{ccl}\frac{\partial\omega^{\epsilon}}{\partial
t}&=&-\left(\rho_{\omega^{\epsilon}}^{T}-\theta_{\epsilon}\right)+\beta\omega^{\epsilon}\\\
\omega^{\epsilon}\left(0\right)&=&\widehat{\omega}_{0}^{\epsilon}\equiv\omega_{0}+\sqrt{-1}k\partial_{B}\overline{\partial}_{B}\chi_{\epsilon}(\epsilon^{2}+\left\|S\right\|_{h}^{2}).\end{array}\right.\end{array}$

Here $k$ is a sufficiently small positive number, $D^{T}=\left[S=0\right]$,
$h$ is a Hermitian metric on $\left[D^{T}\right]$,

$\begin{array}[c]{c}\chi_{\epsilon}\left(\epsilon^{2}+t\right)=\frac{1}{\beta}\int_{0}^{t}\frac{\left(\epsilon^{2}+r\right)^{\beta}-\epsilon^{2\beta}}{r}dr\end{array}$

and

(3.2)
$\begin{array}[c]{c}\theta_{\epsilon}=\left(1-\beta\right)[\omega_{0}+\sqrt{-1}\partial_{B}\overline{\partial}_{B}\log(\left\|S\right\|_{h}^{2}+\epsilon^{2})].\end{array}$

Then the parabolic Monge-Ampere equations for potentials are

(3.3)
$\begin{array}[c]{c}\left\\{\begin{array}[c]{lcl}\frac{\partial\varphi_{\varepsilon}}{\partial
t}&=&\log\frac{\omega_{\varphi_{\varepsilon}}^{n}\wedge\eta_{0}}{\omega_{\varepsilon}^{n}\wedge\eta_{0}}+F_{\varepsilon}+\beta(k\chi+\varphi_{\varepsilon}),\\\
\varphi_{\varepsilon}(0)&=&c_{\varepsilon}(0).\end{array}\right.\end{array}$

Here $c_{\varepsilon}(0)$ are uniformly bounded for $\varepsilon$

$\begin{array}[c]{c}c_{\varepsilon}(0)=\frac{1}{\beta}\int_{0}^{\infty}e^{-\beta
s}||\nabla^{T}u_{\varepsilon}(s)||_{L^{2}}^{2}ds-\frac{1}{V}\int_{M}F_{\varepsilon}d\mu_{\varepsilon}-\frac{k\beta}{V}\int_{M}\chi
d\mu_{\varepsilon}\end{array}$

and

$\begin{array}[c]{c}F_{\varepsilon}=F_{0}+\log[\frac{\omega_{{}_{\varepsilon}}^{n}\wedge\eta_{0}}{\omega_{0}^{n}\wedge\eta_{0}}(|S^{T}|^{2}+\varepsilon^{2})^{1-\beta}]\end{array}$

with $F_{0}$ is the Ricci potential of $\omega_{0}$ so that
$-\mathrm{Ric}^{T}(\omega_{0})+\omega_{0}=\sqrt{-1}\partial_{B}\overline{\partial}_{B}F_{0}$
with $\int e^{-F_{0}}dV_{0}=V.$

It follows from the standard argument that we have the long-time existence of
the solution to the twisted Sasaki-Ricci flow (3.1). Note that
$\omega^{\epsilon}\left(t\right)$ evolves in the same Kähler class
$\left[\omega_{0}\right]$. So $\omega^{\epsilon}\left(t\right)$ could be
expressed in terms of the transverse Kähler potential

$\begin{array}[c]{c}\omega^{\epsilon}\left(t\right)=\widehat{\omega}_{0}^{\epsilon}+\sqrt{-1}\partial_{B}\overline{\partial}_{B}\varphi_{\epsilon}\left(t\right).\end{array}$

Take $u_{\epsilon}\left(t\right)$ to be the transverse twisted Ricci
potential, i.e.

(3.4)
$\begin{array}[c]{c}-\left(\rho_{\omega^{\epsilon}}^{T}-\theta_{\epsilon}\right)+\beta\omega^{\epsilon}=\sqrt{-1}\partial_{B}\overline{\partial}_{B}u_{\epsilon}\left(t\right)\end{array}$

which is normalized by

$\begin{array}[c]{c}\frac{1}{V}\int_{M}\exp\left(-u_{\epsilon}\left(t\right)\right)d\mu_{\epsilon}\left(t\right)=1\end{array}$

where $V=\int_{M}d\mu_{\epsilon}\left(t\right)$ and
$d\mu_{\epsilon}\left(t\right)=\left(\omega^{\epsilon}\left(t\right)\right)^{n}\wedge\eta$.
It’s easy to see $\int_{M}d\mu_{\epsilon}\left(t\right)$ is independent of the
choice of $\epsilon$ and $t$.

######  Proposition 3.1.

We have the equality

$\begin{array}[c]{c}\frac{1}{2}\square_{B}[\left(\Delta_{B}u_{\epsilon}\left(t\right)\right)^{2}]=-\left|\nabla^{T}\Delta_{B}u_{\epsilon}\left(t\right)\right|_{\omega^{\epsilon}\left(t\right)}^{2}+\Delta_{B}u_{\epsilon}\left(t\right)\langle\rho_{\omega^{\epsilon}}^{T}-\theta_{\epsilon},\sqrt{-1}\partial_{B}\overline{\partial}_{B}u_{\epsilon}\left(t\right)\rangle_{\omega^{\epsilon}\left(t\right)}.\end{array}$

Here $\Delta_{B}$ means the basic Laplacian with respect to
$\omega^{\epsilon}\left(t\right)$,$\ \square_{B}$ denotes the operator
$(\frac{\partial}{\partial t}-\Delta_{B})$.

###### Proof.

Due to $\frac{\partial\omega^{\epsilon}}{\partial
t}=\sqrt{-1}\partial_{B}\overline{\partial}_{B}u_{\epsilon}\left(t\right)$,
there is a constant $c\left(t\right)$ which depends only on the parameter $t$
such that
$\overset{\cdot}{\varphi_{\epsilon}}\left(t\right)=u_{\epsilon}\left(t\right)+c\left(t\right)$.
Here after the quantity with the dot $\cdot$ means its time-derivative. From

$\begin{array}[c]{ccl}\sqrt{-1}\partial_{B}\overline{\partial}_{B}\overset{\cdot}{u_{\epsilon}}&=&-\overset{\cdot}{\rho}_{\omega^{\epsilon}}^{T}+\beta\overset{\cdot}{\omega}^{\epsilon}\\\
&=&\sqrt{-1}\partial_{B}\overline{\partial}_{B}u_{\epsilon}\left(t\right)+\sqrt{-1}\partial_{B}\overline{\partial}_{B}\Delta_{B}u_{\epsilon}\left(t\right),\end{array}$

it could be deduced that $\square_{B}u_{\epsilon}=\beta
u_{\epsilon}-a_{\varepsilon}\left(t\right)$ for

(3.5)
$\begin{array}[c]{c}a_{\varepsilon}\left(t\right)=\frac{\beta}{V}\int_{M}u_{\epsilon}\left(t\right)\exp\left(-u_{\epsilon}\left(t\right)\right)d\mu_{\epsilon}\left(t\right).\end{array}$

Because

$\begin{array}[c]{c}\frac{\partial}{\partial
t}\left(\Delta_{B}u_{\epsilon}\right)=\langle\left(\rho_{\omega^{\epsilon}}^{T}-\theta_{\epsilon}\right)-\beta\omega^{\epsilon},\sqrt{-1}\partial_{B}\overline{\partial}_{B}u_{\epsilon}\left(t\right)\rangle_{\omega^{\epsilon}\left(t\right)}+\Delta_{B}\left(\Delta_{B}u_{\epsilon}+\beta
u_{\epsilon}-a_{\varepsilon}\left(t\right)\right),\end{array}$

we obtain

$\begin{array}[c]{c}\square_{B}\left(\Delta_{B}u_{\epsilon}\right)=\langle\left(\rho_{\omega^{\epsilon}}^{T}-\theta_{\epsilon}\right),\sqrt{-1}\partial_{B}\overline{\partial}_{B}u_{\epsilon}\left(t\right)\rangle_{\omega^{\epsilon}\left(t\right)}.\end{array}$

This implies that

$\begin{array}[c]{c}\frac{1}{2}\square_{B}[\left(\Delta_{B}u_{\epsilon}\left(t\right)\right)^{2}]=-\left|\nabla^{T}\Delta_{B}u_{\epsilon}\left(t\right)\right|_{\omega^{\epsilon}\left(t\right)}^{2}+\Delta_{B}u_{\epsilon}\left(t\right)\langle\rho_{\omega^{\epsilon}}^{T}-\theta_{\epsilon},\sqrt{-1}\partial_{B}\overline{\partial}_{B}u_{\epsilon}\left(t\right)\rangle_{\omega^{\epsilon}\left(t\right)}.\end{array}$

∎

######  Proposition 3.2.

The functions $\nabla^{T}\nabla^{T}u_{\epsilon}$,
$\nabla^{T}\overline{\nabla}^{T}u_{\epsilon}$, and
$|\mathrm{Rm}_{\omega^{\epsilon}\left(t\right)}^{T}|_{\omega^{\epsilon}}$ all
lie in $L^{2}\left(M,\omega^{\epsilon}\right)$.

###### Proof.

It suffices to observe the inequality

$\begin{array}[c]{c}\int_{M}\left|\theta_{\epsilon}\right|_{\omega^{\epsilon}\left(t\right)}^{2}d\mu_{\epsilon}\left(t\right)\leq\int_{M}tr_{\omega^{\epsilon}\left(t\right)}\theta_{\epsilon}d\mu_{\epsilon}\left(t\right)=\int_{M}n\theta_{\epsilon}\wedge
d\mu_{\epsilon}\left(t\right)\rightarrow
n\int_{D^{T}}d\mu_{0}\left(t\right)\end{array}$

as $\epsilon\longrightarrow 0^{+}$; hence, we have

$\begin{array}[c]{c}\int_{M}\left|\theta_{\epsilon}\right|_{\omega^{\epsilon}\left(t\right)}^{2}d\mu_{\epsilon}\left(t\right)\leq
C.\end{array}$

∎

######  Remark 3.1.

Actually, the inequality of the first line in the proof as above is the
equality.

