# Triviality results for quasi $k$-Yamabe solitons

Tokura, W. 1,∗ 1 Universidade Estadual de Mato Grosso do Sul, 79150-000, Av.
João Pedro Fernandes, 2101 - Centro, Maracaju, MS, Brazil.
<EMAIL_ADDRESS>1 ,  Batista, E. 2 2 Universidade Federal de
Goiás, IME, 131, 74001-970, Goiânia, GO, Brazil<EMAIL_ADDRESS>2 and
Kai, P. 3 Universidade Federal de Goiás, INF, s/n, 74690-900, Goiânia, GO,
Brazil<EMAIL_ADDRESS>3

###### Abstract.

In this paper, we show that any compact quasi $k$-Yamabe gradient solitons
must have constant $\sigma_{k}$-curvature. Moreover, we provide a certain
condition for a compact quasi $k$-Yamabe soliton to be gradient, and for
noncompact solitons, we present a geometric rigidity under a decaiment
condition on the norm of the soliton field.

###### Key words and phrases:

$\sigma_{k}$-curvature, quasi $k$-Yamabe solitons, Yamabe solitons, invariant
solutions, scalar curvature.

###### 2010 Mathematics Subject Classification:

53C21, 53C50, 53C25

∗ Corresponding author

##  1\. Introduction and main results

A Riemannian manifold $(M^{n},g)$, $n\geqslant 3$, is an Einstein-type
manifold if there exists a vector field $X\in\mathfrak{X}(M)$ and a smooth
function $\lambda:M\rightarrow\mathbb{R}$ such that

(1) $\alpha Ric+\frac{\beta}{2}\mathcal{L}_{X}g+\mu X^{\flat}\otimes
X^{\flat}=(\rho R+\lambda)g,$

for some constants $\alpha$, $\beta$, $\mu$ $\rho\in\mathbb{R}$ with
$(\alpha,\beta,\mu)\neq(0,0,0)$. Here $\mathcal{L}_{X}g$, $X^{\flat}$ and $R$
stand, respectively, for the Lie derivative of $g$ in the direction of $X$,
the 1-form metrically dual to the vector field $X$ and the scalar curvature of
$M$. The concept of Einstein-type manifold was introduced recently by Catino
et al. [9]. It is worth noting that in terms of equation (1), an Einstein-type
manifold unifies various particular cases well studied in the literature, such
as gradient Ricci solitons, gradient almost Ricci solitons, Yamabe solitons,
quasi Yamabe solitons, conformal gradient solitons, quasi Einstein manifolds
and $\rho$-Einstein solitons. Each of them has a particular importance.

Our purpose is to study some cases of the Einstein-type manifolds which were
not addressed in [9]. More precisely, we focus our analysis on the class
$\alpha=0$, $\beta=1$, $\mu=-\frac{1}{m}$ and $\lambda=\rho R-\sigma_{k}+c$,
$c\in\mathbb{R}$ where $\sigma_{k}$ is the $\sigma_{k}$-curvature of $g$. We
recall that, if we denote by $\lambda_{1},\lambda_{2},\dots,\lambda_{n}$ the
eigenvalues of the symmetric endomorphism $g^{-1}A_{g}$, where $A_{g}$ is the
Schouten tensor defined by

$A_{g}=\frac{1}{n-2}\left[Ric_{g}-\frac{R}{2(n-1)}g\right],$

then the $\sigma_{k}$-curvature of $g$ is defined as the $k$-th symmetric
elementary function of $\lambda_{1},\dots,\lambda_{n}$, namely

$\sigma_{k}=\sigma_{k}(g^{-1}A_{g})=\sum_{i_{1}<\dots<i_{k}}\lambda_{i_{1}}\cdot\dots\cdot\lambda_{i_{k}},\quad\text{for}\quad
1\leq k\leq n.$

In this sense, we introduce the following class of manifolds.

######  Definition 1.1.

A Riemannian manifold $(M^{n},g)$, $n\geqslant 3$, is a quasi $k$-Yamabe
soliton if there exists a vector field $X\in\mathfrak{X}(M)$ and two constants
$m$, $\lambda$ (where $m$ is not zero) such that

(2) $\frac{1}{2}\mathcal{L}_{X}g-\frac{1}{m}X^{\flat}\otimes
X^{\flat}=(\sigma_{k}-\lambda)g,$

We will write the soliton in (2) as $(M^{n},g,X,\lambda)$ for the sake of
simplicity. If $X=\nabla f$ for some smooth function
$f:M\rightarrow\mathbb{R}$, we say that $(M^{n},g,f,\lambda)$ is a quasi
$k$-Yamabe gradient soliton. In this case, equation (2) can be rewritten as

(3) $\nabla^{2}f-\frac{1}{m}df\otimes df=(\sigma_{k}-\lambda)g,$

Moreover, when $f$ is a constant or $X=0$ the soliton is called trivial.

It is worth noting that quasi $k$-Yamabe solitons correspond to a large class
of manifolds well studied in the literature. For instance, if $m\to\infty$,
then quasi $k$-Yamabe solitons reduces to $k$-Yamabe solitons [2, 4, 8, 23].
Also, since $\sigma_{1}=\frac{R}{2(n-1)}$, $1$-Yamabe solitons naturally
correspond to Yamabe solitons [5, 6, 7, 10, 11, 12, 13, 15, 17, 22], and quasi
$1$-Yamabe solitons correspond to quasi Yamabe solitons studied in [16, 25].
When $X=\nabla f$ is a gradient vector field, quasi $1$-Yamabe gradient
solitons correspond to an $f$-almost Yamabe soliton [27].

