1. Introduction
An n-dimensional Riemannian manifold 
$(M^n,g)$ is called a gradient Yamabe soliton if there exists a smooth function F on M and a constant 
$\rho\in \mathbb{R}$, such that:
\begin{equation*}\nabla \nabla F=(R-\rho)g,\end{equation*}where R is the scalar curvature on M. If F is constant, M is called trivial.
One of the most interesting problems of the Yamabe soliton is the Yamabe soliton version of Perelman’s conjecture, that is, ‘Any complete (steady) gradient Yamabe soliton with positive scalar curvature under some natural assumption is rotationally symmetric’. The problem was first considered by Daskalopoulos and Sesum [Reference Daskalopoulos and Sesum9]. They showed that any locally conformally flat complete gradient Yamabe soliton with positive sectional curvature is rotationally symmetric. Later, Catino et al. [Reference Catino, Mantegazza and Mazzieri7] and Cao et al. [Reference Cao, Sun and Zhang6] also considered the same problem. In particular, Cao et al. showed that any locally conformally flat complete gradient Yamabe soliton with positive scalar curvature is rotationally symmetric.
To understand the gradient Yamabe soliton, many generalizations of it have been introduced. For example, (1) Almost gradient Yamabe solitons [Reference Barbosa and Ribeiro3], (2) Gradient k-Yamabe solitons [Reference Catino, Mantegazza and Mazzieri7] and (3) h-almost gradient Yamabe solitons [Reference Zeng20].
To consider all these solitons, we consider the conformal gradient soliton defined by Catino et al. [Reference Catino, Mantegazza and Mazzieri7]:
Definition 1 ([Reference Catino, Mantegazza and Mazzieri7])
Let (M, g) be an n-dimensional Riemannian manifold. For smooth functions F and φ on M, 
$(M,g,F,\varphi)$ is called a conformal gradient soliton if it satisfies:
\begin{equation}
\varphi g=\nabla\nabla F.
\end{equation}If F is constant, M is called trivial. The function F is called the potential function.
Remark 1. By the definition, all results in this paper can be applied to gradient Yamabe solitons, gradient k-Yamabe solitons, almost gradient Yamabe solitons and h-almost gradient Yamabe solitons.
We also remark that conformal gradient solitons were studied by Cheeger-Colding ([Reference Cheeger and Colding8], see also [Reference Tashiro18] and [Reference Fujii and Maeta11]).
Complete conformal gradient solitons were classified first by Tashiro [Reference Tashiro18]. In 2012, Catino, Mantegazza and Mazzieri introduced a groundbreaking perspective by focusing on the critical points of the potential function, which led to a classification result for conformal gradient solitons. Notably, their proof has been significantly simplified [Reference Catino, Mantegazza and Mazzieri7]. Recently, the author also provided a classification result of conformal gradient solitons [Reference Maeta15] (see Theorem 3).
Yamabe solitons are self-similar solutions to the Yamabe flow. Therefore, research aimed at determining their structure is of utmost importance. Consequently, numerous studies have been conducted on Yamabe solitons and k-Yamabe solitons. In particular, it has been shown, under certain assumptions, that the scalar curvature and σk-curvature remain constant for Yamabe solitons and k-Yamabe solitons (see, for example [Reference Cunha and Lima1, Reference Bo, Ho and Sheng4, Reference Ma and Miquel14]). Based on their work, this paper investigates these solitons using the most generalized Yamabe solitons, that is, conformal gradient solitons. Moreover, within the framework of natural assumptions that have been explored thus far, the paper fully determines the structure of conformal gradient solitons. All the results in this paper extend to Yamabe solitons and k-Yamabe solitons, allowing for the complete determination of their structures as well.
Theorem 1. Let 
$(M^n,g,F,\varphi)$ be a nontrivial complete conformal gradient soliton. Assume that M satisfies the following (A), (B) or (C).
