11footnotetext: Department of Mathematics, Clark University, Worcester, MA
01610, USA.
E-mail address<EMAIL_ADDRESS>Dipartimento di
Matematica, Università di Bologna, piazza di Porta S.Donato 5, 40126 Bologna,
Italy.
E-mail address<EMAIL_ADDRESS>
#  $\overline{Q}^{\prime}$-curvature flow on Pseudo-Einstein CR manifolds

Ali Maalaoui(1) & Vittorio Martino(2)

Abstract In this paper we consider the problem of prescribing the
$\overline{Q}^{\prime}$-curvature on three dimensional Pseudo-Einstein CR
manifolds. We study the gradient flow generated by the related functional and
we will prove its convergence to a limit function under suitable assumptions.

Keywords: Pseudo-Einstein CR manifolds, $\overline{P}^{\prime}$-operator

2010 MSC. Primary: 58J60, 58J05. Secondary: 58E05, 58E07 .

##  1 Introduction and statement of the results

Let $(M,T^{1,0}M,\theta)$ be a CR three manifold, which we will always assume
smooth and closed. It is known that one can construct a pair $(Q,P_{\theta})$
such that under a conformal change of the contact form
$\hat{\theta}=e^{2u}\theta$, one has

$P_{\theta}u+Q_{\theta}=Q_{\hat{\theta}}e^{4u}$

where the Paneitz operator $P_{\theta}=(\Delta_{b})^{2}+T^{2}+l.o.t.$; in
particular the operator $P_{\theta}$ contains the space of CR pluriharmonic
functions $\mathcal{P}$ in its kernel, moreover the total $Q$-curvature is
always zero [16], hence it does not provide any extra geometric information.
Therefore, one considers another pair $(P^{\prime},Q^{\prime})$, see [3],
where $P^{\prime}$ is a Paneitz type operator satisfying
$P^{\prime}=4(\Delta_{b})^{2}+l.o.t.$ and is defined on the space of
pluriharmonic functions and the $Q^{\prime}$-curvature is defined implicitly
so that

$P^{\prime}_{\theta}u+Q^{\prime}_{\theta}-\frac{1}{2}P_{\theta}(u^{2})=Q^{\prime}_{\hat{\theta}}e^{2u},$

which is equivalent to

$P^{\prime}_{\theta}u+Q^{\prime}_{\theta}=Q^{\prime}_{\hat{\theta}}e^{2u}\text{
mod }\mathcal{P}^{\perp}.$ (1)

In the case of pseudo-Einstein three dimensional CR manifolds (we refer the
reader to the next section for further details), in [10] the authors showed
that the total $Q^{\prime}$-curvature is not always zero and it is invariant
under the conformal change of the contact structure; in particular it is
proportional to the Burns-Epstein invariant $\mu(M)$ (see [5], [13]) and if
$(M,J)$ is the boundary of a strictly pseudo-convex domain $X$, then

$\int_{M}Q^{\prime}\theta\wedge
d\theta=16\pi^{2}\Big{(}\chi(X)-\int_{X}(c_{2}-\frac{1}{3}c_{1}^{2})\Big{)},$

where $c_{1}$ and $c_{2}$ are the first and second Chern forms of the Kähler-
Einstein metric on $X$ obtained by solving Fefferman’s equation.
At this point, in order to avoid the problem of solving orthogonally to the
infinite dimensional space $\mathcal{P}^{\perp}$, in [9] it is introduced on
pseudo-Einstein three dimensional CR manifolds a new couple
$(\overline{P}^{\prime},\overline{Q}^{\prime})$ which comes from the
projection of equation $(\ref{eq1})$ on to the space of $L^{2}$ CR
pluriharmonic functions $\hat{\mathcal{P}}$, the completion of $\mathcal{P}$
under the $L^{2}$-norm. Since the $P^{\prime}$-operator is only defined after
projection on $\mathcal{P}$, if $\Gamma:L^{2}(M)\to\hat{\mathcal{P}}$ denotes
the orthogonal projection and we let $\overline{P}^{\prime}=\Gamma\circ
P^{\prime}$ and $\overline{Q}^{\prime}=\Gamma\circ Q^{\prime}$, then one can
consider the problem of prescribing the $\overline{Q}^{\prime}$-curvature,
under conformal change of the contact structure on pseudo-Einstein CR
manifolds. In particular, for a given a function $f\in\hat{\mathcal{P}}$, one
wants to solve the following equation

$P^{\prime}u+Q^{\prime}=fe^{2u}\text{ mod }\mathcal{P}^{\perp},$ (2)

that is equivalent to

$\overline{P}^{\prime}u+\overline{Q}^{\prime}=\Gamma(fe^{2u}).$

Therefore, if $u$ solves $(\ref{eq2})$, then for $\tilde{\theta}=e^{u}\theta$,
one has $\overline{Q}^{\prime}_{\tilde{\theta}}=f$. Let us explicitly notice
the differences between the two projections, since the space of $L^{2}$ CR
pluriharmonic functions $\hat{\mathcal{P}}$ does not depend on the contact
form. Thus, $\overline{Q}^{\prime}$ is the orthogonal projection of
$Q^{\prime}$ on $\hat{\mathcal{P}}$ with respect to the $L^{2}$-inner product
induced by $\theta$, while $\overline{Q}^{\prime}_{\tilde{\theta}}$ is the
orthogonal projection of $Q^{\prime}_{\tilde{\theta}}$ with respect to the
$L^{2}$-inner product induced by $\tilde{\theta}$; in particular
$\phi\in\hat{\mathcal{P}}_{\theta}$ if and only if
$\phi\in\hat{\mathcal{P}}_{\tilde{\theta}}$ and
$\psi\in\mathcal{P}^{\perp}_{\theta}$ if and only if
$e^{-2u}\psi\in\mathcal{P}^{\perp}_{\tilde{\theta}}$. Therefore, by denoting
$\Gamma_{u}$ the orthogonal projection induced by $\tilde{\theta}$, one has
$\Gamma_{u}(Q^{\prime}_{\tilde{\theta}})=f$. Let us also recall that in [10],
the authors show that the non-negativity of the Paneitz operator $P_{\theta}$
and the positivity of the CR-Yamabe invariant imply that
$\overline{P}^{\prime}$ is non-negative and
$\ker\overline{P}^{\prime}=\mathbb{R}$. Moreover,
$\int_{M}Q^{\prime}=\int_{M}\overline{Q}^{\prime}\leq 16\pi^{2}$ with equality
if an only if $(M,T^{1,0}M,\theta)$ is the standard sphere; in particular the
previously assumptions imply that the $(M,T^{1,0}M,\theta)$ is embeddable (see
[12]). Notice that unlike the Riemannian case, it remains unclear if the non-
negativity of $\overline{P}^{\prime}$ and
$\ker\overline{P}^{\prime}=\mathbb{R}$ is a sufficient condition for
$\int_{M}Q^{\prime}_{\theta}\leq 16\pi^{2}$. In particular, the results
presented in this paper do not fully cover the case $\overline{P}^{\prime}\geq
0$ and $\ker\overline{P}^{\prime}=\mathbb{R}$.
Thus, from now on we will always assume that $(M,T^{1,0}M,\theta)$ is a
pseudo-Einstein CR three dimensional manifold such that
$\overline{P}^{\prime}$ is non-negative and
$\ker\overline{P}^{\prime}=\mathbb{R}$. The problem in (2) was first studied
in [10] for $f$ constant and in the subcritical case, namely
$\int_{M}\overline{Q}^{\prime}<16\pi^{2}$. Then in [21] the problem was solved
for $f>0$ via a probabilistic approach again in the subcritical case; also, a
solution of the problem was provided in [17] for $f>0$ and
$0<\int_{M}\overline{Q}^{\prime}<16\pi^{2}$ via direct minimization.
In this paper we will study the equation (2) allowing $f$ to change sign: our
approach follows closely the methods in [1], where the authors study the
analogous problem in the Riemannian setting. In particular, we will use a
variational approach by defining a suitable functional on an appropriate space
and then we will study the evolution problem along the negative gradient flow
lines: the convergence at infinity will provide a solution to the initial
problem. Indeed, with respect to [1], new technical issues will appear, which
are essentially due to our sub-Riemannian setting: in particular all the
computations and the estimates regarding the convergence along the flow lines
have to be done accordingly to the projection on the space of $L^{2}$ CR
pluriharmonic functions, that we defined earlier. Moreover, some technical
estimates on the sphere will be adapted to the CR setting as we will see in
Section 5 and the Appendix. Therefore, let us define the following functional
$E:H\to\mathbb{R}$, by

$E(u)=\int_{M}u\overline{P}^{\prime}u+2\int_{M}\overline{Q}^{\prime}u$

where $H=\hat{\mathcal{P}}\cap\mathcal{S}^{2}(M)$ and $\mathcal{S}^{2}(M)$ is
the Folland-Stein Sobolev space equipped with the equivalent norm (see section
2), defined by

$\|u\|^{2}=\int_{M}u\overline{P}^{\prime}u+\int_{M}u^{2}.$ (3)

We consider the following space, which will serve as a constraint

$X=\left\\{u\in
H;N(u):=\int_{M}\Gamma\left(fe^{2u}\right)=\int_{M}\overline{Q}^{\prime}\;\right\\};$

we notice that the space is well defined since $e^{u}\in L^{2}$, see [9],
Theorem 3.1.

As in the classical case, we will need the following hypotheses, depending on
the sign of $\int_{M}\overline{Q}^{\prime}$, namely:

$\left\\{\begin{array}[]{lll}(i)&\displaystyle\inf_{x\in M}f(x)<0,&\text{ if
}\displaystyle\int_{M}\overline{Q}^{\prime}<0\\\ \\\
(ii)&\displaystyle\sup_{x\in M}f(x)>0,\;\inf_{x\in M}f(x)<0&\text{ if
}\displaystyle\int_{M}\overline{Q}^{\prime}=0\\\ \\\
(iii)&\displaystyle\sup_{x\in M}f(x)>0,&\text{ if
}0<\displaystyle\int_{M}\overline{Q}^{\prime}\leq
16\pi^{2}.\end{array}\right.$ (4)

In the case when $\int_{M}\overline{Q}^{\prime}=0$, we let $\ell$ the unique
CR pluriharmonic function satisfying
$\overline{P}^{\prime}_{\theta}\ell+\overline{Q}^{\prime}=0$ and
$\int_{M}\ell=0$, see [9], Theorem 1.1. Notice that
$\overline{Q}^{\prime}_{e^{\ell}\theta}=0$. We also recall that in the
critical case $M=S^{3}$, there are some extra compatibility condition of
Kazdan-Warner type that $f$ needs to satisfy in order to be the
$\overline{Q}^{\prime}$-curvature of a contact structure conformal to the
standard one on the sphere (see Theorem 1.3. in [17]).
Now, in order to define the flow equation, we compute the first variation of
$E$, $N$, and their ($\mathcal{S}^{2}$) gradient, respectively:

$\displaystyle\langle\nabla E(u),\phi\rangle$
$\displaystyle=2\int_{M}\left(\overline{P}^{\prime}u+\overline{Q}^{\prime}\right)\phi\;,\forall\phi\in
H\;,$ $\displaystyle\langle\nabla N(u),\phi\rangle$
$\displaystyle=2\int_{M}\Gamma\left(fe^{2u}\right)\phi\;,\forall\phi\in H\;,$
$\displaystyle\nabla E(u)$
$\displaystyle=2\left(\overline{P}^{\prime}+I\right)^{-1}\left(\overline{P}^{\prime}u+\overline{Q}^{\prime}\right)\;,$
$\displaystyle\nabla N(u)$
$\displaystyle=2\left(\overline{P}^{\prime}+I\right)^{-1}\Gamma\left(fe^{2u}\right)\;.$

In addition, since by hypotheses (4), $\nabla N\neq 0$ on $X$, then $X$ is a
regular hypersurface in $H$ and a unit normal vector field on $X$ is given by
$\nabla N/\|\nabla N\|$. Indeed, $\nabla N(u)\not=0$ if and only if
$\Gamma(e^{2u}f)\not=0$. This last identity is clear for the hypothesis $(i)$
and $(iii)$. But for $(ii)$, recall that $f\in\hat{\mathcal{P}}$, so if
$\Gamma(e^{2u}f)=0$, then $\int_{M}e^{2u}f^{2}=0$, leading to a contradiction.
The gradient of $E$ restricted to $X$ is then

$\nabla^{X}E=\nabla E-\left\langle\nabla E,\frac{\nabla N}{\|\nabla
N\|}\right\rangle\frac{\nabla N}{\|\nabla N\|}.$

Finally, the (negative) gradient flow equation is given by

$\left\\{\begin{array}[]{lll}\partial_{t}u=-\nabla^{X}E(u)\\\ \\\
u(0)=u_{0}\in X\end{array}\right.$ (5)

Now we can state our main results.

######  Theorem 1.1.

Let $(M,T^{1,0}M,\theta)$ be a pseudo-Einstein CR three dimensional manifold
such that $\overline{P}^{\prime}$ is non-negative and
$\ker\overline{P}^{\prime}=\mathbb{R}$. Let us assume that
$\displaystyle\int_{M}\overline{Q}^{\prime}<0$ and let $f\in
C(M)\cap\hat{\mathcal{P}}$ as in (4). Then there exists a positive constant
$C_{0}$ depending on $f^{-}=\max\\{-f,0\\}$, $M$ and $\theta$, such that if

$e^{\tau\|u_{0}\|^{2}}\sup_{x\in M}f(x)\leq C_{0}$

for a constant $\tau>1$ depending on $M$ and $\theta$, then as $t\to\infty$,
the flow converges in $H$ to a solution $u_{\infty}$ of $(\ref{eq1})$.
Moreover, there exist constants $B,\beta>0$ such that

$\|u(t)-u_{\infty}\|\leq B(1+t)^{-\beta},$

for all $t\geq 0$.

######  Theorem 1.2.

Let $(M,T^{1,0}M,\theta)$ be a pseudo-Einstein CR three dimensional manifold
such that $\overline{P}^{\prime}$ is non-negative and
$\ker\overline{P}^{\prime}=\mathbb{R}$. Let us assume that
$\displaystyle\int_{M}\overline{Q}^{\prime}=0$ and let $f\in
C(M)\cap\hat{\mathcal{P}}$ as in (4). Then as $t\to\infty$, the flow converges
in $H$ to a function $u_{\infty}$ and there exists a constant $\lambda$ such
that $v=u_{\infty}+\lambda$ satisfies

$\overline{P}^{\prime}v+\overline{Q}^{\prime}=\delta\Gamma(fe^{2v}),$

where $\delta\in\\{+1,0,-1\\}$. Moreover, there exist constants $B,\beta>0$
such that

$\|u(t)-u_{\infty}\|\leq B(1+t)^{-\beta},$

for all $t\geq 0$. If in addition, we assume that $\int_{M}fe^{2\ell}\not=0$,
then $\delta\not=0$.

######  Theorem 1.3.

Let $(M,T^{1,0}M,\theta)$ be a pseudo-Einstein CR three dimensional manifold
such that $\overline{P}^{\prime}$ is non-negative and
$\ker\overline{P}^{\prime}=\mathbb{R}$. Let us assume that
$0<\displaystyle\int_{M}\overline{Q}^{\prime}<16\pi^{2}$ and let $f\in
C(M)\cap\hat{\mathcal{P}}$ as in (4). Then as $t\to\infty$, the flow converges
in $H$ to a solution $u_{\infty}$ of $(\ref{eq1})$. Moreover, there exist
constants $B,\beta>0$ such that

$\|u(t)-u_{\infty}\|\leq B(1+t)^{-\beta},$

for all $t\geq 0$.

Finally, the critical case of the sphere, which is a bit different. We will
consider a group $G$ acting on $S^{3}$ preserving the CR structure. We denote
by $\Sigma$ the set of points fixed by $G$, that is

$\Sigma=\left\\{x\in S^{3};\;g\cdot x=x,\;\forall g\in G\right\\}$

and we will assume $f$ being invariant under $G$, namely $f(g\cdot
x)=f(x),\forall g\in G$. Then we have the following

######  Theorem 1.4.

Let us consider the sphere $M=S^{3}$ equipped with its standard contact
structure and let $f\in C(M)\cap\hat{\mathcal{P}}$ as in (4) and invariant
under $G$. Let us assume that also $u_{0}\in X$ is invariant under $G$. If
$\Sigma=\emptyset$ or

$\sup_{x\in\Sigma}f(x)\leq e^{-\frac{E(u_{0})}{16\pi^{2}}},$

then as $t\to\infty$, the flow converges in $H$ to a solution (invariant under
$G$) $u_{\infty}$ of $(\ref{eq1})$. Moreover, there exist constants
$B,\beta>0$ such that

$\|u(t)-u_{\infty}\|\leq B(1+t)^{-\beta},$

for all $t\geq 0$.