In recent years, many efforts have been devoted to study the geometry of
Yamabe solitons and their generalizations. For instance, Hsu in [15] shown
that any compact gradient $1$-Yamabe soliton is trivial. For $k>1$, the
extension of the previous result was recently investigated. Catino et al. [8]
proved that any compact gradient $k$-Yamabe soliton with nonnegative Ricci
curvature is trivial. Bo et al. [4] also proved that any compact gradient
$k$-Yamabe soliton with negative constant scalar curvature necessarily has
constant $\sigma_{k}$-curvature. In [23] it was shown that any compact
gradient $k$-Yamabe soliton must be trivial. For $m\neq\infty$, Huang and Li
[16] proved that any compact quasi $1$-Yamabe gradient soliton is trivial. Our
first theorem extends all these results at once.

######  Theorem 1.2.

Any compact quasi $k$-Yamabe gradient soliton $(M^{n},g,f,\lambda)$ is
trivial, i.e., has constant $\sigma_{k}$-curvature.

The Hodge-de Rham decomposition theorem shows that any vector field $X$ on a
compact oriented Riemannian manifold $M^{n}$ can be decompose as follows:

(4) $X=\nabla h+Y,$

where $h:M\rightarrow\mathbb{R}$ is a smooth function and $Y$ is a divergence
free vector field on $M^{n}$. In fact, by the Hodge-de Rham theorem [26], we
have that $X^{\flat}$ takes the following form:

$X^{\flat}=d\alpha+\delta\beta+\gamma.$

Taking $Y=(\delta\beta+\gamma)^{\sharp}$ and $(d\alpha)^{\sharp}=\nabla h$ we
arrive at the desired result.

Our next result provide a necessary and sufficient condition for a compact
quasi $k$-Yamabe soliton to be gradient.

######  Theorem 1.3.

The compact quasi $k$-Yamabe soliton $(M^{n},g,X,\lambda)$ is gradient if, and
only if

(5) $\begin{split}&\int_{M}Ric(\nabla
h,Y)+\frac{1}{m}g(\nabla^{2}h,X^{\flat}\otimes
X^{\flat})-\frac{2}{m}g(\nabla^{2}h,dh\otimes dh)+\frac{1}{m^{2}}|dh\otimes
dh|^{2}+\frac{n}{2m}(\sigma_{k}-\lambda)|X|^{2}\\\
&\qquad\qquad\qquad\qquad\qquad\qquad\qquad+\frac{2}{m}(\sigma_{k}-\lambda)|\nabla
h|^{2}+\frac{3}{2m^{2}}|X|^{4}\leqslant 0.\end{split}$

where $h$ and $Y$ are the Hodge–de Rham decomposition components of $X$.

Note that when $m\to\infty$, then Theorem 1.3 correspond to Theorem 1.3 of
[23]. On the other hand, combining Theorem 1.2 and Theorem 1.3, we have the
following result.

######  Corollary 1.4.

Any compact quasi $k$-Yamabe soliton $(M^{n},g,X,\lambda)$ satisfying (5) is
trivial, i.e., has constant $\sigma_{k}$-curvature.

Now we notice that the same result obtained in [1] for compact Ricci solitons
also works for compact quasi $k$-Yamabe solitons. More precisely, we have the
following theorem.

######  Theorem 1.5.

Let $(M^{n},g,X,\lambda)$ be a compact quasi $k$-Yamabe soliton and $X=\nabla
h+Y$ the Hodge–de Rham decomposition of $X$. If

$\int_{M^{n}}g(\nabla h,X)dv_{g}\leqslant 0,$

then $(M^{n},g)$ is trivial, i.e., has constant $\sigma_{k}$-curvature.

The next theorem provide a rigidity result in the scope of noncompact quasi
$k$-Yamabe gradient solitons.

In [18], Ma and Miguel study Liouville type theorem of harmonic functions with
finite weighted Dirichlet integral and use it to prove a rigidity result for
gradient Yamabe solitons with non negative Ricci curvature. We also provide a
rigidity result assuming that the weighted integral of the soliton vector
field is finite.

######  Theorem 1.6.

Let $(M^{n},g,f,\lambda)$ be a complete and noncompact quasi $k$-Yamabe
gradient soliton satisfying

$\int_{M^{n}\setminus B(r)}d(x,x_{0})^{-1}|\nabla f|d\mu<\infty,$

where $d$ is the distance function with respect to $g$, $B(r)$ is the open
ball of radius $r>0$ centered at $x_{0}$ and $d\mu=e^{-\frac{f}{m}}dv_{g}$ .
If $f$ is $f$-subharmonic, then $(M^{n},g)$ has constant
$\sigma_{k}$-curvature.

Before we present the proof of main results, let’s take a look at some
examples. For others examples see Section 3.

######  Example 1.7.

Identities

$Ric_{g_{\mathbb{S}^{n}}}=(n-1)g_{\mathbb{S}^{n}},\quad
R_{g_{\mathbb{S}^{n}}}=n(n-1)\quad\mbox{and}\quad
A_{g_{\mathbb{S}^{n}}}=\frac{1}{2}g_{\mathbb{S}^{n}},$

rule the Ricci tensor, scalar curvature and Schouten tensor, respectively, of
the Euclidean sphere $(\mathbb{S}^{n},g_{\mathbb{S}^{n}})$. Therefore, we have
that

$\sigma_{k}(g_{\mathbb{S}^{n}}^{-1}A_{g_{\mathbb{S}^{n}}})=\frac{1}{2^{k}}\binom{n}{k},\quad
1\leqslant k\leqslant n.$

Then $(\mathbb{S}^{n},g_{\mathbb{S}^{n}})$ is a trivial quasi $k$-Yamabe
gradient soliton with $\sigma_{k}=\lambda$.