(A) The function φ satisfies that
$\int_M|\varphi| \lt +\infty,$ $\int_M{\rm Ric} (\nabla F,\nabla F)\leq0$, and $|\nabla F|$ has at most linear growth on M.(B) The function φ is nonnegative, and
$|\nabla F|$ has at most linear growth on M. Let u be a non-constant solution of:
\begin{equation*}
\left\{
\begin{aligned}
\Delta u+h(u)=0,\\
\int_Mh(u)\langle\nabla u,\nabla F\rangle \geq0,
\end{aligned}
\right.
\end{equation*}for
$h\in C^1(\mathbb{R})$, and the function $|\nabla u|$ satisfies:
\begin{equation*}\int_{B(x_0,R)}|\nabla u|^2=o(\log R),~~~\text{as}~~~R\rightarrow +\infty.\end{equation*}(C) The manifold M is parabolic and nontrivial with
${\rm Ric} (\nabla F,\nabla F)\leq0$ and $|\nabla F|\in L^\infty(M)$.Then,
$(M,g,F,\varphi)$ is:
\begin{equation*}(\mathbb{R}\times N^{n-1},ds^2+a^2\bar g,as+b,0),\end{equation*}where
$a,b\in\mathbb{R}$.
Remark 2. Here we remark that the parabolicity of M is as follows: A Riemannian manifold M is parabolic, if every subharmonic function on M which is bounded from above is constant (see [Reference Grigor’yan12]). Theorem 1 (A) is a generalization of Theorem 1 in [Reference Cunha and Lima1] and Theorem 3 in [Reference Ma and Miquel14]. Theorem 1 (B) is a generalization of Theorem 7 in [Reference Cunha and Lima1]. Theorem 1 (C) is a generalization of Theorem 6 in [Reference Cunha and Lima1].
We also give triviality results of conformal gradient solitons under some assumption considered in [Reference Cunha and Lima1, Reference Bo, Ho and Sheng4, Reference Ma and Miquel14].
Theorem 2. Let 
$(M^n,g,F,\varphi)$ be a complete conformal gradient soliton. If any of the following conditions is satisfied, then it is trivial.
(A) The potential function satisfies that
$F\geq K \gt 0$ for some $K\in \mathbb{R}$, $\varphi\leq0$, and one of the following conditions is satisfied: (i) M is parabolic, (ii) $|\nabla F|\in L^1(M)$, (iii) $F^{-1}\in L^p(M)$ for some p > 1, or (iv) Mn has linear volume growth.(B) The Ricci curvature satisfies that
${\rm Ric} (\nabla F,\nabla F)\leq0$, and $|\nabla F|\in L^{p}(M)$ for some p > 1.(C) The potential function F is nonnegative,
$\varphi\geq0$ and $\int_MF^p \lt +\infty$ for some $p\geq0$.(D) The Ricci curvature is nonnegative,
$\varphi\geq0$ and
\begin{equation*}\int_{M\setminus B(x_0,R_0)}\frac{\exp(F)}{d(x_0,x)^2} \lt +\infty,\end{equation*}for some
$x_0\in M$ and $R_0 \gt 0$.
Remark 3. Theorem 2 (A) is a generalization of Theorem 2 in [Reference Cunha and Lima1]. Theorem 2 (B) is a generalization of Theorem 6 in [Reference Cunha and Lima1]. Theorems 2 (C) and (D) are similar to Theorems 3 and 4 in [Reference Cunha and Lima1], Theorems 4 and 5 in [Reference Ma and Miquel14], and Theorem 1.7 in [Reference Bo, Ho and Sheng4].
As is well known, Yau proved the valuable maximum principle [Reference Yau19]: ‘On a complete Riemannian manifold M, if a nonnegative subharmonic function F satisfies 
$F\in L^p(M)$ for some 
$1 \lt p \lt +\infty$, then F is constant.’ An interesting aspect of Theorem 2 (C) is its resemblance to Yau’s maximum principle.
As a corollary, we have the following:
Corollary 1. Let 
$(M,g,F)$ be a complete steady gradient Yamabe solitons with nonnegative scalar curvature. If a nonnegative potential function F satisfies that 
$\int_MF^p \lt +\infty$ for some 
$p\geq0$, then it is trivial.
To show these theorems, we use the following result shown by the author [Reference Maeta15].