Acknowledgment

The authors wants to express their gratitude to the referee for his/her
careful reading of the paper: the remarks and the suggestions led to a serious
improvement of the paper.

##  2 Some definition in pseudo-Hermitian geometry

We will follow the notations in [10]. Let $M^{3}$ be a smooth, oriented three-
dimensional manifold. A CR structure on $M$ is a one-dimensional complex sub-
bundle $T^{1,0}M\subset T_{\mathbb{C}}M:=TM\otimes\mathbb{C}$ such that
$T^{1,0}M\cap T^{0,1}M=\\{0\\}$ for $T^{0,1}M:=\overline{T^{1,0}M}$. Let
$\mathcal{H}=ReT^{1,0}M$ and let $J\colon\mathcal{H}\to\mathcal{H}$ be the
almost complex structure defined by $J(Z+\bar{Z})=i(Z-\bar{Z})$, for all $Z\in
T^{1,0}M$. The condition that $T^{1,0}M\cap T^{0,1}M=\\{0\\}$ is equivalent to
the existence of a contact form $\theta$ such that $\ker\theta=\mathcal{H}$.
We recall that a 1-form $\theta$ is said to be a contact form if $\theta\wedge
d\theta$ is a volume form on $M^{3}$. Since $M$ is oriented, a contact form
always exists, and is determined up to multiplication by a positive real-
valued smooth function. We say that $(M^{3},T^{1,0}M)$ is strictly pseudo-
convex if the Levi form $d\theta(\cdot,J\cdot)$ on
$\mathcal{H}\otimes\mathcal{H}$ is positive definite for some, and hence any,
choice of contact form $\theta$. We shall always assume that our CR manifolds
are strictly pseudo-convex.
Notice that in a CR-manifold, there is no canonical choice of the contact form
$\theta$. A pseudo-Hermitian manifold is a triple $(M^{3},T^{1,0}M,\theta)$
consisting of a CR manifold and a contact form. The Reeb vector field $T$ is
the vector field such that $\theta(T)=1$ and $d\theta(T,\cdot)=0$. The choice
of $\theta$ induces a natural $L^{2}$-dot product $\langle\cdot,\cdot\rangle$,
defined by

$\langle f,g\rangle=\int_{M}f(x)g(x)\ \theta\wedge d\theta.$

A $(1,0)$-form is a section of $T_{\mathbb{C}}^{\ast}M$ which annihilates
$T^{0,1}M$. An admissible coframe is a non-vanishing $(1,0)$-form $\theta^{1}$
in an open set $U\subset M$ such that $\theta^{1}(T)=0$. Let
$\theta^{\bar{1}}:=\overline{\theta^{1}}$ be its conjugate. Then
$d\theta=ih_{1\bar{1}}\theta^{1}\wedge\theta^{\bar{1}}$ for some positive
function $h_{1\bar{1}}$. The function $h_{1\bar{1}}$ is equivalent to the Levi
form. We set $\\{Z_{1},Z_{\bar{1}},T\\}$ to the dual of
$(\theta^{1},\theta^{\bar{1}},\theta)$. The geometric structure of a CR
manifold is determined by the connection form $\omega_{1}{}^{1}$ and the
torsion form $\tau_{1}=A_{11}\theta^{1}$ defined in an admissible coframe
$\theta^{1}$ and is uniquely determined by

$\left\\{\begin{array}[]{ll}d\theta^{1}=\theta^{1}\wedge\omega_{1}{}^{1}+\theta\wedge\tau^{1},\\\
\omega_{1\bar{1}}+\omega_{\bar{1}1}=dh_{1\bar{1}},\end{array}\right.$

where we use $h_{1\bar{1}}$ to raise and lower indices. The connection forms
determine the pseudo-Hermitian connection $\nabla$, also called the Tanaka-
Webster connection, by

$\nabla Z_{1}:=\omega_{1}{}^{1}\otimes Z_{1}.$

The scalar curvature $R$ of $\theta$, also called the Webster curvature, is
given by the expression

$d\omega_{1}{}^{1}=R\theta^{1}\wedge\theta^{\bar{1}}\mod\theta.$

######  Definition 2.1.

A real-valued function $w\in C^{\infty}(M)$ is CR pluriharmonic if locally
$w=Ref$ for some complex-valued function $f\in C^{\infty}(M,\mathbb{C})$
satisfying $Z_{\bar{1}}f=0$.

Equivalently, [20], $w$ is a CR pluriharmonic function if

$P_{3}w:=\nabla_{1}\nabla_{1}\nabla^{1}w+iA_{11}\nabla^{1}w=0$

for $\nabla_{1}:=\nabla_{Z_{1}}$. We denote by $\mathcal{P}$ the space of all
CR pluriharmonic functions and $\hat{\mathcal{P}}$ the completion of
$\mathcal{P}$ in $L^{2}(M)$, also called the space of $L^{2}$ CR pluriharmonic
functions. Let $\Gamma:L^{2}(M)\to\hat{\mathcal{P}}$ be the orthogonal
projection on the space of $L^{2}$ pluriharmonic functions. If
$S:L^{2}(M)\to\ker\bar{\partial}_{b}$ denotes the Szego kernel, then

$\Gamma=S+\bar{S}+F,$ (6)

where $F$ is a smoothing kernel as shown in [19]. The Paneitz operator
$P_{\theta}$ is the differential operator

$\displaystyle P_{\theta}(w)$ $\displaystyle:=4\text{div}(P_{3}w)$
$\displaystyle=\Delta_{b}^{2}w+T^{2}-4\text{Im}\nabla^{1}\left(A_{11}\nabla^{1}f\right)$

for $\Delta_{b}:=\nabla^{1}\nabla_{1}+\nabla^{\bar{1}}\nabla_{\bar{1}}$ the
sub-Laplacian. In particular, $\mathcal{P}\subset\ker P_{\theta}$. Hence,
$\ker P_{\theta}$ is infinite dimensional. For a thorough study of the
analytical properties of $P_{\theta}$ and its kernel, we refer the reader to
[19, 6, 8]. The main property of the Paneitz operator $P_{\theta}$ is that it
is CR covariant [16]. That is, if $\hat{\theta}=e^{w}\theta$, then
$e^{2w}P_{\hat{\theta}}=P_{\theta}$.

######  Definition 2.2.

Let $(M^{3},T^{1,0}M,\theta)$ be a pseudo-Hermitian manifold. The Paneitz type
operator $P_{\theta}^{\prime}\colon\mathcal{P}\to C^{\infty}(M)$ is defined by

$\displaystyle P_{\theta}^{\prime}f$
$\displaystyle=4\Delta_{b}^{2}f-8\textnormal{Im}\left(\nabla^{\alpha}(A_{\alpha\beta}\nabla^{\beta}f)\right)-4\textnormal{Re}\left(\nabla^{\alpha}(R\nabla_{\alpha}f)\right)$
$\displaystyle\quad+\frac{8}{3}\textnormal{Re}(\nabla_{\alpha}R-i\nabla^{\beta}A_{\alpha\beta})\nabla^{\alpha}f-\frac{4}{3}f\nabla^{\alpha}(\nabla_{\alpha}R-i\nabla^{\beta}A_{\alpha\beta})$
(7)

for $f\in\mathcal{P}$.

The main property of the operator $P_{\theta}^{\prime}$ is its “almost”
conformal covariance as shown in [2, 10]. That is if $(M^{3},T^{1,0}M,\theta)$
is a pseudo-Hermitian manifold, $w\in C^{\infty}(M)$, and we set
$\hat{\theta}=e^{w}\theta$, then

$e^{2w}P_{\hat{\theta}}^{\prime}(u)=P_{\theta}^{\prime}(u)+P_{\theta}\left(uw\right)$
(8)

for all $u\in\mathcal{P}$. In particular, since $P_{\theta}$ is self-adjoint
and $\mathcal{P}\subset\ker P_{\theta}$, we have that the operator
$P^{\prime}$ is conformally covariant, mod $\mathcal{P}^{\perp}$.

######  Definition 2.3.

A pseudo-Hermitian manifold $(M^{3},T^{1,0}M,\theta)$ is pseudo-Einstein if

$\nabla_{\alpha}R-i\nabla^{\beta}A_{\alpha\beta}=0.$

Moreover, if $\theta$ induces a pseudo-Einstein structure then $e^{u}\theta$
is pseudo-Einstein if and only if $u\in\mathcal{P}$. The definition above was
stated in [10], but it was implicitly mentioned in [16]. In particular, if
$(M^{3},T^{1,0}M,\theta)$ is pseudo-Einstein, then $P_{\theta}^{\prime}$ takes
a simpler form:

$P_{\theta}^{\prime}f=4\Delta_{b}^{2}f-8\text{Im}\left(\nabla^{1}(A_{11}\nabla^{1}f)\right)-4\text{Re}\left(\nabla^{1}(R\nabla_{1}f)\right).$

In particular, one has

$\int_{M}uP^{\prime}_{\theta}u\geq
4\int_{M}|\Delta_{b}u|^{2}-C\int_{M}|\nabla_{b}u|^{2}.$

Using the interpolation inequality

$\int_{M}|\nabla_{b}u|^{2}\leq C\|u\|_{L^{2}}\|\Delta_{b}u\|_{L^{2}},$

and $2ab\leq\varepsilon a^{2}+\frac{1}{\epsilon}b^{2}$, we have the existence
of $C_{1}>0$ and $C_{2}>0$, such that

$\int_{M}uP^{\prime}_{\theta}u\geq
C_{1}\int_{M}|\Delta_{b}u|^{2}-C_{2}\int_{M}u^{2}.$

Hence, if $P^{\prime}_{\theta}$ is non-negative, with trivial kernel, one has
the equivalence of the Folland-Stein Sobolev norm and $(\ref{norm})$.

######  Definition 2.4.

Let $(M^{3},T^{1,0}M,\theta)$ be a pseudo-Einstein manifold. The
$Q^{\prime}$-curvature is the scalar quantity defined by

$Q_{\theta}^{\prime}=2\Delta_{b}R-4|A|^{2}+R^{2}.$ (9)

The main equation that we will be dealing with is the change of the
$Q^{\prime}$-curvature under conformal change. Let $(M^{3},T^{1,0}M,\theta)$
be a pseudo-Einstein manifold, let $w\in\mathcal{P}$, and set
$\hat{\theta}=e^{w}\theta$. Hence $\hat{\theta}$ is pseudo-Einstein. Then [2,
10]

$e^{2w}Q_{\hat{\theta}}^{\prime}=Q_{\theta}^{\prime}+P_{\theta}^{\prime}(w)+\frac{1}{2}P_{\theta}\left(w^{2}\right).$
(10)

In particular, $Q_{\theta}^{\prime}$ behaves as the $Q$-curvature for
$P_{\theta}^{\prime}$, mod $\mathcal{P}^{\perp}$. Since we are working modulo
$\mathcal{P}^{\perp}$ it is convenient to project the previously defined
quantities on $\hat{\mathcal{P}}$. So we define the operator
$\overline{P}_{\theta}^{\prime}=\Gamma\circ P_{\theta}^{\prime}$ and the
$\overline{Q}^{\prime}$-curvature by
$\overline{Q}^{\prime}_{\theta}=\Gamma(Q^{\prime}_{\theta})$. Notice that

$\int_{M}Q^{\prime}_{\theta}\ \theta\wedge
d\theta=\int_{M}\overline{Q}^{\prime}_{\theta}\ \theta\wedge d\theta.$

Moreover, the operator $\overline{P}^{\prime}_{\theta}$ has many interesting
analytical properties. Indeed,
$\overline{P}^{\prime}_{\theta}:\mathcal{P}\to\hat{\mathcal{P}}$ is an
elliptic pseudo-differential operator (see [9]) and if we assume that
$\ker\overline{P}^{\prime}_{\theta}=\mathbb{R}$, then its Green’s function $G$
satisfies

$\overline{P}^{\prime}_{\theta}G(\cdot,y)=\Gamma(\cdot,y)-\frac{1}{V},$

where $V=\int_{M}\theta\wedge d\theta$ is the volume of $M$ and
$\Gamma(\cdot,\cdot)$ is the kernel of the projection operator $\Gamma$.
Moreover,

$G(x,y)=-\frac{1}{4\pi^{2}}\ln(|xy^{-1}|)+\mathcal{K}(x,y),$

where $\mathcal{K}$ is a bounded kernel as proved in [7].

##  3 Preliminary results on the flow

First we recall one fundamental inequality that we will be using all along
this paper, namely the CR version of the Beckner-Onofri inequality. This
inequality was first proved in the odd dimensional spheres in [3] and then
naturally extended to pseudo-Einstein 3-manifolds in [9, Theorem 3.1].

######  Theorem 3.1.

Assume that $\overline{P}^{\prime}$ is non-negative and
$\ker\overline{P}^{\prime}=\mathbb{R}$. Then, there exists $C>0$ such that for
all $u\in\hat{\mathcal{P}}\cap\mathcal{S}^{2}(M)$ with
$\displaystyle\int_{M}u=0$, we have

$\frac{1}{16\pi^{2}}\int_{M}u\overline{P}^{\prime}u+C\geq\ln\left(\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{M}e^{2u}\right).$

In the case of the sphere, $C$ can be taken to be $0$ and equality holds if
and only if $u=J(h)$ with $h\in Aut(S^{3})$ and $J(h)=det(Jac(h))$ is the
determinant of the Jacobian determinant of $h$. The dual version of the above
inequality was also investigated in [22], where the existence of extremals was
investigated.
Now, we prove the global existence of solutions of (5):

######  Lemma 3.1.

Let $(M,T^{1,0}M,\theta)$ be a pseudo-Einstein CR three dimensional manifold
such that $\overline{P}^{\prime}$ is non-negative and
$\ker\overline{P}^{\prime}=\mathbb{R}$. Let $f\in C(M)\cap\hat{\mathcal{P}}$
as in (4). Then for any $u_{0}\in X$ there exists a solution $u\in
C^{\infty}([0,\infty),H)$ of problem (5) such that $u(t)\in X$, for all $t\geq
0$. Moreover it holds

$\int_{0}^{t}\|\partial_{s}u(s)\|^{2}ds=E(u_{0})-E(u(t)),$

for all $t\geq 0$.

###### Proof.

Since all the functionals involved are regular, the short time existence of a
solution $u$ for (5) is ensured by the Cauchy-Lipschitz Theorem. In order to
extend it to all $t\geq 0$, we notice that

$\|\partial_{t}u\|=\|\nabla^{X}E(u)\|\leq 2\|\nabla E(u)\|\leq
C_{1}\|u\|+C_{2}.$

Thus, since

$\partial_{t}\|u\|^{2}=2\langle u,\partial_{t}u\rangle\leq
C_{3}\|u\|^{2}+C_{4},$

by Gronwall’s lemma, the solution $u$ exists for all $t\geq 0$. In addition

$\partial_{t}N(u)=\left\langle\nabla
N(u),\partial_{t}u\right\rangle=-\left\langle\nabla
N(u),\nabla^{X}E(u)\right\rangle=0,$

therefore $u(t)\in X$, for all $t\geq 0$. Finally, we have

$\partial_{t}E(u)=\langle\nabla
E(u),\partial_{t}u\rangle=-\|\partial_{t}u\|^{2}.$

Hence, $E$ is decreasing along the flow and the following energy identity
holds

$\int_{0}^{t}\|\partial_{s}u\|^{2}ds=E(u_{0})-E(u(t)).$ (11)

∎

Next we prove the following lemma about the convergence

######  Lemma 3.2.

Let $(M,T^{1,0}M,\theta)$ be a pseudo-Einstein CR three dimensional manifold
such that $\overline{P}^{\prime}$ is non-negative and
$\ker\overline{P}^{\prime}=\mathbb{R}$. Let $f\in C(M)\cap\hat{\mathcal{P}}$
as in (4) and let $u$ be the solution of problem (5) obtained in the previous
Lemma (3.1). If there exists a constant $C>0$ such that $\|u(t)\|\leq C$, for
all $t\geq 0$, then when $t\to\infty$, $u(t)\to u_{\infty}$ in $H$ and
$u_{\infty}$ solves the equation

$\overline{P}^{\prime}u+\overline{Q}^{\prime}u=\lambda\Gamma(fe^{2u}),$

for a certain $\lambda\in\mathbb{R}$. Moreover, there exist constants
$B,\beta>0$ such that

$\|u(t)-u_{\infty}\|\leq B(1+t)^{-\beta},$

for all $t\geq 0$.

