# The CR Almost Schur Lemma and the positivity conditions

Stefan Ivanov  University of Sofia, Faculty of Mathematics and Informatics,
blvd. James Bourchier 5, 1164, Sofia, Bulgaria  and Institute of Mathematics
and Informatics, Bulgarian Academy of Sciences<EMAIL_ADDRESS>and
Alexander Petkov  University of Sofia, Faculty of Mathematics and Informatics,
blvd. James Bourchier 5, 1164, Sofia, Bulgaria<EMAIL_ADDRESS>
###### Abstract.

We establish a new version of the CR almost Schur Lemma which gives an
estimation of the pseudohermitian scalar curvature on a compact strictly
pseudoconvex pseudohermitian manifold to be a constant in terms of the norm of
the traceless Webster Ricci tensor and the pseudohermitian torsion under a
certain positivity condition. In the torsion-free case, i.e. for a compact
Sasakian manifold, our positivity condition coincides with the known one and
we obtain a better estimate.

###### Key words and phrases:

CR structure, pseudohermitian structure, CR Lichnerowicz condition, CR Cordes
estimate, Paneitz operator

###### 2010 Mathematics Subject Classification:

53C21, 58J60, 53C17, 35P15, 53C25

###### Contents

  1. 1 Introduction
  2. 2 Pseudohermitian manifolds and the Tanaka-Webster connection
    1. 2.1 Invariant decompositions
    2. 2.2 The Tanaka-Webster connection
    3. 2.3 The Ricci identities for the Tanaka-Webster connection
  3. 3 The positivity conditions, CR Cordes type inequality
  4. 4 Proof of Theorem 1.3
    1. 4.1 Proof of Corollary 1.5
    2. 4.2 Proof of Corollary 1.7
  5. 5 Proof of Corollary 1.6
  6. 6 Appendix
    1. 6.1 The Greenleaf’s CR-Bochner formula
    2. 6.2 The CR-Paneitz operator

##  1\. Introduction

For a compact Riemannian manifold $(M^{n},g)$ of dimension $n\geq 3$ the
famous Schur lemma states that if $(M^{n},g)$ is Einstein, then it has
constant scalar curvature, $S=Const.$ The metric $g$ is said to be Einstein,
if the Ricci tensor is proportional to the metric, $Ric=\frac{S}{n}g.$ A
generalization of the Schur lemma is the result of De Lellis and Topping [9]
that states as follows.

######  Theorem 1.1.

[9, Almost Schur Lemma] Let $(M^{n},g)$ be a compact Riemannian manifold of
dimension $n\geq 3$ with non–negative Ricci tensor, $Ric\geq 0$. Then the
following inequality holds

$\int_{M}(S-\bar{S})^{2}vol_{g}\leq\frac{4n(n-1)}{(n-2)^{2}}\int_{M}\left|Ric-\frac{S}{n}g\right|_{g}^{2}vol_{g},$

where $\bar{S}$ means the average value of the scalar curvature $S$ of $g$.

The equality holds if and only if the manifold is Einstein.

It is also shown in [9] that the positivity condition $Ric\geq 0$ assumed on
the Ricci tensor is essential and can not be dropped.

In the CR case there are known two positivity conditions written in terms of
the Webster Ricci curvature and the pseudohermitian torsion; one is used for
obtaining a lower bound of the first eigenvalue of the sub-Laplacian (see e.g.
[12]), while the other one appears in the CR Cordes type estimate (see [6]).

A CR version of the almost Schur lemma was first established in [7] in terms
of the Tanaka-Webster connection, its Ricci curvature and the Webster torsion.
The key assumption is the positivity condition (1.1) below. We present this
result in real notations.

######  Theorem 1.2.

[7, Theorem 1.2] Let $n\geq 2$ and $(M,J,\theta)$ be a $(2n+1)$-dimensional
compact strictly pseudoconvex pseudohermitian manifold satisfying the
condition

(1.1) $Ric(X,X)+4A(JX,X)=Rc(X,X)+2(n+1)A(JX,X)\geq 0,X\in H.$

Then the following inequality holds

(1.2)
$\int_{M}(S-\bar{S})^{2}\,Vol_{\theta}\leq\frac{4n(n+1)}{(n-1)(n+2)}\int_{M}|Rc_{0}|^{2}\,Vol_{\theta}-8n\int_{M}\sum_{a,b=1}^{2n}A(e_{a},Je_{b})(\nabla^{2}\varphi)_{[-1]}(e_{a},e_{b})\,Vol_{\theta}.$

If the equality holds, then

(1.3)
$\int_{M}(S-\bar{S})^{2}\,Vol_{\theta}=\frac{4n(n+1)}{(n-1)(n+2)}\int_{M}|Rc_{0}|^{2}\,Vol_{\theta}$

and the manifold is CR equivalent to a pseudo-Einstein space.

In the inequalities (1.1) and (1.2), $Ric$ is the Ricci tensor of the Tanaka-
Webster connection, $Rc$ is the Webster Ricci tensor, $A$ is the
pseudohermitian torsion, $S$ and $\overline{S}=\int_{M}S\,Vol_{\theta}$ are
the pseudohermitian scalar curvature and its average value, respectively, and
$Rc_{0}=Rc-\frac{S}{2n}g,\quad|Rc_{0}|^{2}=\sum_{a,b=1}^{2n}Rc_{0}(e_{a},e_{b})Rc_{0}(e_{a},e_{b})$
are the trace-free part of the Webster Ricci tensor and its horizontal norm,
respectively. Finally, $\varphi$ is the unique solution of the sub-elliptic
equation

$\Delta\varphi=S-\overline{S}\quad\textnormal{with}\quad\int_{M}\varphi\,Vol_{\theta}=0.$

The aim of this note is to present another version of the CR almost Schur
lemma with a different positivity condition. We assume the positivity
condition (1.4) below instead of (1.1). In the torsion-free case, i.e. for
Sasakian manifolds, both positivity conditions coincide and we obtain a better
estimate then that following from Theorem 1.2. Our main result is

######  Theorem 1.3.

Let $n\geq 2$ and $(M,J,\theta)$ be a $(2n+1)$-dimensional compact strictly
pseudoconvex pseudohermitian manifold satisfying the condition

(1.4) $Ric(X,X)+6A(JX,X)=Rc(X,X)+2(n+2)A(JX,X)\geq 0,X\in H.$

Then the following inequality holds

(1.5)
$\int_{M}(S-\bar{S})^{2}\,Vol_{\theta}\leq\frac{2n(2n+3)}{(n-1)(n+3)}\int_{M}|Rc_{0}|^{2}\,Vol_{\theta}-8n\int_{M}\sum_{a,b=1}^{2n}A(e_{a},Je_{b})(\nabla^{2}\varphi)_{[-1]}(e_{a},e_{b})\,Vol_{\theta}.$

If the equality holds, then

(1.6)
$\int_{M}(S-\bar{S})^{2}\,Vol_{\theta}=\frac{2n(2n+3)}{(n-1)(n+3)}\int_{M}|Rc_{0}|^{2}\,Vol_{\theta}$

and the manifold is CR equivalent to a pseudo-Einstein space.

######  Remark 1.4.

Note that the expression in the left-hand side of (1.1) is precisely the CR-
Lichnerowicz condition used to find a lower bound of the first eigenvalue of
the sub-Laplacian (see [12, 18, 8, 4, 5, 14]), while the expression in the
left-hand side of(1.4) appears in the CR Cordes type a priori inequality
between the (horizontal) Hessian and the sub-Laplacian of a function, derived
in [6, Theorem 1], see Theorem 3.1 below.

The second Bianchi identity (4.4) below shows that a compact pseudo-Einstein
space has constant pseudohermitian scalar curvature if and only if the next
condition holds

(1.7) $(\nabla_{e_{b}}\nabla_{e_{a}}A)(e_{a},Je_{b})=0,$

and it seems natural to assume the condition (1.7) in order to have constant
pseudohermitian scalar curvature.

######  Corollary 1.5.

If, in addition to the conditions of Theorem 1.3, we suppose the equality
(1.7) holds, then

(1.8)
$\int_{M}(S-\bar{S})^{2}\,Vol_{\theta}\leq\frac{2n(2n+3)}{(n-1)(n+3)}\int_{M}|Rc_{0}|^{2}\,Vol_{\theta}.$

If we have equality in (1.8) then the compact pseudohermitian manifold is
pseudo-Einstein with constant pseudohermitian scalar curvature.

The next result slightly improves [7, Corollary 1.3].