######  Lemma 3.1.

For any $t\geq 1$, there exists a positive constant $C$ which is independent
of $\epsilon$ and $t$ such that

$\begin{array}[c]{c}\int_{M\times\left[t,t+1\right]}\left|\nabla^{T}\Delta_{B}u_{\epsilon}\left(t\right)\right|_{\omega^{\epsilon}\left(t\right)}^{2}d\mu_{\epsilon}\left(t\right)\leq
C.\end{array}$

###### Proof.

By Proposition 3.1 and the integration by parts, we see that

$\begin{array}[c]{cl}&\int_{M}\left|\nabla^{T}\Delta_{B}u_{\epsilon}\left(t\right)\right|_{\omega^{\epsilon}\left(t\right)}^{2}d\mu_{\epsilon}\left(t\right)\\\
\leq&\int_{M}|\langle\rho_{\omega^{\epsilon}}^{T}-\theta_{\epsilon},\sqrt{-1}\partial_{B}\overline{\partial}_{B}u_{\epsilon}\left(t\right)\rangle_{\omega^{\epsilon}\left(t\right)}|\left|\left(\Delta_{B}u_{\epsilon}\left(t\right)\right)\right|d\mu_{\epsilon}\left(t\right)-\frac{1}{2}\int_{M}\frac{\partial}{\partial
t}\left(\Delta_{B}u_{\epsilon}\left(t\right)\right)^{2}d\mu_{\epsilon}\left(t\right)\\\
\leq&\int_{M}|\langle\rho_{\omega^{\epsilon}}^{T}-\theta_{\epsilon},\sqrt{-1}\partial_{B}\overline{\partial}_{B}u_{\epsilon}\left(t\right)\rangle_{\omega^{\epsilon}\left(t\right)}|\left|\left(\Delta_{B}u_{\epsilon}\left(t\right)\right)\right|d\mu_{\epsilon}\left(t\right)-\frac{1}{2}\frac{d}{dt}\int_{M}\left(\Delta_{B}u_{\epsilon}\left(t\right)\right)^{2}d\mu_{\epsilon}\left(t\right)\\\
&+\frac{1}{2}\int_{M}\left(\Delta_{B}u_{\epsilon}\left(t\right)\right)^{2}|\mathrm{R}_{\omega^{\epsilon}\left(t\right)}^{T}-tr_{\omega^{\epsilon}\left(t\right)}\theta_{\epsilon}-\beta
n|d\mu_{\epsilon}\left(t\right).\end{array}$

Then, integrating both sides over the interval $\left[t,t+1\right]$, the
estimate

$\begin{array}[c]{c}\int_{M\times\left[t,t+1\right]}\left|\nabla^{T}\Delta_{B}u_{\epsilon}\left(t\right)\right|_{\omega^{\epsilon}\left(t\right)}^{2}d\mu_{\epsilon}\left(t\right)\leq
C\end{array}$

holds with the help of Proposition 3.7 and Proposition 3.2. ∎

As in [LZ], the straightforward calculations conclude the following
inequalities:

######  Proposition 3.3.

There exists a universal positive constant $C=C(\omega_{0})$ such that for
$t\geq 1$ and $\epsilon>0$

  1. (1)

$\begin{array}[c]{l}\int_{M}|\nabla^{T}\overline{\nabla}^{T}u_{\epsilon}|_{\omega^{\epsilon}\left(t\right)}^{4}d\mu_{\epsilon}\left(t\right)\\\
\leq
C\int_{M}[|\overline{\nabla}^{T}\nabla^{T}\nabla^{T}u_{\epsilon}|_{\omega^{\epsilon}\left(t\right)}^{2}+|\nabla^{T}\nabla^{T}\overline{\nabla}^{T}u_{\epsilon}|_{\omega^{\epsilon}\left(t\right)}^{2}]d\mu_{\epsilon}\left(t\right).\end{array}$

  2. (2)

$\begin{array}[c]{l}\int_{M}|\nabla^{T}\nabla^{T}u_{\epsilon}|_{\omega^{\epsilon}\left(t\right)}^{4}d\mu_{\epsilon}\left(t\right)\\\
\leq
C\int_{M}[|\overline{\nabla}^{T}\nabla^{T}\nabla^{T}u_{\epsilon}|_{\omega^{\epsilon}\left(t\right)}^{2}+|\nabla^{T}\nabla^{T}\overline{\nabla}^{T}u_{\epsilon}|_{\omega^{\epsilon}\left(t\right)}^{2}+|\nabla^{T}\nabla^{T}\nabla^{T}u_{\epsilon}|_{\omega^{\epsilon}\left(t\right)}^{2}]d\mu_{\epsilon}\left(t\right)\\\
\leq
C\int_{M}[|\nabla^{T}\Delta_{B}u_{\epsilon}|_{\omega^{\epsilon}\left(t\right)}^{2}+|\nabla^{T}\nabla^{T}u_{\epsilon}|_{\omega^{\epsilon}\left(t\right)}^{2}+|\mathrm{Rm}_{\omega^{\epsilon}\left(t\right)}^{T}|_{\omega^{\epsilon}\left(t\right)}^{2}]d\mu_{\epsilon}\left(t\right).\end{array}$

######  Theorem 3.1.

There exists a positive constant $C$ which is independent of $\epsilon$ and
$t$ such that

$\begin{array}[c]{c}\int_{M}\left|\rho_{\omega^{\epsilon}}^{T}-\theta_{\epsilon}\right|_{\omega^{\epsilon}\left(t\right)}^{4}d\mu_{\epsilon}\left(t\right)\leq
C\end{array}$

for any $\epsilon>0$ and $t\geq 1$.

###### Proof.

By the equation (3.4), we only need to justify the uniform boundedness of
$\int_{M}|\sqrt{-1}\partial_{B}\overline{\partial}_{B}u_{\epsilon}\left(t\right)|_{\omega^{\epsilon}\left(t\right)}^{4}d\mu_{\epsilon}\left(t\right)$.
From the equality

$\begin{array}[c]{c}\square_{B}\left(\Delta_{B}u_{\epsilon}\right)=\langle\rho_{\omega^{\epsilon}}^{T}-\theta_{\epsilon},\sqrt{-1}\partial_{B}\overline{\partial}_{B}u_{\epsilon}\left(t\right)\rangle_{\omega^{\epsilon}\left(t\right)}=\beta\Delta_{B}u_{\epsilon}-|\nabla^{T}\overline{\nabla}^{T}u_{\epsilon}|_{\omega^{\epsilon}\left(t\right)}^{2},\end{array}$

we have

$\begin{array}[c]{cl}&\frac{\partial}{\partial
t}\left|\nabla^{T}\Delta_{B}u_{\epsilon}\left(t\right)\right|_{\omega^{\epsilon}\left(t\right)}^{2}\\\
=&-u_{\epsilon}^{j\overline{k}}\left(t\right)\nabla_{j}^{T}\Delta_{B}u_{\epsilon}\nabla_{\overline{k}}^{T}\Delta_{B}u_{\epsilon}+\nabla_{j}^{T}\Delta_{B}\overset{\cdot}{u}_{\epsilon}\nabla_{\overline{j}}^{T}\Delta_{B}u_{\epsilon}+\nabla_{j}^{T}\Delta_{B}u_{\epsilon}\nabla_{\overline{j}}^{T}\Delta_{B}\overset{\cdot}{u}_{\epsilon}\\\
=&-u_{\epsilon}^{j\overline{k}}\left(t\right)\nabla_{j}^{T}\Delta_{B}u_{\epsilon}\nabla_{\overline{k}}^{T}\Delta_{B}u_{\epsilon}+\nabla_{j}^{T}(\Delta_{B}^{2}u_{\epsilon}+\beta\Delta_{B}u_{\epsilon})\nabla_{\overline{j}}^{T}\Delta_{B}u_{\epsilon}\\\
&+\nabla_{j}^{T}\Delta_{B}u_{\epsilon}\nabla_{\overline{j}}^{T}(\Delta_{B}^{2}u_{\epsilon}+\beta\Delta_{B}u_{\epsilon}-|\nabla^{T}\overline{\nabla}^{T}u_{\epsilon}|_{\omega^{\epsilon}\left(t\right)}^{2}-\nabla_{j}^{T}|\nabla^{T}\overline{\nabla}^{T}u_{\epsilon}|_{\omega^{\epsilon}\left(t\right)}^{2})\\\
=&-u_{\epsilon}^{j\overline{k}}\left(t\right)\nabla_{j}^{T}\Delta_{B}u_{\epsilon}\nabla_{\overline{k}}^{T}\Delta_{B}u_{\epsilon}-(\nabla_{j}^{T}|\nabla^{T}\overline{\nabla}^{T}u_{\epsilon}|_{\omega^{\epsilon}\left(t\right)}^{2})\nabla_{\overline{j}}^{T}\Delta_{B}u_{\epsilon}\\\
&-\nabla_{j}^{T}\Delta_{B}u_{\epsilon}(\nabla_{\overline{j}}^{T}|\nabla^{T}\overline{\nabla}^{T}u_{\epsilon}|_{\omega^{\epsilon}\left(t\right)}^{2})+w_{j}w_{\overline{j}i\overline{i}}\\\
&+w_{\overline{j}}\\{w_{ji\overline{i}}-[\left(\theta_{\epsilon}\right)_{j\overline{k}}+\beta\left(g_{\epsilon}\right)_{j\overline{k}}-\left(u_{\epsilon}\right)_{j\overline{k}}]w_{k}\\}+2\beta|\nabla_{j}^{T}\Delta_{B}u_{\epsilon}|_{\omega^{\epsilon}\left(t\right)}^{2}\\\
=&-(\nabla_{j}^{T}|\nabla^{T}\overline{\nabla}^{T}u_{\epsilon}|_{\omega^{\epsilon}\left(t\right)}^{2})\nabla_{\overline{j}}^{T}\Delta_{B}u_{\epsilon}-\nabla_{j}^{T}\Delta_{B}u_{\epsilon}(\nabla_{\overline{j}}^{T}|\nabla^{T}\overline{\nabla}^{T}u_{\epsilon}|_{\omega^{\epsilon}\left(t\right)}^{2})\\\
&+w_{\overline{j}}w_{ji\overline{i}}+w_{j}w_{\overline{j}i\overline{i}}-\left(\theta_{\epsilon}\right)_{j\overline{k}}w_{\overline{j}}w_{k}+\beta\left|\nabla^{T}\Delta_{B}u_{\epsilon}\right|_{\omega^{\epsilon}\left(t\right)}^{2}\\\
=&-(\nabla_{j}^{T}|\nabla^{T}\overline{\nabla}^{T}u_{\epsilon}|_{\omega^{\epsilon}\left(t\right)}^{2})\nabla_{\overline{j}}^{T}\Delta_{B}u_{\epsilon}-\nabla_{j}^{T}\Delta_{B}u_{\epsilon}(\nabla_{\overline{j}}^{T}|\nabla^{T}\overline{\nabla}^{T}u_{\epsilon}|_{\omega^{\epsilon}\left(t\right)}^{2})\\\
&+\Delta_{B}\left|\nabla^{T}\Delta_{B}u_{\epsilon}\left(t\right)\right|_{\omega^{\epsilon}\left(t\right)}^{2}-|\nabla^{T}\nabla^{T}\Delta_{B}u_{\epsilon}|_{\omega^{\epsilon}\left(t\right)}^{2}-|\nabla^{T}\overline{\nabla}^{T}\Delta_{B}u_{\epsilon}|_{\omega^{\epsilon}\left(t\right)}^{2}\\\
&-\left(\theta_{\epsilon}\right)_{j\overline{k}}w_{\overline{j}}w_{k}+\beta\left|\nabla^{T}\Delta_{B}u_{\epsilon}\right|_{\omega^{\epsilon}\left(t\right)}^{2}\end{array}$

for $w=\Delta_{B}u_{\epsilon}$; then, by considering the quantity
$\frac{d}{dt}\int_{M}\left|\nabla^{T}\Delta_{B}u_{\epsilon}\left(t\right)\right|_{\omega^{\epsilon}\left(t\right)}^{2}d\mu_{\epsilon}\left(t\right)$,
it could be found that

(3.6)
$\begin{array}[c]{cl}&\frac{d}{dt}\int_{M}\left|\nabla^{T}\Delta_{B}u_{\epsilon}\left(t\right)\right|_{\omega^{\epsilon}\left(t\right)}^{2}d\mu_{\epsilon}\left(t\right)\\\
\leq&-\frac{1}{2}\int_{M}[|\nabla^{T}\overline{\nabla}^{T}\Delta_{B}u_{\epsilon}|_{\omega^{\epsilon}\left(t\right)}^{2}+|\nabla^{T}\nabla^{T}\Delta_{B}u_{\epsilon}|_{\omega^{\epsilon}\left(t\right)}^{2}]d\mu_{\epsilon}\left(t\right)\\\
&-\int_{M}\left(\theta_{\epsilon}\right)_{j\overline{k}}w_{\overline{j}}w_{k}d\mu_{\epsilon}\left(t\right)+C\int_{M}\left|\nabla^{T}\Delta_{B}u_{\epsilon}\right|_{\omega^{\epsilon}\left(t\right)}^{2}d\mu_{\epsilon}\left(t\right)+C\\\
\leq&C\int_{M}\left|\nabla^{T}\Delta_{B}u_{\epsilon}\right|_{\omega^{\epsilon}\left(t\right)}^{2}d\mu_{\epsilon}\left(t\right)+C\end{array}$

from the fact that
$\sqrt{-1}\partial_{B}\overline{\partial}_{B}\log(\left\|S\right\|_{h}^{2}+\epsilon^{2})\geq
0$ and Proposition 3.3.