Theorem 3 ([Reference Maeta15])
A nontrivial complete conformal gradient soliton 
$(M^n,g,F,\varphi)$ is either:
(1) compact and rotationally symmetric, or
(2) the warped product:
\begin{equation*}(\mathbb{R},ds^2)\times_{|\nabla F|} \left(N^{n-1},\bar g\right),\end{equation*}where the scalar curvature
$\bar R$ of N satisfies:
\begin{equation*}|\nabla F|^2R=\bar R-(n-1)(n-2)\varphi^2-2(n-1)g(\nabla F,\nabla\varphi),\end{equation*}or
(3) rotationally symmetric and equal to the warped product:
\begin{equation*}([0,+\infty),ds^2)\times_{|\nabla F|}(\mathbb{S}^{n-1},{\bar g}_{S}),\end{equation*}where
$\bar g_{S}$ is the round metric on $\mathbb{S}^{n-1}.$
Furthermore, the potential function F depends only on s.
Therefore, to consider the Yamabe soliton version of Perelman’s conjecture, we only have to consider (2) of Theorem 3.
2. Proof of Theorem 1
In this section, we prove Theorem 1, which includes significant advancements over the results presented in [Reference Cunha and Lima1, Reference Ma and Miquel14]. In particular, we completely elucidate the structure of gradient k-Yamabe solitons and gradient Yamabe solitons under the assumption as in [Reference Cunha and Lima1, Reference Ma and Miquel14].
We first show some formulas which will be used later. For any conformal gradient soliton, we have:
\begin{equation*}
\Delta {\nabla}_iF={\nabla}_i\Delta F+R_{ij}{\nabla}_jF,
\end{equation*}
\begin{equation*}
\Delta {\nabla}_iF={\nabla}_k{\nabla}_k{\nabla}_iF={\nabla}_k(\varphi g_{ki})={\nabla}_i\varphi,
\end{equation*}and
\begin{equation*}
{\nabla}_i\Delta F={\nabla}_i(n\varphi)=n{\nabla}_i\varphi.
\end{equation*}Hence, we have,
\begin{equation}
(n-1){\nabla}_i\varphi+R_{ij}{\nabla}_jF=0,
\end{equation}where Rij is the Ricci tensor of M. Therefore, one has:
\begin{equation}
\langle \nabla \varphi,\nabla F\rangle=-\frac{1}{n-1}{\rm Ric} (\nabla F,\nabla F).
\end{equation} By applying 
${\nabla}_l$ to the both sides of (2.1), we obtain:
\begin{equation}
(n-1){\nabla}_l{\nabla}_i\varphi+{\nabla}_lR_{ij} \cdot {\nabla}_jF+R_{ij}{\nabla}_l{\nabla}_jF=0.
\end{equation}Taking the trace, we obtain:
\begin{equation}
(n-1)\Delta{\varphi}+\frac{1}{2}~g(\nabla R,\nabla F)+R \varphi =0.
\end{equation}We observe the following proposition.
Proposition 1. Any compact conformal gradient soliton with:
\begin{equation*}\int_M{\rm Ric} (\nabla F,\nabla F)\leq 0\end{equation*}is trivial.
Proof. By 
$\Delta F=n\varphi$ and (2.2), we have:
\begin{align*}
\int_{M}\varphi^2
=&~\frac{1}{n}\int_{M}\varphi\Delta F\\
=&~-\frac{1}{n}\int_{M}\langle \nabla \varphi, \nabla F\rangle\\
=&\frac{1}{n(n-1)}\int_M{\rm Ric}(\nabla F,\nabla F)\leq0.
\end{align*} Thus, one has φ = 0 and 
$\nabla\nabla F=0$. Hence, we have 
$\Delta F=0$. By the standard maximum principle, we have that M is trivial.
By using the above arguments, we show Theorem 1.
Proof of Theorem 1
(A) If M is compact, by Proposition 1, M is trivial.