######  Corollary 1.6.

If, in addition to the conditions of Theorem 1.2, we suppose the equality
(1.7) holds, then

(1.9)
$\int_{M}(S-\bar{S})^{2}\,Vol_{\theta}\leq\frac{4n(n+1)}{(n-1)(n+2)}\int_{M}|Rc_{0}|^{2}\,Vol_{\theta}.$

If we have equality in (1.9) then the compact pseudohermitian manifold is
pseudo-Einstein with constant pseudohermitian scalar curvature.

In the torsion-free case we get from Corollary 1.5

######  Corollary 1.7.

Let $n\geq 2$ and $(M,J,\theta)$ be a $(2n+1)$-dimensional compact torsion-
free strictly pseudoconvex pseudohermitian manifold, i.e. a Sasakian manifold,
with non-negative Webster Ricci tensor, $Rc\geq 0$. Then the following
inequality holds

(1.10)
$\int_{M}(S-\bar{S})^{2}\,Vol_{\theta}\leq\frac{2n(2n+3)}{(n-1)(n+3)}\int_{M}|Rc_{0}|^{2}\,Vol_{\theta}.$

If the equality in (1.10) holds, then the compact Sasakian space is pseudo-
Einstein with constant pseudohermitian scalar curvature and therefore it is a
Riemannian Sasaki $\eta$-Einstein space D-homothetic to a Riemannian Sasaki-
Einstein space.

######  Remark 1.8.

In the torsion-free case, $A=0$, the positivity assumptions (1.1) and (1.4)
coincide and because the number $\frac{2n(2n+3)}{(n-1)(n+3)}$ is smaller than
the number $\frac{4n(n+1)}{(n-1)(n+2)}$,we get a better estimate for Sasakian
manifolds than the one following from [7, Theorem 1.2], i.e. Theorem 1.2
above.

In the proof of Crollary 1.6 we also present a proof of Theorem 1.2, [7,
Theorem 1.2].

In the Appendix we record for self-sufficiency some of the results of [11, 12,
16] in real variables (see also [14, Appendix]) including the Greenleaf’s CR
Bochner formula [12], the CR Paneitz operator and its non-negativity for $n>1$
[11].

######  Convention 1.9.

  1. a)

We shall use $X,Y,Z,U$ to denote horizontal vector fields, i.e. $X,Y,Z,U\in
H$.

  2. b)

$\\{e_{1},\dots,e_{2n}\\}$ denotes a local orthonormal basis of the horizontal
space $H$.

  3. c)

The summation convention over repeated vectors from the basis
$\\{e_{1},\dots,e_{2n}\\}$ will be used. For example, for a (0,4)-tensor $P$,
the formula $k=P(e_{b},e_{a},e_{a},e_{b})$ means
$k=\sum_{a,b=1}^{2n}P(e_{b},e_{a},e_{a},e_{b}).$

Acknowledgments The research of both authors is partially supported by
Contract DH/12/3/12.12.2017, Contract 80-10-161/05.04.2021 with the Sofia
University ”St. Kliment Ohridski”, and the National Science Fund of Bulgaria,
National Scientific Program ”VIHREN”, Project No. KP-06-DV-7.

##  2\. Pseudohermitian manifolds and the Tanaka-Webster connection

In this section we will briefly review the basic notions of the
pseudohermitian geometry of a CR manifold. Also, we recall some results (in
their real form) from [16, 20, 22, 23], see also [10, 14, 15], which we will
use in this paper.

A CR manifold is a smooth manifold $M$ of real dimension 2n+1, with a fixed
n-dimensional complex sub-bundle $\mathcal{H}$ of the complexified tangent
bundle $\mathbb{C}TM$ satisfying $\mathcal{H}\cap\overline{\mathcal{H}}=0$ and
$[\mathcal{H},\mathcal{H}]\subset\mathcal{H}$. If we let
$H=Re\,\mathcal{H}\oplus\overline{\mathcal{H}}$, the real sub-bundle $H$ is
equipped with a formally integrable almost complex structure $J$. We assume
that $M$ is oriented and there exists a globally defined compatible contact
form $\theta$ such that ${H}=Ker\,\theta$. In other words, the hermitian
bilinear form

$2g(X,Y)=-d\theta(JX,Y)$

is non-degenerate. The CR structure is called strictly pseudoconvex if $g$ is
a positive definite tensor on $H$. The vector field $\xi$ dual to $\theta$
with respect to $g$ satisfying $\xi\lrcorner d\theta=0$ is called the Reeb
vector field. The almost complex structure $J$ is formally integrable in the
sense that

$([JX,Y]+[X,JY])\in{H}$

and the Nijenhuis tensor

$N^{J}(X,Y)=[JX,JY]-[X,Y]-J[JX,Y]-J[X,JY]=0.$

A CR manifold $(M,\theta,g)$ with a fixed compatible contact form $\theta$ is
called _a pseudohermitian manifold_. In this case the 2-form

$d\theta_{|_{{H}}}:=2\omega$

is called the fundamental form. Note that the contact form is determined up to
a conformal factor, i.e. $\bar{\theta}=\nu\theta$ for a positive smooth
function $\nu$ defines another pseudohermitian structure called pseudo-
conformal to the original one.

###  2.1. Invariant decompositions

As usual any endomorphism $\Psi$ of $H$ can be decomposed with respect to the
complex structure $J$ uniquely into its $U(n)$-invariant $(2,0)+(0,2)$ and
$(1,1)$ parts. In short we will denote these components correspondingly by
$\Psi_{[-1]}$ and $\Psi_{[1]}$. Furthermore, we shall use the same notation
for the corresponding $2$-tensor, $\Psi(X,Y)=g(\Psi X,Y)$. Explicitly,
$\Psi=\Psi_{[1]}+\Psi_{[-1]}$, where

(2.1)
${}\Psi_{[1]}(X,Y)=\frac{1}{2}\left[\Psi(X,Y)+\Psi(JX,JY)\right],\qquad\Psi_{[-1]}(X,Y)=\frac{1}{2}\left[\Psi(X,Y)-\Psi(JX,JY)\right].$

The above notation is justified by the fact that the $(2,0)+(0,2)$ and $(1,1)$
components are the projections on the eigenspaces of the operator

$\Upsilon=\ J\otimes J,\quad(\Upsilon\Psi)(X,Y)\overset{def}{=}\Psi(JX,JY),$

corresponding, respectively, to the eigenvalues $-1$ and $1$. Note that both
the metric $g$ and the 2-form $\omega$ belong to the [1]-component, since
$g(X,Y)=g(JX,JY)$ and $\omega(X,Y)=\omega(JX,JY)$. Furthermore, the two
components are orthogonal to each other with respect to $g$.

###  2.2. The Tanaka-Webster connection

The Tanaka-Webster connection [20, 22, 23] is the unique linear connection
$\nabla$ with torsion $T$ preserving a given pseudohermitian structure, i.e.,
it has the properties

(2.2) $\nabla\xi=\nabla J=\nabla\theta=\nabla g=0,$ (2.3) $\displaystyle
T(X,Y)=d\theta(X,Y)\xi=2\omega(X,Y)\xi,\quad T(\xi,X)\in{H},$ $\displaystyle
g(T(\xi,X),Y)=g(T(\xi,Y),X)=-g(T(\xi,JX),JY).$

It is well known that the endomorphism $T(\xi,\cdot)$ is the obstruction for a
pseudohermitian manifold to be Sasakian. The symmetric endomorphism
$T_{\xi}:{H}\longrightarrow{H}$ is denoted by $A$, $A(X,Y):=T(\xi,X,Y)$, and
it is called _the torsion of the pseudohermitian manifold_ or _pseudohermitian
torsion._ The pseudohermitian torsion $A$ is completely trace-free satisfying

(2.4) $A(e_{a},e_{a})=A(e_{a},Je_{a})=0,\quad A(X,Y)=A(Y,X)=-A(JX,JY).$

Let $R$ be the curvature of the Tanaka-Webster connection. The pseudohermitian
Ricci tensor $Ric$, the pseudohermitian scalar curvature $S$ and the
pseudohermitian Ricci 2-form $\rho$ are defined by

$Ric(A,B)=R(e_{a},A,B,e_{a}),\quad
S=Ric(e_{a},e_{a}),\quad\rho(A,B)=\frac{1}{2}R(A,B,e_{a},Je_{a}).$

We summarize below the well known properties of the curvature $R$ of the
Tanaka-Webster connection [22, 23, 16] using real expression, see also [10,
15, 14, 13].