Finally, for any $\epsilon>0$ and $t\geq 1$, the uniform boundedness of

$\begin{array}[c]{c}\int_{M}\left|\rho_{\omega^{\epsilon}}^{T}-\theta_{\epsilon}\right|_{\omega^{\epsilon}\left(t\right)}^{4}d\mu_{\epsilon}\left(t\right)\leq
C\end{array}$

follows from (3.6), Lemma 3.1 and Proposition 3.3. ∎

######  Remark 3.2.

It is worth to note that it is may not true for the boundedness of
$\theta_{\epsilon}$ in general; however, we are able to control its integral
bound!

###  3.2. Longtime Behavior under the Twisted Sasaki-Ricci Flow

All transverse quantities on Sasakian manifolds such as the transverse
Perelman’s $\mathcal{W}$-functional can be viewed as their Kähler counterparts
restricted on basic forms and transverse Kähler structure. Hence, under the
twisted or conic Sasaki-Ricci flow, the Reeb vector field and the transverse
holomorphic structure are both invariant, and the metrics are bundle-like. One
can apply integration by parts, the expressions involved behave essentially
the same as in the Kähler case.

In this section, the proofs are given here, primarily along the lines of the
arguments in [CCHLW] for the conic Sasaki-Ricci flow.

Firstly, we observe a long-time solution to the conic Sasaki-Ricci flow from
the twisted Sasaki-Ricci flow (3.3) as following:

######  Proposition 3.4.

([LZ, Theorem 4.1 and Proposition 4.2]) There exists a sequence
$\\{\varepsilon_{i}\\}$ satisfying $\varepsilon_{i}\rightarrow 0$ such that
the solution $\varphi_{\varepsilon_{i}}(t)$ of the twisted Sasaki-Ricci flow
(3.1) converges to the unique solution $\varphi(t)$ of the conic Sasaki-Ricci
flow (1.1) in $C_{B}^{\infty}$-topology on $[0,\infty)\times(M\backslash
D^{T})$. That is,
$\omega\left(t\right)=\widehat{\omega}_{0}+\sqrt{-1}\partial_{B}\overline{\partial}_{B}\varphi\left(t\right)$
is a long-time solution to the conic Sasaki-Ricci flow. Moreover, for any
$t\in[0,\infty),$ there exists a positive constant $\alpha<1$ such that the
basic function $\varphi(t)$ is Hőlder continuous with respect to $\omega_{0}$
on $M$. And $||\varphi(t)||_{C_{B}^{\alpha}}$ is bounded by a constant $C$
which depends only on $\beta,$ $\omega_{0}$,
$||\varphi(t)||_{C_{B}^{0}(M\backslash D^{T})}$ and
$||\overset{.}{\varphi}(t)||_{C_{B}^{0}(M\backslash D^{T})}$.

Next we will show that the transverse twisted Ricci potential
$u_{\epsilon}\left(t\right)$ which is a basic function behaves very well as
$t\rightarrow\infty$ under the twisted Sasaki-Ricci flow. This implies that
the limit of twisted Sasaki-Ricci flow should be the conic Sasaki-Ricci
soliton in the $L^{2}$-topology as in the next subsection. In fact, it follows
from the estimate of previous subsection and the twisted transverse Perelman’s
$\mathcal{W}_{\theta_{\epsilon}}^{T}$-functional below that

######  Proposition 3.5.

([CCHLW, Theorem 7 and Theorem 8]) Under the twisted Sasaki-Ricci flow

  1. (1)

(3.7)
$\begin{array}[c]{c}\int_{M}|\nabla^{T}\nabla^{T}u_{\varepsilon}|^{2}(\omega^{\epsilon})^{n}\wedge\eta_{0}\rightarrow
0\end{array}$

as $t\rightarrow\infty.$

  2. (2)

(3.8)
$\begin{array}[c]{c}\int_{M}(\Delta_{B}u_{\varepsilon}-|\nabla^{T}u_{\varepsilon}|^{2}+u_{\varepsilon}-a_{\varepsilon})^{2}(\omega^{\epsilon})^{n}\wedge\eta_{0}\rightarrow
0\end{array}$

as $t\rightarrow\infty.$ Here
$a_{\varepsilon}\left(t\right)=\frac{\beta}{V}\int_{M}u_{\epsilon}\left(t\right)\exp\left(-u_{\epsilon}\left(t\right)\right)d\mu_{\epsilon}\left(t\right)$
as in (3.5).

Next we define the twisted transverse Perelman’s
$\mathcal{W}_{\theta_{\epsilon}}^{T}$-functional on a compact Sasakian
$(2n+1)$-manifold by

(3.9) $\begin{array}[c]{l}\mathcal{W}_{\theta_{\epsilon}}^{T}(g^{T},f,\tau)\\\
=(4\pi\tau)^{-n}\int_{M}(\tau(\mathrm{R}^{T}(g(t))-tr_{g(t)}\theta_{\epsilon}+|\nabla^{T}f|^{2})+\beta
f-2n)e^{-f}(\omega^{\epsilon})^{n}\wedge\eta_{0},\end{array}$

for $f\in C_{B}^{\infty}(M;\mathbb{R})$, $\tau>0.$ Also denote

$\begin{array}[c]{c}\lambda_{\varepsilon}^{T}(g^{T},\tau)=\inf\\{\mathcal{W}_{\theta_{\epsilon}}^{T}(g^{T},f,\tau):f\in
C_{B}^{\infty}(M;\mathbb{R});\text{
}\int_{M}(4\pi\tau)^{-n}e^{-f}(\omega^{\epsilon})^{n}\wedge\eta_{0}=V\\}.\end{array}$

Then we have

$-\infty<\lambda_{\varepsilon}^{T}(g^{T},\tau)\leq C$

and there exists $f_{\tau}\in C_{B}^{\infty}(M;\mathbb{R})$ so that

$\mathcal{W}_{\theta_{\epsilon}}^{T}(g^{T},f_{\tau},\tau)=\lambda_{\varepsilon}^{T}(g^{T},\tau).$

Note that $\mathcal{W}_{\theta}^{T}$-functional can be expressed as

$\begin{array}[c]{ccl}\mathcal{W}_{\theta_{\epsilon}}^{T}(g^{T},f)&=&\mathcal{W}^{T}(g^{T},f,\frac{1}{2})+(2\pi)^{-n}(2n)V\\\
&=&(2\pi)^{-n}\int_{M}(\frac{1}{2}(\mathrm{R}^{T}-tr_{g(t)}\theta_{\epsilon}+|\nabla^{T}f|^{2})+\beta
f)e^{-f}(\omega^{\epsilon})^{n}\wedge\eta_{0},\end{array}$

where $(g^{T},f)$ satisfies
$\int_{M}e^{-f}(\omega^{\epsilon})^{n}\wedge\eta_{0}=V$ and
$\tau=\frac{1}{2}.$ Again

$\begin{array}[c]{c}\lambda_{\varepsilon}^{T}(g^{T})=\inf\\{\mathcal{W}_{\theta_{\epsilon}}^{T}(g^{T},f):f_{\tau}\in
C_{B}^{\infty}(M;\mathbb{R})\text{ {with}
}(2\pi)^{-n}\int_{M}e^{-f}(\omega^{\epsilon})^{n}\wedge\eta_{0}=V\\}.\end{array}$

######  Proposition 3.6.

([L]) Under the twisted Sasaki-Ricci flow, we have

  1. (1)

(3.10)
$\begin{array}[c]{ccl}\frac{d}{dt}\lambda_{\varepsilon}^{T}(g^{T})&=&\int_{M}|\mathrm{Ric}^{T}-\theta_{\epsilon}+\overline{\nabla}^{T}\nabla^{T}f-\beta
g^{T}|^{2}e^{-f}\omega^{n}\wedge\eta_{0}\\\
&&+\frac{1}{2}\int_{M}\theta_{\epsilon}(\nabla^{T}f,\overline{\nabla}^{T}f)e^{-f}\omega^{n}\wedge\eta_{0}+\int_{M}|\nabla^{T}\nabla^{T}f|^{2}e^{-f}\omega^{n}\wedge\eta_{0}.\end{array}$

Here $f$ are minimizering solutions of $\lambda_{\varepsilon}^{T}(g^{T})$
associated to metrics $g^{T}$ and $\sigma^{T}$ is the family of transverse
diffeomorphisms of $M$ generated by the time-dependent vector field
$\nabla_{g^{T}}^{T}f.$

  2. (2)

$g^{T}$ is a critical point of $\lambda_{\varepsilon}^{T}(g^{T})$ if and only
if $g^{T}$ is a gradient shrinking twisted Sasaki-Ricci soliton

(3.11)
$\begin{array}[c]{c}\mathrm{Ric}^{T}-\theta_{\epsilon}+\nabla^{T}\overline{\nabla}^{T}f=\beta
g^{T}\text{ {and} }\nabla^{T}\nabla^{T}f=0.\end{array}$

Here $f$ is a minimizer of $\mathcal{W}^{T}(g^{T},\cdot).$

Now based on Perelman’s non-collapsing theorem, we have

######  Proposition 3.7.

([LZ], [Co1], [CHLW]) Under the twisted Sasaki-Ricci flow, for any $t\geq 1,$
$\varepsilon>0$ and

$\begin{array}[c]{c}-\mathrm{Ric}^{T}(\omega_{\varepsilon}(t))+\beta\omega_{\varepsilon}(t)+\theta_{\varepsilon}=\sqrt{-1}\partial_{B}\overline{\partial}_{B}u_{{}_{\varepsilon}}(t),\end{array}$

we have

  1. (1)

$\mathrm{Vol}_{g_{\varepsilon}(t)}(B_{\xi,g_{\varepsilon}(t)}(x,r))\geq
Cr^{2n},$

  2. (2)

$|(\mathrm{R}^{T}(g_{\varepsilon}(t))-tr_{g_{\varepsilon}(t)}\theta_{\varepsilon})|\leq
C,$

  3. (3)

$||u_{\varepsilon}(t)||_{C^{1}(g_{\varepsilon}(t))}\leq C\ \ \mathrm{and}\ \
\mathrm{diam}(M,g_{\varepsilon}(t))\leq C.$

As a consequence,

######  Corollary 3.1.

([LZ], [Co2]) Under the twisted Sasaki-Ricci flow, we have

(3.12)
$\begin{array}[c]{l}\left(\int_{M}f^{\frac{2(2n+1)}{2n-1}}d\mu_{\varepsilon}\right)^{\frac{2n-1}{2n+1}}\leq
A\int_{M}||\nabla^{T}f||_{g_{\varepsilon}(t)}^{2}d\mu_{\varepsilon}+B\int_{M}f^{2}d\mu_{\varepsilon}.\end{array}$