Note that, according to Theorem 1.2, the Euclidean sphere does not admit any
non-trivial structure of quasi $k$-Yamabe gradient soliton.

######  Example 1.8.

Consider the hyperbolic space
$\mathbb{H}^{n+1}=\mathbb{R}\times_{e^{t}}\mathbb{R}^{n}$ furnished with the
warped product metric [20]:

$g=dt^{2}+e^{2t}g_{\mathbb{R}^{n}}^{2}.$

It is well known that the horospheres of the hyperbolic space are totally
umbilical hypersurfaces isometric to $\mathbb{R}^{n}$ and correspond to slices
${\\{t_{0}\\}}\times\mathbb{R}^{n}$, $t_{0}\in\mathbb{R}$. Hence, taking the
inclusion

$i:\\{t_{0}\\}\times\mathbb{R}^{n}\rightarrow\mathbb{R}\times\mathbb{R}^{n}\quad(t_{0},x)\mapsto(t_{0},x),$

we deduce that the height function from the immersion satisfies $f(x)=t_{0}$,
and then the standard Euclidean space $\\{t_{0}\\}\times\mathbb{R}^{n}$ is a
trivial quasi $k$-Yamabe gradient soliton immersed into hyperbolic space with
potential $f(x)=t_{0}$ and $\lambda=0$.

######  Example 1.9.

Consider $M^{n}=\\{(x_{1},\dots,x_{n})\in\mathbb{R}^{n}\;;\;x_{n}>0\\}$,
$g_{ij}=(k_{0}x_{n})^{-2}\delta_{ij}$, $k_{0}\in(0,\infty)$ and the potential
function

$f(x_{1},\dots,x_{n})=-m\log\left(\frac{k_{1}}{x_{n}}\right),\qquad
k_{1}\in(0,\infty).$

By a direct computation, we deduce that

$\sigma_{k}=-\frac{n!}{k!(n-k)!}(-1)^{k-1}\left(\frac{k_{0}^{2}}{2}\right)^{k},\qquad\nabla^{2}f-\frac{1}{m}df\otimes
df=-mk_{0}^{2}g.$

Then, $(M^{n},g)$ is a quasi $k$-Yamabe gradient soliton with

$\lambda=-\frac{n!}{k!(n-k)!}(-1)^{k-1}\left(\frac{k_{0}^{2}}{2}\right)^{k}+mk_{0}^{2}.$

######  Example 1.10.

Consider the Euclidean subset
$M^{n}=\\{(x_{1},\dots,x_{n})\in\mathbb{R}^{n}\;;\;x_{1}+\dots+x_{n}>0\\}$
furnished with the metric tensor $g_{ij}=\delta_{ij}$ and potential function
given by

$f(x_{1},\dots,x_{n})=-m\log\left(x_{1}+\cdots+x_{n}\right).$

Since $Ric_{g}=0$, $R_{g}=0$, we have that $A_{g}=0$. Consequently
$\sigma_{k}=0$ for all $k\in\\{1,\cdots,n\\}$. On the other hand, from the
potential function $f$, we deduce that $\nabla^{2}f=\frac{1}{m}df\otimes df$.
So, $(M^{n},g)$ is a quasi $k$-Yamabe gradient soliton with $\lambda=0$.

Example 1.10 shows that for a noncompact quasi $k$-Yamabe gradient soliton,
the condition of constant $\sigma_{k}$-curvature does not imply that the
potential function $f$ is constant.

######  Example 1.11.

Consider
$M^{n}=\\{(x_{1},\dots,x_{n})\in\mathbb{R}^{n}\;;\;x_{1}+\dots+x_{n}>0\\}$,
$g_{ij}=(x_{1}+\dots+x_{n})^{-2}\delta_{ij}$ and the potential function

$f(x_{1},\dots,x_{n})=-m\log\left(\frac{1}{x_{1}+\dots+x_{n}}\right).$

By a direct computation, we deduce that

$\sigma_{k}=-\frac{n!}{k!(n-k)!}(-1)^{k-1}\left(\frac{n}{2}\right)^{k},\qquad\nabla^{2}f-\frac{1}{m}df\otimes
df=-mng.$

Then, $(M^{n},g)$ is a quasi $k$-Yamabe gradient soliton with

$\lambda=-\frac{n!}{k!(n-k)!}(-1)^{k-1}\left(\frac{n}{2}\right)^{k}+mn.$

Example 1.9, Example 1.10 and Example 1.11 show that the compactness of the
manifold can not be discarded in the prove of Theorem 1.2.