(2.5) $\displaystyle Ric(X,Y)=Ric(Y,X),\quad
Ric(X,Y)-Ric(JX,JY)=4(n-1)A(X,JY),$ (2.6) $\displaystyle
2\rho(X,JY)=-Ric(X,Y)-Ric(JX,JY)=R(e_{a},Je_{a},X,JY),$ (2.7) $\displaystyle
2(\nabla_{e_{a}}Ric)(e_{a},X)=dS(X).\hskip 180.67499pt$

The equalities (2.5) and (2.6) imply

(2.8) $Ric(X,Y)=\rho(JX,Y)+2(n-1)A(JX,Y),$

i.e. $\rho$ is the $(1,1)$-part of the pseudohermitian Ricci tensor, while the
$(2,0)+(0,2)$-part is given by the pseudohermitian torsion $A$.

The Webster Ricci tensor $Rc$ is defined by

(2.9) $Rc(X,Y)=\rho(JX,Y)=Rc(JX,JY)=Rc(Y,X).$

The Webster Ricci tensor $Rc$ is symmetric, of type (1,1) with respect to $J$
and shares the same trace with the Ricci tensor $Ric$,
$S=Ric(e_{a},e_{a})=\rho(Je_{a},e_{a})=Rc(e_{a},e_{a})$ due to (2.8).

The trace-free part $Rc_{0}$ of the Webster Ricci tensor is given by

(2.10) $Rc_{0}(X,Y)=Rc(X,Y)-\frac{S}{2n}g(X,Y).$

We recall that a pseudohermitian manifold is called pseudo-Einstein if the
trace-free part of the Webster Ricci tensor vanishes.

###  2.3. The Ricci identities for the Tanaka-Webster connection

Let $f$ be a smooth function on a pseudohermitian manifold $M$ with $\nabla f$
its horizontal gradient, $g(\nabla f,X)=df(X)$. The sub-Laplacian (or
horizontal Laplacian) $\triangle f$ and the norm of the horizontal gradient
$\nabla f=df(e_{a})e_{a}$ of a smooth function $f$ on $M$ are defined
respectively by

(2.11) $\triangle f\ =-\ tr^{g}_{H}(\nabla^{2}f)\ =-\
\nabla^{2}f(e_{a},e_{a}),\qquad|\nabla f|^{2}\ =\ df(e_{a})\,df(e_{a}).$

We have the next Ricci identities for the Tanaka-Webster connection following
from the general Ricci identities for a connection with torsion applying the
properties of the pseudohermitian torsion listed in (2.3):

(2.12) $\displaystyle\nabla^{2}f(X,Y)-\nabla^{2}f(Y,X)=-2\omega(X,Y)df(\xi)$
$\displaystyle\nabla^{2}f(X,\xi)-\nabla^{2}f(\xi,X)=A(X,\nabla f)$
$\displaystyle\nabla^{3}f(X,Y,Z)-\nabla^{3}f(Y,X,Z)=-R(X,Y,Z,\nabla
f)-2\omega(X,Y)\nabla^{2}f(\xi,Z)$
$\displaystyle\nabla^{3}f(X,Y,Z)-\nabla^{3}f(Z,Y,X)=-R(X,Y,Z,\nabla
f)-R(Y,Z,X,\nabla f)-2\omega(X,Y)\nabla^{2}f(\xi,Z)$
$\displaystyle-2\omega(Y,Z)\nabla^{2}f(\xi,X)+2\omega(Z,X)\nabla^{2}f(\xi,Y)+2\omega(Z,X)A(Y,\nabla
f).$

An important consequence of the first Ricci identity is the following
fundamental formula

(2.13) $g(\nabla^{2}f,\omega)=\nabla^{2}f(e_{a},Je_{a})=-2n\,df(\xi).$

On the other hand, it follows from (2.11) that the trace with respect to the
metric is the sub-Laplacian:

$g(\nabla^{2}f,g)=\nabla^{2}f(e_{a},e_{a})=-\triangle f.$

We also recall the horizontal divergence theorem [20]. Let $(M,g,\theta)$ be a
pseudohermitian manifold of dimension $2n+1$. For a fixed local 1-form
$\theta$ the form $Vol_{\theta}=\theta\wedge\omega^{n}$ is a globally defined
volume form since $Vol_{\theta}$ is independent on the local 1-form $\theta$.
The (horizontal) divergence of a horizontal vector field/one-form
$\sigma\in\Lambda^{1}\,(H)$ is defined by $\nabla^{*}\,\sigma\
=-tr|_{H}\nabla\sigma=\ -(\nabla_{e_{a}}\sigma)e_{a}.$ If the manifold is
compact it is well known [20] that

$\int_{M}(\nabla^{*}\sigma)Vol_{\theta}\ =\ 0.$

##  3\. The positivity conditions, CR Cordes type inequality

In this section we discuss the importance of the positivity conditions (1.1)
and (1.4), recover the CR Cordes type inequality esteblished in [6] giving
more information in the equality case.

We recall from [16], (see also [2] and [1]), that if $n\geq 2$, a function
$f\in\mathcal{C}^{3}(M)$ on a compact $(2n+1)$\- dimensional strictly
pseudoconvex pseudohermitian manifold is CR-pluriharmonic, i.e., locally it is
the real part of a CR holomorphic function, if and only if the non-negative CR
Paneitz operator of $f$ vanishes, i.e. the (1,1) trace-free part of the
horizontal Hessian of $f$ is zero, $(\nabla^{2}f)_{[1][0]}=0$. In fact, as
shown in [11], only one fourth-order equation $Cf=0$ suffices for
$(\nabla^{2}f)_{[1][0]}=0$ to hold.

Now, we recover the CR Cordes estimate proved in [6, Theorem 1] and give a bit
more information in the equality case for dimensions bigger than three.

######  Theorem 3.1.

[6] On a compact strictly pseudoconvex pseudohermitian manifold of dimension
$2n+1$, for any real-valued function $f$ and $n>1$ we have the inequality

(3.1) $\frac{n+2}{n}\int_{M}(\triangle
f)^{2}\,Vol_{\theta}\geq\int_{M}\Big{[}Rc(\nabla f,\nabla f)+2(n+2)A(J\nabla
f,\nabla
f)+\left|(\nabla^{2}f)_{[1]}\right|^{2}+\left|(\nabla^{2}f)_{[-1]}\right|^{2}\Big{]}\,Vol_{\theta}.$

The equality is achieved only for CR-pluriharmonic functions.

If, moreover, the positivity condition (1.4) holds then

$\frac{n+2}{n}\int_{M}(\triangle
f)^{2}\,Vol_{\theta}\geq\int_{M}\Big{[}\left|(\nabla^{2}f)_{[1]}\right|^{2}+\left|(\nabla^{2}f)_{[-1]}\right|^{2}\Big{]}\,Vol_{\theta}.$

In the case $n=1$, if we assume in addition that the CR-Paneitz operator is
non-negative, then the inequality (3.1) holds also for $n=1$.

###### Proof.

Insert (6.16) into (6.18) and apply (2.8) and (2.9) to obtain

(3.2) $\frac{n+2}{n}\int_{M}(\triangle
f)^{2}\,Vol_{\theta}=\int_{M}\Big{[}Rc(\nabla f,\nabla f)+2(n+2)A(J\nabla
f,\nabla f)\Big{]}\,Vol_{\theta}\\\
+\int_{M}\Big{[}\left|(\nabla^{2}f)_{[1]}\right|^{2}+\left|(\nabla^{2}f)_{[-1]}\right|^{2}-\frac{2}{n}P(\nabla
f)\Big{]}\,Vol_{\theta}.$

Due to Lemma 6.3 the last term in (3.2) is non-negative, $-P(\nabla f)\geq 0$,
and the inequality (3.1) follows from (3.2). Moreover, the equality in (3.1)
can be achieved only when $P(\nabla f)=0=(\nabla f)^{2}_{[1][0]}$ due again to
Lemma 6.3. The assertion follows from the discussion above the formulation of
the theorem. ∎

######  Remark 3.2.

It was pointed out in [5] that in the original Greenleaf’s CR-Bochner formula
from [12] the coefficient in front of $A$ is not correct. In the proof of [6,
Theorem 1] it was used that incorrect CR-Bochner formula and therefore the
coefficient in front of the pseudohermitian torsion $A$ in (3.1) differs a bit
from the coefficient in front of $A$ in the formula (22) of [6, Theorem 1].