Therefore, we assume that M is noncompact. By 
$\Delta F=n\varphi$ and (2.2), we have:
\begin{align*}
\int_{B(x_0,R)}\varphi^2
=&~\frac{1}{n}\int_{B(x_0,R)}\varphi\Delta F\\
=&~-\frac{1}{n}\left\{\int_{B(x_0,R)}\langle \nabla \varphi, \nabla F\rangle +\int_{\partial B(x_0,R)}\varphi\langle \nu,\nabla F\rangle \right\}\\
=&~-\frac{1}{n}\left\{\int_{B(x_0,R)}-\frac{1}{n-1}{\rm Ric}(\nabla F,\nabla F) +\int_{\partial B(x_0,R)}\varphi\langle \nu,\nabla F\rangle \right\}\\
\leq&~\frac{1}{n(n-1)}\int_{B(x_0,R)}{\rm Ric}(\nabla F,\nabla F)
+CR\int_{\partial B(x_0,R)}|\varphi|,
\end{align*} where ν is the outward unit normal to the boundary 
$\partial B(x_0,R)$. Since 
$\int_M|\varphi| \lt +\infty$, by taking 
$R\nearrow +\infty,$ one has:
\begin{equation*}CR\int_{\partial B(x_0,R)}|\varphi|\rightarrow 0.\end{equation*}Therefore, by the assumption, we have:
\begin{equation*}\int_M\varphi^2=0,\end{equation*}hence, φ = 0.
By Theorem 3, we have three types of conformal gradient solitons.
Case 1. M is compact. This case cannot happen.
Case 2. M is the warped product:
\begin{equation*}(\mathbb{R},ds^2)\times_{|\nabla F|} \left(N^{n-1},\bar g\right).\end{equation*} Since 
$\nabla \nabla F=0,$ we have:
\begin{equation*}\nabla |\nabla F|^2=2\nabla_j\nabla_iF\nabla_iF=0.\end{equation*} Hence, 
$\nabla F$ is a constant vector field. Therefore, we have 
$F(s)=as+b$.
Case 3. M is rotationally symmetric and equal to the warped product:
\begin{equation*}([0,+\infty),ds^2)\times_{|\nabla F|}(\mathbb{S}^{n-1},{\bar g}_{S}).\end{equation*} By the same argument as in Case 2, we have that 
$|\nabla F|$ is constant. Since 
$F'(0)=0$ (see the Proof of Theorem 1.1 in [Reference Maeta15]), F is constant.
(B) We will use the logarithmic cutoff argument developed by Farina et al. [Reference Farina, Mari and Valdinoci10].
\begin{equation}
\eta(r)=
\left\{
\begin{aligned}
&1 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ r\leq R,\\
&2-\frac{\log r}{\log R}\ \ \ \ r\in [R,R^2],\\
&0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ r\geq R^2.
\end{aligned}
\right.
\end{equation}By the soliton equation, we have:
\begin{align}
\int_M\varphi|\nabla u|^2\eta^2
=&\int_{B(x_0,R^2)} \nabla_i\nabla_jF\nabla_iu\nabla_ju\eta^2\\
=&-\int_{B(x_0,R^2)} \nabla_iF\nabla_j\nabla_iu\nabla_ju\eta^2
+\nabla_iF\Delta u\nabla_ju\eta^2\notag\\
&-\int_{B(x_0,R^2)}\nabla_iF\nabla_iu\nabla_ju\nabla_j\eta^2.\notag
\end{align}Substituting
\begin{equation*}-\int_{B(x_0,R^2)} \nabla_iF\nabla_j\nabla_iu\nabla_ju\eta^2
=\frac{1}{2}\int_{B(x_0,R^2)} n\varphi|\nabla u|^2\eta^2+|\nabla u|^2\langle \nabla F,\nabla \eta^2\rangle,\end{equation*}into (2.6), we have:
\begin{align*}
\int_M\varphi|\nabla u|^2\eta^2
=&\frac{1}{2}\int_{B(x_0,R^2)} n\varphi|\nabla u|^2\eta^2
+2\eta|\nabla u|^2\langle \nabla F,\nabla \eta\rangle\\
&+\int_{B(x_0,R^2)} \langle \nabla F,\nabla u\rangle h\eta^2
-2\eta\langle \nabla F,\nabla u\rangle \langle \nabla u,\nabla \eta\rangle.