###### Proof.

Since $\|u\|\leq C$, we have that

$|E(u)|\leq 2\|u\|^{2}+C_{2}.$

Therefore, by the previous energy estimate

$\int_{0}^{\infty}\|\partial_{t}u\|^{2}dt<\infty.$

So there exists a sequence $t_{k}\to\infty$ such that

$\|\partial_{t}u(t_{k})\|=\|\nabla^{X}E(u(t_{k}))\|\to 0.$

Now, from the boundedness of $\|u\|$, we also have the convergence
$u(t_{k})\to u_{\infty}$ strongly in $L^{2}(M)$ and weakly in
$\mathcal{S}^{2}(M)$. From Theorem 3.1, we have that $e^{2u(t_{k})}\in
L^{p}(M)$, with $p\geq 1$, and $\|e^{2u(t_{k})}\|_{L^{p}}$ is uniformly
bounded. Thus by Egorov’s lemma, we can deduce that

$\left\|fe^{2u(t_{k})}-fe^{2u_{\infty}}\right\|_{L^{p}}\to 0,\;1\leq
p<\infty.$

Indeed, we fix $\varepsilon>0$. Then, there exists a set $A$ with
$Vol(A)<\varepsilon$ such that $fe^{2u(t_{k})}$ converges uniformly to
$fe^{2u_{\infty}}$ on $M\setminus A$. Therefore,

$\left\|fe^{2u(t_{k})}-fe^{2u_{\infty}}\right\|_{L^{p}}\leq
C\left\|fe^{2u(t_{k})}-fe^{2u_{\infty}}\right\|_{L^{\infty}(M\setminus
A)}+\Big{(}\left\|fe^{2u(t_{k})}\right\|_{L^{\tilde{p}}}+\left\|fe^{2u_{\infty}}\right\|_{L^{\tilde{p}}}\Big{)}Vol(A)^{\frac{1}{p}-\frac{1}{\tilde{p}}},$

for p¡$\tilde{p}<\infty$. So the conclusion follows from the uniform
boundedness of $\|e^{2u(t_{k})}\|_{L^{p}}$, for all $1\leq p<\infty$.

Thus $u_{\infty}\in X$. Now we have that

$\nabla
N(u(t_{k}))=2(\overline{P}^{\prime}+I)^{-1}\Gamma\left(fe^{2u(t_{k})}\right),$

and since $fe^{2u(t_{k})}$ converges strongly in $L^{p}(M)$ and $\Gamma$ maps
continuously $L^{p}(M)$ to $L^{p}(M)$ (this follows from (6) and [24]), we
have by the compactness of $\overline{P}^{\prime}+I$ that $\nabla N(u(t_{k})$
converges strongly to $\nabla N(u_{\infty})$. Also,

$\nabla
E(u(t))=2(\overline{P}^{\prime}+I)^{-1}(\overline{P}^{\prime}u+\overline{Q}^{\prime})=2u(t)+2(\overline{P}^{\prime}+I)^{-1}(\overline{Q}^{\prime}-u).$

Thus, since $\partial_{t}u(t_{k})\to 0$ in $\mathcal{S}^{2}(M)$, we have that
$u(t_{k})$ converges strongly to $u_{\infty}$ and $\nabla^{X}E(u_{\infty})=0$.
Moreover, we have

$(\overline{P}^{\prime}+I)^{-1}(\overline{P}^{\prime}u_{\infty}+\overline{Q}^{\prime})=\lambda(u_{\infty})(\overline{P}^{\prime}+I)^{-1}\Gamma(fe^{2u_{\infty}})$

where

$\lambda(u_{\infty})=\frac{\langle\nabla E(u_{\infty}),\nabla
N(u_{\infty})\rangle}{\|\nabla N(u_{\infty})\|^{2}}.$

Now by integration we have that

$\int_{M}\overline{Q}^{\prime}=\lambda(u_{\infty})\int_{M}\Gamma\left(fe^{2u_{\infty}}\right),$

and since $u_{\infty}\in X$, we have if
$\displaystyle\int_{M}\overline{Q}^{\prime}\not=0$, that
$\lambda(u_{\infty})=1$ and hence $u_{\infty}$ solves the desired equation. On
the other hand, if $\displaystyle\int_{M}\overline{Q}^{\prime}=0$, either
$\lambda(u_{\infty})=0$ and thus

$\overline{P}^{\prime}u_{\infty}+\overline{Q}^{\prime}=0,$

or $\lambda(u_{\infty})>0$, thus setting
$v=u_{\infty}+\frac{1}{2}\ln(\lambda(u_{\infty}))$ we have

$\overline{P}^{\prime}v+\overline{Q}^{\prime}=\Gamma(fe^{2v}),$

and similarly if $\lambda(u_{\infty})<0$, we have a function
$v=u_{\infty}+\frac{1}{2}\ln(-\lambda(u_{\infty}))$ such that

$\overline{P}^{\prime}v+\overline{Q}^{\prime}=-\Gamma(fe^{2v}).$

In particular, if we assume that $\int_{M}fe^{2\ell}\not=0$ in the case
$\lambda(u_{\infty})=0$, we have $u_{\infty}-\ell$ is constant. Hence,

$0=\int_{M}fe^{2u_{\infty}}=e^{2(u_{\infty}-\ell)}\int_{M}fe^{2\ell}\not=0,$

which is a contradiction.
The polynomial convergence of the flow can be deduced from the Lojasiewicz-
Simon inequality following Theorem 3 in [25] and Lemma 3.2 in [1]. Let
$\eta:H\to T_{u_{\infty}}X$ be the natural projection, where $T_{u_{\infty}}X$
denotes the tangent space of the manifold $X$ at the point $u_{\infty}$. We
have, for $v\in T_{u_{\infty}}X$

$\left(\nabla^{X}\right)^{2}E(u_{\infty})v=\eta\left(\nabla^{2}E(u_{\infty})v-\frac{\langle\nabla
E(u_{\infty}),\nabla N(u_{\infty})\rangle}{\|\nabla
N(u)\|^{2}}\nabla^{2}N(u_{\infty})v+R^{\perp}v\right)$

where $R^{\perp}v$ is the component along $\nabla N(u_{\infty})$. Thus, since
$\eta\left(R^{\perp}v\right)=0$, we have that

$\left(\nabla^{X}\right)^{2}E(u_{\infty})v=2\left(I-\eta(\overline{P}^{\prime}+1)\right)v-4\frac{\langle\nabla
E(u_{\infty}),\nabla N(u_{\infty})\rangle}{\|\nabla
N(u)\|^{2}}(\overline{P}^{\prime}+1)^{-1}\Gamma(fe^{2u_{\infty}}v).$

It can be checked that $(\nabla^{X})^{2}E(u_{\infty}):T_{u_{\infty}}X\to
T_{u_{\infty}}X$ is a Fredholm operator, then there exists a constant
$\delta>0$ and $0<\kappa<\frac{1}{2}$ such that if
$\|u(t)-u_{\infty}\|<\delta$, it holds

$\|\nabla^{X}E(u)\|\geq(E(u(t))-E(u_{\infty}))^{1-\kappa}.$

We note that if $E(u(t_{0}))=E(u_{\infty})$ for some $t_{0}\geq 0$, then the
flow is stationary and the estimate is trivially satisfied. So we can assume
that $E(u(t))-E(u_{\infty})>0$, for all $t\geq 0$. Since
$\lim_{n\to\infty}\|u(t_{n})-u_{\infty}\|=0$, for a given $\varepsilon>0$,
there exists $n_{0}>0$ such that for $n\geq n_{0}$ we have,

$\|u(t_{n})-u_{\infty}\|<\frac{\varepsilon}{2}$

and

$\frac{1}{\kappa}(E(u(t_{n}))-E(u_{\infty}))^{\kappa}<\frac{\varepsilon}{2}.$

We set $\varepsilon=\frac{\delta}{2}$ and

$T:=\sup\left\\{t\geq
t_{n_{0}};\|u(s)-u_{\infty}\|<\delta;s\in[t_{n_{0}},t]\right\\},$

and we assume for the sake of contradiction that $T<\infty$. Now we have

$-\partial_{t}[E(u(t))-E(u_{\infty})]^{\kappa}=-\kappa\partial_{t}E(u(t))[E(u(t))-E(u_{\infty})]^{\kappa-1},$

but

$-\partial_{t}E(u(t))=-\langle
E(u),\partial_{t}u\rangle=\|\nabla^{X}E(u)\|\|\partial_{t}u\|.$

Thus, for $t\in[t_{n_{0}},T]$ we have

$-\partial_{t}[E(u(t))-E(u_{\infty})]^{\kappa}\geq\kappa\|\partial_{t}u\|,$

and since $E$ is non-increasing along the flow, we have after integration in
the interval $[t_{n_{0}},T]$

$\|u(T)-u(t_{n_{0}})\|\leq\int_{t_{n_{0}}}^{T}\|\partial_{s}u\|ds\leq\frac{1}{\kappa}[E(u(t_{n_{0}}))-E(u_{\infty})]^{\kappa}<\frac{\varepsilon}{2}.$

Hence,

$\|u(T)-u_{\infty}\|\leq\|u(T)-u(t_{n_{0}})\|+\|u(t_{n_{0}})-u_{\infty}\|<\varepsilon=\frac{\delta}{2}$

which is a contradiction and so $T=+\infty$. We set now
$g(t)=E(u(t))-E(u_{\infty})$, for $t\in[t_{n_{0}},+\infty)$. Then we have

$g^{\prime}(t)=-\|\nabla^{X}E(u)\|^{2}\geq g^{2\kappa-1}(t).$

By integration we have

$g^{2\kappa-1}(t)\geq g^{2\kappa-1}(t_{n_{0}})+(1-2\kappa)(t-t_{n_{0}}).$

Since $2\kappa-1<0$, we have

$g(t)\leq[g^{2\kappa-1}(t_{n_{0}})+(1-2\kappa)(t-t_{n_{0}})]^{\frac{1}{2\kappa-1}}\leq
Ct^{\frac{1}{2\kappa-1}}.$

Now, by taking $t^{\prime}>t$, we have

$\|u(t)-u(t^{\prime})\|\leq\int_{t}^{t^{\prime}}\|\partial_{s}u\|ds\leq\frac{1}{\theta}[E(u(t))-E(u_{\infty})]^{\kappa}\leq\frac{1}{\kappa}g^{\kappa}(t)\leq
Ct^{\frac{\kappa}{2\kappa-1}}.$

For $t^{\prime}=t_{n}$, letting $n\to\infty$ and setting
$\beta=\frac{\kappa}{1-2\kappa}$, we get that for $t>t_{n_{0}}$

$\|u(t)-u_{\infty}\|\leq Ct^{-\beta}$

Therefore, since $\|u(t)-u_{\infty}\|$ is bounded for $t>t_{n_{0}}$, we have
the existence of $B>0$ such that for all $t\geq 0$

$\|u(t)-u_{\infty}\|\leq B(1+t)^{-\beta}.$

∎

######  Corollary 3.1.

Let $(M,T^{1,0}M,\theta)$ be a pseudo-Einstein CR three dimensional manifold
such that $\overline{P}^{\prime}$ is non-negative and
$\ker\overline{P}^{\prime}=\mathbb{R}$. Let $f\in C(M)\cap\hat{\mathcal{P}}$
as in (4) and let $u$ be the solution of problem (5) obtained in the Lemma
(3.1). If $\bar{u}=\displaystyle\frac{1}{V}\displaystyle\int_{M}u$ is
uniformly bounded then the flow converges. Here
$V=\displaystyle\int_{M}\theta\wedge d\theta$ is the volume of $M$.

###### Proof.

From the energy identity (11) we have that

$\int_{M}u\overline{P}^{\prime}u+2\int_{M}\overline{Q}^{\prime}u\leq
E(u_{0}),$

but we also have from the Poincaré-type inequality (or the non-negativity of
the operator $\overline{P}^{\prime}$), that

$\int_{M}u\overline{P}^{\prime}u\geq\lambda_{1}\int_{M}(u-\bar{u})^{2}.$

Here $\lambda_{1}$ is the first non-zero eigenvalue of the operator
$\overline{P}^{\prime}$. In particular, from Young’s inequality, we have that

$\int_{M}u\overline{P}^{\prime}u\leq
E(u_{0})+\varepsilon\int_{M}(u-\bar{u})^{2}+C(\varepsilon)\|\overline{Q}^{\prime}\|_{L^{2}}^{2}-2\bar{u}\int_{M}\overline{Q}^{\prime}$

Hence, for $\varepsilon$ small enough, we have that

$\int_{M}u\overline{P}^{\prime}u\leq C,$

since $\bar{u}$ is uniformly bounded, then the uniform boundedness of $\|u\|$
and the conclusion follows from Lemma 3.2. ∎

Therefore, in the rest of the paper, we will show the uniform boundedness of
$\bar{u}$ along the flow, in order to have convergence at infinity.

##  4 The sub-critical case

Along all this section we will assume that $\overline{P}^{\prime}$ is non-
negative and $\ker\overline{P}^{\prime}=\mathbb{R}$. Next we consider the
three separate cases in which
$\displaystyle\int_{M}\overline{Q}^{\prime}<16\pi^{2}$. Also we let
$V=\displaystyle\int_{M}\theta\wedge d\theta$ be the volume of $M$.

###  4.1 Case $\displaystyle\int_{M}\overline{Q}^{\prime}<0$ and proof of
Theorem 1.1

######  Lemma 4.1.

There exists a positive constant $C>0$ depending on $M$ and $\theta$ such that
for any measurable subset $K\subset M$ with $Vol(K)>0$, we have

$\int_{M}u\leq|E(u_{0})|+\frac{C}{Vol(K)}+\frac{4V}{Vol(K)}\max\left(\int_{K}u,0\right)$

###### Proof.

Without loss of generality we can assume that $\displaystyle\int_{M}u>0$
otherwise the inequality is trivially satisfied. First we have that

$\int u\overline{P}^{\prime}u\leq E(u_{0})-2\int_{M}\overline{Q}^{\prime}u$

and

$\|u-\bar{u}\|_{L^{2}}^{2}\leq\frac{1}{\lambda_{1}}\int_{M}u\overline{P}^{\prime}u.$

Hence,

$\int_{M}u^{2}\leq\frac{1}{\lambda_{1}}E(u_{0})-\frac{2}{\lambda_{1}}\int_{M}\overline{Q}^{\prime}u+\frac{1}{V}\left(\int_{M}u\right)^{2}$

Now if $\displaystyle\int_{K}u\leq 0$, then we have

$\left(\int_{M}u\right)^{2}\leq\left(\int_{K^{c}}u\right)^{2}\leq
Vol(K^{c})\int_{M}u^{2},$

hence

$\frac{Vol(K)}{V}\int_{M}u^{2}\leq\frac{1}{\lambda_{1}}E(u_{0})-\frac{2}{\lambda_{1}}\int_{M}\overline{Q}^{\prime}u.$

Again using Young’s inequality we have

$\int_{M}u^{2}\leq\frac{2V}{\lambda_{1}Vol(K)}E(u_{0})+\frac{4\|\overline{Q}^{\prime}\|_{L^{2}}^{2}V^{2}}{\lambda_{1}^{2}Vol(K)^{2}},$

but

$\displaystyle\left(\int_{M}u\right)^{2}$ $\displaystyle\leq
V\int_{M}u^{2}\leq\frac{2V^{2}}{\lambda_{1}Vol(K)}E(u_{0})+\frac{4\|\overline{Q}^{\prime}\|_{L^{2}}^{2}V^{3}}{\lambda_{1}^{2}Vol(K)^{2}}$
$\displaystyle\leq|E(u_{0})|^{2}+\frac{V^{4}}{\lambda_{1}^{2}Vol(K)^{2}}+\frac{4\|\overline{Q}^{\prime}\|_{L^{2}}^{2}V^{3}}{\lambda_{1}^{2}Vol(K)^{2}},$

which yields

$\int_{M}u\leq|E(u_{0})|+\frac{C}{Vol{K}}.$

We assume now that $\displaystyle\int_{K}u>0$. Then one has

$\int_{M}u^{2}\leq\frac{1}{\lambda_{1}}E(u_{0})-\frac{2}{\lambda_{1}}\int_{M}\overline{Q}^{\prime}u+\frac{1}{V}\left(\left(\int_{K}u\right)^{2}+\left(\int_{K^{c}}u\right)^{2}+2\int_{K^{c}}u\int_{K}u\right),$

and

$\frac{2}{V}\int_{K^{c}}u\int_{K}u\leq\frac{2Vol(K^{c})}{Vol(K)V}\left(\int_{K}u\right)^{2}+\frac{Vol(K)}{2V}\int_{M}u^{2}.$

Hence,

$\frac{Vol(K)}{2V}\int_{M}u^{2}\leq\frac{1}{\lambda_{1}}E(u_{0})-\frac{2}{\lambda_{1}}\int_{M}\overline{Q}^{\prime}u+\frac{3}{Vol(K)}\left(\int_{K}u\right)^{2}.$