Applying (6.17) to the identity (3.2), taking into account (6.9), we also get

(3.3) $\frac{n+1}{n}\int_{M}(\triangle
f)^{2}\,Vol_{\theta}=\int_{M}\Big{[}Rc(\nabla f,\nabla f)+2(n+1)A(J\nabla
f,\nabla f)\Big{]}\,Vol_{\theta}\\\
+\int_{M}\Big{[}\left|(\nabla^{2}f)_{[1][0]}\right|^{2}+\left|(\nabla^{2}f)_{[-1]}\right|^{2}-\frac{3}{2n}P(\nabla
f)\Big{]}\,Vol_{\theta}.$

######  Remark 3.3.

Note that the expression in the right-hand side of the first line in (3.3) is
precisely the CR-Lichnerowicz positivity condition used to find a lower bound
of the first eigenvalue of the sub-Laplacian (see [12, 4, 5, 14]). Indeed, if
one assumes the positivity condition (1.1) to be in the form

$Rc(X,X)+2(n+1)A(JX,X)\geq k_{0}g(X,X)$

and $\triangle f=\lambda f$, one easily gets $\lambda\geq\frac{n}{n+1}k_{0}$
which is the Greenleaf’s estimation of the first eigenvalue of the sub-
Laplacian.

##  4\. Proof of Theorem 1.3

We follow the approach of [9] in the Riemannian case and [7] in the CR case.

Denote by $\bar{S}$ the average value of the scalar curvature,

$\bar{S}=\int_{M}S\,Vol_{\theta}.$

Let $\varphi$ be the unique solution of the following PDE:

(4.1) $\triangle\varphi=S-\bar{S},\quad\int_{M}\varphi\,Vol_{\theta}=0.$

We obtain

(4.2)
$\int_{M}(S-\bar{S})^{2}\,Vol_{\theta}=\int_{M}(S-\bar{S})\triangle\varphi\,Vol_{\theta}=\int_{M}dS(\nabla\varphi)\,Vol_{\theta},$

where we used an integration by parts to achieve the last equality.

We write, using (2.10), the equality (2.8) in the form

(4.3) $Ric(X,Y)=Rc_{0}(X,Y)+\frac{S}{2n}g(X,Y)+2(n-1)A(JX,Y).$

In view of (4.3), the second Bianchi identity (2.7) takes the form

$2(\nabla_{e_{a}}Rc_{0})(e_{a},X)+\frac{1}{n}dS(X)+4(n-1)(\nabla_{e_{a}}A)(e_{a},JX)=dS(X),$

which yields

(4.4)
$dS(X)=\frac{2n}{n-1}(\nabla_{e_{a}}Rc_{0})(e_{a},X)+4n(\nabla_{e_{a}}A)(e_{a},JX).$

Substitute (4.4) into (4.2) to get after an integration by parts, applying
(2.4) and (2.9), the following relations

(4.5)
$\int_{M}(S-\bar{S})^{2}\,Vol_{\theta}=\int_{M}dS(\nabla\varphi)\,Vol_{\theta}\\\
=\frac{2n}{n-1}\int_{M}(\nabla_{e_{a}}Rc_{0})(e_{a},\nabla\varphi)\,Vol_{\theta}+4n\int_{M}(\nabla_{e_{a}}A)(e_{a},J\nabla\varphi)\,Vol_{\theta}\\\
=-\frac{2n}{n-1}\int_{M}Rc_{0}(e_{a},e_{b})\nabla^{2}\varphi(e_{a},e_{b})\,Vol_{\theta}-4n\int_{M}A(e_{a},Je_{b})\nabla^{2}\varphi(e_{a},e_{b})\,Vol_{\theta}\\\
=-\frac{2n}{n-1}\int_{M}Rc_{0}(e_{a},e_{b})(\nabla^{2}\varphi)_{[1][0]}(e_{a},e_{b})\,Vol_{\theta}-4n\int_{M}A(e_{a},Je_{b})(\nabla^{2}\varphi)_{[-1]}(e_{a},e_{b})\,Vol_{\theta}.$

Applying the Young’s inequality $2ab\leq\alpha a^{2}+\frac{1}{\alpha}b^{2}$,
we get from (4.5) that

(4.6) $\int_{M}(S-\bar{S})^{2}\,Vol_{\theta}\\\
\leq\frac{n}{n-1}\int_{M}\Big{[}\alpha|Rc_{0}|^{2}+\frac{1}{\alpha}|(\nabla^{2}\varphi)_{[1][0]}|^{2}\Big{]}\,Vol_{\theta}-4n\int_{M}A(e_{a},Je_{b})(\nabla^{2}\varphi)_{[-1]}(e_{a},e_{b})\,Vol_{\theta}.$

At this point we need to evaluate the norm of the (1,1) trace-free part of the
horizontal Hessian. To do this, we involve the positivity condition (1.4). We
have

######  Proposition 4.1.

On a compact strictly pseudoconvex pseudohermitian manifold of dimension
$2n+1$ for $n\geq 2$ and for any smooth function $f$ we have

(4.7) $\int_{M}(\triangle
f)^{2}\,Vol_{\theta}=\frac{2n}{2n+3}\int_{M}\Big{[}Rc(\nabla f,\nabla
f)+2(n+2)A(J\nabla f,\nabla f)\Big{]}\,Vol_{\theta}\\\
+\int_{M}\Big{[}\frac{2n(n+3)}{(n-1)(2n+3)}\left|(\nabla^{2}f)_{[1][0]}\right|^{2}+\frac{2n}{2n+3}\left|(\nabla^{2}f)_{[-1]}\right|^{2}+\frac{4n^{2}}{2n+3}(df(\xi))^{2}\Big{]}\,Vol_{\theta}.$

###### Proof.

It follows directly from (3.2), Lemma 6.3 and (6.9). ∎

The positivity condition (1.4) and (4.7) yield the inequality

$\int_{M}\left|(\nabla^{2}f)_{[1][0]}\right|^{2}\,Vol_{\theta}\leq\frac{(n-1)(2n+3)}{2n(n+3)}\int_{M}(\triangle
f)^{2}\,Vol_{\theta},$

which taken with respect to the function $\varphi$, then substituted into
(4.6) and applying (4.1) lead to the inequalty

(4.8)
$\Big{(}1-\frac{2n+3}{2\alpha(n+3)}\Big{)}\int_{M}(S-\bar{S})^{2}\,Vol_{\theta}\\\
\leq\frac{\alpha
n}{n-1}\int_{M}|Rc_{0}|^{2}\,Vol_{\theta}-4n\int_{M}A(e_{a},Je_{b})(\nabla^{2}\varphi)_{[-1]}(e_{a},e_{b})\,Vol_{\theta}.$

Take $\alpha=\frac{2n+3}{n+3}$ to obtain from (4.8) the inequality (1.5),
which completes the proof of the first part of Theorem 1.3.

Suppose we have equality in (1.5). Take $\alpha=\frac{2n+3}{n+3}$ into (4.6)
and then use the expression for $\left|(\nabla^{2}f)_{[1][0]}\right|^{2}$ from
(4.7) substituted into (4.6) to get, taking into account (4.1), that

(4.9)
$0\leq-\frac{n}{2n+3}\int_{M}\Big{[}Rc(\nabla\varphi,\nabla\varphi)+2(n+2)A(J\nabla\varphi,\nabla\varphi)+\left|(\nabla^{2}\varphi)_{[-1]}\right|^{2}+2n(d\varphi(\xi))^{2}\Big{]}\,Vol_{\theta}.$

The positivity condition (1.4) and (4.9) imply

(4.10)
$Rc(\nabla\varphi,\nabla\varphi)+2(n+2)A(J\nabla\varphi,\nabla\varphi)=0;\quad(\nabla^{2}\varphi)_{[-1]}=0;\quad(d\varphi(\xi))^{2}=0.$

The equality in (1.5) and the second equality in (4.10) yield (1.6).

Substitute (4.10) into (4.7) and use (1.6) to get

(4.11)
$\int_{M}\left|(\nabla^{2}\varphi)_{[1][0]}\right|^{2}\,Vol_{\theta}=\frac{(n-1)(2n+3)}{2n(n+3)}\int_{M}(\triangle\varphi)^{2}\,Vol_{\theta}=\frac{(2n+3)^{2}}{(n+3)^{2}}\int_{M}|Rc_{0}|^{2}\,Vol_{\theta}.$

Applying (1.6) and (4.10) to (4.5) yields

(4.12)
$\int_{M}(S-\bar{S})^{2}\,Vol_{\theta}=\frac{2n(2n+3)}{(n-1)(n+3)}\int_{M}|Rc_{0}|^{2}\,Vol_{\theta}\\\
=-\frac{2n}{n-1}\int_{M}Rc_{0}(e_{a},e_{b})(\nabla^{2}\varphi)_{[1][0]}(e_{a},e_{b})\,Vol_{\theta},$

which together with (4.11) implies that we have equality in the Young’s
inequality. This shows that

(4.13)
$Rc_{0}(e_{a},e_{b})=-\frac{n+3}{2n+3}(\nabla^{2}\varphi)_{[1][0]}(e_{a},e_{b}),$

which implies that the contact form
$\bar{\theta}=exp(-\frac{2n+3}{(n+3)(n+2)}\varphi)\theta$ will be pseudo-
Einstein by [10, Proposition 5.9]. This completes the proof of Theorem 1.3.