##  2\. Proofs

### Proof of Theorem 1.2 :

If $k=1$, then $(M^{n},g)$ is a quasi Yamabe gradient soliton and the result
is well known from [16]. Now, consider $k\geq 2$ and suppose by contradiction
that $f$ is nonconstant. Set $u=e^{-\frac{f}{m}}$. Then

$\nabla u=-\frac{u}{m}\nabla f,\qquad\nabla^{2}u=\frac{u}{m^{2}}df\otimes
df-\frac{u}{m}\nabla^{2}f,$

and (3) can be rewritten as follows:

(6) $\nabla^{2}u=-\frac{u}{m}(\sigma_{k}-\lambda)g.$

Since $f$ is nonconstant, $u$ is also nonconstant, so from Theorem 1.1 of [8],
we obtain that $(M^{n},g)$ is rotationally symmetric and
$M^{n}\setminus\\{N,S\\}$ is locally conformally flat. Here $N,S$ corresponds
to the extremal points of $u$ in $M$. From (6), we know that $\nabla u$ is a
conformal Killing vector field; hence, we can apply Theorem 5.2 of [24] (see
also Theorem 1 of [14]) to deduce

(7) $0=\int_{M^{n}\setminus\\{N,S\\}}g(\nabla\sigma_{k},\nabla
u)dv_{g}=\int_{M^{n}}g(\nabla\sigma_{k},\nabla
u)dv_{g}=\frac{n}{m}\int_{M^{n}}u\sigma_{k}(\sigma_{k}-\lambda)dv_{g},$

where in the last equality we have used the divergence theorem. On the other
hand, again from the divergence theorem, we get

(8) $0=\int_{M^{n}}\Delta
udv_{g}=-\frac{n}{m}\int_{M^{n}}u(\sigma_{k}-\lambda)dv_{g}.$

Combining equations (7) and (8) we arrive at

$\frac{n}{m}\int_{M^{n}}u(\sigma_{k}-\lambda)^{2}dv_{g}=0,$

which implies that $\sigma_{k}=\lambda$ and $u$ is harmonic. Since $M^{n}$ is
compact, $u$ is a constant, which leads to a contradiction. This proves that
$f$ is constant.

$\square$

### Proof of Theorem 1.3 :

From the Hodge-de Rham decomposition $X=\nabla h+Y$, we deduce that

(9)
$\frac{1}{2}\mathcal{L}_{Y}g=\frac{1}{2}\mathcal{L}_{X}g-\frac{1}{2}\mathcal{L}_{\nabla
h}g,$

and

(10) $T_{m}:=\frac{1}{2}\mathcal{L}_{Y}g-\frac{1}{m}X^{\flat}\otimes
Y^{\flat}-\frac{1}{m}Y^{\flat}\otimes
dh=(\sigma_{k}-\lambda)g-\nabla^{2}h+\frac{1}{m}dh\otimes dh.$

Therefore, to prove that $(M^{n},g)$ admits a quasi $k$-Yamabe gradient
soliton structure, it is necessary and sufficient to show that $T_{m}=0$. From
(9) we arrive at

(11)
$\begin{split}\int_{M}|T_{m}|^{2}&=\int_{M}n(\sigma_{k}-\lambda)^{2}-2(\sigma_{k}-\lambda)\Delta
h+2(\sigma_{k}-\lambda)\frac{|\nabla
h|^{2}}{m}+|\nabla^{2}h|^{2}-\frac{2}{m}g(\nabla^{2}h,dh\otimes dh)\\\
&\qquad+\frac{1}{m^{2}}|dh\otimes dh|^{2}\\\
&=\int_{M}|\nabla^{2}h|^{2}-n(\sigma_{k}-\lambda)^{2}+2(\sigma_{k}-\lambda)\frac{|\nabla
h|^{2}-|X|^{2}}{m}-\frac{2}{m}g(\nabla^{2}h,dh\otimes dh)\\\
&\qquad+\frac{1}{m^{2}}|dh\otimes dh|^{2}.\end{split}$

We are going to compute the right-hand side of above equation using the
following integral identity

(12) $\int_{M^{n}}2Ric(\nabla h,Y)dv_{g}=\int_{M^{n}}\left[Ric(X,X)-Ric(\nabla
h,\nabla h)-Ric(Y,Y)\right]dv_{g}.$

Taking the divergence of (9), we get

(13)
$\begin{split}\frac{1}{2}div(\mathcal{L}_{Y}g)(Y)&=\frac{1}{2}div(\mathcal{L}_{X}g)(Y)-\frac{1}{2}div(\mathcal{L}_{\nabla
h}g)(Y)\\\ &=div(\sigma_{k}-\lambda)(Y)+\frac{1}{m}div(X^{\flat}\otimes
X^{\flat})(Y)-\frac{1}{2}div(\mathcal{L}_{\nabla h}g)(Y)\\\
&=g(\nabla\sigma_{k},Y)+\frac{1}{m}div(X^{\flat}\otimes
X^{\flat})(Y)-\frac{1}{2}div(\mathcal{L}_{\nabla h}g)(Y).\end{split}$

From the Bochner formula (see Lemma 2.1 of [21]), we can express (13) as
follows

$\frac{1}{2}\Delta|Y|^{2}-|\nabla
Y|^{2}+Ric(Y,Y)=2g(\nabla\sigma_{k},Y)+\frac{2}{m}div(X^{\flat}\otimes
X^{\flat})(Y)-2Ric(Y,\nabla h)-2g(\nabla\Delta h,Y),$

and using the compactness of $M^{n}$, we arrive at equation

(14) $\int_{M}Ric(Y,Y)=\int_{M}\frac{2}{m}div(X^{\flat}\otimes
X^{\flat})(Y)-2Ric(Y,\nabla h)+|\nabla Y|^{2}.$

On the other hand, the same argument as above shows that

(15) $\frac{1}{2}\Delta|X|^{2}-|\nabla
X|^{2}+Ric(X,X)+(n-2)g(\nabla\sigma_{k},X)+\frac{1}{m}\nabla_{X}|X|^{2}=\frac{2}{m}div(X^{\flat}\otimes
X^{\flat})(X).$