Combining these estimates of Proposition 4.1 and Corollary 3.1, it follows
that

######  Corollary 3.2.

There exist positive constants $A,$ $B,$ $C$ such that, for any $\varepsilon$
and $t,$

  1. (1)

$\begin{array}[c]{c}||\overset{\cdot}{\phi}_{\varepsilon}(t)||_{C_{B}^{0}}\leq
C.\end{array}$

  2. (2)

(3.13)
$\begin{array}[c]{c}osc(\phi_{\varepsilon}(t))\leq\frac{A}{V}\int_{M}\phi_{\varepsilon}(t)d\mu_{0}+B.\end{array}$

By using (3.13), (4.3) and (4.4) below, it follows that

######  Proposition 3.8.

Suppose that the twisted transverse Mabuchi $K$-energy
$K_{\eta_{0},\theta_{\varepsilon}}$ are uniformly proper on

$\mathcal{H}(\omega_{0})=\\{\phi\in
C^{\infty}(M)|\omega_{0}+\sqrt{-1}\partial_{B}\overline{\partial}_{B}\phi>0\\},$

that is

(3.14) $K_{\eta_{0},\theta_{\varepsilon}}(\phi)\geq f(J_{\eta_{0}}(\phi)),$

for any $\varepsilon$ and $\phi\in\mathcal{H}(\omega_{0}),$ where $f(t)$ is a
monotone increasing function with $\lim_{t\rightarrow\infty}f(t)=\infty.$ Then

$||\phi_{\varepsilon}(t)||_{C^{0}}\leq C$

for any $\varepsilon$ and $t.$

###  3.3. Regularity of the Limit Space and Its Smooth Convergence

Firstly, we will apply our previous results plus Cheeger-Colding-Tian
structure theory for Kähler manifolds ([CHLW], [CCT], [TW2] and [TZ,
Theorem2.3]) to study the structure of desired limit spaces.

Since $(M,\eta,\xi,\Phi,g,\omega)$ is a compact regular Sasakian manifold. By
the first structure theorem on Sasakian manifolds, $M$ is a principal
$\mathbb{S}^{1}$-bundle over $Z$ which is also a $\mathbb{Q}$-factorial,
polarized, projective variety such that there is a Riemannian submersion$\
\pi:(M,g,\omega)\rightarrow(Z,h,\omega_{h})$ with $g=g^{T}+\eta\otimes\eta,$
$g^{T}=\pi^{\ast}(h)$ and $\frac{1}{2}d\eta=\pi^{\ast}(\omega_{h}).$ The orbit
$\xi_{x}$ is compact for any $x\in M,$ we then define the transverse distance
function as

$\begin{array}[c]{c}d^{T}(x,y)\triangleq
d_{g}(\xi_{x},\xi_{y})=d_{h}(\pi(x),\pi(y)),\end{array}$

where $d$ is the distance function defined by the Sasaki metric $g.$ We define
a transverse ball $B_{\xi,g}(x,r)$ as follows:

$\begin{array}[c]{c}B_{\xi,g}(x,r)=\left\\{y:d^{T}(x,y)<r\right\\}=\left\\{y:d_{h}(\pi(x),\pi(y))<r\right\\}.\end{array}$

Note that when $r$ small enough, $B_{\xi,g}(x,r)$ is a trivial
$\mathbb{S}^{1}$-bundle over the geodesic ball $B_{h}(\pi(x),r)$.

It follows from our previous work [CCHLW] and [CHT] ([TW2] and [TZ,
Theorem2.3]) that

######  Theorem 3.2.

Let $(M_{i},(1-\beta)D^{T},\eta_{i},\xi,\Phi_{i},g_{i},\omega_{i})$ be a
sequence of log transverse Fano regular Sasakian $(2n+1)$-manifolds with conic
Sasaki metrics $g_{i}=g_{i}^{T}+\eta_{i}\otimes\eta_{i}$ such that for basic
potentials $\varphi_{i}$

$\begin{array}[c]{c}\eta_{i}=\eta+d_{B}^{C}\varphi_{i}\end{array}$

and

$\begin{array}[c]{c}d\eta_{i}=d\eta+\sqrt{-1}\partial_{B}\overline{\partial}_{B}\varphi_{i}.\end{array}$

We denote that $(Z_{i},(1-\beta)D,h_{i},J_{i},\omega_{h_{i}})$ are a sequence
of projective Kähler manifolds which are the corresponding foliation leave
space with respect to
$(M_{i},(1-\beta)D^{T},\eta_{i},\xi,\Phi_{i},g_{i},\omega_{i})$ such that

$\begin{array}[c]{c}\omega_{g_{i}^{T}}=\frac{1}{2}d\eta_{i}=\pi^{\ast}(\omega_{h_{i}});\text{
}\Phi_{i}=\pi^{\ast}(J_{i}).\end{array}$

Suppose that $(M_{i},(1-\beta)D^{T},\eta_{i},\xi,\Phi_{i},g_{i},\omega_{i})$
satisfies

$\begin{array}[c]{c}\int_{M}|\mathrm{Ric}_{g_{i}^{T}}^{T}-(1-\beta)[D^{T}]|^{p}\omega_{i}^{n}\wedge\eta\leq\Lambda,\end{array}$

and

$\begin{array}[c]{c}\mathrm{Vol}(\emph{B}_{\xi,g_{i}^{T}}(x_{i},1))\geq\upsilon\end{array}$

for some $p>n,$ $\Lambda>0,$ $\upsilon>0$. Then passing to a subsequence if
necessary, $(M_{i},(1-\beta)D^{T},\Phi_{i},g_{i},x_{i})$ converges in the
Cheeger-Gromov sense to limit length spaces
$(M_{\infty},(1-\beta)D_{\infty}^{T},\Phi_{\infty},d_{\infty},x_{\infty})$ and
the corresponding leave space $(Z_{i},(1-\beta)D,J_{i},h_{i},\pi(x_{i}))$
converges to
$(Z_{\infty},(1-\beta)D_{\infty},J_{\infty},h_{\infty},\pi(x_{\infty}))$ such
that

  1. (1)

for any $r>0$ and $p_{i}\in M_{i}$ with $p_{i}\rightarrow p_{\infty}\in
M_{\infty},$

$\begin{array}[c]{c}\mathrm{Vol}(B_{h_{i}}(\pi(p_{i}),r))\rightarrow\mathcal{H}^{2n}(B_{h_{\infty}}(\pi(p_{\infty}),r))\end{array}$

and

$\begin{array}[c]{c}\mathrm{Vol}(B_{\xi,g_{i}^{T}}(p_{i},r))\rightarrow\mathcal{H}^{2n}(B_{\xi,g_{\infty}^{T}}(p_{\infty},r)).\end{array}$

Moreover,

$\begin{array}[c]{c}\mathrm{Vol}(B(p_{i},r))\rightarrow\mathcal{H}^{2n+1}(B(p_{\infty},r)),\end{array}$

where $\mathcal{H}^{m}$ denotes the $m$-dimensional Hausdorff measure.

  2. (2)

$M_{\infty}$ is a $\mathbb{S}^{1}$-orbibundle over the normal projective
variety $Z_{\infty}=M_{\infty}/\mathcal{F}_{\xi}.$

  3. (3)

$Z_{\infty}=\mathcal{R}\cup\mathcal{S}$ is a normal projective variety and
$\mathcal{S=}\overline{\mathcal{S}}\cup D_{\infty}$ such that
$\overline{\mathcal{S}}$ is a closed singular set of at least codimension two
and $\mathcal{R}$ consists of points whose tangent cones are
$\mathbb{R}^{2n}.$

  4. (4)

The convergence on the regular part of $M_{\infty}$ which is a
$\mathbb{S}^{1}$-principle bundle over $\mathcal{R}$ in the
$(C_{B}^{\alpha}\cap L_{B}^{2,p})$-topology for any $0<\alpha<2-\frac{n}{p}$.

Now by using the convergence theorem in Theorem 3.2, we have the regularity of
the limit space. We define a family $g_{i}^{T}(t)$ of conic Sasaki-Ricci flow
by

$(M,g_{i}^{T}(t))=(M,g^{T}(t_{i}+t))$

for $t\geq-1$ and $t_{i}\rightarrow\infty$ and for
$g_{i}^{T}(t)=\pi^{\ast}(h_{i}(t))$

$(Z,h_{i}(t))=(M,h_{i}(t_{i}+t)).$

Now for the associated transverse Ricci potential $u_{i}(t)$ as in Proposition
3.7, we have

$||u_{i}(t)||_{C^{0}}+||\nabla^{T}u_{i}(t)||_{C^{0}}+||\Delta_{B}u_{i}(t)||_{C^{0}}\leq
C.$

Moreover, it follows from (3.7) that

$\begin{array}[c]{c}\int_{M}|\nabla^{T}\nabla^{T}u_{i}|^{2}\omega(t)^{n}\wedge\eta_{0}\rightarrow
0\end{array}$

as $i\rightarrow\infty.$ Furthermore, by Theorem 3.1 and Theorem 3.2, passing
to a subsequence if necessary, we have at $t=0,$

$(M_{i},(1-\beta)D^{T},g_{i}^{T}(0))\rightarrow(M_{\infty},(1-\beta)D^{T},g_{\infty}^{T},d_{\infty}^{T})$

such that

$(Z_{i},(1-\beta)D,h_{i}(0))\rightarrow(Z_{\infty},(1-\beta)D,h_{\infty},d_{h_{\infty}})$

in the Cheeger-Gromov sense. Moreover,

(3.15) $\begin{array}[c]{c}(g_{i}^{T}(0),u_{i}(0))\overset{C_{B}^{\alpha}\cap
L_{B}^{2,p}}{\rightarrow}(g_{\infty}^{T},u_{\infty})\end{array}$

on $(M_{\infty})_{reg}$ which is a $\mathbb{S}^{1}$-principle bundle over
$\mathcal{R}.$ The convergence of $u_{i}(0)$ on the regular part follows from
the elliptic regularity ([Co1], [Co2]) of

$\Delta_{B}u_{i}(0)=n-\mathrm{R}^{T}(g_{i}^{T}(0))\in L_{B}^{p}.$

######  Theorem 3.3.

([CCHLW, Theorem 10]) Suppose (3.15) holds, then $g_{\infty}^{T}$ is smooth
and satisfies the conic Sasaki-Ricci soliton equation

(3.16)
$\mathrm{Ric}^{T}(g_{\infty}^{T})-(1-\beta)[D^{T}]+\mathrm{Hess}^{T}(u_{\infty})=\beta
g_{\infty}^{T}$

on $(M_{\infty})_{reg}$ which is a $\mathbb{S}^{1}$-bundle over $\mathcal{R}.$
Moreover, $\Phi_{\infty}$ is smooth and $g_{\infty}^{T}$ is Kähler with
respect to $\Phi_{\infty}=\pi^{\ast}(J_{\infty}).$

###### Proof.

Since all $g_{\infty}^{T}$ and $u_{\infty}$ are basic, in the basic harmonic
foliation normal coordinates $\\{x,x^{1},x^{2},\cdots,x^{2n}\\}$ with
$z^{i}=x^{i}+\sqrt{-1}x^{n+i}$, the conic Sasaki-Ricci soliton equation (3.16)
is equivalent to