###  4.1. Proof of Corollary 1.5

Two integration by parts yield

$\int_{M}A(e_{a},Je_{b})(\nabla^{2}\varphi)_{[-1]}(e_{a},e_{b})\,Vol_{\theta}=\int_{M}A(e_{a},Je_{b})\nabla^{2}\varphi(e_{a},e_{b})\,Vol_{\theta}=-\int_{M}(\nabla_{e_{a}}A)(e_{a},J\nabla\varphi)\,Vol_{\theta}\\\
=\int_{M}\varphi(\nabla_{e_{b}}\nabla_{e_{a}}A)(e_{a},Je_{b})\,Vol_{\theta}=0$

due to (1.7), which proves the first part.

To show the second part, suppose we have an equality in (1.8). Then (4.10) and
(4.11) hold true. We obtain the following chain of equalities by integration
by parts, using the second equality in (4.10):

(4.14)
$\int_{M}A(J\nabla\varphi,\nabla\varphi)\,Vol_{\theta}=-\int_{M}\varphi(\nabla_{e_{a}}A)(e_{a},J\nabla\varphi)\,Vol_{\theta}-\int_{M}\varphi
A(e_{a},Je_{b})\nabla^{2}\varphi(e_{a},e_{b})\,Vol_{\theta}\\\
=-\int_{M}\varphi(\nabla_{e_{a}}A)(e_{a},J\nabla\varphi)\,Vol_{\theta}-\int_{M}\varphi
A(e_{a},Je_{b})(\nabla^{2}\varphi)_{[-1]}(e_{a},e_{b})\,Vol_{\theta}\\\
=-\int_{M}\varphi(\nabla_{e_{a}}A)(e_{a},J\nabla\varphi)\,Vol_{\theta}=\frac{1}{2}\int_{M}\varphi^{2}(\nabla_{e_{b}}\nabla_{e_{a}}A)(e_{a},Je_{b})\,Vol_{\theta}=0.$

Applying (4.14), Lemma 6.3 and the third equality in (4.10) to (6.17), we
obtain using (4.11) that

(4.15)
$0=2\int_{M}A(\nabla\varphi,J\nabla\varphi)\,Vol_{\theta}=\int_{M}\Big{[}\frac{1}{2n}(\triangle\varphi)^{2}-\frac{1}{n-1}\left|(\nabla^{2}\varphi)_{[1][0]}\right|^{2}\Big{]}\,Vol_{\theta}\\\
=\int_{M}\Big{[}\frac{1}{2n}(\triangle\varphi)^{2}-\frac{2n+3}{2n(n+3)}(\triangle\varphi)^{2}\Big{]}\,Vol_{\theta}=-\frac{1}{2(n+3)}\int_{M}(\triangle\varphi)^{2}\,Vol_{\theta}.$

Now, (4.15) shows that $\triangle\varphi=0$ and the manifold is pseudo-
Einstein with constant pseudohermitian scalar curvature, which completes the
proof of Corollary 1.5.

The equality

$\Big{|}Rc-\frac{\bar{S}}{2n}g\Big{|}^{2}=\Big{|}Rc-\frac{S}{2n}g\Big{|}^{2}+\frac{1}{2n}(S-\bar{S})^{2}$

together with Corollary 1.5 imply

######  Corollary 4.2.

In the conditions of Corollary 1.5 one has

$\int_{M}\Big{|}Rc-\frac{\bar{S}}{2n}g\Big{|}^{2}\,Vol_{\theta}\leq\frac{n(n+4)}{(n-1)(n+3)}\int_{M}\Big{|}Rc-\frac{S}{2n}g\Big{|}^{2}\,Vol_{\theta}.$

###  4.2. Proof of Corollary 1.7

It remains to show only the last part of Corollary 1.7. In the torsion-free
case, i.e. in Sasakian case, it is well known that the Ricci tensor $Ric^{h}$
of the Riemannian metric $h=g+\eta\otimes\eta$ and the Webster Ricci tensor
$Rc$ are connected by (see e.g. [10])

(4.16) $Ric^{h}(X,X)=Rc(X,X)-2g(X,X),\qquad Ric^{h}(\xi,\xi)=2n.$

If we have pseudo-Einstein Sasakian structure, we obtain from (4.16)

$Ric^{h}(X,X)=\frac{S-4n}{2n}g(X,X),\qquad
Ric^{h}=\frac{S-4n}{2n}h+\Big{(}2n-\frac{S-4n}{2n}\Big{)}\eta\otimes\eta$

showing that the Riemannian Sasaki structure is $\eta$-Einstein [19] and the
Tanno’s D-homothetic deformation
$\bar{\eta}=a\eta,\quad\bar{\xi}=\frac{1}{a}\xi,\quad\bar{h}=ah+a(a-1)\eta\otimes\eta$
makes it a Sasaki-Einstein space for a suitable constant $a$ [21] c.f. also
[3].

##  5\. Proof of Corollary 1.6

For completeness and a better understanding of the proof of Corollary 1.6, we
give a proof of [7, Theorem 1.2], Theorem 1.2.

The next result involves the positivity condition (1.1). We have

######  Proposition 5.1.

On a compact strictly pseudoconvex pseudohermitian manifold of dimension
$2n+1$ for $n\geq 2$ and for any smooth function $f$ we have

(5.1) $\int_{M}(\triangle
f)^{2}\,Vol_{\theta}=\frac{n}{n+1}\int_{M}\Big{[}Rc(\nabla f,\nabla
f)+2(n+1)A(J\nabla f,\nabla f)\Big{]}\,Vol_{\theta}\\\
+\int_{M}\Big{[}\frac{n(n+2)}{n^{2}-1}\left|(\nabla^{2}f)_{[1][0]}\right|^{2}+\frac{n}{n+1}\left|(\nabla^{2}f)_{[-1]}\right|^{2}\Big{]}\,Vol_{\theta}.$

###### Proof.

It follows directly from (3.3) and Lemma 6.3. ∎

The positivity condition (1.1) and (5.1) yield the inequality

$\int_{M}\left|(\nabla^{2}f)_{[1][0]}\right|^{2}\,Vol_{\theta}\leq\frac{n^{2}-1}{n(n+2)}\int_{M}(\triangle
f)^{2}\,Vol_{\theta},$

which taken with respect to the function $\varphi$, then substituted into
(4.6) and applying (4.1) lead to the inequalty

(5.2)
$\Big{(}1-\frac{n+1}{\alpha(n+2)}\Big{)}\int_{M}(S-\bar{S})^{2}\,Vol_{\theta}\leq\frac{\alpha
n}{n-1}\int_{M}|Rc_{0}|^{2}\,Vol_{\theta}-4n\int_{M}A(e_{a},Je_{b})(\nabla^{2}\varphi)_{[-1]}(e_{a},e_{b})\,Vol_{\theta}.$

Taking $\alpha=\frac{2(n+1)}{n+2}$, we obtain from (5.2) the inequality (1.2),
which completes the proof of the first part of Theorem 1.2 from [7].

The second part follows similarly to the proof of the second part of Theorem
1.3. Indeed, suppose we have equality in (1.2). Take
$\alpha=\frac{2(n+1)}{n+2}$ into (4.6), then use the expression for
$\left|(\nabla^{2}f)_{[1][0]}\right|^{2}$ from (5.1) substituted into (4.6) to
get, taking into account (4.1), that

(5.3)
$0\leq-\frac{n}{2(n+1)}\int_{M}\Big{[}Rc(\nabla\varphi,\nabla\varphi)+2(n+1)A(J\nabla\varphi,\nabla\varphi)+\left|(\nabla^{2}\varphi)_{[-1]}\right|^{2}\Big{]}\,Vol_{\theta}.$

The positivity condition (1.1) and (5.3) imply

(5.4)
$Rc(\nabla\varphi,\nabla\varphi)+2(n+1)A(J\nabla\varphi,\nabla\varphi)=0;\quad(\nabla^{2}\varphi)_{[-1]}=0.$

The equality in (1.2) and the second equality in (5.4) yield (1.3).

