Rigid properties of generalized $\tau$-quasi Ricci-harmonic metrics

TL;DR

This paper investigates the properties and rigidity of compact generalized $ au$-quasi Ricci-harmonic metrics, providing conditions for harmonic-Einstein status and establishing gap theorems with specific criteria.

Contribution

It offers new characterization and rigidity results for generalized $ au$-quasi Ricci-harmonic metrics, including conditions for harmonic-Einstein metrics and gap theorems.

Findings

Conditions under which these metrics are harmonic-Einstein

Rigidity results for compact $( au, ho)$-quasi Ricci-harmonic metrics

Necessary and sufficient conditions for metrics to be harmonic-Einstein

Abstract

In this paper, we study compact generalized $\tau$-quasi Ricci-harmonic metrics. In the first part, we explore conditions under which generalized $\tau$-quasi Ricci-harmonic metrics are harmonic-Einstein and give some characterization results for it. In the second part, we obtain some rigidity results for compact $(\tau, \rho)$-quasi Ricci-harmonic metrics which are special case of generalized $\tau$-quasi Ricci-harmonic metrics. In the third part, we shall give two gap theorems for compact $\tau$-quasi Ricci-harmonic metrics by showing some necessary and sufficient conditions for the metrics to be harmonic-Einstein.

Equations

$$\left\{\begin{array}[]{l}Ric_{f,\tau}-\alpha\nabla\phi\otimes\nabla\phi=\lambda g,\\ \tau_{g}\phi=\langle\nabla\phi,\nabla f\rangle,\end{array}\right.$$

$$Ri{c}_{f,\tau }=Ric+{\nabla }^{2}f-\frac{1}{\tau }df\otimes df.$$

$$Ri{c}_{\varphi }:=Ric-\alpha \nabla \varphi \otimes \nabla \varphi$$

$$R_{\varphi }:=trace(Ri{c}_{\varphi })=R-\alpha |\nabla \varphi {|}^2$$

$$\left\{\begin{array}[]{l}Ric_{f,\tau}-\alpha\nabla\phi\otimes\nabla\phi=(\rho R_{\phi}+\mu)g,\\ \tau_{g}\phi=\langle\nabla\phi,\nabla f\rangle.\end{array}\right.$$

$$\left\{\begin{array}[]{l}Ric_{f,\tau}-\alpha\nabla\phi\otimes\nabla\phi=\lambda g,\\ \tau_{g}\phi=\langle\nabla\phi,\nabla f\rangle.\end{array}\right.$$

$$\left\{\begin{array}[]{l}Ric_{f}-\alpha\nabla\phi\otimes\nabla\phi=\lambda g,\\ \tau_{g}\phi=\langle\nabla\phi,\nabla f\rangle,\end{array}\right.$$

$$Ri{c}_f=Ric+{\nabla }^{2}f.$$

$$\left\{\begin{array}[]{l}\frac{\partial g}{\partial t}(t)=-2Ric_{g(t)}+2\alpha(t)\nabla\phi(t)\otimes\nabla\phi(t),~{}~{}g(0)=g_{0},\\ \frac{\partial\phi}{\partial t}(t)=\tau_{g(t)}\phi(t),~{}~{}\phi(0)=\phi_{0},\end{array}\right.$$

$$\left\{\begin{array}[]{l}Ric-\alpha\nabla\phi\otimes\nabla\phi=\lambda g,\\ \tau_{g}\phi=0.\end{array}\right.$$

$$\underset{M^m}{\int }⟨\nabla R_{\varphi },\nabla f⟩e^{-\frac{f}{\tau }}d\upsilon \le 0,$$

$$\mu <0,R_{\varphi }\le \frac{m(m-1)\mu }{\tau +(m-1)(1-m\rho )},$$

$$\mu >0,R_{\varphi }\ge \frac{m(m-1)\mu }{\tau +(m-1)(1-m\rho )},$$

$$(R_{\varphi }{)}_{max}-m\lambda \le (\tau -1)\left(\frac{2}{m(m+\tau -1)}+\frac{1}{\tau }\right)\frac{1}{V_{\tau ,f}}\underset{M^m}{\int }|\nabla f{|}^2{e}^{-\frac{f}{\tau }}d\upsilon ,$$

$$Ri{c}_{\varphi }\ge (1-\delta )\lambda g$$

$$0\le \delta <\min \left\{\frac{1}{6},\frac{\underset{M^m}{\int }|\nabla f{|}^2{e}^{-\frac{f}{\tau }}d\upsilon }{3m\tau \lambda \underset{M^m}{\int }{e}^{-\frac{f}{\tau }}d\upsilon }\right\}.$$