(3.17)
$\begin{array}[c]{ll}&(g^{T})^{\alpha\beta}\frac{\partial^{2}g_{\gamma\delta}^{T}}{\partial
x^{\beta}\partial x^{\alpha}}\\\ =&\frac{\partial^{2}u_{\infty}}{\partial
x^{\gamma}\partial x^{\delta}}+Q(g^{T},\partial
g^{T})_{\gamma\delta}+T(g^{-1},\partial g^{T},\partial
u)_{\gamma\delta}-(1-\beta)\overset{0}{g}_{\gamma\delta}^{T}-\beta
g_{\gamma\delta}^{T}.\end{array}$

By (3.7), (3.17) holds in $L_{B}^{2}((M_{\infty})_{reg})$. But
$g_{\infty}^{T}$ and $u_{\infty}$ are $L_{B}^{2,p}$, then (3.17) holds in
$L_{B}^{p}((M_{\infty})_{reg})$ too. On the other hand, by (3.8) and (3.16),
we have that

(3.18)
$\begin{array}[c]{c}g_{\alpha\beta}^{T}\frac{\partial^{2}u_{\infty}}{\partial
x^{\beta}\partial x^{\alpha}}=(g^{T})^{\alpha\beta}\frac{\partial
u_{\infty}}{\partial x^{\alpha}}\frac{\partial u_{\infty}}{\partial
x^{\alpha}}-2u_{\infty}+2a_{\infty}\end{array}$

in $L_{B}^{p}((M_{\infty})_{reg}).$ Then a bootstrap argument as in ([Pe]) to
the elliptic systems (3.17) and (3.18) shows that $g_{\infty}^{T}$ and
$u_{\infty}$ are smooth on $(M_{\infty})_{reg}$. The elliptic regularity shows
that $\Phi_{\infty}$ is smooth since
$\nabla_{g_{\infty}^{T}}^{T}\Phi_{\infty}=0.$ ∎

By applying the argument as in [TZ, Theorem 1.2] (also [CCHLW]) to the normal
orbifold variety $Z_{\infty}$ which is mainly depended on the Perelman’s
pseudolocality theorem ([P1]), we have the smooth convergence of the conic
Sasaki-Ricci flow on the regular set $(M_{\infty})_{reg}$ which is a
$\mathbb{S}^{1}$-principle bundle over $\mathcal{R}$ and $\mathcal{R}$ is the
regular set of $Z_{\infty}$:

######  Theorem 3.4.

The limit $(M_{\infty},(1-\beta)D^{T},d_{\infty})$ is smooth on the regular
set $(M_{\infty})_{reg}$ which is a $\mathbb{S}^{1}$-principle bundle over the
regular set $\mathcal{R}$ of $Z_{\infty}$ and $d_{\infty}^{T}$ is induced by a
smooth conic Sasaki-Ricci soliton $g_{\infty}^{T}$ and $g^{T}(t_{i})$ converge
to $g_{\infty}^{T}$ in the $C^{\infty}$-topology on $(M_{\infty})_{reg}.$
Moreover, the singular set $\overline{\mathcal{S}}$ of $Z_{\infty}$ is the
codimension two orbifold singularities which are klt.

The proofs are straightforward from [CCHLW]. We give the proof here for
completeness.

###### Proof.

Note that Since $g^{T}$ is a transverse Kähler metric and basic. It is evolved
by the conic Sasaki-Ricci flow, then it follows from the standard computations
as in [Co3] and [LZ] that

$\begin{array}[c]{c}\frac{\partial}{\partial
t}\mathrm{Rm}^{T}=\Delta_{B}\mathrm{Rm}^{T}+\mathrm{Rm}^{T}\ast\mathrm{Rm}^{T}+\mathrm{Rm}^{T}.\end{array}$

Now by Perelman’s pseudolocality theorem ([P1], [TZ]) with Proposition 3.7,
there exists $\varepsilon_{0},$ $\delta_{0},$ $r_{0}$ which depend on $p$ as
in the Theorem 3.2 such that for any $(x_{0},t_{0})$, if

(3.19)
$\begin{array}[c]{c}\mathrm{Vol}(B_{\xi,g_{i}^{T}(t_{0})}(x_{0},r))\geq(1-\varepsilon_{0})\mathrm{Vol}(B(0,r))\end{array}$

for some $r\leq r_{0},$ where $\mathrm{Vol}(B(0,r))$ denotes the volume of
Euclidean ball of radius $r$ in $\mathbb{R}^{2n},$ then we have the following
curvature estimate

(3.20)
$\begin{array}[c]{c}|\mathrm{Rm}^{T}|_{g_{i}^{T}}(x,t)\leq\frac{1}{t-t_{0}}\end{array}$

for all $x\in B_{\xi,g_{i}^{T}(t)}(x_{0},\varepsilon_{0}r)$ and $t_{0}<t\leq
t_{0}+\varepsilon_{0}^{2}r^{2}$ and the volume estimate

(3.21)
$\begin{array}[c]{c}\mathrm{Vol}(B_{\xi,g_{i}^{T}(t)}(x_{0},\delta_{0}\sqrt{t-t_{0}}))\geq(1-\eta)\mathrm{Vol}(B(0,\delta_{0}\sqrt{t-t_{0}}))\end{array}$

for $t_{0}<t\leq t_{0}+\varepsilon_{0}^{2}r^{2}$ and $\eta\leq\varepsilon_{0}$
is the constant such that the $C^{\alpha}$ harmonic radius at $x_{0}$ is
bounded below by $\delta_{0}\sqrt{t-t_{0}}.$

As in Theorem 3.2, the metric $g_{i}^{T}(0)$ converges to $g_{\infty}^{T}$ in
the $(C^{\alpha}\cap L_{B}^{2,p})$-topology on $\mathcal{R}$. Now it is our
goal to show that the metric $g_{i}^{T}(0)$ converges smoothly to
$g_{\infty}^{T}$. For $0<r\leq r_{0}$ and $t\geq-1,$ define

$\Omega_{r,i,t}:=\\{x\in M\text{ }|\text{ (\ref{e1} ) {holds on}
}B_{\xi,g_{i}^{T}(t)}(x,t)\\}.$

Then (3.21) implies that

$\Omega_{r,i,t}\subset\Omega_{\delta_{0}\sqrt{s},i,t+s}$

for $0<s\leq\varepsilon_{0}^{2}r^{2}.$

Let $r_{j}$ to be a decreasing sequence of radii such that $r_{j}\rightarrow
0$ and $t_{j}=-\varepsilon_{0}r_{j}$. Then by applying [TZ, (3.42)], we may
assume that

$\Omega_{r_{j},i,t_{j}}\subset\Omega_{r_{j+1},i,t_{j+1}}.$

Then by (3.20)

(3.22)
$\begin{array}[c]{c}||\mathrm{Rm}^{T}||_{g_{i}^{T}(t)}(x,t)\leq\frac{1}{t-t_{j}}\end{array}$

for all $(x,t)$ with

$\begin{array}[c]{c}d_{g_{i}^{T}(t)}^{T}(x,\Omega_{r_{j},i,t_{j}})\leq\varepsilon_{0}r_{j},\text{
}t_{j}<t\leq 0.\end{array}$

By Shi’s derivative estimate [Shi] to the curvature, there exists a sequence
of constants $C_{k,j,i}$ such that

(3.23)
$\begin{array}[c]{c}||(\nabla^{T})^{k}\mathrm{Rm}^{T}||_{g_{i}^{T}(0)}(x,t)\leq
C_{k,j,i}\end{array}$

on $\Omega_{r_{j},i,t_{j}}.$ Then passing to a subsequence if necessary, one
can find a subsequence $\\{i_{j}\\}$ of $\\{j\\}$ such that

$\begin{array}[c]{c}(\Omega_{r_{j},i_{j},t_{j}},g_{i_{j}}^{T}(t_{j}))\overset{C^{\alpha}}{\rightarrow}(\overline{\Omega},g_{\overline{\Omega}}^{T})\end{array}$

and

$\begin{array}[c]{c}(\Omega_{r_{j},i_{j},t_{j}},g_{i_{j}}^{T}(0))\overset{C^{\infty}}{\rightarrow}(\Omega,g_{\Omega}^{T}),\end{array}$

where $(\overline{\Omega},g_{\overline{\Omega}}^{T})$ and
$(\Omega,g_{\Omega}^{T})$ are smooth Riemannian manifolds and

$(\Omega,g_{\Omega}^{T})\text{ {is isometric to}
}((M_{\infty})_{reg},d_{\infty}^{T}).$

On the other hand, as in Theorem 3.2, we may also have

$\begin{array}[c]{c}(M,g_{i_{j}}^{T}(t_{j}))\overset{d_{G,H}^{T}.}{\rightarrow}(\overline{M}_{\infty},\overline{d}_{\infty}^{T})\end{array}$

with $\overline{Z}_{\infty}=\overline{\mathcal{R}}\cup\overline{\mathcal{S}}.$
Then

(3.24) $(M_{\infty})_{reg}\text{ is the }\mathbb{S}^{1}\text{-principle bundle
over }\mathcal{R}$

and

(3.25) $(\overline{M}_{\infty})_{reg}\text{ is the
}\mathbb{S}^{1}\text{-principle bundle over }\overline{\mathcal{R}}.$

Moreover, by the continuity of volume under the Cheeger-Gromov convergence
([TZ, Claim 3.7]), we have

(3.26) $(\overline{\Omega},g_{\overline{\Omega}}^{T})\text{ is isometric to
}((\overline{M}_{\infty})_{reg},\overline{d}_{\infty}^{T})$

and then follows from (3.22) as in [TZ, Claim 3.8] that

(3.27) $(\overline{\Omega},g_{\overline{\Omega}}^{T})\text{ is also isometric
to }(\Omega,g_{\Omega}^{T}).$

Finally (3.24), (3.25), (3.26) and (3.27) imply the metric $g_{i}^{T}(0)$
converges smoothly to $g_{\infty}^{T}$ on $\mathcal{R}$. ∎

Now we work on the partial $C^{0}$-estimate under the conical Sasaki-Ricci
flow. For the solution $(M,\xi_{0},\eta_{0},\omega(t),g^{T}(t))$ of the
conical Sasaki-Ricci flow and the line bundle $(L,h(t)),$

$L:=(K_{M}^{T})^{-1}-(1-\beta)[D^{T}]$

with the basic Hermitian metric $h(t)=\omega^{n}(t),$ we work on the evolution
of the basic transverse holomorphic line bundle $(L^{m},h^{m}(t))$ for a large
integer $m$ such that $L^{m}$ is very ample. We consider the basic embedding
([CCHLW]) which is $\mathbb{S}^{1}$-equivariant with respect to the weighted
$\mathbb{C}^{\ast}$-action in $\mathbb{C}^{N_{m}+1}$

$\Psi:M\rightarrow(\mathbb{CP}^{N_{m}},\omega_{FS})$

defined by the orthonormal basic transverse holomorphic section
$\\{s_{0},s_{1},\cdots,s_{N}\\}$ in $H_{B}^{0}(M,L^{m})$ with $N_{m}=\dim
H_{B}^{0}(M,L^{m})-1$ with

$\begin{array}[c]{c}\int_{M}(S_{i},S_{j})_{h_{t}^{m}}\omega_{t}^{n}\wedge\eta_{0}=\delta_{ij}.\end{array}$

Define

(3.28)
$\begin{array}[c]{c}\mathcal{F}_{m}(x,t):=\sum_{\alpha=0}^{N_{m}}||S(x)||_{h_{t}^{m}}^{2}.\end{array}$