Substitute (5.4) into (5.1) and use (1.3) to get

(5.5)
$\int_{M}\left|(\nabla^{2}\varphi)_{[1][0]}\right|^{2}\,Vol_{\theta}=\frac{n^{2}-1}{n(n+2)}\int_{M}(\triangle\varphi)^{2}\,Vol_{\theta}=\frac{(2n+2)^{2}}{(n+2)^{2}}\int_{M}|Rc_{0}|^{2}\,Vol_{\theta}.$

Applying (1.3) and (5.4) to (4.5) yields

$\int_{M}(S-\bar{S})^{2}\,Vol_{\theta}=\frac{4n(n+1)}{(n-1)(n+2)}\int_{M}|Rc_{0}|^{2}\,Vol_{\theta}=-\frac{2n}{n-1}\int_{M}Rc_{0}(e_{a},e_{b})(\nabla^{2}\varphi)_{[1][0]}(e_{a},e_{b})\,Vol_{\theta},$

which, together with (5.5), implies that we have equality in the Young’s
inequality. This shows that

$Rc_{0}(e_{a},e_{b})=-\frac{n+2}{2n+2}(\nabla^{2}\varphi)_{[1][0]}(e_{a},e_{b}),$

yielding that the contact form $\bar{\theta}=exp(-\frac{1}{n+1}\varphi)\theta$
will be pseudo-Einstein by [10, Proposition 5.9] which completes the proof of
Theorem 1.2 of [7].

To finish the proof of Corollary 1.6, it remains to show only that if we have
equality in (1.9) then the manifold is pseudo-Einstein with constant
pseudohermitian scalar curvature.

Indeed, in the equality case we have that (5.4) and (5.5) are valid. The
second identity in (5.4) implies that (4.14) holds true. Apply (4.14), Lemma
6.3 and the second equality in (5.4) to (6.17) to get, using (5.5), that

(5.6)
$0=2\int_{M}A(\nabla\varphi,J\nabla\varphi)\,Vol_{\theta}=\int_{M}\Big{[}-\frac{1}{2n}g(\nabla^{2}\varphi,\omega)^{2}+\frac{1}{2n}(\triangle\varphi)^{2}-\frac{1}{n-1}\left|(\nabla^{2}\varphi)_{[1][0]}\right|^{2}\Big{]}\,Vol_{\theta}\\\
=\int_{M}\Big{[}-\frac{1}{2n}g(\nabla^{2}\varphi,\omega)^{2}+\frac{1}{2n}(\triangle\varphi)^{2}-\frac{n+1}{n(n+2)}(\triangle\varphi)^{2}\Big{]}\,Vol_{\theta}\\\
=-\int_{M}\Big{[}\frac{1}{2(n+2)}(\triangle\varphi)^{2}+\frac{1}{2n}g(\nabla^{2}\varphi,\omega)^{2}\Big{]}\,Vol_{\theta}.$

Now, (5.6) shows that $\triangle\varphi=0$ and the manifold is pseudo-Einstein
with constant pseudohermitian scalar curvature which completes the proof.

##  6\. Appendix

The purpose of this section is to record for self-sufficiency proofs of some
of the results of [12, 16, 11] in real variables (see also [14, Appendix])
including the Greenleaf’s CR Bochner formula [12], the CR Paneitz operator and
its non-negativity for $n>1$ [11], etc.

###  6.1. The Greenleaf’s CR-Bochner formula

######  Theorem 6.1.

[12] On a strictly pseudoconvex pseudohermitian manifold of dimension $2n+1$,
$n\geq 1$, the following Bochner-type identity holds

(6.1) $-\frac{1}{2}\triangle|\nabla f|^{2}=-g(\nabla(\triangle f),\nabla
f)+Ric(\nabla f,\nabla f)+2A(J\nabla f,\nabla f)+|\nabla^{2}f|^{2}+4\nabla
df(\xi,J\nabla f).$

###### Proof.

By definition we have

(6.2) $-\frac{1}{2}\triangle|\nabla
f|^{2}=\nabla^{3}f(e_{a},e_{a},e_{b})df(e_{b})+\nabla^{2}f(e_{a},e_{b})\nabla^{2}f(e_{a},e_{b})=\nabla^{3}f(e_{a},e_{a},e_{b})df(e_{b})+|\nabla^{2}f|^{2}.$

To evaluate the first term in the right-hand side of (6.2), we use the Ricci
identities (2.12). Taking into account $(\nabla_{X}T)(Y,Z)=0$ and applying
successively the Ricci identities (2.12), we obtain

(6.3) $\nabla^{3}f(e_{a},e_{a},e_{b})df(e_{b})=-g(\nabla(\triangle f),\nabla
f)+Ric(\nabla f,\nabla f)+2A(J\nabla f,\nabla f)+4\nabla^{2}f(\xi,J\nabla f).$

A substitution of (6.3) into (6.2) completes the proof of (6.1). ∎

The next integral formula was originally proved in [12], we take it in real
notations from [14].

######  Lemma 6.2.

[12] On a compact strictly pseudoconvex pseudohermitian manifold of dimension
$2n+1$, $n\geq 1$, we have the identity

(6.4) $\int_{M}\nabla^{2}f(\xi,J\nabla
f)\,Vol_{\theta}=-\int_{M}\Big{[}2n(df(\xi))^{2}+A(J\nabla f,\nabla
f)\Big{]}\,Vol_{\theta}.$

###### Proof.

We consider the horizontal 1-form defined by $D(X)=df(JX)df(\xi),$ whose
divergence is, taking into account the second formula of (2.12),

(6.5)
$(\nabla_{e_{a}}D)(e_{a})=\nabla^{2}f(e_{a},Je_{a})df(\xi)-\nabla^{2}f(\xi,J\nabla
f)-A(J\nabla f,\nabla f).$

Integrating (6.5) over $M$ and using (2.13) implies (6.4), which completes the
proof of the lemma. ∎

###  6.2. The CR-Paneitz operator

The famous CR-Paneitz operator is defined as follows [17, 11].

Given a function $f$ we define the one form,

(6.6) $P(X)\equiv
P_{f}(X)=\nabla^{3}f(X,e_{b},e_{b})+\nabla^{3}f(JX,e_{b},Je_{b})+4nA(X,J\nabla
f),$

and also a fourth order differential operator (the so called CR-Paneitz
operator in [8]),

(6.7)
$Cf=(\nabla_{e_{a}}P)({e_{a}})=\nabla^{4}f(e_{a},e_{a},e_{b},e_{b})+\nabla^{4}f(e_{a},Je_{a},e_{b},Je_{b})-4n\nabla^{*}A(J\nabla
f)-4n\,g(\nabla^{2}f,JA).$

According to (2.1), the horizontal Hessian $\nabla^{2}f$ splits into two
parts, $\nabla^{2}f=(\nabla^{2}f)_{[1]}+(\nabla^{2}f)_{[-1]}$ , where

(6.8)
$\begin{split}(\nabla^{2}f)_{[1]}(X,Y)=\frac{1}{2}\left[(\nabla^{2}f)(X,Y)+(\nabla^{2}f)(JX,JY)\right],\\\
(\nabla^{2}f)_{[-1]}(X,Y)=\frac{1}{2}\left[(\nabla^{2}f)(X,Y)-(\nabla^{2}f)(JX,JY)\right].\end{split}$

In view of (2.13), the trace-free part $(\nabla^{2}f)_{[1][0]}$ of the $(1,1)$
component of the horizontal Hessian is given by

(6.9) $\begin{split}(\nabla^{2}f)_{[1][0]}=(\nabla^{2}f)_{[1]}+\frac{\triangle
f}{2n}g(X,Y)+df(\xi)\,\omega(X,Y);\\\
|(\nabla^{2}f)_{[1][0]}|^{2}=|(\nabla^{2}f)_{[1]}|^{2}-\frac{(\triangle
f)^{2}}{2n}-2n(df(\xi))^{2}.\end{split}$

One of the basic results relating the above defined operators is the next
lemma proved in [11], which we present in real notations, see e.g. [14].

######  Lemma 6.3.

[11] On a compact strictly pseudoconvex pseudohermitian manifold of dimension
$2n+1$, $n\geq 1$, the following identities hold true

(6.10)
$\displaystyle\nabla_{e_{a}}(\nabla^{2}f)_{[1][0]}(e_{a},X)=\frac{n-1}{2n}P_{f}(X),$
(6.11)
$\displaystyle\int_{M}|(\nabla^{2}f)_{[1][0]}|^{2}\,Vol_{\theta}=-\frac{n-1}{2n}\int_{M}P_{f}(\nabla
f)\,Vol_{\theta}=\frac{n-1}{2n}\int_{M}f(Cf)\,Vol_{\theta}.$

In particular, if $n>1$ the CR-Paneitz operator is non-negative,
$\int_{M}f(Cf)\,Vol_{\theta}\geq 0$.