$${\Delta }_{\tau ,f}(u)=$$

$$\underset{M^m}{\int }u{\Delta }_{\tau ,f}(v)e^{-\frac{f}{\tau }}d\upsilon =-\underset{M^m}{\int }⟨\nabla u,\nabla v⟩e^{-\frac{f}{\tau }}d\upsilon =\underset{M^m}{\int }v{\Delta }_{\tau ,f}(u)e^{-\frac{f}{\tau }}d\upsilon .$$

$${\Delta }_{\tau ,f}(f)=\Delta f-\frac{1}{\tau }|\nabla f{|}^2=m\lambda -R_{\varphi },$$

$$\frac{1}{2}\nabla R_{\varphi }=(m-1)\nabla \lambda +\frac{\tau -1}{\tau }Ri{c}_{\varphi }(\nabla f)+\frac{1}{\tau }[R_{\varphi }-(m-1)\lambda ]\nabla f,$$

$$\frac{1}{2}\Delta R_{\varphi }=(m-1)\Delta \lambda -\frac{2(m-1)}{\tau }⟨\nabla f,\nabla \lambda ⟩+\frac{\tau +2}{2\tau }⟨\nabla f,\nabla R_{\varphi }⟩$$

$$-\frac{m+\tau -1}{m\tau }(R_{\varphi }-m\lambda )\left(R_{\varphi }-\frac{m(m-1)}{m+\tau -1}\lambda \right)$$

$$\displaystyle\qquad\qquad-\frac{\tau-1}{\tau}\bigg{|}Ric_{\phi}-\frac{R_{\phi}}{m}g\bigg{|}^{2}-\frac{\tau-1}{\tau}\alpha(\tau_{g}\phi)^{2}.$$

$$\nabla R=2divRic$$

$$\nabla |\nabla \varphi {|}^2=2{\nabla }_{\nabla \varphi }\nabla \varphi$$

$$div(\nabla \varphi \otimes \nabla \varphi )={\tau }_g\varphi \nabla \varphi +{\nabla }_{\nabla \varphi }\nabla \varphi$$

$$\nabla R_{\varphi }=$$

$$=$$

$$div{\nabla }^{2}f=Ric(\nabla f)+\nabla \Delta f$$

$$div(\nabla f\otimes \nabla f)=\Delta f\nabla f+{\nabla }_{\nabla f}\nabla f,$$

$$\nabla R_{\varphi }=$$

$$=$$

Full text

This work was supported by Nanhu Scholars Program for Young Scholars of XYNU

1. Introduction

In this paper, we investigate the rigid properties for generalized $\tau$-quasi Ricci-harmonic metrics and a special class of generalized $\tau$-quasi Ricci-harmonic metrics. To be precise, let us first give our notation. Throughout this paper, let $(M,g)$ and $(N,h)$ be two static complete Riemannian manifolds of dimension $m$ and $n$, respectively. We denote by $Ric$ and $R$ the Ricci tensor (with respect to $g$) and scalar curvature of $M$, respectively. We denote by $\nabla$, $\Delta$ and $\nabla^{2}$ the gradient, the Laplacian and the Hessian on $(M,g)$, respectively. Let $\otimes$, $\langle,\rangle$ and $|\cdot|$ be the tensorial product, the metric $g$ and its associated norm, respectively. Let $\phi:(M,g)\to(N,h)$ be a smooth map between $(M,g)$ and $(N,h)$, $f:M\to\mathbb{R}$ a smooth function on $M$.

First we give the precise definition of a generalized $\tau$-quasi Ricci-harmonic metric.

Definition 1.1**.**

For $\tau>0$, we call a metric $g$ of $M$ generalized $\tau$-quasi Ricci-harmonic (with respect to $h$), if for some map $\phi:(M,g)\to(N,h)$, some potential function $f:M\to\mathbb{R}$ and a soliton function $\lambda:M\to\mathbb{R}$ and some constant $\alpha\geq 0$, $g$ satisfies the following coupled system

[TABLE]

where $\nabla\phi\otimes\nabla\phi:=\phi^{\ast}h$ is the pull-back of the metric $h$ via $\phi$ and $\tau_{g}\phi=trace\nabla d\phi$ denotes the tension field of $\phi$ with respect to $g$ [8], and $Ric_{f,\tau}$ denotes the $\tau$-Bakry-Émery Ricci tensor

[TABLE]

For simplicity, we put

[TABLE]

and call it a generalized Ricci curvature. We denote by

[TABLE]

and call it a generalized scalar curvature.

Note that, if $(N,h)=(\mathbb{R},dr^{2})$ and $\phi:(M,g)\to(\mathbb{R},dr^{2})$ is a constant function in (1.1), then the generalized $\tau$-quasi Ricci-harmonic metric is exactly a generalized $\tau$-quasi Einstein metric, see [3, 6, 7, 12, 13, 17, 25].