\end{align*}Therefore, we have:
\begin{align*}
\frac{n-2}{2}\int_M\varphi|\nabla u|^2\eta^2
=&-\int_{B(x_0,R^2)} \eta|\nabla u|^2\langle \nabla F,\nabla \eta\rangle\\
&-\int_{B(x_0,R^2)} \langle \nabla F,\nabla u\rangle h\eta^2
+2\int_{B(x_0,R^2)} \eta\langle \nabla F,\nabla u\rangle \langle \nabla u,\nabla \eta\rangle\\
\leq&~3\int_{B(x_0,R^2)} \eta|\nabla u|^2 |\nabla F| |\nabla \eta|
-\int_{B(x_0,R^2)} \langle \nabla F,\nabla u\rangle h\eta^2\\
\leq&~\frac{C}{\log R}\int_{B(x_0,R^2)\setminus B(x_0,R)} |\nabla u|^2
-\int_{B(x_0,R^2)} \langle \nabla F,\nabla u\rangle h\eta^2,
\end{align*} where the last inequality follows from 
$|\nabla F|\leq Cr$ near infinity and the definition of the cut off function η. Take 
$R\nearrow +\infty.$ From this and the assumption, we have:
\begin{align*}
\frac{n-2}{2}\int_M\varphi|\nabla u|^2\eta^2
\leq -\int_M \langle \nabla F,\nabla u\rangle h\eta^2\leq0.
\end{align*} Since u is a non-constant solution, we have φ = 0. Therefore, we have 
$\nabla\nabla F=0$.
By Theorem 3, we have 3 types of conformal gradient solitons.
Case 1. M is compact and rotationally symmetric.
Since M is compact, by the standard Maximum principle, we have that F is constant. Therefore, M is trivial.
Cases 2 and 3 are considered by the same argument as in (A).
(C) Since (2.2) and
$\Delta F=n\varphi$, by a direct computation,
\begin{align*}
\frac{1}{2}\Delta |\nabla F|^2
=&~|\nabla\nabla F|^2+{\rm Ric} (\nabla F,\nabla F)+\langle \nabla F,\nabla \Delta F\rangle\\
=&~|\nabla\nabla F|^2-\frac{1}{n-1}{\rm Ric} (\nabla F,\nabla F)\\
\geq&~0,
\end{align*}where, the last inequality follows from the assumption. Since M is parabolic and
${\rm sup}|\nabla F| \lt +\infty$, we have that $|\nabla F|$ is constant. Therefore, we have $\nabla \nabla F=0$. By the same argument as in (B), we complete the proof.
3. Proof of Theorem 2
To show Theorem 2, we show the following lemma.
Lemma 1. Let 
$(M,g,F,\varphi)$ be a complete conformal gradient soliton. If the nonnegative potential function F satisfies that 
$\nabla F$ is a constant vector field, then it is trivial.
Proof. By Theorem 3, we have three cases.
Case 1. M is compact and rotationally symmetric.
Since M is compact, by the standard Maximum principle, we have that F is constant.
Case 2. M is the warped product:
\begin{equation*}(\mathbb{R},ds^2)\times_{|\nabla F|} \left(N^{n-1},\bar g\right).\end{equation*} Since F depends only on 
$s\in \mathbb{R}$, one can get that:
\begin{equation*}\nabla F=F'(s)\partial _s,\end{equation*} where 
$F'(s)$ is a constant, say a. If 
$a\not=0$, one has 
$F(s)=as+b\geq0 $ on 
$\mathbb{R}$, which cannot happen. Hence, F is constant.
Case 3. M is rotationally symmetric and equal to the warped product:
\begin{equation*}([0,+\infty),ds^2)\times_{|\nabla F|}(\mathbb{S}^{n-1},{\bar g}_{S}).\end{equation*} The potential function satisfies that 
$F'(0)=0$. Combining this with the assumption, F is constant.
Proof of Theorem 2
(A) If M is compact, by
$\Delta (-F)=-n\varphi \geq0$, the standard Maximum principle shows that F is constant. Therefore, we assume that M is noncompact.
(i) Since
$-F\leq-K$, and $\varphi\leq0$, we have:
\begin{equation*}\Delta(-F)=-n\varphi\geq0.\end{equation*}Since M is parabolic, −F is constant. Therefore, M is trivial.