By using that

$\left|\frac{2}{\lambda_{1}}\int_{M}\overline{Q}^{\prime}u\right|\leq\frac{Vol(K)}{4V}\int_{M}u^{2}+\frac{4V\|\overline{Q}^{\prime}\|_{L^{2}}^{2}}{\lambda_{1}^{2}Vol(K)},$

we have,

$\int_{M}u^{2}\leq\frac{4V}{\lambda_{1}Vol(K)}|E(u_{0})|+\frac{16V^{2}\|\overline{Q}^{\prime}\|_{L^{2}}^{2}}{\lambda_{1}^{2}Vol(K)^{2}}+\frac{12V}{Vol(K)^{2}}\left(\int_{M}u\right)^{2}.$

Hence,

$\left(\int_{M}u\right)^{2}\leq\frac{4V^{2}}{\lambda_{1}Vol(K)}|E(u_{0})|+\frac{16V^{3}\|\overline{Q}^{\prime}\|_{L^{2}}^{2}}{\lambda_{1}^{2}Vol(K)^{2}}+\frac{12V^{2}}{Vol(K)^{2}}\left(\int_{M}u\right)^{2},$

and therefore

$\int_{M}u\leq|E(u_{0})|+\frac{C}{Vol(K)}+\frac{4V}{Vol(K)}\int_{K}u.$

∎

######  Lemma 4.2.

Let $K$ be a measurable subset of $M$ such that $Vol(K)>0$. Then there exists
a constant $\alpha>1$ depending on $M$ and $\theta$ and a constant $C_{K}>1$
depending on $Vol(K)$ such that

$\int_{M}e^{2u}\leq
C_{K}e^{\alpha\|u_{0}\|^{2}}\max\left(\left(\int_{K}e^{2u}\right)^{\alpha},1\right).$

###### Proof.

Recall that from Theorem 3.1 one has the existence of $C>0$ such that

$\int_{M}e^{2u}\leq
C\exp\left(\frac{1}{16\pi^{2}}\int_{M}u\overline{P}^{\prime}u+\frac{2}{V}\int_{M}u\right).$

Again, by the energy identity (11) and Young’s inequality, we have

$\displaystyle\int_{M}u\overline{P}^{\prime}u$ $\displaystyle\leq
E(u_{0})-2\int_{M}\overline{Q}^{\prime}(u-\bar{u})-2\bar{u}\int_{M}\overline{Q}^{\prime}$
$\displaystyle\leq
E(u_{0})-2\bar{u}\int_{M}\overline{Q}^{\prime}+\frac{1}{\varepsilon}\|\overline{Q}^{\prime}\|_{L^{2}}^{2}+\frac{\varepsilon}{\lambda_{1}}\int_{M}u\overline{P}^{\prime}u.$

Thus, for $\varepsilon=\frac{\lambda_{1}}{2}$,

$\frac{1}{2}\int_{M}u\overline{P}^{\prime}u\leq
E(u_{0})-2\bar{u}\int_{M}\overline{Q}^{\prime}+\frac{2}{\lambda_{1}}\|\overline{Q}^{\prime}\|_{L^{2}}^{2}.$

Therefore

$\int_{M}e^{2u}\leq
C\exp\left(\frac{1}{8\pi^{2}}E(u_{0})+\frac{\|\overline{Q}^{\prime}\|_{L^{2}}^{2}}{4\lambda_{1}\pi^{2}}+\left(2-\frac{1}{4\pi^{2}}\int_{M}\overline{Q}^{\prime}\right)\bar{u}\right).$

Now we notice that
$E(u_{0})\leq\|u_{0}\|^{2}+\|\overline{Q}^{\prime}\|_{L^{2}}^{2}$, hence there
exist constants $C_{1}$ and $C_{2}$ such that

$\int_{M}e^{2u}\leq
C_{1}\exp\left(\frac{1}{8\pi^{2}}\|u_{0}\|^{2}+C_{2}\int_{M}u\right).$

By using Lemma 4.1, we get

$\int_{M}e^{2u}\leq\bar{C}_{K}\exp\left(A_{1}\|u_{0}\|^{2}+\frac{A_{2}}{Vol(K)}\max\left(\int_{K}u,0\right)\right),$

where $\bar{C}_{K}$ depends on $Vol(K)$. Now, we set
$\alpha=\max\left(A_{1},\frac{A_{2}}{2},2\right)>1$, and we get

$\int_{M}e^{2u}\leq\bar{C}_{K}\exp\left(\alpha\|u_{0}\|^{2}+\frac{\alpha}{Vol(K)}\max\left(\int_{K}2u,0\right)\right).$

But Jensen’s inequality yields

$\exp\left(\frac{1}{Vol(K)}\displaystyle\int_{K}2u\right)\leq\frac{1}{Vol(K)}\int_{K}e^{2u},$

in particular

$\exp\left(\frac{\alpha}{Vol(K)}\max\left(\int_{K}u,0\right)\right)\leq\max\left(\left(\frac{1}{Vol(K)}\int_{K}e^{2u}\right)^{\alpha},1\right).$

Therefore, by adjusting the constant eventually

$\int_{M}e^{2u}\leq
C_{K}e^{\alpha\|u_{0}\|^{2}}\max\left(\left(\int_{K}e^{2u}\right)^{\alpha},1\right)$

which completes the proof. ∎

Next we move to the proof of Theorem 1.1. We set

$K=\left\\{x\in M;f(x)\leq\frac{1}{2}\inf_{x\in M}f(x)\right\\}.$

From the compatibility condition $(i)$ in (4), we have that $Vol(K)>0$, and
since

$\int_{M}\overline{Q}^{\prime}=\int_{M}fe^{2u_{0}},$

we have

$\frac{\int_{M}\overline{Q}^{\prime}}{\displaystyle\inf_{x\in
M}f(x)}\leq\int_{M}e^{2u_{0}}.$

Thus, there exists $C>0$ (we will assume $C>1$ actually) such that

$\int_{M}e^{2u_{0}}\leq
C\exp\left[C\left(\int_{M}u_{0}\overline{P}^{\prime}u_{0}+\int_{M}u_{0}^{2}\right)\right]=Ce^{C\|u_{0}\|^{2}}.$

Hence,

$\frac{\int_{M}\overline{Q}^{\prime}}{\displaystyle\inf_{x\in M}f(x)}\leq
Ce^{C\|u_{0}\|^{2}}.$ (12)

Next we will prove the following

######  Lemma 4.3.

Let $C_{K}$ and $\alpha$ be the constants found in Lemma 4.2. Let

$r=C_{K}(8C)^{\alpha}e^{(C+1)\alpha\|u_{0}\|^{2}},$

and let us assume that

$e^{\tau\|u_{0}\|^{2}}\sup_{x\in M}f(x)\leq C_{0},$

where $\tau=\alpha(C+1)-C$ and

$C_{0}=-\frac{\displaystyle\inf_{x\in M}f(x)}{8^{\alpha}C_{K}C^{\alpha-1}}.$

Then for all $t\geq 0$, it holds

$\int_{M}e^{2u}\leq 2r.$

###### Proof.

Let

$T=\sup\left\\{s\geq 0;\int_{M}e^{2u}\leq 2r\text{ in }[0,s]\right\\}$

and let us assume for the sake of contradiction that $T<\infty$. We notice
that by continuity, we have that

$\int_{M}e^{2u(T)}=2r.$

We assume first that

$\int_{M}f^{+}e^{2u(T)}\leq\frac{1}{2}\int_{M}f^{-}e^{2u(T)},$

where $f^{+}:=\max\\{f,0\\}$ and $f^{-}=f^{+}-f$ denote the positive and
negative part of $f$ respectively. Then we have

$\int_{M}f^{-}e^{2u(T)}\leq-2\int_{M}fe^{2u(T)}=-2\int_{M}\overline{Q}^{\prime}\leq-4\int_{M}\overline{Q}^{\prime}.$

Since in $K$ we have $f^{-}(x)\geq-\frac{1}{2}\displaystyle\inf_{x\in M}f(x)$,
we have

$\int_{K}e^{2u(T)}\leq\frac{8\int_{M}\overline{Q}^{\prime}}{\displaystyle\inf_{x\in
M}f(x)}$

which combined with (12) gives

$\int_{K}e^{2u(T)}\leq 8Ce^{C\|u_{0}\|^{2}}.$

But from Lemma 4.2, we have

$\int_{M}e^{2u(T)}\leq
C_{K}e^{\alpha\|u_{0}\|^{2}}\max\left(\left(\int_{K}e^{2u}\right)^{\alpha},1\right).$

Thus

$\int_{M}e^{2u(T)}\leq
C_{K}e^{\alpha\|u_{0}\|^{2}}\left(8Ce^{C\|u_{0}\|^{2}}\right)^{\alpha}=r,$

which is a contradiction.
So we move to the next case, where

$\int_{M}f^{+}e^{2u(T)}>\frac{1}{2}\int_{M}f^{-}e^{2u(T)}.$

Then we have

$-\frac{1}{2}\inf_{x\in
M}f(x)\int_{K}e^{2u(T)}\leq\int_{M}f^{-}e^{2u(T)}<2\int_{M}f^{+}e^{2u(T)}\leq
4r\sup_{x\in M}f(x).$

Hence,

$\int_{K}e^{2u(T)}\leq-\frac{8r\displaystyle\sup_{x\in
M}f(x)}{\displaystyle\inf_{x\in M}f(x)}.$

By using our assumption, we have that

$\int_{K}e^{2u(T)}\leq-\frac{8re^{-\tau\|u_{0}\|^{2}}C_{0}}{\displaystyle\inf_{x\in
M}f(x)},$

and by Lemma 4.2, we have

$\int_{M}e^{2u(T)}\leq
C_{K}e^{\alpha\|u_{0}\|^{2}}\left(\frac{8re^{-\tau\|u_{0}\|^{2}}C_{0}}{-\displaystyle\inf_{x\in
M}f(x)}\right)^{\alpha}\leq r,$

leading again to a contradiction. Hence $T=+\infty$ and
$\displaystyle\int_{M}e^{2u}$ is uniformly bounded. ∎

Now, by Jensen’s inequality we have

$\exp{\left(\frac{1}{V}\int_{M}2u\right)}\leq\frac{1}{V}\int_{M}e^{2u}\leq\frac{2r}{V},$

thus $\bar{u}$ is bounded from above. Now again using the energy identity
(11), we have

$\int_{M}u\overline{P}^{\prime}u+2\int_{M}\overline{Q}^{\prime}(u-\bar{u})+2\bar{u}\int_{M}\overline{Q}^{\prime}\leq
E(u_{0}),$

and

$\int_{M}u\overline{P}^{\prime}u+2\int_{M}\overline{Q}^{\prime}(u-\bar{u})\geq\frac{1}{2}\int_{M}u\overline{P}^{\prime}u-\frac{2\|\overline{Q}^{\prime}\|^{2}_{L^{2}}}{\lambda_{1}}\geq-
C_{3}.$

Therefore

$2\bar{u}\int_{M}\overline{Q}^{\prime}\leq E(u_{0})+C_{3},$

and since $\displaystyle\int_{M}\overline{Q}^{\prime}<0$ we have that
$\bar{u}$ is uniformly bounded from below which finishes the proof of Theorem
1.1.

###  4.2 Case $\displaystyle\int_{M}\overline{Q}^{\prime}=0$ and proof of
Theorem 1.2

Since $\displaystyle\int_{M}\overline{Q}^{\prime}=0$, we have that

$\langle\nabla E(u),1\rangle=2\int_{M}\overline{P}^{\prime}u=0$

and

$\langle\nabla
N(u),1\rangle=2\int_{M}\Gamma\left(fe^{2u}\right)=2\int_{M}\overline{Q}^{\prime}=0$

Hence,

$0=\int_{M}\partial_{t}u=\partial_{t}\int_{M}u,$

which means that the average value of $u$ is preserved. Therefore
$\bar{u}=\bar{u}_{0}$ and by Corollary 3.1, we have the convergence of the
flow. This ends the proof of Theorem 1.2.

###  4.3 Case $0<\displaystyle\int_{M}\overline{Q}^{\prime}<16\pi^{2}$ and
proof of Theorem 1.3

First, we have again from the energy identity (11)

$\int_{M}u\overline{P}^{\prime}u+2\int_{M}\overline{Q}^{\prime}(u-\bar{u})+2\bar{u}\int_{M}\overline{Q}^{\prime}\leq
E(u_{0}).$ (13)

Hence

$2\bar{u}\int_{M}\overline{Q}^{\prime}\leq
E(u_{0})-\frac{1}{2}\int_{M}u\overline{P}^{\prime}u+\frac{2}{\lambda_{1}}\|\overline{Q}^{\prime}\|_{L^{2}}^{2}$

and then $\bar{u}$ is bounded from above; we will need a bound from below.
Since $u\in X$, we get

$\int_{M}\overline{Q}^{\prime}=\int_{M}fe^{2u}\leq\|f\|_{\infty}\int_{M}e^{2u},$

and therefore

$\ln\left(\frac{\displaystyle\int_{M}\overline{Q}^{\prime}}{\|f\|_{\infty}}\right)\leq\ln\left(\int_{M}e^{2u}\right).$

Now again from Theorem 3.1 we have

$\ln\left(\frac{\displaystyle\int_{M}\overline{Q}^{\prime}}{\|f\|_{\infty}}\right)\leq
C+\frac{1}{16\pi^{2}}\int_{M}u\overline{P}^{\prime}u+\frac{2}{V}\int_{M}u.$
(14)

Let $\delta>0$ to be determined later, we sum equation (13) and $-\delta$
times equation (14), obtaining

$\ln\left(\frac{\displaystyle\int_{M}\overline{Q}^{\prime}}{\|f\|_{\infty}}\right)-\delta
E(u_{0})\leq
C+\left(\frac{1}{16\pi^{2}}-\delta\right)\int_{M}u\overline{P}^{\prime}u+2\left(1-\delta\int_{M}\overline{Q}^{\prime}\right)\bar{u}-2\delta\int_{M}\overline{Q}^{\prime}(u-\bar{u}).$

Since $\displaystyle\int_{M}\overline{Q}^{\prime}<16\pi^{2}$, we choose
$\delta$ such that
$\displaystyle\int_{M}\overline{Q}^{\prime}<\frac{1}{\delta}<16\pi^{2}$, and
we set

$c_{1}=2\left(1-\delta\int_{M}\overline{Q}^{\prime}\right),\qquad
c_{2}=\delta-\frac{1}{16\pi^{2}}.$

We have

$\ln\left(\frac{\displaystyle\int_{M}\overline{Q}^{\prime}}{\|f\|_{\infty}}\right)-\delta
E(u_{0})-C+c_{2}\int_{M}u\overline{P}^{\prime}u+2\delta\int_{M}\overline{Q}^{\prime}(u-\bar{u})\leq
c_{1}\bar{u}.$

Now we notice that

$c_{2}\int_{M}u\overline{P}^{\prime}u+2\delta\int_{M}\overline{Q}^{\prime}(u-\bar{u})\geq(c_{2}\lambda_{1}-\delta\varepsilon)\|u-\bar{u}\|_{L^{2}}^{2}-\frac{\delta}{\varepsilon}\|\overline{Q}^{\prime}\|_{L^{2}}^{2},$

therefore for $\varepsilon$ small enough we have that

$c_{2}\int_{M}u\overline{P}^{\prime}u+2\delta\int_{M}\overline{Q}^{\prime}(u-\bar{u})\geq-
c_{3}.$

It follows that $\bar{u}$ is bounded from below and therefore from Corollary
3.1 this finishes the proof.