In the special case $\lambda=\rho R_{\phi}+\mu$ in (1.1) with $\rho,\mu$ two real constants, $g$ becomes a $(\tau,\rho)$-quasi Ricci-harmonic metric, which is defined as follows.

Definition 1.2**.**

For $\tau>0$, we call a metric $g$ of $M$ $(\tau,\rho)$-quasi Ricci-harmonic (with respect to $h$), if there exists some map $\phi$, some potential function $f$ on $(M,g)$ and three real constants $\alpha\geq 0,\rho,\mu$ such that $g$ satisfies the following coupled system

[TABLE]

Note that, if $(N,h)=(\mathbb{R},dr^{2})$ and $\phi:(M,g)\to(\mathbb{R},dr^{2})$ is a constant function in (1.2), then the $(\tau,\rho)$-quasi Ricci-harmonic metric is exactly a $(\tau,\rho)$-quasi Einstein metric, see [15, 18, 21, 28].

In the special case $\lambda$ is a constant in (1.1), $g$ becomes a $\tau$-quasi Ricci-harmonic metric, which was given in [26, 27].

Definition 1.3**.**

For $\tau>0$, we call a metric $g$ of $M$ $\tau$-quasi Ricci-harmonic (with respect to $h$), if for some map $\phi$, some potential function $f$ and a soliton constant $\lambda$ and some constant $\alpha\geq 0$, $g$ satisfies the following coupled system

[TABLE]

Note that, if $(N,h)=(\mathbb{R},dr^{2})$ and $\phi:(M,g)\to(\mathbb{R},dr^{2})$ is a constant function in (1.3), then the $\tau$-quasi Ricci-harmonic metric is exactly a $\tau$-quasi Einstein metric, see [5, 11, 29]. We know from [2] that $\tau$-quasi Einstein metrics are closely relative to the existence of warped product Einstein manifolds for any positive integer $\tau$.

It is important to point out that if $\tau=\infty$, the equation (1.3) becomes gradient Ricci-harmonic soliton metric, which was introduced by Müller [19].

Definition 1.4**.**

We call a metric $g$ of $M$ gradient Ricci-harmonic soliton metric (with respect to $h$), if for some map $\phi$, some potential function $f$ and a soliton constant $\lambda$ and some constant $\alpha\geq 0$, $g$ satisfies the following coupled system

[TABLE]

where $Ric_{f}$ denotes the $\infty$-Bakry-Émery curvature defined by:

[TABLE]

We call a gradient Ricci-harmonic soliton metric shrinking, steady or expanding, if $\lambda>0$, $\lambda=0$ or $\lambda<0$, respectively.

As pointed out in [19, 20], a gradient Ricci-harmonic soliton arises from the the Ricci-harmonic flow. Let $(M,g(t))$ be a family of complete Riemannian manifolds with Riemannian metrics $g(t)$ evolving by the Ricci-harmonic flow

[TABLE]

where $\alpha(t)\geq 0$ is a non-negative time-dependent coupling constant, $\phi(t):(M,g(t))\to(N,h)$ is a family of smooth maps between $(M,g(t))$ and a fixed complete Riemannian manifold $(N,h)$ and $\nabla\phi(t)\otimes\nabla\phi(t):=\phi(t)^{\ast}h$ is the pull-back of the metric $h$ via $\phi(t)$. And as before, $\tau_{g}\phi=trace\nabla d\phi$ denotes the tension field of $\phi$ with respect to $g(t)$.

Note that, if $(N,h)=(\mathbb{R},dr^{2})$ and $\phi:(M,g)\to(\mathbb{R},dr^{2})$ is a constant function in (1.4), then the gradient Ricci-harmonic soliton metric is exactly a gradient Ricci soliton metric. The gradient Ricci soliton metrics play a very important role in Hamilton’s Ricci flow as they correspond to the self-similar solutions and often arise as singularity models, for a survey in this subject we refer to the work due to Cao in [4].

When $f$ is constant, the $\tau$-quasi Ricci-harmonic metric and gradient Ricci-harmonic soliton metric defined in (1.3) and (1.4) are called harmonic-Einstein, which satisfy the following coupled system [23]:

[TABLE]

We point out that if $\lambda=\lambda(x)$ is a smooth function on $(M,g)$, then gradient Ricci-harmonic soliton and harmonic-Einstein are almost Ricci-harmonic soliton and almost harmonic-Einstein, respectively [1]. Obviously, harmonic-Einstein and almost harmonic-Einstein are natural generalizations of Einstein metrics.