We say that, under the conic Sasaki-Ricci flow,
$\\{(M(t),(1-\beta)D^{T},\omega(t),g^{T}(t)):0\leq t<\infty\\}$ has a partial
$C^{0}$-estimate if $L^{m}$ is very ample for a large integer $m$ and
$F_{m}(x,t)$ is a uniformly bounded below function on $M\times[0,\infty).$

As in our previous estimate ([CCHLW]), based by the uniform bound of the
Sobolev constant (3.12) for the basic function along the conic Sasaki-Ricci
flow and Hörmander’s $L^{2}$-estimate to $\overline{\partial}_{B}$-operator on
basic $(0,1)$-forms ([CHLL, Proposition 2.1]) on $H_{B}^{0}(M.L^{m})$, it
follows straightforward from Theorem 3.2, Theorem 3.4, ([DS]), ([T5, (5.4)]),
([D2]) and ([D2]) that

######  Theorem 3.5.

([CCHLW]) Suppose (3.15) holds, we have

(3.29) $\begin{array}[c]{c}\inf_{t_{i}}\inf_{x\in
M}\mathcal{F}_{m}(x,t_{i})>0\end{array}$

for a sequence of $m\rightarrow\infty.$

As a consequence of the first structure theorem for Sasakian manifolds and
Theorem 3.5, the Gromov-Hausdorff limit $Z_{\infty}$ is a orbifold variety
embedded in some $\mathbb{CP}^{N}$ and the singular set
$\overline{\mathcal{S}}$ is a subvariety ([DS], [TW1]). Then one can refine
the regularity of Theorem 3.4 as following:

######  Corollary 3.3.

Let $(M,(1-\beta)D^{T},\xi,\eta_{0},g_{0})$ be a compact transverse log Fano
regular Sasakian manifold of dimension five and
$(Z_{0}=M/\mathcal{F}_{\xi},(1-\beta)D,h_{0},\omega_{h_{0}})$ denote the space
of leaves of the characteristic foliation which is a projective smooth
variety. Then under the conic Sasaki-Ricci flow, $(M_{\infty},(1-\beta)D^{T})$
is a $\mathbb{S}^{1}$-orbidbundle over $(Z_{\infty},(1-\beta)D)$ which is a
normal projective variety and the singular subvariety $\overline{\mathcal{S}}$
of $Z_{\infty}$ is a codimension two orbifold klt singularities.

Finally, Theorem 1.1 follows easily from Theorem 3.2, Theorem 3.3, and
Corollary 3.3.

##  4\. The Transverse Log K-Polystable

###  4.1. Log Sasaki–Donaldson-Futaki Invariant

All transverse quantities on Sasakian manifolds such as log Sasaki-Donaldson-
Futaki invariant, log Sasaki-Mabuchi $K$-energy can be viewed as their Kähler
counterparts restricted on basic forms and transverse Kähler structure. That
is, all the integrands are only involved with the transverse Kähler structure.
Hence, under the twisted or conic Sasaki-Ricci flow, the Reeb vector field and
the transverse holomorphic structure are both invariant, and the metrics are
bundle-like. Furthermore, when one applies integration by parts, the
expressions involved behave essentially the same as in the Kähler case.

We will adapt the notions as in [D2], [Li] and [LZ] in our Sasakian setting.
Recall that, for the Hamiltonian holomorphic vector field $V,$
$d\pi_{\alpha}(V)$ is a holomorphic vector field on $V_{\alpha}$ and the
complex valued Hamiltonian function $u_{V}:=\sqrt{-1}\eta(V)$ satisfies

$\begin{array}[c]{c}\overline{\partial}_{B}u_{V}=-\sqrt{-1}i_{V}(\frac{1}{2}d\eta).\end{array}$

######  Definition 4.1.

The ordinary Sasaki-Futaki invariant is defined by

$\begin{array}[c]{c}f_{M}(V):=-n\int_{M}u_{V}(\mathrm{Ric}_{\omega}^{T}-\omega)\omega^{n-1}\wedge\eta\end{array}$

or

$\begin{array}[c]{c}f_{M}(V):=-\int_{M}u_{V}(\mathrm{R}_{\omega}^{T}-n)\omega^{n}\wedge\eta.\end{array}$

Let $(M,\eta,\left(1-\beta\right)D^{T},\widehat{\omega})$ be a regular
Sasakian $(2n+1)$-manifold with the transverse cone angle $2\pi\beta$ along
the divisor $D^{T}\thicksim-K_{M}^{T}.$ Then the scalar curvature of
$\widehat{\omega}$ on $M\backslash D^{T}$ is defined by

$\begin{array}[c]{c}\mathrm{R}^{T}(\widehat{\omega})=\widehat{g}^{i\overline{j}}\widehat{\mathrm{R}}_{i\overline{j}}^{T}=n\int_{M}\mathrm{Ric}^{T}(\widehat{\omega})\wedge\widehat{\omega}^{n-1}\wedge\eta/\int_{M}\widehat{\omega}^{n}\wedge\eta\end{array}$

with

$\begin{array}[c]{c}\widehat{\omega}^{n}\wedge\eta=\frac{\omega_{0}^{n}\wedge\eta}{||S^{T}||^{2(1-\beta)}}\end{array}$

for some smooth Sasaki metric $\omega_{0}$ in $c_{1}^{B}(M)$. Here $S^{T}$ is
the defining section of $D^{T}.$ Define a norm $||s^{T}||$ for any basic
section $s^{T}\in\Gamma(M,L^{T},h)$ given by

$||s^{T}||_{h}=\sqrt{h_{\alpha}}|s_{\alpha}^{T}|$

for the orthonormal CR-holomorphic basic section $\\{s_{\alpha}^{T}\\}$ in
$H_{B}^{0}(M,L^{T})$ on $U_{\alpha}$ and

$c_{1}^{B}(L^{T},h)=-\sqrt{-1}\partial_{B}\overline{\partial}_{B}\log
h_{\alpha}\text{\ }\mathrm{on}\text{ }U_{\alpha}.$

Hence by the CR Poincáre-Lelong formula

$\sqrt{-1}\partial_{B}\overline{\partial}_{B}\log||s^{T}||_{h}^{2}=-c_{1}^{B}(L^{T},h)+[Z_{s^{T}}]$

for any CR-holomorphic line bundle $(L^{T},h).$ In particular

$\sqrt{-1}\partial_{B}\overline{\partial}_{B}\log||s^{T}||^{2}=-c_{1}^{B}([D^{T}])+|D^{T}|.$

Hence

(4.1)
$\begin{array}[c]{ccl}\mathrm{Ric}^{T}(\widehat{\omega})&=&\mathrm{Ric}^{T}(\omega_{0})+\sqrt{-1}\partial_{B}\overline{\partial}_{B}\log||S^{T}||^{2(1-\beta)}\\\
&=&\mathrm{Ric}^{T}(\omega_{0})-(1-\beta)c_{1}^{B}([D^{T}])+(1-\beta)|D^{T}|\end{array}$

and the scalar curvature of $\widehat{\omega}$ on $M\backslash D^{T}$

$\begin{array}[c]{c}\mathrm{R}^{T}(\widehat{\omega})=n\int_{M}[\mathrm{Ric}^{T}(\omega_{0})-(1-\beta)c_{1}^{B}([D^{T}])]\wedge\widehat{\omega}^{n-1}\wedge\eta/\int_{M}\widehat{\omega}^{n}\wedge\eta.\end{array}$

Set

$\begin{array}[c]{c}\kappa:=\int_{M}[c_{1}^{B}(M)-(1-\beta)c_{1}^{B}([D^{T}])]\wedge
c_{1}^{B}(M)^{n-1}\wedge\eta/\int_{M}c_{1}^{B}(M)^{n}\wedge\eta\end{array}$

which is only depends on basic cohomological classes. We define

$\begin{array}[c]{c}\mathrm{Vol}(M):=\int_{M}\frac{c_{1}^{B}(M)^{n}\wedge\eta}{n!}\text{
\ }\mathrm{and}\text{ \
}\mathrm{Vol}(D^{T}):=\int_{D^{T}}\frac{c_{1}^{B}(M)^{n-1}\wedge\eta}{(n-1)!}.\end{array}$

Moreover, since $\mathrm{Vol}(D^{T})=n\mathrm{Vol}(M)$, it follows from [Li]
that if $R^{T}(\widehat{\omega})$ is constant $n\kappa$:

$\begin{array}[c]{ccl}n\kappa&=&n\int_{M}[c_{1}^{B}(M)-(1-\beta)c_{1}^{B}([D^{T}])]\wedge
c_{1}^{B}(M)^{n-1}\wedge\eta/\int_{M}c_{1}^{B}(M)^{n}\wedge\eta\\\
&=&\\{n\int_{M}c_{1}^{B}(M)^{n}\wedge\eta-(1-\beta)n\int_{M}[c_{1}^{B}([D^{T}])\wedge
c_{1}^{B}(M)^{n-1}\wedge\eta]\\}/\int_{M}c_{1}^{B}(M)^{n}\wedge\eta\\\
&=&n-\frac{(1-\beta)\mathrm{Vol}(D^{T})}{\mathrm{Vol}(M)}\end{array}$

and then

$\kappa=\beta\text{.}$

######  Definition 4.2.

If $\widehat{\omega}$ is a conic Sasaki metric with the transverse conic angle
$2\pi\beta$ along the divisor $D^{T}\thicksim-K_{M}^{T}$, one can define log
Sasaki-Futaki invariant

$\begin{array}[c]{c}f_{M,(1-\beta)D^{T}}(V):=-n\int_{M}\widehat{u}_{V}(\mathrm{Ric}_{\overset{\wedge}{\omega}}^{T}-\beta\widehat{\omega})\wedge\widehat{\omega}^{n-1}\wedge\eta,\end{array}$

or

$\begin{array}[c]{c}f_{M,(1-\beta)D^{T}}(V):=-\int_{M}\widehat{u}_{V}(\mathrm{R}_{\widehat{\omega}}^{T}-n\beta)\wedge\widehat{\omega}^{n}\wedge\eta,\end{array}$

or

$\begin{array}[c]{c}f_{M,(1-\beta)D^{T}}(V):=\int_{M}V(u_{\widehat{\omega}})\widehat{\omega}^{n}\wedge\eta.\end{array}$

One can also calculate the ordinary Futaki invariant using the conic metric
$\widehat{\omega}$. Indeed, for choosing the smooth transverse Kähler metric
$\omega$, then it follows from (4.1) that we can recapture the log Sasaki-
Futaki invariant defined by Donaldson:

######  Definition 4.3.

([D2], [Li]) The log Sasaki-Donaldson-Futaki invariant is defined by

$\begin{array}[c]{l}f_{M,(1-\beta)D^{T}}(V):=f_{M}(V)+(1-\beta)[\int_{D^{T}}u_{V}\omega^{n-1}\wedge\eta-n\int_{M}u_{V}\omega^{n}\wedge\eta].\end{array}$

###  4.2. Log Sasaki-Mabuchi $K$-energy

For $d\mu_{\phi}=(d\eta_{\phi})^{n}\wedge\eta_{0}$ and
$\eta_{\phi}=\eta_{0}+d_{B}^{C}\phi$ with $\phi_{t}$ is any path with
$\phi_{0}=c$ and $\phi_{1}=\phi,$ we recall that

$\begin{array}[c]{c}I_{\eta_{0}}(\phi):=\frac{1}{V}\int_{M}\phi(d\mu_{0}-d\mu_{\phi})\end{array}$

and

$\begin{array}[c]{c}J_{\eta_{0}}(\phi):=\frac{1}{V}\int_{0}^{1}\int_{M}\overset{\cdot}{\phi}_{t}(d\mu_{0}-d\mu_{\phi_{t}})dt.\end{array}$