##  5 The critical case and proof of Theorem 1.4

Here we will study the case
$\displaystyle\int_{M}\overline{Q}^{\prime}=16\pi^{2}$, where $M=S^{3}$ is the
sphere equipped with its standard contact structure. We will see $S^{3}$ as a
subset of $\mathbb{C}^{2}$ with coordinates $(\zeta_{1},\zeta_{2})$ such that

$S^{3}=\left\\{(\zeta_{1},\zeta_{2})\in\mathbb{C}^{2}\;;\;|\zeta_{1}|^{2}+|\zeta_{2}|^{2}=1\right\\}.$

Recall that the set $Aut(\mathbb{H}^{1})$ of conformal transformation of the
Heisenberg group $\mathbb{H}^{1}$ is generated by left translations,
dilations. rotations and inversion. Using the Cayley transform
$C:\mathbb{H}^{1}\to S^{3}\\{(0,-i)\\}$ one has a clear description of the set
$Aut(S^{3})$:

$Aut(S^{3})=\\{C\circ h\circ C^{-1};h\in Aut(\mathbb{H}^{1})\\}.$

For $p\in S^{3}$ and $r\geq 1$, we will write $h_{p,r}$ the element of
$Aut(S^{3})$ corresponding to a Cayley transform centered at $p$ and a
dilation of size $r$. Now, for $u\in X$ we set

$v_{p,r}=u\circ h_{p,r}+\frac{1}{2}\ln(J(h_{p,r})),$

where we denoted $J(h)=det(Jac(h))$, the Jacobian determinant of $h$. We have

$E(v_{p,r})=E(u)\leq E(u_{0}),$

and since $u\in X$

$\int_{S^{3}}f\circ h_{p,r}e^{2v_{p,r}}=\int_{S^{3}}fe^{2u},$

hence

$\int_{S^{3}}e^{2v_{p,r}}\geq\frac{16\pi^{2}}{\displaystyle\sup_{x\in
S^{3}}f(x)}.$

From [3, page 38], we know that for all $t\geq 1$ there exists $r(t)\geq 1$
and $p(t)\in S^{3}$ such that

$\int_{S^{3}}\xi_{i}e^{2v_{p(t),r(t)}}=0,\;i=1,2.$

So we let $v(t)=v_{p(t),r(t)}$ and $h(t)=h_{p(t),r(t)}$. Then using Corollary
A.2 in the Appendix, one has the existence of
$a<\displaystyle\frac{1}{16\pi^{2}}$ and a constant $C_{1}$ such that

$a\int_{S^{3}}v(t)\overline{P}^{\prime}v(t)+2\int_{S^{3}}v(t)-\ln\left(\int_{S^{3}}e^{2v(t)}\right)+C_{1}\geq
0.$

Since $E(v(t))\leq E(u_{0})$, we find that

$\int_{S^{3}}v(t)\overline{P}^{\prime}v(t)\leq C,$

and

$\left|\int_{S^{3}}v(t)\right|\leq C.$

In particular we have that for all $p\geq 1$

$\int_{S^{3}}e^{2|pv(t)|}\leq C_{p},$

and hence

$\int_{S^{3}}v^{2}(t)\leq C$

leading to the boundedness of $v(t)$ in $H$. We need the following
concentration-compactness lemma in order to prove uniform boundedness.

######  Lemma 5.1.

Either

$(i)\quad\|u(t)\|\leq C$, for some constant $C$;
or

$(ii)\quad$ there exists a sequence $t_{n}\to\infty$ and a point $p_{0}\in
S^{3}$ such that for all $r>0$

$\lim_{n\to\infty}\int_{B_{r}(p_{0})}fe^{2u(t_{n})}=16\pi^{2}.$

Moreover, for any $\tilde{x}\in S^{3}\setminus\\{p_{0}\\}$, and any
$r<d(\tilde{x},p_{0})$, we have

$\lim_{n\to\infty}\int_{B_{r}(\tilde{x})}fe^{2u(t_{n})}=0.$

###### Proof.

We assume first that $r(t)$ is bounded. Then we have that

$0<C_{1}\leq J(h_{p(t),r(t)})\leq C_{2}.$

Thus, from the uniform boundedness of $v(t)$ we have

$\int_{S^{3}}|u(t)|\leq C.$

Therefore, from Lemma 3.1, it follows that $\|u(t)\|$ is uniformly bounded.
So now we assume that $r(t)$ is not bounded, then there exists a sequence
$t_{n}\to\infty$ such that $r(t_{n})\to\infty$ and without loss of generality,
by compactness of $S^{3}$ we can assume that $p(t_{n})\to p_{0}$. From the
uniform boundedness of $v(t)$, we can also assume that $v(t_{n})\to
v_{\infty}$ strongly in $L^{2}(S^{3})$ and weakly in $H$. We let then $r>0$
and set $K_{n}=h(t_{n})^{-1}(B_{r}(p_{0}))$. Then we have

$\left|\int_{S^{3}}f\circ h(t_{n})e^{2v(t_{n})}-\int_{K_{n}}f\circ
h(t_{n})e^{2v(t_{n})}\right|\leq\left(\sup_{x\in
S^{3}}f(x)\right)\left(Vol(K_{n}^{c})\int_{S^{3}}e^{4|v(t_{n})|}\right)^{\frac{1}{2}}.$

Since $h(t_{n})(x)\to p_{0}$ a.e. then
$\displaystyle\lim_{n\to\infty}Vol(K_{n})=V,$ and thus

$\int_{B_{r}(p_{0})}fe^{2u(t_{n})}=\int_{K_{n}}f\circ
h(t_{n})e^{2v(t_{n})}=\int_{S^{3}}f\circ h(t_{n})e^{2v(t_{n})}+o(1).$

We have also

$\int_{S^{3}}f\circ h(t_{n})e^{2v(t_{n})}=16\pi^{2},$

and then

$\lim_{n\to\infty}\int_{B_{r}(p_{0})}fe^{2u(t_{n})}=16\pi^{2}.$

Now if we consider $\tilde{x}\in S^{3}\setminus\\{p_{0}\\}$ and
$r<d(p_{0},\tilde{x})$ we have that $h(t_{n})(x)\not\in B_{r}(\tilde{x})$ for
$n$ big enough, since $\displaystyle\lim_{n\to\infty}h(t_{n})(x)=p_{0}$ a.e.;
in particular

$\lim_{n\to\infty}\chi_{h(t_{n})^{-1}(B_{r}(\tilde{x}))}=0,$

where $\chi$ is the characteristic function. Therefore

$\lim_{n\to\infty}\int_{B_{r}(\tilde{x})}fe^{2u(t_{n})}=\lim_{n\to\infty}\int_{h(t_{n})^{-1}(B_{r}(\tilde{x}))}f\circ
h(t_{n})e^{2v(t_{n})}=0.$

∎

Let us assume now that $\Sigma=\emptyset$. By using the previous lemma, if
$\|u(t)\|$ is not uniformly bounded, then there exists $p_{0}\in S^{3}$ such
that

$\lim_{n\to\infty}\int_{B_{r}(p_{0})}fe^{2u(t_{n})}=16\pi^{2},$

and if $p_{1}\not=p_{0}$ and $r<d(p_{0},p_{1})$, then

$\lim_{n\to\infty}\int_{B_{r}(p_{1})}fe^{2u(t)}=0.$

Since $\Sigma=\emptyset$, then there exists $g\in G$ such that $p_{1}=g\cdot
p_{0}\not=p_{0}$. But

$16\pi^{2}=\lim_{n\to\infty}\int_{B_{r}(p_{0})}fe^{2u(t)}=\lim_{n\to\infty}\int_{B_{r}(g\cdot
p_{0})}fe^{2u(t)}=\lim_{n\to\infty}\int_{B_{r}(p_{1})}fe^{2u(t)}=0$

which is a contradiction. Hence $\|u(t)\|$ is uniformly bounded.
Now we assume that $\Sigma\neq\emptyset$ and that $\|u(t)\|$ is not uniformly
bounded. We have that the concentration point $p_{0}\in\Sigma$, otherwise we
reach a contradiction arguing as in the previous case. So we have

$\int_{B_{r}(p_{0})}fe^{2u(t_{n})}\leq\sup_{x\in
B_{r}(p_{0})}f(x)\int_{B_{r}(p_{0})}e^{2u(t_{n})}\leq\max\left(\sup_{x\in
B_{r}(p_{0})}f(x),0\right)\int_{S^{3}}e^{2u(t_{n})}.$

By using the the sphere version of Theorem 3.1, proved in [3], we have that

$\frac{1}{V}\int_{S^{3}}e^{2u(t_{n})}\leq e^{\frac{E(u(t_{n}))}{V}}.$

Thus

$\int_{B_{r}(p_{0})}fe^{2u(t_{n})}<\max\left(\sup_{x\in
B_{r}(p_{0})}f(x),0\right)Ve^{\frac{E(u_{0})}{V}}.$

Now we first let $n\to\infty$, then $r\to\infty$ and we get

$16\pi^{2}<V\max(f(p_{0}),0)e^{\frac{E(u_{0})}{V}}.$

Therefore $f(p_{0})>0$ and

$1<f(p_{0})e^{\frac{E(u_{0})}{16\pi^{2}}},$

hence

$f(p_{0})>e^{-\frac{E(u_{0})}{16\pi^{2}}},$

which leads to a contradiction of the assumption in Theorem 1.4. Therefore we
have the uniform boundedness of $\|u\|$ also in this case, which yields the
convergence of the flow and it ends the proof.

##  Appendix A Appendix: Improved Moser-Trudinger Inequality

In what follows we will consider $S^{3}$ as a subset of $\mathbb{C}^{2}$ with
coordinates $(\zeta_{1},\zeta_{2})$ such that
$|\zeta_{1}|^{2}+|\zeta_{2}|^{2}=1$. We recall here the Improved Moser-
Trudinger inequality introduced in [3] in order to prove the existence of a
minimizer:

######  Proposition A.1.

([3], Proposition 3.4) Given $\frac{1}{2}<a<1$, there exist constants
$C_{1}(a)$, $C_{2}(a)$ such that for $u\in H$ with
$\int_{S^{3}}\zeta_{i}e^{2u}=0$, $i=1,2$, it holds:

$\frac{a}{16\pi^{2}}\int_{S^{3}}uP^{\prime}u+2\int_{S^{3}}u-\ln\left(\int_{S^{3}}e^{2u}\right)+C_{1}(a)\|(-\Delta_{b})^{\frac{3}{4}}u\|_{2}^{2}+C_{2}(a)\geq
0$

This improved estimate will not be useful to us in our setting since it
contains the term $C_{1}(a)\|(-\Delta_{b})^{\frac{3}{4}}u\|_{2}^{2}$ that we
cannot bound along the flow. Notice that in [3], the authors exploit Ekeland’s
principle to exhibit a good minimizing Palais-Smale sequence that allows the
control of this extra term. In our setting, we will prove a result that can be
seen as intermediate between Proposition A.1 and the usual Moser-Trudinger
inequality in Theorem 3.1. In fact in [3] the authors gave hints on how to
prove this result, knowing that this method only works in dimension 3 and 5.
We will follow a technique used in [11], since it is simpler and it allows
even more improved estimates.

We set

$P_{k}:=\Big{\\{}\text{polynomials of }\mathbb{C}^{2}\text{ with degree at
most }k\Big{\\}}$

and

$P_{k,0}:=\Big{\\{}f\in P_{k};\int_{S^{3}}f=0\Big{\\}}.$

For a given $m\in\mathbb{N}$ we let

$\mathcal{N}_{m}:=\left\\{\begin{array}[]{lc}N\in\mathbb{N};\exists
x_{1},\cdots,x_{N}\in S^{3},\nu_{1},\cdots,\nu_{N}\in\mathbb{R}^{+}\text{ with
}\sum_{k=1}^{N}\nu_{k}=1\\\ \text{ and for any }f\in
P_{m,0};\sum_{k=1}^{N}\nu_{k}f(x_{k})=0\end{array}\right\\}.$

We let then $N_{m}=\min\mathcal{N}_{m}$. As it was shown in [11], one has
$N_{1}=2$ and $N_{2}=4$. We recall from [3] that one has the following
inequality on the standard sphere:
There exists a constant $A_{2}>0$ such that

$\int_{S^{3}}\exp\Big{[}A_{2}\frac{|u-\bar{u}|^{2}}{\|\Delta_{b}u\|_{L^{2}}^{2}}\Big{]}\leq
C_{0}.$

In fact the sharp constant $A_{2}$ was explicitly computed in [3] and it has
the value $A_{2}=32$. With this result we can easily deduce that if
$u\in\mathcal{S}^{2}(S^{3})$ then $e^{2u}\in L^{p}(S^{3})$ for all $1\leq
p<\infty$.

######  Lemma A.1.

Consider a sequence of functions $u_{k}\in\mathcal{S}^{2}(S^{3})$ such that

$\bar{u}_{k}=0,\quad\|\Delta_{b}u_{k}\|_{L^{2}}\leq 1$

and suppose that $u_{k}\rightharpoonup u$ weakly in $\mathcal{S}^{2}(S^{3})$
and

$|\Delta_{b}u_{k}|^{2}\rightharpoonup|\Delta_{b}u|^{2}+\sigma\text{ in measure
},$

where $\sigma$ is a measure on $S^{3}$. Let $K\subset S^{3}$ be a compact set
with $\sigma(K)<1$, then for all $1\leq p<\frac{1}{\sigma(K)}$ we have

$\sup_{k}\int_{K}\exp\Big{[}pA_{2}u_{k}^{2}\Big{]}<\infty.$

###### Proof.

Let $\varphi$ be a fixed smooth compactly supported function on $S^{3}$. We
set $v_{k}=u_{k}-u$. Then $v_{k}\to 0$ strongly in $L^{2}$ and weakly in
$\mathcal{S}^{2}(S^{3})$. Now we compute

$\displaystyle\int_{S^{3}}|\Delta_{b}(\varphi v_{k})|^{2}$
$\displaystyle=\int_{S^{3}}(\varphi\Delta_{b}v_{k}+v_{k}\Delta_{b}\varphi+2\nabla_{H}\varphi\nabla_{H}v_{k})^{2}$
$\displaystyle=\int_{S^{3}}\varphi^{2}(\Delta_{b}v_{k})^{2}+v_{k}^{2}(\Delta_{b}\varphi)^{2}+4|\nabla_{H}v_{k}\nabla_{H}\varphi|^{2}+2\varphi
v_{k}\Delta_{b}\varphi\Delta_{b}v_{k}+$ (15)
$\displaystyle\qquad+4\varphi(\nabla_{H}\varphi\nabla_{H}v_{k})\Delta_{b}v_{k}+4v_{k}(\nabla_{H}v_{k}\nabla_{H}\varphi)\Delta_{b}\varphi.$

Hence,

$\int_{S^{3}}|\Delta_{b}(\varphi
v_{k})|^{2}\to\int_{S^{3}}\varphi^{2}d\sigma.$

Assume that $1\leq p_{1}<\frac{1}{\sigma(K)}$ and take $\varphi$ so that
$\varphi_{|K}=1$, and $\int_{S^{3}}\varphi^{2}d\sigma<\frac{1}{p_{1}}$. Then
we have for $k$ large,

$\|\Delta_{b}(\varphi v_{k})\|_{L^{2}}^{2}<\frac{1}{p_{1}}.$

Therefore,

$\int_{K}\exp\Big{[}p_{1}A_{2}(v_{k}-\overline{\varphi
v_{k}})^{2}\Big{]}\leq\int_{S^{3}}\exp\Big{[}p_{1}A_{2}(\varphi
v_{k}-\overline{\varphi
v_{k}})^{2}\Big{]}\leq\int_{S^{3}}\exp\Big{[}A_{2}\frac{(\varphi
v_{k}-\overline{\varphi v_{k}})^{2}}{\|\Delta_{b}\varphi
v_{k}\|_{L^{2}}^{2}}\Big{]}\leq C_{0}.$

Thus, if we fix $\varepsilon>0$, we can write

$\displaystyle u_{k}^{2}$ $\displaystyle=(v_{k}-\overline{\varphi
v_{k}}+u+\overline{\varphi v_{k}})^{2}$
$\displaystyle=(v_{k}-\overline{\varphi v_{k}})^{2}+2(v_{k}-\overline{\varphi
v_{k}})(u+\overline{\varphi v_{k}})+(u+\overline{\varphi v_{k}})^{2}$
$\displaystyle\leq(1+\varepsilon)(v_{k}-\overline{\varphi
v_{k}})^{2}+2(1+\frac{1}{\varepsilon})u^{2}+2(1+\frac{1}{\varepsilon})^{2}\overline{\varphi
v_{k}}^{2}.$

Hence, given $p<\frac{1}{\sigma(K)}$ we can take
$p_{1}\in(p,\frac{1}{\sigma(K)})$ such that

$\int_{K}e^{A_{2}p_{1}u_{k}^{2}}<C_{0},$

which finishes the proof. ∎

######  Corollary A.1.

We consider the same assumptions as in Lemma A.1 and we let $\ell=\max_{x\in
S^{3}}\sigma(\\{x\\})\leq 1.$ Then the following hold

  * •

If $\ell<1$, then for any $1\leq p<\frac{1}{\ell}$, $e^{A_{2}u_{k}^{2}}$ is
bounded in $L^{p}(S^{3})$. In particular $e^{A_{2}u_{k}^{2}}\to
e^{A_{2}u^{2}}$ in $L^{1}$.

  * •

If $\ell=1$, then there exists $x_{0}\in S^{3}$ such that
$\sigma=\delta_{x_{0}}$, $u=0$ and after passing to a subsequence if
necessary, we have

$e^{A_{2}u_{k}^{2}}\rightharpoonup 1+c_{0}\delta_{x_{0}},$

for some $c_{0}\geq 0$.