(ii) By a direct computation,
(3.1)\begin{equation}
{\rm div}\nabla (-F)=\Delta (-F)=-n\varphi\geq0.
\end{equation}By Theorem 3, we have three types of conformal gradient solitons.
Case 1. M is compact and rotationally symmetric.
This case cannot happen.
To consider Cases 2 and 3, let us recall the following:
Lemma 2. ([Reference Caminha, Sousa and Camargo5])
Let X be a smooth vector field on a complete noncompact Riemannian manifold, such that, 
${\rm div} X$ does not change the sign on M. If 
$|X|\in L^1({M}),$ then 
${\rm div}_MX=0$.
By Lemma 2, we have 
$\Delta F=0.$ Hence, we have φ = 0, and 
$\nabla\nabla F=0$.
Case 2. M is the warped product:
\begin{equation*}(\mathbb{R},ds^2)\times_{|\nabla F|} \left(N^{n-1},\bar g\right).\end{equation*} Since 
$\nabla\nabla F=0$, we have:
\begin{equation*}\nabla |\nabla F|^2=2\nabla_j\nabla_iF\nabla_iF=0.\end{equation*} Hence, 
$\nabla F$ is a constant vector field. Set 
$|\nabla F|=a$. If 
$a\not=0$,
\begin{equation*}a{\rm Vol}\,(\mathbb{R}\times N^{n-1})=+\infty.\end{equation*}From this and the assumption, we have a = 0. Therefore, F is constant, and M is trivial.
Case 3. M is rotationally symmetric and equal to the warped product:
\begin{equation*}([0,+\infty),ds^2)\times_{|\nabla F|}(\mathbb{S}^{n-1},{\bar g}_{S}).\end{equation*}By the same argument as in Case 2, we have that F is constant.
(iii) Since
$\Delta (-F)\geq0,$ one has:
\begin{equation*}\Delta F^{-1}=2F^{-3}|\nabla F|^2-\Delta F F^{-2}\geq0.\end{equation*}By the Yau’s Maximum principle, one has that F −1 is constant.
(iv) By Lemma 6.3 in [Reference Schoen and Yau17] and the assumption, we have:
(3.2)\begin{align}
\int_{B(x_0,R)}|\nabla F^{-1}|^2
\leq&\frac{C}{R^2}\int_{B(x_0,2R)}F^{-2}\\
\leq&\frac{C}{R^2K^2}{\rm Vol}(B(x_0,2R))\notag\\
\leq&\frac{\bar C}{RK^2}.\notag
\end{align}Take
$R\nearrow +\infty$. The right-hand side of (3.2) goes to 0. Therefore, we have that F is constant.
(B) By
$\Delta F=n\varphi$ and (2.2), we have:
(3.3)\begin{align}
\frac{1}{2}\Delta |\nabla F|^2
=&|\nabla\nabla F|^2+{\rm Ric}(\nabla F,\nabla F)+\langle \nabla F,\nabla \Delta F\rangle\\
=&|\nabla\nabla F|^2-\frac{1}{n-1}{\rm Ric}(\nabla F,\nabla F).\notag
\end{align}
From this and 
${\rm Ric}(\nabla F,\nabla F)\leq 0$, we have:
\begin{equation*}\Delta |\nabla F|^2\geq0.\end{equation*} If M is compact, by the standard Maximum principle, we have that 
$|\nabla F|$ is constant.
Assume that M is noncompact. By the Yau’s Maximum principle, 
$|\nabla F|$ is constant.
By (3.3),
\begin{equation*}|\nabla\nabla F|^2-\frac{1}{n-1}{\rm Ric}(\nabla F,\nabla F)=0.\end{equation*} Therefore, we have 
$\nabla\nabla F=0$.
By Theorem 3, we have three types of conformal gradient solitons.
Case 1. M is compact and rotationally symmetric.
Since 
$\Delta F=0$ and M is compact, by the standard Maximum principle, we have that F is constant.
Cases 2 and 3 are considered by the same argument as in (A)-(ii).
(C) Case 1. M is compact and rotationally symmetric.