Since

$\begin{split}\int_{M^{n}}|\nabla
X|^{2}dv_{g}&=\int_{M^{n}}\big{[}|\nabla^{2}h|^{2}+|\nabla
Y|^{2}+2g(\nabla\nabla h,\nabla Y)\big{]}dv_{g}\\\
&=\int_{M^{n}}\big{[}|\nabla^{2}h|^{2}+|\nabla Y|^{2}-2g(\nabla\Delta
h+Ric(\nabla h),Y)\big{]}dv_{g}\\\
&=\int_{M^{n}}\big{[}|\nabla^{2}h|^{2}+|\nabla Y|^{2}-2Ric(\nabla
h,Y)\big{]}dv_{g},\\\ \end{split}$

we may integrate (15) over $M^{n}$ to deduce

(16) $\begin{split}\int_{M^{n}}Ric(X,X)&=\int_{M^{n}}|\nabla
X|^{2}+\frac{2}{m}div(X^{\flat}\otimes
X^{\flat})(X)-(n-2)g(\nabla\sigma_{k},X)-\frac{1}{m}\nabla_{X}|X|^{2}\\\
&=\int_{M^{n}}|\nabla^{2}h|^{2}+|\nabla Y|^{2}-2Ric(\nabla
h,Y)+\frac{2}{m}div(X^{\flat}\otimes X^{\flat})(X)\\\
&\qquad+(n-2)\sigma_{k}\left[(\sigma_{k}-\lambda)n+\frac{1}{m}|X|^{2}\right]-\frac{1}{m}\nabla_{X}|X|^{2}\\\
&=\int_{M^{n}}|\nabla^{2}h|^{2}+|\nabla Y|^{2}-2Ric(\nabla
h,Y)+\frac{2}{m}div(X^{\flat}\otimes X^{\flat})(X)\\\
&\qquad+n(n-2)(\sigma_{k}-\lambda)^{2}-(n-2)\lambda\frac{1}{m}|X|^{2}+\frac{(n-2)}{m}\sigma_{k}|X|^{2}-\frac{1}{m}\nabla_{X}|X|^{2}.\\\
\end{split}$

Again, the same argument based on Lemma 2.1 of [21], allow us to deduce that

(17) $\int_{M^{n}}Ric(\nabla h,\nabla
h)dv_{g}=\int_{M^{n}}n^{2}(\sigma_{k}-\lambda)^{2}+\frac{2n}{m}(\sigma_{k}-\lambda)|X|^{2}+\frac{1}{m^{2}}|X|^{4}-|\nabla^{2}h|^{2}.$

Replacing (14), (16) and (17) back into (12), we get

(18) $\begin{split}\int_{M^{n}}2Ric(\nabla
h,Y)dv_{g}&=\int_{M^{n}}2|\nabla^{2}h|^{2}+\frac{2}{m}div(X^{\flat}\otimes
X^{\flat})(X-Y)-2n(\sigma_{k}-\lambda)^{2}\\\
&\qquad-\frac{(n+2)}{m}(\sigma_{k}-\lambda)|X|^{2}-\frac{1}{m}\nabla_{X}|X|^{2}-\frac{1}{m^{2}}|X|^{4}.\end{split}$

Since

$\begin{split}div((X^{\flat}\otimes X^{\flat})(\nabla
h))&=div(X^{\flat}\otimes X^{\flat})(\nabla h)+g(\nabla^{2}h,X^{\flat}\otimes
X^{\flat})\\\ &=divX.g(X,\nabla h)+g(\nabla_{X}X,\nabla
h)+g(\nabla^{2}h,X^{\flat}\otimes X^{\flat}),\end{split}$

we have that (18) can be rewritten as follows

(19) $\begin{split}\int_{M^{n}}2Ric(\nabla
h,Y)dv_{g}&=\int_{M^{n}}2|\nabla^{2}h|^{2}-\frac{2}{m}g(\nabla^{2}h,X^{\flat}\otimes
X^{\flat})-2n(\sigma_{k}-\lambda)^{2}\\\
&\qquad-\frac{(n+2)}{m}(\sigma_{k}-\lambda)|X|^{2}-\frac{1}{m}\nabla_{X}|X|^{2}-\frac{1}{m^{2}}|X|^{4}.\end{split}$

Combining (11) with (19) we produce the desired result.

$\square$

### Proof of Theorem 1.5 :

Since the Hodge-de Rham decomposition is orthogonal on $L^{2}(M)$, we get

$\int_{M^{n}}g(\nabla h,X)dv_{g}=\int_{M^{n}}g(\nabla h,\nabla
h+Y)dv_{g}=\int_{M^{n}}|\nabla h|^{2}dv_{g}.$

Therefore, if

$\int_{M^{n}}g(\nabla h,X)dv_{g}\leq 0,$

we obtain that $\nabla h=0$ and, consequently, $X=Y$. Now, since $Y$ is a
divergence free vector field, we deduce

$0=divY=divX=n(\sigma_{k}-\lambda)+\frac{1}{m}|X|^{2},$

which implies that

$n(\sigma_{k}-\lambda)=-\frac{1}{m}|X|^{2}.$

On the other hand, from the fundamental equation (2), we have

$\begin{split}\langle\nabla_{X}X,X\rangle&=(\sigma_{k}-\lambda)|X|^{2}+\frac{1}{m}|X|^{4}=-\frac{1}{nm}|X|^{4}+\frac{1}{m}|X|^{4}=\frac{n-1}{nm}|X|^{4}.\end{split}$

Note that

$\begin{split}div(|X|^{2}X)&=|X|^{2}divX+\langle\nabla|X|^{2},X\rangle\\\
&=|X|^{2}divX+\nabla_{X}|X|^{2}\\\
&=|X|^{2}divX+2\langle\nabla_{X}X,X\rangle\\\
&=2\langle\nabla_{X}X,X\rangle.\end{split}$

Hence

$0=\int_{M}div(|X|^{2}X)=2\int_{M}\frac{n-1}{nm}|X|^{4}.$

Therefore $X=0$ and $\sigma_{k}=\lambda$.