###### Proof.

Assume that $\ell<1$ and let $1\leq p<\frac{1}{\ell}$. Then for all $x\in
S^{3}$, $\sigma(\\{x\\})<\frac{1}{p}$. By continuity, there exists $r_{x}>0$
such that $\sigma(\overline{B_{r_{x}}(x)})<\frac{1}{p}$. Since $S^{3}$ is
compact we can find a finite collection of balls of the form
$B_{r_{i}}(x_{i})$ such that

$S^{3}=\bigcup_{i=1}^{N}\overline{B_{r_{i}}(x_{i})}.$

So using Lemma A.1, we have

$\sup_{k}\int_{\overline{B_{r_{i}}(x_{i})}}\exp\Big{[}pA_{2}u_{k}^{2}\Big{]}<\infty.$

Thus,

$\sup_{k}\int_{S^{3}}\exp\Big{[}pA_{2}u_{k}^{2}\Big{]}<\infty.$

We assume now that $\ell=1$. Since $\|\Delta_{b}u_{k}\|^{2}\leq 1$ we have
that $\|\Delta_{b}u\|^{2}+\sigma(S^{3})\leq 1$. Therefore, we have $u=0$ and
there exists $x_{0}\in S^{3}$ such that $\sigma=\delta_{x_{0}}$. Hence, for
$r$ small, we have that

$\sup_{k}\int_{S^{3}\setminus
B_{r}(x_{0})}\exp\Big{[}qA_{2}u_{k}^{2}\Big{]}<\infty,$

for all $q\geq 1$. Therefore, $e^{A_{2}u_{k}^{2}}\to 1$ in
$L^{1}(S^{3}\setminus B_{r}(x_{0}))$ for every $r>0$ and small. Hence, after
passing to a subsequence if necessary we have that
$e^{A_{2}u_{k}^{2}}\rightharpoonup 1+c_{0}\delta_{x_{0}}$ in measure. ∎

######  Proposition A.2.

Let $\alpha>0$ and consider a sequence $m_{k}\to\infty$ and
$u_{k}\in\mathcal{S}^{2}(S^{3})$ such that $\overline{u_{k}}=0$ and
$\|\Delta_{b}u_{k}\|_{L^{2}}=1$ such that $u_{k}\rightharpoonup u$ weakly in
$\mathcal{S}^{2}(S^{3})$ and
$(\Delta_{b}u_{k})^{2}\rightharpoonup(\Delta_{b}u)^{2}+\sigma$ in measure. We
assume moreover that

$\ln\left(\int_{S^{3}}e^{2m_{k}u_{k}}\right)\geq\alpha m_{k},$

and

$\frac{e^{2m_{k}u_{k}}}{\displaystyle\int_{S^{3}}e^{2m_{k}u_{k}}}\rightharpoonup\nu\text{
in measure}.$

We set $R=\Big{\\{}x\in S^{3};\sigma(\\{x\\})\geq
A_{2}\alpha\Big{\\}}=\\{x_{1},\cdots,x_{N}\\}$. Then
$\nu=\sum_{i=1}^{N}\nu_{i}\delta_{x_{i}}$ with $\nu_{i}\geq 0$ and
$\sum_{i}\nu_{i}=1$.

###### Proof.

Let $K\subset S^{3}$ such that $\sigma(K)<A_{2}\alpha$. By continuity, we can
find a compact set $K_{1}$ such that $K\subset int(K_{1})$ and
$\sigma(K_{1})<A_{2}\alpha$. Now given
$\frac{1}{A_{2}\alpha}<p<\frac{1}{\sigma(K_{1})}$, we have

$\sup_{k}\int_{K_{1}}e^{pA_{2}u_{k}^{2}}\leq C_{0}.$

Since $2m_{k}u_{k}\leq pA_{2}u_{k}^{2}+\frac{m_{k}^{2}}{pA_{2}}$, we have

$\int_{K_{1}}e^{2m_{k}u_{k}}\leq Ce^{\frac{m_{k}^{2}}{A_{2}p}}.$

Therefore,

$\frac{\displaystyle\int_{K_{1}}e^{2m_{k}u_{k}}}{\displaystyle\int_{S^{3}}e^{2m_{k}u_{k}}}\leq
Ce^{\Big{(}\frac{1}{A_{2}p}-\alpha\Big{)}m_{k}^{2}}.$

So $\nu(K)\leq\nu(K_{1})=0$ and $\nu(K)=0$. Thus, if
$\sigma(\\{x\\})<A_{2}\alpha$, then there exists $r_{x}>0$ small enough so
that $\sigma(\overline{B_{r_{x}}(x)})<A_{2}\alpha$. Hence,
$\nu(\overline{B_{r_{x}}(x)})=0$. We deduce then that $\nu(S^{3}\setminus
R)=0.$
Therefore

$\nu=\sum_{k=1}^{N}\nu_{k}\delta_{x_{k}},$

with $\nu_{k}\geq 0$ and $\sum_{k=1}^{N}\nu_{k}=1$. ∎

Let $f_{1},\cdots,f_{\ell}\in C(S^{3})$. We define

$\mathcal{S}_{f}=\left\\{u\in\mathcal{S}^{2}(S^{3});\overline{u}=0;\int_{S^{3}}f_{k}e^{2u}=0;k=1,\cdots,\ell\right\\}.$

We assume that the inequality

$\ln\left(\int_{S^{3}}e^{2u}\right)\leq\alpha\|\Delta_{b}u\|^{2}_{2}+C$

does not hold for $u\in\mathcal{S}_{f}$. Then there exists a sequence
$u_{k}\in\mathcal{S}_{f}$ such that

$\ln\left(\int_{S^{3}}e^{2u_{k}}\right)-\alpha\|\Delta
u_{k}\|_{L^{2}}^{2}\to\infty.$

Therefore, it follows that $\int_{S^{3}}e^{2u_{k}}\to\infty$ and
$\|\Delta_{b}u_{k}\|_{L^{2}}\to\infty$. So we let
$m_{k}=\|\Delta_{b}u_{k}\|_{L^{2}}$ and $v_{k}=\frac{u_{k}}{m_{k}}$. Then
$m_{k}\to\infty$, $\|\Delta_{b}v_{k}\|^{2}_{L^{2}}=1$. Hence, after passing to
a subsequence, we have

$\left\\{\begin{array}[]{lll}v_{k}\rightharpoonup v\text{ weakly in
}\mathcal{S}^{2}(S^{3}),\\\ \\\
|\Delta_{b}v_{k}|^{2}\rightharpoonup|\Delta_{b}v|^{2}+\sigma\text{ in
measure},\\\ \\\
\frac{e^{2m_{k}v_{k}}}{\displaystyle\int_{S^{3}}e^{2m_{k}v_{k}}}\rightharpoonup\nu\text{
in measure}.\end{array}\right.$

So we let $R=\\{x\in S^{3};\sigma(\\{x\\})\geq
A_{2}\alpha\\}=\\{x_{1},\cdots,x_{N}\\}$. It follows from Proposition A.2 that
$\nu=\sum_{j=1}^{N}\nu_{j}\delta_{x_{j}}$, with $\sum_{j=1}^{N}\nu_{j}=1$ and
$\nu_{j}\geq 0$.
But since $u_{k}\in\mathcal{S}_{f}$, we have

$\int_{S^{3}}f_{j}d\nu=0.$

Therefore,

$\sum_{i=1}^{N}\nu_{i}f_{j}(x_{i})=0,\text{ for all }1\leq j\leq\ell.$

On the other hand, $A_{2}\alpha N\leq 1$. In particular, if $f_{j}\in
P_{m,0}$, we have that $N\in\mathcal{N}_{m}$. Therefore,

$\alpha\leq\frac{1}{A_{2}N}\leq\frac{1}{A_{2}N_{m}}.$

Hence, if $\alpha=\frac{1}{A_{2}N_{m}}+\varepsilon$ we get a contradiction and
the result holds. Therefore, if we define

$\mathcal{S}_{0}=\left\\{u\in\mathcal{S}^{2}(S^{3});\overline{u}=0;\int_{S^{3}}fe^{2u}=0\text{
for all }f\in P_{1,0}\right\\},$

the following corollary holds

######  Corollary A.2.

There exist $a<\frac{1}{16\pi^{2}}$ and $C>0$ such that for all
$u\in\mathcal{P}\cap\mathcal{S}_{0}$, we have

$a\int_{S^{3}}u\overline{P}^{\prime}u+2\int_{M}u-\ln\left(\int_{M}e^{2u}\right)\geq-C.$

Indeed, this corollary follows from the fact that

$\int_{S^{3}}u\overline{P}^{\prime}u\geq\int_{S^{3}}|2\Delta_{b}u|^{2}$

for all $u\in\hat{\mathcal{P}}$ and $8A_{2}>16\pi^{2}$.

## References

  * [1] Baird, P., Fardoun, A., Regbaoui, R.; _Q-curvarure flow on 4-manifolds_ , Calc. Var. 27 (2006) 75-104.
  * [2] Branson, T., Gover, A.; _Conformally invariant operators, differential forms, cohomology and a generalisation of Q-curvature_ , Comm. Partial Differential Equations, 30(10-12): (2005), 1611-1669.
  * [3] Branson, T., Fontana, L., Morpurgo, C., _Moser-Trudinger and Beckner-Onofri’s inequalities on the CR sphere_ , Ann. of Math. (2), 177(1): (2013), 1-52.
  * [4] Brendle, S.; _Convergence of the Q-curvature flow on $S^{4}$_, Adv. Math. 205, (2006), 1–32.
  * [5] Burns,D., Epstein,C. L.; _A global invariant for three-dimensional CR-manifolds_ , Invent. Math., 92(2): (1988), 333-348.
  * [6] Case, J.S., Chanillo, S., Yang, P.; _A remark on the kernel of the CR Paneitz operator_ , Nonlinear Analysis Volume 126, (2015), 153-158.
  * [7] Case, J.S., Cheng, J.H., Yang, P.; _An integral formula for the q-prime curvature in 3-dimensional cr geometry_ , Proceedings of the American Mathematical Society, 147(4), (2018), 1577-1586.
  * [8] Case, J.S., Hsiao, C.Y., Yang, P.C.; _Extremal metrics for the $\overline{Q}^{\prime}$-curvature in three dimensions_. Comptes Rendus Mathematique, Vol 354 (4), (2016), 407-410.
  * [9] Case, J.S., Hsiao, C.Y., Yang, P.C.; _Extremal metrics for the $Q^{\prime}$-curvature in three dimensions_. Journal of the European Mathematical Society, Vol 21 (2), (2019), 585-626.
  * [10] Case, J.S., Yang, P.; _A Paneitz-type operator for CR pluriharmonic functions_ , Bull. Inst. Math. Acad. Sin. (N.S.), 8(3): (2013), 285-322.
  * [11] Chang, S-Y., Fengbo, H.; _Improved Moser-Trudinger-Onofri inequality under constraints_ , Preprint, arXiv:1909.00431
  * [12] Chanillo, S., Chiu, H.L., Yang, P.; _Embeddability for $3$-dimensional Cauchy-Riemann manifolds and CR Yamabe invariants_, Duke Math. J., 161(15): (2012), 2909-2921.
  * [13] Cheng, J.H., Lee, J.M.; _The Burns-Epstein invariant and deformation of CR structures_. Duke Math. J. 60, no. 1, (1990), 221-254.
  * [14] Frank, R.L., Lieb, E.H.; _Sharp constants in several inequalities on the Heisenberg group. Ann. of Math._ (2) 176, no. 1, 349-381, (2012).
  * [15] Graham, C.R., Jenne, R., Mason, L., Sparling, G.; _Embeddability for $3$-dimensional Cauchy-Riemann manifolds and CR Yamabe invariants Conformally invariant powers of the Laplacian. I. Existence_, J. London Math. Soc. (2) 46, (1992), 557-565.
  * [16] Hirachi, K.; _Scalar pseudo-Hermitian invariants and the Szegö kernel on three - dimensional CR manifolds_ , In Complex geometry (Osaka, 1990), volume 143 of Lecture Notes in Pure and Appl. Math., pages 67-76. Dekker, New York, 1993.
  * [17] Ho, P.T.; _Prescribing the $\overline{Q}^{\prime}$-curvature in three dimension_, Discrete Contin. Dyn. Syst. 39, (2019), 2285-2294.
  * [18] Ho, P.T.; _Prescribed Q-curvature flow on $S^{n}$_, J. Geom. Phys. 62, (2012), 1233-1261.
  * [19] Hsiao, C.Y.; _On CR Paneitz operators and CR pluriharmonic functions_ , Math. Ann. 362: (2015), 903-929.
  * [20] Lee, J.M.; _Pseudo-Einstein structures on CR manifolds_ , Amer. J. Math., 110, (1988), 157-178.
  * [21] Maalaoui, A.; _Prescribing the $\overline{Q}^{\prime}$-Curvature on Pseudo-Einstein CR 3-Manifolds_, Preprint.
  * [22] Maalaoui, A.; _Logarithmic Hardy-Littlewood-Sobolev Inequality on Pseudo-Einstein 3-manifolds and the Logarithmic Robin Mass_ , Preprint.
  * [23] Malchiodi, A., Struwe, M.; _Q-curvature flow on $S^{4}$_, J. Differential Geom. 73, (2006), 1-44.
  * [24] Phong, D. H., Stein, E. M.; _Estimates for the Bergman and Szego projections on strongly pseudo-convex domains._ Duke Math. J. 44 (1977), no. 3, 695-704.
  * [25] Simon, L.; _Asymptotics for a class of nonlinear evolution equations, with applications to geometric problems_. Ann. of Math. (2) 118, no. 3, (1983), 525-571.


11institutetext: Departamento de Física, Facultad de ciencias básicas,
Universidad de Antofagasta, Casilla 170, Antofagasta, Chile.

# A note of the first law of thermodynamics by gravitational decoupling

Milko Estrada _e-mail:_<EMAIL_ADDRESS>Reginaldo Prado _e-mail:_
<EMAIL_ADDRESS>
(Received: date / Revised version: date)

###### Abstract

We provide a way of decoupling the first law of thermodynamics in two sectors
: the standard first law of thermodynamics and the quasi first law of
thermodynamics. It is showed that both sectors share the same thermodynamics
volume and the same entropy. However, the total thermodynamics pressure, the
total temperature and the total local energy correspond to a simple sum of the
thermodynamics contributions of each sector. Furthermore, it is showed a
simple example, where there is a phase transition due to the behavior of the
temperature at the quasi sector.

###### pacs:

PACS-keydiscribing text of that key and PACS-keydiscribing text of that key

##  1 Introduction

Finding new solutions of physical interest to the Einstein field equations is
not an easy task due to the highly nonlinear behavior of its equations. In
this regard, in (2017) was proposed the Gravitational Decoupling Method
Ovalle:2017fgl , which represents an easy algorithm to decouple gravitational
sources in General Relativity. This algorithm involves a Minimal Geometric
Deformation (MGD) to the metric tensor together with a decoupling of sources.
The method was described in reference Ovalle:2017wqi  as follows: “given two
gravitational sources: a source A and an extra source B, standard Einstein’s
equations are first solved for A, and then a simpler set of quasi-Einstein
equations are solved for B. Finally, the two solutions can be combined in
order to derive the complete solution for the total system”. The source A
represents to the seed sector and the source B represents to the extra sector.
Since its appearance, this method has served to find several new solution of
physical interest, as for example well behaved solutions that could represent
stellar distributions Ovalle:2017wqi ; Gabbanelli:2018bhs ; Estrada:2018zbh ;
Heras:2018cpz ; Sharif:2018toc ; Morales:2018nmq ; Morales:2018urp ;
Estrada:2018vrl ; Maurya:2019wsk , black hole solutions Ovalle:2018umz ;
Contreras:2018nfg ; Contreras:2019iwm , $f({\mathcal{G}})$ gravity
Sharif:2018tiz , Einstein Klein Gordon System Ovalle:2018ans , Pure Lovelock
gravity Estrada:2019aeh , $f(R,T)$ gravity Maurya:2019hds , $f(R)$ gravity
Sharif:2019zan . See other applications Contreras:2018gzd ;
Panotopoulos:2018law ; Contreras:2019fbk ; Contreras:2019mhf ; Heras:2019ibr ;
Gabbanelli:2019txr ; Hensh:2019rtb ; Leon:2019abq ; Torres:2019mee ;
Rincon:2019jal ; Casadio:2019usg ; Sharif:2019mzv ; Singh:2019ktp ;
Abellan:2020wjw

On the other hand, the fact that the black holes, due to quantum fluctuations,
emit as black bodies where its temperature is related to its surface gravity
Hawking:1974sw ; Bekenstein:1972tm ; Bekenstein:1973ur ; Hawking:1974rv ,
shows that in these objects the geometry and thermodynamics are directly
connected. In this regard, the simplest version of the first law of
thermodynamics is (see for example Wald:1984rg ):

$dM=TdS+\Omega dJ+\phi dQ,$ (1)

where $M$, $T$, $S$, $J$ and $Q$ represent the mass, temperature, entropy,
angular momentum and electric charge, respectively. In equation (1) one can
notice the absence of the terms of pressure and volume. In that respect, one
way to address this problem was showed in reference Kastor:2009wy , where the
cosmological constant was associated with the thermodynamics pressure of the
system, $P=-\frac{\Lambda}{8\pi}$, whereas the thermodynamics volume
corresponds to the thermodynamic variable conjugate to $P$. So, the mass
parameter is interpreted as the enthalpy. However, other way to address this
problem was showed in reference Kothawala:2007em , identifying the
thermodynamics pressure with the radial pressure of the energy momentum
tensor. As indicates this reference, it is showed an interesting “ analogy
between the gravitational dynamics of the horizons and thermodynamics,
specifically, it is showed that it is possible to interpret the field
equations near any spherically symmetric horizon as a thermodynamic identity
$dU=TdS-PdV$” for the non rotating and non charged case. In this way, the term
$-PdV$ can represent the macroscopic work done by the system. This analogy has
been used for example for the study of the evolution of the regular black
holes with a cosmological horizon in theories of Lovelock with a unique ground
state Aros:2019auf , the study of regular black hole thermodynamics in
Einstein Hilbert theory Dymnikova:2010zz , stationary axis-symmetric space
time and time dependent evolving horizons Kothawala:2007em , generic Lovelock
theory Kothawala:2009kc  . See also Sheykhi:2010zz ; Padmanabhan:2002sha ;
Cai:2007bh .