###### Proof.

Taking into account the third and the fourth Ricci identity in (2.12), we have

$\displaystyle\nabla^{3}f(e_{a},e_{a},X)$
$\displaystyle=\nabla^{3}f(X,e_{a},e_{a})+Ric(X,\nabla
f)+4\nabla^{2}f(\xi,JX)+2A(JX,\nabla f),$
$\displaystyle\nabla^{3}f(e_{a},Je_{a},JX)$
$\displaystyle=\frac{1}{2}\Big{(}\nabla^{3}f(e_{a},Je_{a},JX)-\nabla^{3}f(Je_{a},e_{a},JX)\Big{)}=-\rho(JX,\nabla
f)-2n\nabla^{2}f(\xi,JX).$

The sum of the above equalities gives

(6.12) $2\nabla_{e_{a}}(\nabla^{2}f)_{[1]}(e_{a},X)\\\
=\nabla^{3}f(X,e_{a},e_{a})+Ric(X,\nabla f)-\rho(JX,\nabla
f)+(4-2n)\nabla^{2}f(\xi,JX)+2A(JX,\nabla f).$

The equality (2.13) yields
$\nabla^{2}f(JX,\xi)=-\frac{1}{2n}\nabla^{3}f(JX,e_{a},Je_{a})$, which
together with the second Ricci identity in (2.12) imply

(6.13) $\nabla^{2}f(\xi,JX)=\nabla^{2}f(JX,\xi)-A(JX,\nabla
f)=-\frac{1}{2n}\nabla^{3}f(JX,e_{a},Je_{a})-A(JX,\nabla f).$

Therefore, using (2.8) and (6.13) we get from (6.12) that

(6.14)
$2\nabla_{e_{a}}(\nabla^{2}f)_{[1]}(e_{a},X)=\nabla^{3}f(X,e_{a},e_{a})+\frac{n-2}{n}\nabla^{3}f(JX,e_{a},Je_{a})+4(n-1)A(X,J\nabla
f).$

The divergence of the trace part of $(\nabla^{2}f)_{[1]}$ is computed as
follows

(6.15) $\nabla_{e_{a}}\left(-\frac{1}{2n}\triangle f\cdot
g-df(\xi)\omega\right)(e_{a},X)=\frac{1}{2n}\nabla^{3}f(X,e_{a},e_{a})+\nabla^{2}f(JX,\xi)\\\
=\frac{1}{2n}\nabla^{3}f(X,e_{a},e_{a})-\frac{1}{2n}\nabla^{3}f(JX,e_{a},Je_{a}).$

Now, the identities (6.14) and (6.15) imply (6.10). The second identity (6.11)
follows by an integration by parts from (6.10). ∎

The next result, essentially proved in [4], involves the CR-Paneitz operator.
We present it in real notations from [14, Lemma 8.7]:

######  Lemma 6.4.

On a strictly pseudoconvex pseudohermitian manifold of dimension $2n+1$,
$n\geq 1$, we have the identity

$\nabla^{2}f(\xi,Z)=\frac{1}{2n}\nabla^{3}f(Z,Je_{a},e_{a})-A(Z,\nabla f).$

Additionally, in the compact case, the following integral identity holds true:

(6.16) $\int_{M}\nabla^{2}f(\xi,J\nabla
f)Vol_{\theta}=\int_{M}\Big{[}-\frac{1}{2n}\left(\triangle
f\right)^{2}+A(J\nabla f,\nabla f)-\frac{1}{2n}P_{f}(\nabla
f)\Big{]}\,Vol_{\theta}.$

###### Proof.

We compute using the first two Ricci identities in (2.12)

$2\nabla^{3}f(Z,Je_{a},e_{a})=\nabla^{3}f(Z,Je_{a},e_{a})-\nabla^{3}f(Z,e_{a},Je_{a})=-2\omega(Je_{a},e_{a})\nabla^{2}f(Z,\xi)\\\
=4n\left(\nabla^{2}f(\xi,Z)+A(Z,\nabla f)\right),$

which proves the first formula. The second identity (6.16) follows from the
first formula, the definition (6.6) of $P_{f}$ and an integration by parts. ∎

Combining (6.16) with (6.4), one gets

(6.17) $2\int_{M}A(J\nabla f,\nabla
f)\,Vol_{\theta}=\int_{M}\Big{[}-\frac{1}{2n}g(\nabla^{2}f,\omega)^{2}+\frac{1}{2n}(\triangle
f)^{2}+\frac{1}{2n}P_{f}(\nabla f)\Big{]}\,Vol_{\theta}.$

Integrating the Bochner type formula (6.1) on a compact $M$ gives

(6.18) $0=\int_{M}\Big{[}-(\triangle
f)^{2}+\left|(\nabla^{2}f)_{[1]}\right|^{2}+\left|(\nabla^{2}f)_{[-1]}\right|^{2}\\\
+Ric(\nabla f,\nabla f)+2A(J\nabla f,\nabla f)+4\nabla^{2}f(\xi,J\nabla
f)\Big{]}\,Vol_{\theta}.$

We use Lemma 6.4 to represent the last term, which turns the identity (6.18)
into the following

(6.19) $0=\int_{M}\Big{[}-(\triangle
f)^{2}+\left|(\nabla^{2}f)_{[1]}\right|^{2}+\left|(\nabla^{2}f)_{[-1]}\right|^{2}+Ric(\nabla
f,\nabla f)+6A(J\nabla f,\nabla f)\\\ -\frac{2}{n}(\triangle
f)^{2}-\frac{2}{n}P_{f}(\nabla f)\Big{]}\,Vol_{\theta}.$

After a substitution of (6.17) in (6.19), taking into account (6.9), we get

(6.20) $0=\int_{M}\Big{[}Ric(\nabla f,\nabla f)+4A(J\nabla f,\nabla
f)-\frac{n+1}{n}(\triangle f)^{2}\Big{]}\,Vol_{\theta}\\\
+\int_{M}\Big{[}\left|(\nabla^{2}f)_{[1][0]}\right|^{2}+\left|(\nabla^{2}f)_{[-1]}\right|^{2}-\frac{3}{2n}P_{f}(\nabla
f)\Big{]}\,Vol_{\theta}.$

The above identities (6.19) and (6.20) are the key identities relating the CR-
Paneitz operator and the Greenleaf’s CR-Bochner formula (6.18) on a compact
manifold, and were essentially proved in [4]. We presented these identities in
real notations from [14, (8.13), (8.15)].

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# Explain to me like I am five - Sentence Simplification Using Transformers

Aman Agarwal
Luddy School of Informatics, Computing, and Engineering
Indiana University, Bloomington, IN 47408
<EMAIL_ADDRESS>

###### Abstract

Sentence simplification aims at making the structure of text easier to read
and understand while maintaining its original meaning. This can be helpful for
people with disabilities, new language learners, or those with low literacy.
Simplification often involves removing difficult words and rephrasing the
sentence. Previous research have focused on tackling this task by either using
external linguistic databases for simplification or by using control tokens
for desired fine-tuning of sentences. However, in this paper we purely use
pre-trained transformer models. We experiment with a combination of GPT-2 and
BERT models, achieving the best SARI score of 46.80 on the Mechanical Turk
dataset, which is significantly better than previous state-of-the-art results.
The code can be found at https://github.com/amanbasu/sentence-simplification.

##  1 Introduction

Sentence Simplification (SS) involves generating a simpler version of a
complex sentence while maintaining its meaning. This can be helpful for people
with disabilities like dyslexia, aphasia or autism, language learners, or
those with low literacy. For example, aphasic patients face challenges in
following long sentences with convoluted structure, or, new language learners
are generally unaware of rare words and phrases. Simplification, thereby,
typically involves removing unnecessary or complex words and paraphrasing for
simplicity. This improves the text’s accessibility and readability, making it
easier for a wider audience to understand and retain the content.

In this paper, we propose an approach for the task of SS using fine-tuned
Transformers Vaswani et al. (2017). With the advent of Transformer models, the
whole field of Natural Language Processing has been revolutionized. Unlike
auto-regressive models like RNN and LSTM, these feed-forward and fully-
connected models are based on the idea of self-attention, where the model
attends to different parts of the input sequence to generate an output.
Transformer models can be pre-trained on large corpora, such as Wikipedia or
the entire Internet’s text, to capture a wide range of knowledge and language
patterns. Subsequently, the models can then be fine-tuned on a specific tasks,
such as language translation or sentence simplification, to further improve
their performance.