$\square$

### Proof of Theorem 1.6 :

As we already know, the fundamental equation

$\nabla^{2}f-\frac{1}{m}df\otimes df=(\sigma_{k}-\lambda)g,$

leads to

(20) $L(f):=e^{\frac{f}{m}}\textup{div}\,(e^{-\frac{f}{m}}\nabla f)=\Delta
f-\frac{1}{m}|\nabla f|^{2}=n(\sigma_{k}-\lambda),$

and because we suppose that $L(f)\geqslant 0$ we must then admit that
$\sigma_{k}-\lambda\geqslant 0$. So, if we now take a cut-off function
$\psi:M\rightarrow\mathbb{R}$ satisfying

$0\leqslant\psi\leqslant 1\mbox{ on }M,\quad\psi\equiv 1\hskip
5.69046pt\text{in}\hskip 5.69046ptB(r),\quad\textup{supp}{(\psi)}\subset
B(2r)\quad\text{and}\quad|\nabla\psi|\leqslant\frac{K}{r},$

where $K>0$ is a real constant, we are in place to conclude that

$\begin{split}n\int_{B(r)}\left(\sigma_{k}-\lambda\right)d\mu&=\int_{B(r)}n\psi\left(\sigma_{k}-\lambda\right)d\mu\leqslant\int_{B(2r)}n\psi\left(\sigma_{k}-\lambda\right)d\mu=\int_{B(2r)}\psi
L(f)d\mu=\\\ &=-\int_{B(2r)}g(\nabla\psi,\nabla
f)d\mu\leqslant\int_{B(2r))\setminus B(r)}|\nabla\psi||\nabla
f|d\mu\leqslant\\\ &\leqslant K\int_{B(2r)\setminus B(r)}\frac{|\nabla
f|}{r}d\mu\leqslant 2K\int_{M\setminus B(r)}\dfrac{|\nabla
f|}{d(x,x_{0})}d\mu,\end{split}$

from what it follows that

$\displaystyle 0\leqslant\int_{M}\left(\sigma_{k}-\lambda\right)d\mu$
$\displaystyle=\lim_{r\to\infty}\int_{B(r)}\left(\sigma_{k}-\lambda\right)d\mu\leqslant\frac{2K}{n}\lim_{r\to\infty}\int_{M\setminus
B(r)}\frac{|\nabla f|}{d(x,x_{0})}d\mu=0.$

Henceforth, we have that $L(f)=\sigma_{k}-\lambda=0$ which proves the theorem.

$\square$

##  3\. Invariant by translation examples of quasi $k$-Yamabe solitons

In this section, we provide a method to construct examples of conformally flat
quasi $k$-Yamabe gradient solitons. We focus our attention on solitons whose
solutions are invariant under the action of translation group. More precisely,
we consider the Riemannian metric

$\delta=\sum_{i=1}^{n}dx_{i}\otimes dx_{i},$

in coordinates $x=(x_{1},\dots,x_{n})$ of $\mathbb{R}^{n}$, where $n\geq 3$.
For an arbitrary choice of non zero vector
$\alpha=(\alpha_{1},\dots,\alpha_{n})$ we define the translation function
$\xi:\mathbb{R}^{n}\rightarrow\mathbb{R}$ by

$\xi(x_{1},\dots,x_{n})=\alpha_{1}x_{1}+\dots+\alpha_{n}x_{n}.$

Next, we consider that $\mathbb{R}^{n}$ admits a group of symmetries
consisting of translations [19] and we then look for smooth functions
$\varphi,f:(a,b)\subset\mathbb{R}\rightarrow\mathbb{R}$, with $\varphi>0$,
such that the compositions

$f=f\circ\xi:\xi^{-1}(a,b)\longrightarrow\mathbb{R},\qquad\varphi=\varphi\circ\xi:\xi^{-1}(a,b)\longrightarrow\mathbb{R},$

satisfies

(21) $\nabla^{2}f-\frac{1}{m}df\otimes
df=(\sigma_{k}-\lambda)g\quad\text{with}\quad g=\varphi^{-2}\delta$

What has been said above is summed up in the next result.

######  Theorem 3.1.

Let $(\mathbb{R}^{n},\delta)$ and $f=f\circ\xi$, $\varphi=\varphi\circ\xi$ as
above. Then $\delta_{ij}\varphi^{-2}$ is a quasi $k$-Yamabe gradient soliton
metric with potential function $f=-m\log u$ if, and only if,

(22) $u^{\prime\prime}+2\frac{u^{\prime}\varphi^{\prime}}{\varphi}=0,$ (23)
$b_{n,k}\left[k\varphi\varphi^{\prime\prime}-\frac{n}{2}(\varphi^{\prime})^{2}\right](\varphi^{\prime})^{2(k-1)}||\alpha||^{2k}-m\varphi\varphi^{\prime}\frac{u^{\prime}}{u}||\alpha||^{2}=\lambda,$

where

$b_{n,k}=\frac{(n-1)!}{k!(n-k)!}(-1)^{k-1}\frac{1}{2^{k-1}}.$

### Proof of Theorem 3.1 :

It is well known that for the conformal metric $\bar{g}=\varphi^{-2}\delta$,
the Ricci curvature is given by [3]:

(24)
$Ric_{\bar{g}}=\frac{1}{\varphi^{2}}\Big{\\{}(n-2)\varphi\nabla^{2}_{\delta}\varphi+[\varphi\Delta_{\delta}\varphi-(n-1)|\nabla_{\delta}\varphi|^{2}]\delta\Big{\\}},$

and consequently, the scalar curvature $R_{\bar{g}}$ on conformal metric is
given by

(25)
$R_{\bar{g}}=(n-1)(2\varphi\Delta_{\delta}\varphi-n|\nabla_{\delta}\varphi|^{2}).$

In order to compute the Schouten Tensor on the conformal geometry
$A_{\bar{g}}$ we evoke the expression

$A_{\bar{g}}=\frac{1}{n-2}\left[Ric_{\bar{g}}-\frac{R_{\bar{g}}}{2(n-1)}\bar{g}\right].$

Therefore, from (24) and (25) we deduce that

$A_{\bar{g}}=\frac{\nabla^{2}_{\delta}\varphi}{\varphi}-\frac{|\nabla_{\delta}\varphi|^{2}}{2\varphi^{2}}\delta.$

Denote by $\varphi_{x_{i}}$, $\varphi_{x_{i},x_{j}}$ the derivative of
$\varphi$ with respect the variables $x_{i}$ and $x_{i}x_{j}$, respectively.
That being said, since we are assuming that $\varphi(\xi)$ and $f(\xi)$ are
functions of $\xi=\alpha_{1}x_{1}+\dots+\alpha_{n}x_{n}$, we get

$\varphi_{,x_{i}}=\varphi^{\prime}\alpha_{i},\qquad
f_{,x_{i}}=f^{\prime}\alpha_{i},\qquad\hskip
5.69046pt\varphi_{,x_{i}x_{j}}=\varphi^{\prime\prime}\alpha_{i}\alpha_{j},\qquad
f_{,x_{i}x_{j}}=f^{\prime\prime}\alpha_{i}\alpha_{j}.\hskip 5.69046pt$

Hence

$(\bar{g}^{-1}A_{\bar{g}})_{ij}=\varphi\varphi^{\prime\prime}\alpha_{i}\alpha_{j}-\frac{1}{2}(\varphi^{\prime})^{2}||\alpha||^{2}\delta_{ij}.$

The eigenvalues of $\bar{g}^{-1}A_{\bar{g}}$ are
$\theta=-\frac{1}{2}(\varphi^{\prime})^{2}||\alpha||^{2}$ with multiplicity
$(n-1)$, and
$\mu=(\varphi\varphi^{\prime\prime}-\frac{1}{2}(\varphi^{\prime})^{2})||\alpha||^{2}$
with multiplicity $1$. The formula for $\sigma_{k}$ can be found easily by the
binomial expansion of $(x-\theta)^{n-1}(x-\mu)$

(26) $\displaystyle\sigma_{k}$ $\displaystyle=$
$\displaystyle\frac{(n-1)!}{k!(n-k)!}\left[(n-k)\theta+k\mu\right]\theta^{k-1}$
$\displaystyle=$
$\displaystyle\frac{(n-1)!}{k!(n-k)!}(-1)^{k-1}\frac{1}{2^{k-1}}\left[k\varphi\varphi^{\prime\prime}-\frac{n}{2}(\varphi^{\prime})^{2}\right](\varphi^{\prime})^{2(k-1)}||\alpha||^{2k}.$

Now, in order to compute the Hessian of $u$ relatively to $\bar{g}$ we evoke
the expression

$(\nabla_{\bar{g}}^{2}u)_{ij}=u_{x_{i},x_{j}}-\sum_{k=1}^{n}\Gamma_{ij}^{k}u_{x_{k}},$

where the Christoffel symbol $\Gamma_{ij}^{k}$ for distinct $i,j,k$ are given
by

$\Gamma_{ij}^{k}=0,\ \Gamma_{ij}^{i}=-\frac{\varphi_{x_{j}}}{\varphi},\
\Gamma_{ii}^{k}=\frac{\varphi_{x_{k}}}{\varphi}\;\ \mbox{and}\;\
\Gamma_{ii}^{i}=-\frac{\varphi_{x_{i}}}{\varphi}.$

Therefore,

(27) $\displaystyle(\nabla_{\bar{g}}^{2}u)_{ij}$ $\displaystyle=$
$\displaystyle
u_{x_{i},x_{j}}+\varphi^{-1}(\varphi_{x_{i}}u_{,x_{j}}+\varphi_{x_{j}}u_{x_{i}})-\delta_{ij}\sum_{k}\varphi^{-1}\varphi_{x_{k}}u_{x_{k}}$
$\displaystyle=$
$\displaystyle\alpha_{i}\alpha_{j}u^{\prime\prime}+(2\alpha_{i}\alpha_{j}-\delta_{ij}||\alpha||^{2})\varphi^{-1}\varphi^{\prime}u^{\prime}.$

If we make the change $u=e^{-\frac{f}{m}}$, then the fundamental soliton
equation in (21) can be rewritten as follows:

(28) $\nabla^{2}u=-\frac{u}{m}(\sigma_{k}-\lambda)g.$

Substituting (26) and (27) into (28) and considering $i\neq j$ we obtain

$\alpha_{i}\alpha_{j}\left(u^{\prime\prime}+2\frac{\varphi^{\prime}u^{\prime}}{\varphi}\right)=0.$

If there exist $i,j$, $i\neq j$ such that $\alpha_{i}\alpha_{j}\neq 0$, then
we get

$u^{\prime\prime}+2\frac{u^{\prime}\varphi^{\prime}}{\varphi}=0,$

which provides equation (22). For $i=j$, substituting (26) and (27) into (28)
we obtain (23).