In this work, inspired by the gravitational decoupling method, where the
equations of motion are characterized by the standard Einstein’s equations and
the quasi Einstein equations, we write the equations of motion as a
thermodynamics identity $dU=TdS-PdV$, which, in analogous way to the mentioned
method, will be characterized by two sectors, namely, the standard first law
and the quasi first law. We will determine, as consequence of the application
of the algorithm, which thermodynamics quantities are shared for both sectors
and which quantities of the total system correspond to a simple sum of the
quantities of each sector. Furthermore, by mean of an example, where the
energy density of the seed sector correspond to the Hayward model
Hayward:2005gi  and the energy density of the extra sector correspond to the
Dymnikova model Dymnikova:1992ux , we will test the thermodynamics behavior of
each sector and the total system.

##  2 The Einstein Field Equations for multiple sources

In this section, we will show the Einstein field equations for multiple
sources. The seed energy momentum tensor $\bar{T}_{\mu\nu}$ is deformed by an
additional source $\theta_{\mu\nu}$, whose coupling is proportional to the
constant $\alpha$ and causes anisotropic effects on the self gravitating
system Estrada:2018vrl . As indicates reference Ovalle:2017wqi , this
additional source can contain new fields, like scalar, vector and tensor
fields. So, the energy momentum is:

$T_{\mu\nu}=\bar{T}_{\mu\nu}+\alpha\theta_{\mu\nu},$ (2)

where the conservation law is:

$\nabla_{\nu}T^{\mu\nu}=0.$ (3)

Taken account an energy momentum tensor of the form
$T^{\mu}_{\nu}=\mbox{diag}(-\rho,p_{r},p_{\theta},p_{\theta},...)$ an a seed
energy momentum tensor of the form
$\bar{T}^{\mu}_{\nu}=\mbox{diag}(-\bar{\rho},\bar{p}_{r},\bar{p}_{\theta},\bar{p}_{\theta},...)$,
from equation (2) it is straightforward to note that:

$\displaystyle\rho$ $\displaystyle=\bar{\rho}-\alpha\theta^{0}_{0}$ (4)
$\displaystyle p_{r}$ $\displaystyle=\bar{p}_{r}+\alpha\theta^{1}_{1}$ (5)
$\displaystyle p_{\theta}$
$\displaystyle=\bar{p}_{\theta}+\alpha\theta^{2}_{2}$ (6)

The Einstein field equations, in natural units, correspond to

$G^{\mu}_{\nu}=8\pi T^{\mu}_{\nu}.$ (7)

We will study the following spherically symmetric space time:

$ds^{2}=-\mu(r)dt^{2}+\mu(r)^{-1}dr^{2}+r^{2}d\Omega^{2}_{2}.$ (8)

It is worth to mention that this metric satisfies the Einstein field equations
under the following constraint to the energy momentum tensor:

$\displaystyle T^{t}_{t}$ $\displaystyle=T^{r}_{r}$ (9) $\displaystyle
T^{\theta}_{\theta}$ $\displaystyle=T^{\phi}_{\phi},$ (10)

which implies that

$\displaystyle\rho$ $\displaystyle=-p_{r}$ (11) $\displaystyle p_{\theta}$
$\displaystyle=p_{\phi}.$ (12)

To accomplish the condition (11) it is imposed in arbitrarily way that

$\displaystyle\bar{\rho}$ $\displaystyle=-\bar{p}_{r}\mbox{ ,and}$ (13)
$\displaystyle\theta^{0}_{0}$ $\displaystyle=\theta^{1}_{1},$ (14)

on the other hand , the condition $T^{\theta}_{\theta}=T^{\phi}_{\phi}$ arises
from spherical symmetry.

In this way, the $(t-t)$ and $(r-r)$ components of the Einstein field
equations are similar and are given by:

$8\pi\bar{\rho}-8\pi\alpha\theta^{0}_{0}=\dfrac{1-\mu}{r^{2}}-\dfrac{\mu^{\prime}}{r},$
(15)

where ′ denotes derivation respect to the radial coordinate.

On the other hand, the conservation law has the form:

$\displaystyle\bar{p}_{r}^{\prime}+\frac{2}{r}(\bar{p}_{r}-\bar{p}_{t})$
$\displaystyle+\alpha\Big{(}(\theta^{1}_{1})^{\prime}+\frac{2}{r}(\theta^{1}_{1}-\theta^{2}_{2})\Big{)}=0.$
(16)

As indicates the reference Ovalle:2017fgl , due that, in the limit $\alpha\to
0$, the geometry of the line element (8) is associate with the geometry of the
seed perfect fluid, and its respective Bianchi identities are satisfied, the
seed energy momentum tensor is conserved, i.e. :
$\nabla_{\nu}\bar{T}^{\mu\nu}=0$:

$\bar{p}_{r}^{\prime}+\frac{2}{r}(\bar{p}_{r}-\bar{p}_{t})=0,$ (17)

and, by inserting equation (17) into equation (2) , it is easily to see that
the extra source is also conserved, i.e. $\nabla_{\nu}\theta^{\mu\nu}=0$:

$(\theta^{1}_{1})^{\prime}+\frac{2}{r}(\theta^{1}_{1}-\theta^{2}_{2})=0,$ (18)

thus, each source is separately conserved.

##  3 Decoupling the first law of thermodynamics

In order to analyse the thermodynamics, we will call $r=a$ to the generic
horizon, where $\mu(a)=0$. The first step, following reference
Kothawala:2007em , is, identify the thermodynamics pressure with the radial
pressure of the fluid perfect,

$\displaystyle P_{tot}=p_{r}=$ $\displaystyle\bar{p}_{r}+\alpha\theta^{1}_{1}$
$\displaystyle P_{tot}=$ $\displaystyle P_{s}+\alpha P_{q},$ (19)

where $\bar{p}_{r}=P_{s}$ and $\theta^{0}_{0}=P_{q}$, and, using the
conditions (13) and (14), evaluating the equations of motion (15) at $r=a$:

$4\pi a^{2}\left(P_{s}+\alpha
P_{q}\right)=\dfrac{u^{\prime}}{2}a-\frac{1}{2}.$ (20)

Next, also following Kothawala:2007em , we consider two horizons whose values
are $a$ and $a+da$, thus, multiplying equation (20) by $da$:

$(P_{s}+\alpha P_{q})d\left(\frac{4}{3}\pi
a^{3}\right)=\left(\dfrac{\mu^{\prime}}{4\pi}\right)d\left(\dfrac{4\pi
a^{2}}{4}\right)-d\left(\dfrac{a}{2}\right).$ (21)

To test the role of the constant $\alpha$, we will apply the MGD to the
geometry. Turning on $\alpha$, the effects of the source $\theta_{\mu\nu}$
appear on the seed solution. These effects are encoded in the geometric
deformation undergone by the seed fluid geometry (8) as follows:

$\mu(r)\to\bar{\mu}(r)+\alpha g(r).$ (22)

Thus, it is direct to see that the equation (15) can be written as:

$\mu=\bar{\mu}+\alpha g=1-\dfrac{2}{r}\Big{(}m_{s}(r)+\alpha m_{q}(r)\Big{)},$
(23)

where

$\displaystyle m_{s}(r)=$ $\displaystyle\int 4\pi r^{2}\bar{\rho}dr,$ (24)
$\displaystyle m_{q}(r)=$ $\displaystyle-\int 4\pi r^{2}\theta^{0}_{0}dr,$
(25)

and

$\displaystyle\bar{\mu}(r)=$ $\displaystyle 1-\dfrac{2}{r}m_{s}(r)$ (26)
$\displaystyle g(r)=$ $\displaystyle-\dfrac{2}{r}m_{q}(r).$ (27)

To ensure an asymptotically flat behavior, must be imposed that:

$\displaystyle\lim_{r\to\infty}m_{s}(r)=M_{s}$ (28)
$\displaystyle\lim_{r\to\infty}m_{q}(r)=M_{q},$ (29)

also

$\displaystyle\lim_{r\to\infty}\frac{d}{dr}m_{s}(r)=0$ (30)
$\displaystyle\lim_{r\to\infty}\frac{d}{dr}m_{q}(r)=0,$ (31)

Thus, replacing (23) at $r=a$, where $\mu(a)=0$:

$\dfrac{a}{2}=m_{s}(a)+\alpha\cdot m_{q}(a).$ (32)

Thus, inserting equations (22) and (32) into equation (21), this last equation
takes the following form:

$\displaystyle(P_{s}+\alpha P_{q})d\left(\frac{4}{3}\pi a^{3}\right)=$
$\displaystyle\left(\dfrac{\bar{\mu}^{\prime}}{4\pi}+\alpha\dfrac{g^{\prime}}{4\pi}\right)d\left(\dfrac{4\pi
a^{2}}{4}\right)$ $\displaystyle-d\Big{(}m_{s}(a)+\alpha\cdot
m_{q}(a)\Big{)}.$ (33)

Thus, the system splits into the following sets of equations:

  * •

The standard firs law of thermodynamics of order $\alpha^{0}$, which
corresponds to the standard Einstein equations:

$P_{s}d\left(\frac{4}{3}\pi
a^{3}\right)=\left(\dfrac{\bar{\mu}^{\prime}}{4\pi}\right)d\left(\dfrac{4\pi
a^{2}}{4}\right)-d\Big{(}m_{s}(a)\Big{)}.$ (34)

This equation has the form of the first law of thermodynamics $PdV=TdS-dU$,
where each term is identified as:

$\displaystyle\mbox{Thermodynamics Pressure}=$ $\displaystyle P_{s}$ (35)
$\displaystyle\mbox{Volume}=$ $\displaystyle\frac{4}{3}\pi a^{3}$ (36)
$\displaystyle\mbox{Temperature}=$ $\displaystyle
T_{s}=\frac{\bar{\mu}^{\prime}}{4\pi}$ (37) $\displaystyle\mbox{Entropy}=$
$\displaystyle\dfrac{4\pi a^{2}}{4}=\dfrac{Area}{4}$ (38)
$\displaystyle\mbox{Local Energy}=$ $\displaystyle U_{s}^{loc}=m_{s}(a)$ (39)

It is worth to mention that our thermodynamics variables obtained coincide
with the previously known in literature. The thermodynamics volume coincide
with the geometric volume, the temperature is the well known expression and
the entropy follows the area’s law. The energy $m_{s}(a)$ corresponds to a
local definition of energy, at $r=a$, given by reference Aros:2019auf . In the
vacuum case, where $\bar{\rho}=\theta^{\mu}_{\nu}=0$, the solution (23) has
the form $\mu=1-2M/r$ and the local energy coincide with the energy of the
Schwarzschild black hole, $a/2=M$.

These thermodynamics variables $P_{s}$,$T_{s}$ and $U_{s}^{loc}$ represent the
contribution of the standard sector to the total pressure, total temperature
and total local energy of the black hole. These ones do not represent the
pressure, temperature and local energy of an independent black hole.

  * •

The quasi first law of thermodynamics of order $\alpha$, which corresponds to
the quasi Einstein equations:

$P_{q}d\left(\frac{4}{3}\pi
a^{3}\right)=\left(\dfrac{g^{\prime}}{4\pi}\right)d\left(\dfrac{4\pi
a^{2}}{4}\right)-d\Big{(}m_{q}(a)\Big{)}.$ (40)

This equation also has the form of the first law of thermodynamics, $PdV=TdS-
dU$, and each term is identified as:

$\displaystyle\mbox{Thermodynamics Pressure}=$ $\displaystyle P_{q}$ (41)
$\displaystyle\mbox{Volume}=$ $\displaystyle\frac{4}{3}\pi a^{3}$ (42)
$\displaystyle\mbox{Temperature}=$ $\displaystyle
T_{q}=\frac{g^{\prime}}{4\pi}$ (43) $\displaystyle\mbox{Entropy}=$
$\displaystyle\dfrac{4\pi a^{2}}{4}=\dfrac{Area}{4}$ (44)
$\displaystyle\mbox{Local Energy}=$ $\displaystyle U_{q}^{loc}=m_{q}(a),$ (45)

where, in a similar way to the previous case, all the thermodynamics variables
coincide with the previously known in the literature.

Also, these thermodynamics variables $P_{q}$,$T_{q}$ and $U_{q}^{loc}$
represent the contribution of the quasi sector to the total pressure, total
temperature and total local energy of the black hole. These ones do not
represent the pressure, temperature and local energy of an independent black
hole.

###  3.1 Some Remarks

From the previous analysis one can remark some properties:

  * •

The total thermodynamics pressure, total temperature and total local energy
correspond to a simple sum of the thermodynamics contributions of each sector

$\displaystyle P_{tot}$ $\displaystyle=P_{s}+P_{q}$ (46) $\displaystyle
T_{tot}$ $\displaystyle=T_{s}+T_{q}$ (47) $\displaystyle U_{tot}^{loc}$
$\displaystyle=U_{s}^{loc}+U_{q}^{loc}$ (48)

So, as was said above, it is worth to mention that the temperatures $T_{s}$
and $T_{q}$ represent the contribution of each sector (standard and quasi) to
the total temperature of black hole and, do not represent the temperature of
one independent black hole. The same occurs with the pressure and energy.

  * •

Both sectors share the same thermodynamics volume and entropy.

As was said above, $U_{tot}^{loc}$ corresponds to a local definition of
energy, located at $r=a$ Aros:2019auf . However, the total energy of the
system must be computed by the Noether charge Aros:1999kt , by mean of the
Komar formula Kastor:2008xb .

$\displaystyle E\propto$ $\displaystyle\displaystyle\lim_{r\to\infty}K(\xi)=$
$\displaystyle=$
$\displaystyle\lim_{r\to\infty}\frac{1}{8\pi}\frac{d}{dr}\mu(r)r^{2}\int
d\Omega_{2}$ $\displaystyle=$
$\displaystyle\dfrac{1}{2}\Big{(}M_{s}+\alpha\cdot M_{q}\Big{)},$ (49)

i.e the total energy is proportional to the Komar formula Kastor:2008xb .
After a regularization, based on the inclusion of the boundary terms in the
action Aros:1999kt , can be obtained that:

$E=M_{s}+\alpha\cdot M_{q}.$ (50)

At infinity, our local definitions of energy coincide with the total energy.
Thus, other important result of our decoupling of the first law of
thermodynamics is that: The total energy corresponds to the contribution of
the Noether charge of each sector.

##  4 A simple example

As a simple example, we choose a seed and a extra source, such that, near the
origin, the mass functions behave as:

$\displaystyle m_{s}(r)\Big{|}_{r\approx 0}\approx C_{1}r^{3}$ (51)
$\displaystyle m_{q}(r)\Big{|}_{r\approx 0}\approx C_{2}r^{3}.$ (52)

This one implies that near the origin the geometry behaves as a de Sitter
space time, where the invariants of curvature have finite values, and thus,
unlike the singular Schwarzschild solution, the geometry is regular near the
origin. These models are called regular black holes.

The seed source is given by the Hayward model Hayward:2005gi :

$\rho(r)=\frac{1}{2\pi}\frac{3Q^{2}M_{s}^{2}}{(2Q^{2}M_{s}+r^{3})^{2}},$ (53)

where $Q$ is a constant. The mass function is:

$m(r)=\frac{M_{s}r^{3}}{r^{3}+2M_{s}Q^{2}}.$ (54)

On the other hand, the extra source is given by the Dymnikova model
Dymnikova:2010zz :

$\theta^{0}_{0}(r)=-\frac{M_{q}\exp(-r^{3}/R^{3})}{(4/3)\pi R^{3}},$ (55)

where $R$ is a constant. The mass function is:

$m(r)=M_{q}\Big{(}1-\exp(-r^{3}/R^{3})\Big{)}.$ (56)

It is direct to check that the conditions 51 and 52 are fulfilled, an thus,
this model is suited to represent a regular black hole.