######  Definition 4.4.

Let $u_{\omega_{0}}$ be the transverse twisted Ricci potential of
$\omega_{0}$, that is

$\begin{array}[c]{c}-\mathrm{Ric}^{T}(\omega_{0})+\beta\omega_{0}+\theta=\sqrt{-1}\partial_{B}\overline{\partial}_{B}u_{\omega_{0}}.\end{array}$

  1. (1)

We recall that the twisted Sasaki-Mabuchi $K$-energy
$K_{\omega_{0},\theta}(\phi)$ on a compact transverse log Fano Sasakian
$(2n+1)$-manifold $(M,D^{T})$ is defined by

$\begin{array}[c]{ccl}K_{\omega_{0},\theta}(\phi)&:=&-\beta(I_{\eta_{0}}(\phi)-J_{\eta_{0}}(\phi))-\frac{1}{V}\int_{M}u_{\omega_{0}}(d\mu_{0}-d\mu_{\phi})\\\
&&+\frac{1}{V}\int_{M}\log\frac{\omega_{{}_{\phi}}^{n}\wedge\eta_{0}}{\omega_{0}^{n}\wedge\eta_{0}}d\mu_{\phi}.\end{array}$

  2. (2)

We also denote that
$K_{\omega_{0},\theta}(\phi)=K_{\omega_{0},\theta}(\omega_{1}).$ As in (4.2),
we also denote that

$K_{\omega_{1},\theta}(\omega_{2}):=K_{\omega_{1},(1-\beta)D^{T}}(\omega_{2}).$

It follows easily from the definition that

######  Lemma 4.1.

([LZ], [Li], [CJ])

  1. (1)

$K_{\omega_{0},(1-\beta)D^{T}}$ is independent of the path $\phi_{t}.$
Furthermore, it satisfies the cocycle condition

$\begin{array}[c]{c}K_{\omega_{1},(1-\beta)D^{T}}(\omega_{2})+K_{\omega_{2},(1-\beta)D^{T}}(\omega_{3})=K_{\omega_{1},(1-\beta)D^{T}}(\omega_{3}).\end{array}$

  2. (2)

We also can integrate the log Sasaki-Futaki invariant to get the log Sasaki-
Mabuchi $K$-energy. That is, for $\theta=(1-\beta)D^{T}$ and
$\omega_{t}=\omega+\sqrt{-1}\partial_{B}\overline{\partial}_{B}\phi_{t}$

(4.2) $\begin{array}[c]{ccl}\frac{\partial}{\partial
t}K_{\eta_{0},(1-\beta)D^{T}}(\phi_{t})&=&-\int_{M}\overset{.}{\phi}_{t}[\mathrm{R}^{T}(\omega_{t})-n]\omega_{t}^{n}\wedge\eta\\\
&&+(1-\beta)\int_{D^{T}}\overset{.}{\phi}_{t}\omega_{t}^{n-1}\wedge\eta\\\
&&-n(1-\beta)\int_{M}\overset{.}{\phi}_{t}\omega_{t}^{n}\wedge\eta\\\
&=&-\int_{M}\overset{\cdot}{\phi}_{t}(R_{\phi_{t}}^{T}-\beta
n-tr_{\phi_{t}}\theta)\omega_{t}^{n}\wedge\eta.\end{array}$

  3. (3)

$\begin{array}[c]{c}\frac{\partial}{\partial
t}I_{\eta_{0}}(\phi_{t})=\frac{1}{V}\int_{M}\overset{\cdot}{\phi}_{t}(d\mu_{0}-d\mu_{\phi_{t}})-\frac{1}{V}\int_{M}\phi_{t}\Delta_{B}\overset{\cdot}{\phi}_{t}d\mu_{\phi_{t}}.\end{array}$

  4. (4)

$\begin{array}[c]{c}\frac{\partial}{\partial
t}J_{\eta_{0}}(\phi_{t})=\frac{1}{V}\int_{M}\overset{\cdot}{\phi}_{t}(d\mu_{0}-d\mu_{\phi_{t}}).\end{array}$

Now by definition and Lemma 4.1 with the straightforward computation as same
as in [CJ], we have

######  Proposition 4.1.

Under the twisted Sasaki-Ricci flow (3.3), for
$\phi_{\varepsilon}(t)=\varphi_{\varepsilon}(t)+k\chi_{\epsilon}(\epsilon^{2}+\left\|S\right\|_{h}^{2})$

  1. (1)

$\begin{array}[c]{c}\frac{d}{dt}K_{\eta_{0},\theta_{\varepsilon}}(\phi_{\varepsilon}(t))=-\frac{1}{V}\int_{M}|\nabla^{T}u_{\varepsilon}(t)|_{g_{\varepsilon}(t)L}^{2}d\mu_{\phi_{\varepsilon}},\end{array}$

  2. (2)

(4.3)
$\begin{array}[c]{c}\frac{1}{nV}\int_{M}\phi_{\varepsilon}(t)d\mu_{\phi_{\varepsilon
t}}-C\leq
J_{\eta_{0}}(\phi_{\varepsilon}(t))\leq\frac{1}{V}\int_{M}\phi_{\varepsilon}(t)d\mu_{0}+C,\end{array}$

  3. (3)

(4.4)
$\begin{array}[c]{c}\frac{1}{V}\int_{M}\phi_{\varepsilon}(t)d\mu_{0}\leq\frac{n}{V}\int_{M}(-\phi_{\varepsilon}(t))d\mu_{\phi_{\varepsilon
t}}-(n+1)K_{\eta_{0},\theta_{\varepsilon}}(\phi_{\varepsilon}(t))+C.\end{array}$

###  4.3. Transverse Log K-Polystable

By the CR Kodaira embedding theorem for transverse log Fano quasi-regular
Sasakian manifolds, there exists an embedding

$\Psi:M\rightarrow(\mathbb{CP}^{N},\omega_{FS})$

defined by the basic transverse holomorphic section
$\\{s_{0},s_{1},\cdots,s_{N}\\}$ of $H_{B}^{0}(M.(L^{T})^{m})$ which is
$\mathbb{S}^{1}$-equivariant with respect to the weighted
$\mathbb{C}^{\ast}$-action in $\mathbb{C}^{N+1}$ with $N=\dim
H_{B}^{0}(M.(L^{T})^{m})-1$ for a large positive integer $m.$ Here

$L^{T}:=(K_{M}^{T})^{-1}-(1-\beta)D^{T}.$

Since $Z$ is also log Fano, there is an embedding

$\psi_{|mL_{\emph{Z}}|}:Z\rightarrow\mathcal{P(}H^{0}(Z,(L_{\emph{Z}})^{m})).$

Define

$\Psi_{|m(L^{T})|}=\psi_{|mL_{\emph{Z}}|}\circ\pi$

such that

$\Psi_{|m(L^{T})|}:M\rightarrow\mathcal{P(}H_{B}^{0}(M,(L^{T})^{m}).$

We define

$\mathrm{Diff}^{T}(M)=\\{\sigma\in\mathrm{Diff}(M)\text{\ }|\text{
}\sigma_{\ast}\xi=\xi\text{\ {and}
}\sigma^{\ast}g^{T}=(\sigma^{\ast}g)^{T}\\}$

and

$SL^{T}(N+1,\mathbb{C})=SL(N+1,\mathbb{C})\cap\mathrm{Diff}^{T}(M).$

Any other basis of $H_{B}^{0}(M,(L^{T})^{m})$ gives an embedding of the form
$\sigma^{T}\circ\Psi_{|m(L^{T})|}$ with $\sigma^{T}\in
SL^{T}(N+1,\mathbb{C}).$ Now for any subgroup of the weighted
$\mathbb{C}^{\ast}$-action $G_{0}=\\{\sigma^{T}(t)\\}_{t\in\mathbb{C}^{\ast}}$
of $SL^{T}(N+1,\mathbb{C}),$ there is a unique limiting

$\begin{array}[c]{c}M_{\infty}=\lim_{t\rightarrow
0}\sigma^{T}(t)(M)\subset\mathbb{CP}^{N}\end{array}$

and

$\begin{array}[c]{c}D_{\infty}^{T}=\lim_{t\rightarrow
0}\sigma^{T}(t)(D^{T}).\end{array}$

If $M_{\infty}$ has its leave space $Z_{\infty}=M_{\infty}/\mathcal{F}_{\xi}$
which is a log Fano projective Kähler orbifold with at worst codimension two
orbifold singularities $\overline{S}.$ then for an integer $l>0$ sufficiently
large such that $(L_{\infty}^{T})^{l}$ is very-ample with the
$\mathbb{S}^{1}$-equivariant embedding with respective the weighted
$\mathbb{C}^{\ast}$-action in $\mathbb{C}^{N+1}$

$i:M_{\infty}\rightarrow(\mathbb{CP}^{N},\omega_{FS})$

with the Bergman metric
$\overline{\omega}_{\infty}=\frac{1}{l}i^{\ast}(\omega_{FS}).$

Let $V$ be a Hamiltonian holomorphic vector field whose real part generates
the action by $\sigma^{T}(e^{-s})$. As the previous discussion
$(M_{\infty},(1-\beta)D_{\infty}^{T})$ is transverse log Fano, there is a log
Sasaki-Futaki invariant defined by

$\begin{array}[c]{l}f_{M_{\infty},(1-\beta)D_{\infty}^{T}}(V):=f_{M_{\infty}}(V)+(1-\beta)[\int_{D_{\infty}^{T}}u_{V}\overline{\omega}_{\infty}^{n-1}\wedge\eta_{0}-n\int_{M_{\infty}}u_{V}\overline{\omega}_{\infty}^{n}\wedge\eta_{0}].\end{array}$

Thus one can introduce the Sasaki analogue of the log $K$-polystable on Kähler
manifolds. In particular, if $M$ admits an Sasaki-Einstein, then
$\operatorname{Re}f_{M_{\infty},(1-\beta)D_{\infty}^{T}}(V)$ is nonnegative.

######  Definition 4.5.

Let $(M,\xi,\eta,g,\omega,(1-\beta)D^{T})$ be a compact transverse log Fano
regular Sasakian manifold and its leave space $(Z,h,\omega_{h},(1-\beta)D)$ be
a log Fano smooth projective variety. We say that $M$ is transverse log
$K$-polystable with respect to $(L^{T})^{m}$ if the log Sasaki-Donaldson-
Futaki invariant

$\begin{array}[c]{c}\operatorname{Re}f_{M_{\infty},(1-\beta)D_{\infty}^{T}}(V)\geq
0\text{\ \ {or} \ }\operatorname{Re}f_{Z_{\infty},(1-\beta)D_{\infty}}(X)\geq
0\end{array}$

for any weighted $\mathbb{C}^{\ast}$-action
$G_{0}=\\{\sigma^{T}(t)\\}_{t\in\mathbb{C}^{\ast}}$ of
$SL^{T}(N+1,\mathbb{C})$ with a log Fano normal variety
$Z_{\infty}=M_{\infty}/\mathcal{F}_{\xi}$ and the equality holds if and only
if $M_{\infty}$ is transverse biholomorphic to $M.$ We say that $M$ is
transverse log $K$-polystable if it is transverse log $K$-polystable for all
large positive integer $m$.

With Definition 4.5 in mind, we are ready to show that the log Sasaki-Marbuchi
$K$-energy is bounded from below under the conic Sasaki-Ricci flow.