We use two popular transformer models, BERT Devlin et al. (2018) and GPT-2
Radford et al. (2019). BERT (Bidirectional Encoder Representations from
Transformers) is a natural language processing (NLP) model developed by
Google. It is trained in a bidirectional fashion, meaning that it can take
into account the context of the words in a sentence from both the left and the
right sides, rather than just the left side of a word. GPT-2 is another
popular transformer model from OpenAI. The key difference between the two is
that GPT-2 is a unidirectional model that only considers the left context when
processing input data. This may intuitively make BERT better at understanding
the meaning of words in a sentence, however, GPT-2 is faster and more
efficient at generating text.

After training these models on a huge text corpus, they could be fine-tuned to
do any particular activity, such as question answering, summarizing,
translating, reading comprehension, etc. We show that fine-tuning a large
language model (LM) like BERT or GPT-2 is good enough for SS, rather than
involving additional linguistic knowledge like used in the past. Our SARI
score Xu et al. (2016) of 46.80 on the Mechanical Turk dataset beats the
state-of-the-art (SOTA) by a huge margin of 3.49.

##  2 Related Work

Recent work on sentence simplification using transformer models has focused on
using these models to automatically generate simpler versions of complex
sentences. These models have been shown to be effective at generating simpler
versions of sentences while maintaining their meaning Fang and Stevens (2019);
Zhao et al. (2018); Martin et al. (2019); Omelianchuk et al. (2021); Sheang
and Saggion (2021); Štajner et al. (2022).

Some of these approaches have also explored using additional linguistic
knowledge to improve the quality of the generated simplified sentences.
Especially, through the use of Simple Paraphrase Database (SimplePPDB) Pavlick
and Callison-Burch (2016). It contains a collection of around 4.5 million
rules for text reduction in English. SimplePPDB is generally used to provide
the model with a set of potential reduction rules for a given complex
sentence. These rules can be incorporated in the model’s loss function during
training Fang and Stevens (2019); Zhao et al. (2018).

Additionally, researchers have also explored controllable sentence
simplification Martin et al. (2019); Sheang and Saggion (2021); Štajner et al.
(2022). Controllable sentence simplification is the process of using control
tokens while training transformer models to generate simpler versions of
sentences. This allows the model to be more flexible in generating a wide
range of simplifications, depending on the specific control tokens provided.
For example, a model trained with control tokens for “simplicity” and
“fluency” can generate simpler sentences with a higher level of fluency.
Similarly, a model trained with control tokens for “words” would generate
sentences that use smaller, and thereby, less complicated words.

During training, the model is provided with a set of control tokens and their
corresponding target simplifications. The model then learns to generate the
target simplification based on the provided control tokens. During inference,
the model can be provided with a new set of control tokens and the original
complex sentence. The model then generates a simplification based on the
control tokens and the underlying structure and meaning of the complex
sentence. Overall, controllable sentence simplification using control tokens
allows the model to generate a wider range of simplifications, tailored to
specific control tokens and requirements.

Furthermore, some studies like Omelianchuk et al. (2021) have tried to pose
simplification as a tagging problem, similar to Named Entity Recognition. The
model takes a complex sentence and suggests some edits (through tags) to make
it simpler. For example, convert verb to its simple/base form from its third
person singular present tense form; append a word; or replace a complex word
with its simpler alternative. This process is then repeated for a fixed number
of times as some of the edits can be dependent on the other edits. The process
eventually leads to a simpler sentence.

Overall, the use of transformer models in sentence simplification has shown
promising results, with a potential for further improvement through fine-
tuning, control tokens, and incorporating existing linguistic knowledge.

##  3 Methods

###  3.1 Data Source

We used the WikiLarge dataset created by Zhang and Lapata (2017). It was
formed by merging data from three different sources, the major one being from
Coster and Kauchak (2011). Coster and Kauchak (2011) data contains 137K
aligned sentence pairs from 111http://en.wikipedia.org/English Wikipedia and
corresponding simplified sentences from 222http://simple.wikipedia.orgSimple
English Wikipedia. The total size of WikiLarge by merging the three sources is
296K.

For the validation and test purpose, data from WikiSmall and the
simplifications provided by Amazon Mechanical Turk workers were used Xu et al.
(2016). The validation and test set contains 2000 and 359 sentence pairs
respectively, along with 8 reference pairs. This has sort of become a standard
dataset for SS task and is used in numerous studies. Furthermore, all the
studies compared in this paper use the same data.

###  3.2 Model

We have used the EncoderDecoderModel provided by
333https://huggingface.coHuggingFace’s transformers library with a combination
of BertConfig and GPT2Config as encoder or decoder. The model is implemented
in PyTorch and trained on Nvidia V100 GPU cluster system. BERT uses a
vocabulary size of $30522$ while GPT-2 uses $50257$. Both of them uses
embedding dimension of $768$ and $12$ attention heads. They have a a total of
$12$ hidden layers connected via GeLU non-linearity. The maximum length of the
sentence tokens have been fixed at 80.

For training, AdamW optimizer was used with the initial learning rate of
$1e-4$. It was controlled by a one-cycle learning rate scheduler with the
maximum rate of $1e-3$ and warm start. We used early stopping based on the
SARI score to prevent overfitting of the model. The best model required a
total of 20 epochs and around 24 hours for fine-tuning.

###  3.3 Evaluation Metric

SARI - System output against references and against the input sentence, is the
most common metric for evaluation of SS. It compares the simplified sentence
with both source and reference sentences. Using ngrams, SARI calculates the
scores for three conditions: additions, keeps, and deletions. In general, it
gives a high score when a simple word is added that is absent in the source
sentence but present in the reference sentence, when a complex word is removed
that is present in the source sentence but absent from the reference, or when
the word is retained in both the source and reference sentences. These
separate scores are averaged to determine the result.

##  4 Results & Discussion

We evaluate the results of our models on SARI and compare them against some of
the past studies. SARI is an average of three components: addition, deletion
and keep score as compared between the source, target and the reference
sentences. We report all of these scores, see Table 1 for details. In addition
to using BERT and GPT-2 as encoder and decoder, we also tried a mix of both.
BERT+GPT-2 means that the encoder was made of BERT and the decoder was GPT-2,
and vice versa for GPT-2+BERT.

Figure 1: Distribution of SARI score for all test samples across different
transformer models.

With a SARI score of 46.80, BERT emerged as the top model overall. It may be
as a result of the network’s bidirectional nature. In order to decide whether
or not to modify a word, it may be helpful to consider its context from both
sides. The distribution of test scores in Figure 1 also helps in showing the
large number of samples getting a high SARI score of around 0.5 in BERT.
Following closely after in second place was the GPT-2 model. The lower
performance of GPT-2 can perhaps be due to its unidirectional nature. Lastly,
the combination of BERT and GPT-2 encoder and decoder didn’t seem to perform
very well. It may be inferred that rather than mixing, both of these models
perform best when used within the common architecture.

Model | SARI $\uparrow$ | ADD $\uparrow$ | DELETE $\uparrow$ | KEEP $\uparrow$
---|---|---|---|---
Zhao et al. (2018) | 40.42 | 5.72 | 42.23 | 73.41
Martin et al. (2019) | 41.87 | - | - | -
Omelianchuk et al. (2021) | 41.46 | 6.96 | 47.87 | 69.56
Sheang and Saggion (2021) | 43.31 | - | - | -
Štajner et al. (2022) | 43.30 | - | - | -
[3pt/2pt] BERT+GPT-2 | 42.31 | 11.07 | 62.82 | 53.93
GPT-2+BERT | 42.35 | 10.74 | 62.37 | 54.05
GPT-2 | 46.35 | 12.60 | 66.64 | 59.73
BERT | 46.80 | 12.13 | 67.16 | 61.22

Table 1: Comparison of evaluation scores on Mechanical Turk dataset.

##  5 Conclusion & Future Work

Sentence simplification is the process of creating a simpler version of a
complex sentence and can be useful for people with disabilities, language
learners, or those with low literacy. Simplification often involves removing
complex words and paraphrasing to make the text easier to understand and read.
In this paper, we propose using fine-tuned transformer models for sentence
simplification. We use a combination of transformer encoder and decoder
models, BERT and GPT-2 to be precise. The BERT model proved to be the best of
all the models used in this study and other previous studies. A SARI score of
46.80 on the Mechanical Turk dataset beats the state-of-the-art by a huge
margin. In future, we would like to explore these models on the task of
controlled sentence simplification.

## Acknowledgments

Professor Francis M. Tyers for giving this opportunity and Indiana University
for providing the GPU computing resources.

## References

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