On the other hand, also it is direct to check that the conditions 28 and 29
are fulfilled. This one ensures that the total energy corresponds to the
contribution of the Noether charge of each sector, equation (50).

In the figure 1 one can check the existence of one critical value of
$M_{s}=M_{*}$, which corresponds to the minimum value on the curve, where
there is an extremal black hole, where the internal horizon $a=r_{-}$ and
black hole horizon $a=r_{+}$ coincide. On the other hand, for $M_{s}>M_{*}$
there are two horizons, namely, the internal horizon $r_{-}$ and the black
hole horizon $r_{+}$. This behavior is generic for other values of the
parameters.

Figure 1: Behavior of $M_{s}$ parameter for $\alpha=Q=R1$, $M_{q}=0.7$

Figures 2 (a), 2(b) and 2(c) display the behavior of the contributions $T_{s}$
and $T_{q}$ and the total temperature $T_{tot}$, respectively. This behavior
is generic for other values of the parameters. One can check that the
temperature vanishes at the extremal case Chirco:2010rq . $T_{s}$ and $T_{q}$
are contributions to the total temperature $T_{tot}$, where, this last is
always positive. The total temperature has a local maximum, located at
$r_{+}=r_{max}$. One can check that this inflection point is due to the
behavior of the contribution of the quasi sector, $T_{q}$, which vanishes
after this point. Using the standard definition of the heat capacity:

$C=\frac{\partial S}{\partial r_{+}}\left(\frac{\partial T_{tot}}{\partial
r_{+}}\right)^{-1},$ (57)

it is direct to see that the sign of the specific heat depend only on the
factor $\dfrac{\partial T_{tot}}{\partial r_{+}}$. So in the point
$r_{+}=r_{max}$, where this derivative vanishes, there is a phase transition.
At the left side of $r_{+}=r_{max}$ this derivative is positive and the heat
capacity is positive, i.e the black hole is stable. However, at the right side
of $r_{+}=r_{max}$ this derivative is negative and the heat capacity is
negative, i.e the black hole is unstable.

(a) $T_{s}$ . (b) $T_{q}$ (c) $T_{tot}$

Figure 2: Behavior of $T_{s}$,$T_{q}$ and $T_{tot}$ for $\alpha=Q=R1$,
$M_{q}=0.7$ .

##  5 Conclusion and Discussion

In this work is showed that, a direct consequence of the application of the
gravitational decoupling method to the spherically symmetric space time is the
decoupling of the first law of thermodynamics in two sectors, called, the
standard first law of thermodynamics, and the quasi first law of
thermodynamics. This is achieved, following the approximation of the reference
Kothawala:2007em , where the equations of motion are evaluated at the horizon
$r=a$.

Each sector is written as a thermodynamics identity $PdV=TdS-dU$. In this
respect, both sectors share the same thermodynamics volume and the same
entropy. The total thermodynamics pressure, total temperature and total local
energy correspond to a simple sum of the thermodynamics contributions of each
sector, equations (46), (47) and (48). Other interesting result is that the
total energy corresponds to the contribution of the Noether charge of each
sector, equation 50.

In both sectors, the terms corresponding to the identity $PdV=TdS-dU$ coincide
with the previously known in literature. The thermodynamics volume coincide
with the geometric volume, the temperature is the well known expression and
the entropy follows the area’s law. The local definition of energy coincide
with the definition of the reference Aros:2019auf .

Finally, it is provided one example, where the seed source and the extra
sources correspond to the Hayward model Hayward:2005gi  and the Dymnikova
model Dymnikova:1992ux , respectively. It is showed that the total temperature
has a inflection point where there is a phase transition between
stable/unstable black hole. This phase transition is due to the behavior of
the quasi sector contribution, where this contribution vanishes after this
point.

## References

  * (1) J. Ovalle, Decoupling gravitational sources in general relativity: from perfect to anisotropic fluids, Phys. Rev. D95 (10) (2017) 104019.  arXiv:1704.05899, doi:10.1103/PhysRevD.95.104019.
  * (2) J. Ovalle, R. Casadio, R. da Rocha, A. Sotomayor, Anisotropic solutions by gravitational decoupling, Eur. Phys. J. C78 (2) (2018) 122.  arXiv:1708.00407, doi:10.1140/epjc/s10052-018-5606-6.
  * (3) L. Gabbanelli, A. Rincon, C. Rubio, Gravitational decoupled anisotropies in compact stars, Eur. Phys. J. C78 (5) (2018) 370.  arXiv:1802.08000, doi:10.1140/epjc/s10052-018-5865-2.
  * (4) M. Estrada, F. Tello-Ortiz, A new family of analytical anisotropic solutions by gravitational decoupling, Eur. Phys. J. Plus 133 (11) (2018) 453.  arXiv:1803.02344, doi:10.1140/epjp/i2018-12249-9.
  * (5) C. L. Heras, P. Leon, Using MGD gravitational decoupling to extend the isotropic solutions of Einstein equations to the anisotropical domain, Fortsch. Phys. 66 (7) (2018) 1800036.  arXiv:1804.06874, doi:10.1002/prop.201800036.
  * (6) M. Sharif, S. Sadiq, Gravitational Decoupled Charged Anisotropic Spherical Solutions, Eur. Phys. J. C78 (5) (2018) 410.  arXiv:1804.09616, doi:10.1140/epjc/s10052-018-5894-x.
  * (7) E. Morales, F. Tello-Ortiz, Charged anisotropic compact objects by gravitational decoupling, Eur. Phys. J. C78 (8) (2018) 618.  arXiv:1805.00592, doi:10.1140/epjc/s10052-018-6102-8.
  * (8) E. Morales, F. Tello-Ortiz, Compact Anisotropic Models in General Relativity by Gravitational Decoupling, Eur. Phys. J. C78 (10) (2018) 841.  arXiv:1808.01699, doi:10.1140/epjc/s10052-018-6319-6.
  * (9) M. Estrada, R. Prado, The Gravitational decoupling method: the higher dimensional case to find new analytic solutions, Eur. Phys. J. Plus 134 (4) (2019) 168.  arXiv:1809.03591, doi:10.1140/epjp/i2019-12555-8.
  * (10) S. K. Maurya, F. Tello-Ortiz, Generalized relativistic anisotropic compact star models by gravitational decoupling, Eur. Phys. J. C79 (1) (2019) 85.  doi:10.1140/epjc/s10052-019-6602-1.
  * (11) J. Ovalle, R. Casadio, R. d. Rocha, A. Sotomayor, Z. Stuchlik, Black holes by gravitational decoupling, Eur. Phys. J. C78 (11) (2018) 960.  arXiv:1804.03468, doi:10.1140/epjc/s10052-018-6450-4.
  * (12) E. Contreras, P. Bargueño, Minimal Geometric Deformation in asymptotically (A-)dS space-times and the isotropic sector for a polytropic black hole, Eur. Phys. J. C78 (12) (2018) 985.  arXiv:1809.09820, doi:10.1140/epjc/s10052-018-6472-y.
  * (13) E. Contreras, A. Rincon, P. Bargueño, A general interior anisotropic solution for a BTZ vacuum in the context of the Minimal Geometric Deformation decoupling approach, Eur. Phys. J. C79 (3) (2019) 216.  arXiv:1902.02033, doi:10.1140/epjc/s10052-019-6749-9.
  * (14) M. Sharif, S. Saba, Gravitational decoupled anisotropic solutions in $f({\mathcal{G}})$ gravity, Eur. Phys. J. C78 (11) (2018) 921.  arXiv:1811.08112, doi:10.1140/epjc/s10052-018-6406-8.
  * (15) J. Ovalle, R. Casadio, R. da Rocha, A. Sotomayor, Z. Stuchlik, Einstein-Klein-Gordon system by gravitational decoupling, EPL 124 (2) (2018) 20004.  arXiv:1811.08559, doi:10.1209/0295-5075/124/20004.
  * (16) M. Estrada, A way of decoupling gravitational sources in pure Lovelock gravity, Eur. Phys. J. C79 (11) (2019) 918.  arXiv:1905.12129, doi:10.1140/epjc/s10052-019-7444-6.
  * (17) S. K. Maurya, F. Tello-Ortiz, Charged anisotropic compact star in $f(R,T)$ gravity: A minimal geometric deformation gravitational decoupling approach, Phys. Dark Univ. 27 (2020) 100442.  arXiv:1905.13519, doi:10.1016/j.dark.2019.100442.
  * (18) M. Sharif, A. Waseem, Effects of Charge on Gravitational Decoupled Anisotropic Solutions in f(R) Gravity, Chin. J. Phys. 60 (2019) 426–439.  arXiv:1906.07559, doi:10.1016/j.cjph.2019.05.016.
  * (19) E. Contreras, Minimal Geometric Deformation: the inverse problem, Eur. Phys. J. C78 (8) (2018) 678.  arXiv:1807.03252, doi:10.1140/epjc/s10052-018-6168-3.
  * (20) G. Panotopoulos, A. Rincon, Minimal Geometric Deformation in a cloud of strings, Eur. Phys. J. C78 (10) (2018) 851.  arXiv:1810.08830, doi:10.1140/epjc/s10052-018-6321-z.
  * (21) E. Contreras, Gravitational decoupling in $2+1$ dimensional space–times with cosmological term, Class. Quant. Grav. 36 (9) (2019) 095004.  arXiv:1901.00231, doi:10.1088/1361-6382/ab11e6.
  * (22) E. Contreras, P. Bargueño, Extended gravitational decoupling in 2 + 1 dimensional space-times, Class. Quant. Grav. 36 (21) (2019) 215009.  arXiv:1902.09495, doi:10.1088/1361-6382/ab47e2.
  * (23) C. Las Heras, P. León, New algorithms to obtain analytical solutions of Einstein’s equations in isotropic coordinates, Eur. Phys. J. C79 (12) (2019) 990.  arXiv:1905.02380, doi:10.1140/epjc/s10052-019-7507-8.
  * (24) L. Gabbanelli, J. Ovalle, A. Sotomayor, Z. Stuchlik, R. Casadio, A causal Schwarzschild-de Sitter interior solution by gravitational decoupling, Eur. Phys. J. C79 (6) (2019) 486.  arXiv:1905.10162, doi:10.1140/epjc/s10052-019-7022-y.
  * (25) S. Hensh, Z. Stuchlík, Anisotropic Tolman VII solution by gravitational decoupling, Eur. Phys. J. C79 (10) (2019) 834.  arXiv:1906.08368, doi:10.1140/epjc/s10052-019-7360-9.
  * (26) P. León, A. Sotomayor, Braneworld Gravity under gravitational decoupling, Fortsch. Phys. 67 (12) (2019) 1900077.  arXiv:1907.11763, doi:10.1002/prop.201900077.
  * (27) V. A. Torres-Sánchez, E. Contreras, Anisotropic neutron stars by gravitational decoupling, Eur. Phys. J. C79 (10) (2019) 829.  arXiv:1908.08194, doi:10.1140/epjc/s10052-019-7341-z.
  * (28) A. Rincon, L. Gabbanelli, E. Contreras, F. Tello-Ortiz, Minimal geometric deformation in a Reissner–Nordström background, Eur. Phys. J. C79 (10) (2019) 873.  arXiv:1909.00500, doi:10.1140/epjc/s10052-019-7397-9.
  * (29) R. Casadio, E. Contreras, J. Ovalle, A. Sotomayor, Z. Stuchlick, Isotropization and change of complexity by gravitational decoupling, Eur. Phys. J. C79 (10) (2019) 826.  arXiv:1909.01902, doi:10.1140/epjc/s10052-019-7358-3.
  * (30) M. Sharif, S. Sadiq, 2+1-dimensional gravitational decoupled anisotropic solutions, Chin. J. Phys. 60 (2019) 279–289.  doi:10.1016/j.cjph.2019.05.018.
  * (31) K. Singh, S. K. Maurya, M. K. Jasim, F. Rahaman, Minimally deformed anisotropic model of class one space-time by gravitational decoupling, Eur. Phys. J. C79 (10) (2019) 851.  doi:10.1140/epjc/s10052-019-7377-0.
  * (32) G. Abellán, V. Torres, E. Fuenmayor, E. Contreras, Regularity condition on the anisotropy induced by gravitational decoupling in the framework of MGD, Eur. Phys. J. C80 (2) (2020) 177.  arXiv:2001.08573, doi:10.1140/epjc/s10052-020-7749-5.
  * (33) S. W. Hawking, Particle Creation by Black Holes, Commun. Math. Phys. 43 (1975) 199–220, [,167(1975)].  doi:10.1007/BF02345020,10.1007/BF01608497.
  * (34) J. D. Bekenstein, Black holes and the second law, Lett. Nuovo Cim. 4 (1972) 737–740.  doi:10.1007/BF02757029.
  * (35) J. D. Bekenstein, Black holes and entropy, Phys. Rev. D7 (1973) 2333–2346.  doi:10.1103/PhysRevD.7.2333.
  * (36) S. W. Hawking, Black hole explosions, Nature 248 (1974) 30–31.  doi:10.1038/248030a0.
  * (37) R. M. Wald, General Relativity, Chicago Univ. Pr., Chicago, USA, 1984.  doi:10.7208/chicago/9780226870373.001.0001.
  * (38) D. Kastor, S. Ray, J. Traschen, Enthalpy and the Mechanics of AdS Black Holes, Class. Quant. Grav. 26 (2009) 195011.  arXiv:0904.2765, doi:10.1088/0264-9381/26/19/195011.
  * (39) D. Kothawala, S. Sarkar, T. Padmanabhan, Einstein’s equations as a thermodynamic identity: The Cases of stationary axisymmetric horizons and evolving spherically symmetric horizons, Phys. Lett. B652 (2007) 338–342.  arXiv:gr-qc/0701002, doi:10.1016/j.physletb.2007.07.021.
  * (40) M. Estrada, R. Aros, Regular black holes with $\Lambda>0$ and its evolution in Lovelock gravity, Eur. Phys. J. C79 (10) (2019) 810.  arXiv:1906.01152, doi:10.1140/epjc/s10052-019-7316-0.
  * (41) I. Dymnikova, M. Korpusik, Regular black hole remnants in de Sitter space, Phys. Lett. B685 (2010) 12–18.  doi:10.1016/j.physletb.2010.01.044.
  * (42) D. Kothawala, T. Padmanabhan, Thermodynamic structure of Lanczos-Lovelock field equations from near-horizon symmetries, Phys. Rev. D79 (2009) 104020.  arXiv:0904.0215, doi:10.1103/PhysRevD.79.104020.
  * (43) A. Sheykhi, Thermodynamics of apparent horizon and modified Friedmann equations, Eur. Phys. J. C69 (2010) 265–269.  arXiv:1012.0383, doi:10.1140/epjc/s10052-010-1372-9.
  * (44) T. Padmanabhan, Classical and quantum thermodynamics of horizons in spherically symmetric space-times, Class. Quant. Grav. 19 (2002) 5387–5408.  arXiv:gr-qc/0204019, doi:10.1088/0264-9381/19/21/306.
  * (45) R.-G. Cai, Thermodynamics of apparent horizon in brane world scenarios, Prog. Theor. Phys. Suppl. 172 (2008) 100–109.  arXiv:0712.2142, doi:10.1143/PTPS.172.100.
  * (46) S. A. Hayward, Formation and evaporation of regular black holes, Phys. Rev. Lett. 96 (2006) 031103.  arXiv:gr-qc/0506126, doi:10.1103/PhysRevLett.96.031103.
  * (47) I. Dymnikova, Vacuum nonsingular black hole, Gen. Rel. Grav. 24 (1992) 235–242.  doi:10.1007/BF00760226.
  * (48) R. Aros, M. Contreras, R. Olea, R. Troncoso, J. Zanelli, Conserved charges for even dimensional asymptotically AdS gravity theories, Phys. Rev. D62 (2000) 044002\.  arXiv:hep-th/9912045, doi:10.1103/PhysRevD.62.044002.
  * (49) D. Kastor, Komar Integrals in Higher (and Lower) Derivative Gravity, Class. Quant. Grav. 25 (2008) 175007.  arXiv:0804.1832, doi:10.1088/0264-9381/25/17/175007.
  * (50) G. Chirco, S. Liberati, T. P. Sotiriou, Gedanken experiments on nearly extremal black holes and the Third Law, Phys. Rev. D82 (2010) 104015.  arXiv:1006.3655, doi:10.1103/PhysRevD.82.104015.