Now, we need to consider the case $\alpha_{k_{0}}=1$, $\alpha_{k}=0$ for
$k\neq k_{0}$. In this case, substituting (27) into (28) we obtain

(29)
$-\frac{u}{m}(\sigma_{k}-\lambda)\frac{1}{\varphi^{2}}=-\frac{\varphi^{\prime}u^{\prime}}{\varphi},$

for $i\neq k_{0}$, that is, $\alpha_{i}=0$ with $i=j$, and

(30)
$-\frac{u}{m}(\sigma_{k}-\lambda)\frac{1}{\varphi^{2}}=u^{\prime\prime}+(2-1)\frac{\varphi^{\prime}u^{\prime}}{\varphi},$

for $i=k_{0}$, that is, $\alpha_{k_{0}}=1$ with $i=j=k_{0}$.

However, (29) and (30) are equivalent to equations (22) and (23). This
completes the demonstration.

$\square$

In what follows, we provide examples illustrating Theorem 3.1. We pointed out
that Example 1.9, Example 1.10 and Example 1.11 are contemplated in the next
examples.

######  Example 3.2.

In Theorem 3.1 consider an arbitrary direction
$\alpha=(\alpha_{1},\alpha_{2},\dots,\alpha_{n})$. Therefore the functions

$f(x_{1},\dots,x_{n})=-m\log\left[k_{1}(\alpha_{1}x_{1}+\dots+\alpha_{n}x_{n})\right],\qquad\varphi(x_{1},\dots,x_{n})=k_{0},\qquad
k_{0},k_{1}\in(0,\infty),$

provide a family of quasi $k$-Yamabe gradient soliton on
$\\{(x_{1},\dots,x_{n})\in\mathbb{R}^{n}\;;\;\alpha_{1}x_{1}+\dots+\alpha_{n}x_{n}>0\\}$
with soliton constant $\lambda=0$.

######  Example 3.3.

In Theorem 3.1 consider an arbitrary direction
$\alpha=(\alpha_{1},\alpha_{2},\dots,\alpha_{n})$. Therefore the functions

$f(x_{1},\dots,x_{n})=-m\log\left(\frac{k_{1}}{\alpha_{1}x_{1}+\dots+\alpha_{n}x_{n}}\right),\quad\varphi(x_{1},\dots,x_{n})=k_{0}(\alpha_{1}x_{1}+\dots+\alpha_{n}x_{n}),$

where $k_{0},k_{1}\in(0,\infty)$ provide a family of quasi $k$-Yamabe gradient
soliton on the Euclidean subset
$\\{(x_{1},\dots,x_{n})\in\mathbb{R}^{n}\;;\;\alpha_{1}x_{1}+\dots+\alpha_{n}x_{n}>0\\}$
with soliton constant

$\lambda=-\frac{n!}{k!(n-k)!}(-1)^{k-1}\left(\frac{k_{0}^{2}||\alpha||^{2}}{2}\right)^{k}+mk_{0}^{2}||\alpha||^{2}.$

######  Example 3.4.

In Theorem 3.1 consider the direction $\alpha=(0,0,\dots,0,1)$. Therefore the
functions

$f(x_{1},\dots,x_{n})=-m\log\left(\frac{k_{1}}{x_{n}}\right),\qquad\varphi(x_{1},\dots,x_{n})=k_{0}x_{n},\quad
k_{0},k_{1}\in(0,\infty),$

provide a family of quasi $k$-Yamabe gradient soliton on
$\\{(x_{1},\dots,x_{n})\in\mathbb{R}^{n}\;;\;x_{n}>0\\}$ with soliton constant

$\lambda=-\frac{n!}{k!(n-k)!}(-1)^{k-1}\left(\frac{k_{0}^{2}}{2}\right)^{k}+mk_{0}^{2}.$

######  Example 3.5.

In Theorem 3.1 consider an arbitrary direction
$\alpha=(\alpha_{1},\alpha_{2},\dots,\alpha_{2k})$. Therefore the functions

$f(x_{1},\dots,x_{2k})=k_{0}\qquad\varphi(x_{1},\dots,x_{2k})=k_{1}e^{\alpha_{1}x_{1}+\cdots+\alpha_{2k}x_{2k}},\quad
k_{0},k_{1}\in(0,\infty),$

provide a family of quasi $k$-Yamabe gradient soliton on $\mathbb{R}^{2k}$
with soliton constant $\lambda=0$.

######  Example 3.6.

In Theorem 3.1 consider an arbitrary direction
$\alpha=(\alpha_{1},\alpha_{2},\dots,\alpha_{n})$. Therefore the functions

$f(x_{1},\dots,x_{n})=k_{0}\qquad\varphi(x_{1},\dots,x_{n})=k_{2}\\{k_{1}k+(2k-n)\left[\alpha_{1}x_{1}+\cdots+\alpha_{n}x_{n}\right]\\}^{\frac{2k}{2k-n}},\quad
k_{0},k_{1},k_{2}\in(0,\infty),$

provide a family of quasi $k$-Yamabe gradient soliton on $\mathbb{R}^{n}$ with
soliton constant $\lambda=0$.

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