######  Theorem 4.1.

Let $(M,\xi,\eta_{0},g_{0},(1-\beta)D^{T})$ be a compact regular log
transverse Fano Sasakian manifold of dimension five and
$(Z_{0}=M/\mathcal{F}_{\xi},h_{0},\omega_{h_{0}},(1-\beta)D)$ denote the space
of leaves of the characteristic foliation which is a log Fano projective
Kähler manifold. If $(M,(1-\beta)D^{T},L^{T})$ is transverse log
$K$-polystable, then the log Sasaki-Marbuchi $K$-energy is bounded from below
under the conic Sasaki-Ricci flow

(4.5)
$\begin{array}[c]{c}K_{\omega_{0},(1-\beta)D^{T}}(\omega_{t})\geq-C(g_{0}).\end{array}$

###### Proof.

Firstly, it follows from Proposition 4.1 that
$K_{\omega_{0},(1-\beta)D^{T}}(\omega_{t})$ is non-increasing in $t$. So it
suffices to show a uniform lower bound of

(4.6)
$\begin{array}[c]{c}K_{\omega_{0},(1-\beta)D^{T}}(\omega_{t_{i}})\geq-C.\end{array}$

Now if $(M,(1-\beta)D^{T},L^{T})$ is transverse log $K$-polystable, we fix an
integer $m>0$ sufficiently large such that $(L^{T})^{m}$ is very-ample. We can
define the $\mathbb{S}^{1}$-equivariant embedding
$\Psi_{i}:M\rightarrow(\mathbb{CP}^{N_{m}},\omega_{FS})$ with respective the
weighted $\mathbb{C}^{\ast}$-action in $\mathbb{C}^{N_{m}+1}$ with
$\overline{\omega}_{i}=\frac{1}{m}\Psi_{i}^{\ast}(\omega_{FS})$ so that for
any $i\geq 1,$ there exists a $G_{i}\in SL^{T}(N_{m}+1,\mathbb{C})$ such that
$\Psi_{i}=G_{i}\circ\Psi_{1}.$ It follows from a result by S. Paul that

(4.7)
$\begin{array}[c]{c}K_{\overline{\omega}_{1},(1-\beta)D^{T}}(\overline{\omega}_{i})\geq-C.\end{array}$

By the cocycle condition of the log Sasaki-Mabuchi $K$-energy,

$\begin{array}[c]{l}K_{\omega_{0},(1-\beta)D^{T}}(\omega_{i})+K_{\omega_{i},(1-\beta)D^{T}}(\overline{\omega}_{i})\\\
=K_{\omega_{0},(1-\beta)D^{T}}(\overline{\omega}_{i})\\\
=K_{\omega_{0},(1-\beta)D^{T}}(\overline{\omega}_{1})+K_{\overline{\omega}_{1},(1-\beta)D^{T}}(\overline{\omega}_{i})\end{array}$

and then

$\begin{array}[c]{c}K_{\omega_{0},(1-\beta)D^{T}}(\omega_{i})+K_{\omega_{i},(1-\beta)D^{T}}(\overline{\omega}_{i})\geq-C.\end{array}$

Hence, to show (4.6), we only need to get an upper bound for

(4.8)
$\begin{array}[c]{c}K_{\omega_{i},(1-\beta)D^{T}}(\overline{\omega}_{i})\leq
C.\end{array}$

Let $H_{\omega_{0}}$ be the conic Ricci potential of $\omega_{0}$, that is,
$-\mathrm{Ric}^{T}(\omega_{0})+\beta\omega_{0}+(1-\beta)D^{T}=\sqrt{-1}\partial_{B}\overline{\partial}_{B}H_{\omega_{0}}.$
We recall that the log Sasaki-Mabuchi $K$-energy
$K_{\omega_{0},(1-\beta)D^{T}}(\phi)$ on a compact transverse log Fano
Sasakian $(2n+1)$-manifold $(M,(1-\beta)D^{T})$ is defined by

$\begin{array}[c]{ccl}K_{\omega_{0},(1-\beta)D^{T}}(\phi)&:=&-\beta(I_{\eta_{0}}(\phi)-J_{\eta_{0}}(\phi))-\frac{1}{V}\int_{M}H_{\omega_{0}}(d\mu_{0}-d\mu_{\phi})\\\
&&+\frac{1}{V}\int_{M}\log\frac{\omega_{{}_{\phi}}^{n}\wedge\eta_{0}}{\omega_{0}^{n}\wedge\eta_{0}}d\mu_{\phi}.\end{array}$

For a fixed $m,$ we define

$\begin{array}[c]{c}\overline{\rho}_{i}(x):=\frac{1}{m}\rho_{t_{i},m}(x):=\frac{1}{m}\mathcal{F}_{m}(x,t_{i}).\end{array}$

Here $\mathcal{F}_{m}(x,t_{i})$ as in (3.28). Then

$\begin{array}[c]{c}\omega_{i}=\overline{\omega}_{i}+\sqrt{-1}\partial_{B}\overline{\partial}_{B}\overline{\rho}_{i}.\end{array}$

It follows that the log Sasaki-Mabuchi $K$-energy has the following explicit
expression

(4.9)
$\begin{array}[c]{ll}&K_{\omega_{i},(1-\beta)D^{T}}(\overline{\omega}_{i})\\\
=&\int_{M}\log\frac{\overline{\omega}_{i}^{n}}{\omega_{i}^{n}}\overline{\omega}_{i}^{n}\wedge\eta_{0}+\int_{M}H_{\omega_{i}}(\overline{\omega}_{i}^{n}-\omega_{i}^{n})\wedge\eta_{0}\\\
&-\sqrt{-1}\sum_{k=0}^{n-1}\frac{n-k}{n+1}\int_{M}(\partial_{B}\rho_{i}\wedge\overline{\partial}_{B}\overline{\rho}_{i}\wedge\omega_{i}^{k}\wedge\overline{\omega}_{i}^{n-k-1})\wedge\eta_{0}\\\
\leq&\int_{M}\log\frac{\overline{\omega}_{i}^{n}}{\omega_{i}^{n}}\overline{\omega}_{i}^{n}\wedge\eta_{0}+\int_{M}H_{\omega_{i}}(\overline{\omega}_{i}^{n}-\omega_{i}^{n})\wedge\eta_{0}.\\\
\leq&\int_{M}\log\frac{\overline{\omega}_{i}^{n}}{\omega_{i}^{n}}\overline{\omega}_{i}^{n}\wedge\eta_{0}+C.\end{array}$

Here $H_{\overline{\omega}_{i}}$ is the transverse conic Sasaki-Ricci
potential under the conic Sasaki-Ricci flow and we have used the Perelman
estimate that $|H_{\omega_{i}}|\leq C.$ On the other hand, it follows from the
partial $C^{0}$-estimate and applying the gradient estimate that
$\overline{\omega}_{i}\leq C\omega_{i}.$Therefore (4.9) implies (4.8) and then
we are done. ∎

Now in view of the Kähler counterparts restricted on basic functions and
transverse Kähler structure, we can have some a priori estimates for the
minimizing solution $f_{t}$ of (3.11) under the conic Sasaki-Ricci flow. We
refer to the details estimates as in [L], [TZhu2, Proposition 4.2] and [PSSWe]
which improved the estimates as in Proposition 3.7 if the transverse Sasaki-
Mabuchi $K$-energy is bounded from below.

######  Lemma 4.2.

([L]) If in addition the log Sasaki-Marbuchi $K$-energy is bounded from below
on the space of transverse Kähler potentials, then we have

  1. (1)

$\begin{array}[c]{c}\lim_{t\rightarrow\infty}|u(t)|_{C^{0}(M\backslash(1-\beta)D^{T})}=0,\end{array}$

  2. (2)

$\begin{array}[c]{c}\lim_{t\rightarrow\infty}||\nabla^{T}u(t)||_{C_{B}^{0}(M\backslash(1-\beta)D^{T},g_{t}^{T})}=0,\end{array}$

  3. (3)

$\begin{array}[c]{c}\lim_{t\rightarrow\infty}|\Delta_{B}u(t)|_{C^{0}(M\backslash(1-\beta)D^{T})}=0.\end{array}$

######  Corollary 4.1.

If the log Sasaki-Marbuchi $K$-energy is bounded from below on the space of
transverse Kähler potentials, then there exists a sequence of
$t_{i}\in[i,i+1]$ such that

  1. (1)

$\begin{array}[c]{c}\lim_{t_{i}\rightarrow\infty}|\Delta_{B}f_{t_{i}}|_{L_{B}^{2}(M\backslash(1-\beta)D^{T},g_{t_{I}}^{T})}=0,\end{array}$

  2. (2)

$\begin{array}[c]{c}\lim_{t_{i}\rightarrow\infty}||\nabla^{T}f_{t_{i}}||_{L_{B}^{2}(M\backslash(1-\beta)D^{T},g_{t_{I}}^{T})}=0,\end{array}$

  3. (3)

$\begin{array}[c]{c}\lim_{t_{i}\rightarrow\infty}\int_{M}f_{t_{i}}e^{-f_{t_{i}}}\omega_{g_{t_{I}}^{T}}^{n}\wedge\eta_{0}=0.\end{array}$

Moreover, we have

(4.10)
$\begin{array}[c]{c}\lim_{t\rightarrow\infty}\lambda^{T}(g_{t}^{T})=(2\pi)^{-n}(2n)V=\sup\\{\lambda^{T}(\overline{g}^{T}):\overline{g}^{T}\in
c_{1}^{B}(M,(1-\beta)D^{T})\\}.\end{array}$

It follows from (3.11), Proposition 3.8, Corollary 4.1 and Theorem 4.1 that
$M_{\infty}$ must be conic Sasaki-Einstein. Then

######  Theorem 4.2.

$(M,\xi,\eta_{0},g_{0},(1-\beta)D^{T})$ be a compact transverse log Fano
regular Sasakian manifold of dimension five and
$(Z_{0}=M/\mathcal{F}_{\xi},h_{0},\omega_{h_{0}},(1-\beta)D)$ be the space of
leaves of the characteristic foliation which is a log Fano projective Kähler
manifold. If $(M,(1-\beta)D^{T},L^{T})$ is transverse log $K$-polystable, then
there exists a conic Sasaki-Einstein metric on $(M,(1-\beta)D^{T})$.

###### Proof.

The Lie algebra of all Hamiltonian holomorphic vector fields is reductive
([FOW], [CDS1], [CDS2], [CDS3], [T5]). If $M_{\infty}$ is not equal to $M$,
there is a family of the weighted $\mathbb{C}^{\ast}$-action
$\\{G(s)\\}_{s\in\mathbb{C}^{\ast}}\subset SL^{T}(N_{m}+1,\mathbb{C})$ such
that $\Psi_{s}=G(s)\circ\Psi_{1}$ converges to the
$\mathbb{S}^{1}$-equivariant embedding of $M_{\infty}$ with respect to the
weighted $\mathbb{C}^{\ast}$action in $(\mathbb{CP}^{N_{m}},\omega_{FS}).$
Then the log Sasaki-Donaldson-Futaki invariant
$f_{M_{\infty},(1-\beta)D_{\infty}^{T}}(V)$ of $M_{\infty}$ vanishes. On the
other hand, by the assumption that $M$ is transverse log $K$-polystable, if
$M_{\infty}$ is not equal to $M$, then

$\begin{array}[c]{c}\operatorname{Re}f_{M_{\infty},(1-\beta)D_{\infty}^{T}}>0.\end{array}$

This is a contradiction. Hence $M_{\infty}=M$. Therefore there is a conic
Sasaki-Einstein metric on $(M,(1-\beta)D^{T}).$ ∎

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