Since 
$\Delta F=n\varphi\geq0$ and M is compact, by the standard Maximum principle, we have that F is constant.
Case 2. M is the warped product:
\begin{equation*}(\mathbb{R},ds^2)\times_{|\nabla F|} \left(N^{n-1},\bar g\right).\end{equation*} Since F depends only on 
$s\in \mathbb{R}$, one can get that:
\begin{equation*}\nabla F=F'(s)\partial _s.\end{equation*} Since the potential function F is nonnegative and 
$F' \gt 0$, we have 
$F(s) \gt a \gt 0$ on some interval 
$(s_0,+\infty)$. If 
$F''=0$ on 
$\mathbb{R}$, then Fʹ is constant. However it cannot happen because of Lemma 1. Hence 
$F'' \gt 0$ at some point and 
$F' \gt b \gt 0$ on some interval 
$(s_1,+\infty)$. The volume form of the metric 
$ds^2+|F'(s)|^2\bar g$ is given by 
$|F'(s)|^{n-1}ds\wedge d\mu_N$, where 
$d\mu_N$ is the volume form of N (see for example Page 33 in [Reference Petersen16]). Therefore, one has:
\begin{align*}
\int_MF^p
& \gt \int_{s_0}^{+\infty}\int_N {a}^p (F')^{n-1}ds\wedge d\mu_N\\
& \gt {a}^p b^{n-1}\int_{max\{s_0,s_1\}}^{+\infty}\int_N ds\wedge d\mu_N
=+\infty,
\end{align*}which is a contradiction.
Case 3. M is rotationally symmetric and equal to the warped product:
\begin{equation*}([0,+\infty),ds^2)\times_{|\nabla F|}(\mathbb{S}^{n-1},{\bar g}_{S}).\end{equation*}By the similar argument as in Case 2, we have a contradiction.
(D) If M is compact, by the standard maximum principle, F is constant.
Assume that M is a noncompact complete manifold. Since the Ricci curvature is nonnegative, we can take a cut off function η on M satisfying that:
\begin{equation}
\left\{
\begin{aligned}
&0\leq\eta(x)\leq1\ \ \ (x\in M),\\
&\eta(x)=1\ \ \ \ \ \ \ \ \ (x\in B(x_0,R)),\\
&\eta(x)=0\ \ \ \ \ \ \ \ \ (x\not\in B(x_0,2R)),\\
&|\nabla\eta|\leq\frac{C}{R}\ \ \ \ \ \ \ (x\in M),\ \ \ \textrm{for some constant}\ C\ \textrm{independent of}\ R,\\
&\Delta\eta \leq\frac{C}{R^2}\ \ \ \ \ \ \ (x\in M),\ \ \ \textrm{for some constant}\ C\ \textrm{independent of}\ R,
\end{aligned}
\right.
\end{equation} where 
$B(x_0,R)$ and 
$B(x_0,2R)$ are the balls centred at a fixed point 
$x_0\in M$ with radius R and 2R, respectively (cf. [Reference Bianchi and Setti2, Reference Ma and Miquel14]). By 
$\Delta F=n\varphi$, one has:
\begin{align}
0\leq
&\int_{B(x_0,2R)}\eta(|\nabla F|^2+n\varphi)e^F\\
=&\int_{B(x_0,2R)}\eta\Delta e^F\notag\\
=&\frac{1}{n}\int_{B(x_0,2R)\setminus B(x_0,R)}\Delta\eta e^F\notag\\
\leq&\frac{C}{nR^2}\int_{B(x_0,2R)\setminus B(x_0,R)} e^F.\notag
\end{align} By the assumption, 
$|\nabla F|^2+n\varphi=0.$ Thus, F is constant.□
Acknowledgements
The author would like to express his deep gratitude to the referee for his/her valuable comments and suggestions. Theorems 1 and 2 were improved by his/her comments and suggestions. The author is partially supported by the Grant-in-Aid for Young Scientists, No.19K14534, Japan Society for the Promotion of Science, and Grant-in-Aid for Scientific Research (C), No.23K03107, Japan Society for the Promotion of Science.
Competing interests
There is no conflict of interest in the manuscript.