In recent years, gradient Ricci-harmonic soliton metrics and quasi Ricci-harmonic metrics have been extensively studied by many mathematicians. For instance, Yang and Shen [30] gave a volume growth estimate for complete non-compact domain manifolds of shrinking Ricci-harmonic solitons. Tadano [22] gave some gap theorems for Ricci-harmonic solitons with compact domain manifolds by showing some necessary and sufficient conditions for the solitons to be harmonic-Einstein. Zhu [32] prove that when $\alpha>0$ and the sectional curvature of $N$ is bounded from above by $\frac{\alpha}{m}$, any shrinking or steady Ricci-harmonic soliton must be a Ricci soliton. Wang [27] studied $f$-non-parabolic ends and connectivity respectively for quasi Ricci-harmonic metrics, for more details see [10, 14, 24, 31].

In [3, 17], the authors got some rigid properties for generalized $\tau$-quasi-Einstein metrics by establishing some integral formulas. In the first part of this paper, we derive an integral formula with conditions on the curvature to produce sufficient conditions to conclude that the compact generalized $\tau$-quasi Ricci-harmonic metric is harmonic-Einstein.

Theorem 1.1**.**

Let $(M^{m},g)$ be an $m$-dimensional compact manifold with a metric $g$ satisfying generalized $\tau$-quasi Ricci-harmonic metric equation (1.1) with $m\geq 3$. We also assume that

[TABLE]

where $d\upsilon$ is the volume form of $(M^{m},g)$. Then, $(M^{m},g)$ is harmonic-Einstein.

In [15, 28], the authors proved that a given condition forces $(\tau,\rho)$-quasi Einstein metrics to be a standard Einstein metric. In the second part of this paper, we find rigid properties for $(\tau,\rho)$-quasi Ricci-harmonic metrics under various conditions on $R_{\phi}$, $\rho$ and $\tau$.

Theorem 1.2**.**

Let $(M^{m},g)$ be an $m$-dimensional compact manifold with a metric $g$ satisfying $(\tau,\rho)$-quasi Ricci-harmonic metric equation (1.2) with $m\geq 3$. Then, we have

1. (1)

If $\rho\geq\frac{1}{m}$, then $(M^{m},g)$ is harmonic-Einstein; 2. (2)

If $\frac{1}{2(m-1)}\leq\rho<\frac{1}{m}$ and $\tau\geq 1$, then either

[TABLE]

or $(M^{m},g)$ is harmonic-Einstein; 3. (3)

If $\rho\leq\frac{1}{2(m-1)}$ and $\tau\geq 1$, then either

[TABLE]

or $(M^{m},g)$ is harmonic-Einstein; 4. (4)

If $\rho=\frac{1}{2(m-1)}$ and $\tau\geq 1$, then $(M^{m},g)$ is harmonic-Einstein.

In the third part of this paper, we shall extend gap theorems for compact gradient Ricci solitons [9], for compact Ricci-harmonic solitons [22] and for compact $\tau$-quasi-Einstein metrics [29] to the case of compact $\tau$-quasi Ricci-harmonic metrics.

Theorem 1.3**.**

Let $(M^{m},g)$ be an $m$-dimensional compact manifold with a metric $g$ satisfying $\tau$-quasi Ricci-harmonic metric equation (1.3) with $\lambda>0$ and $\tau>1$. Then

[TABLE]

if and only if $(M^{m},g)$ is harmonic-Einstein, where $(R_{\phi})_{max}$ denotes the maximal value of $R_{\phi}$ on $M$ and $V_{\tau,f}=\int\limits_{M^{m}}\,e^{-\frac{f}{\tau}}d\upsilon$.

We point out that if $(M^{m},g)$ be an $m$-dimensional compact manifold with a metric $g$ satisfying generalized $\tau$-quasi Ricci-harmonic metric equation (1.1) with $\tau>1$, $\lambda>0$ and $\Delta\lambda\leq 0(\geq 0)$, then Theorem 1.3 remains valid by Hopf lemma. For the almost Ricci-harmonic soliton case a result corresponding to the referred theorem has been obtained in [1].

Theorem 1.4**.**

Let $(M^{m},g)$ be an $m$-dimensional compact manifold with a metric $g$ satisfying $\tau$-quasi Ricci-harmonic metric equation (1.3) with $\lambda>0$ and $\tau>64m$. If

[TABLE]

holds for any $\delta$ satisfying

[TABLE]

Then, $(M^{m},g)$ is harmonic-Einstein.

This paper is organized as follows. In Section 2, we shall give some lemmas playing important roles in proving Theorems 1.1-1.4. In Section 3, a proof of Theorem 1.1 shall be given. In Section 4, a proof of Theorem 1.2 shall be given. Theorems 1.3-1.4 are proved in Sections 5, respectively.

2. Preliminaries

In this section, we first derive some basic formulas which will be used later. Moreover, $\rho,\mu$ always stand for two real constants in this paper unless a special explanation. Recall that for smooth function $f$, the following operator:

[TABLE]

is self-adjoint with respect to the $L^{2}$ inner product under the measure $e^{-\frac{f}{\tau}}d\upsilon$ (see Lemma 3.1 in [16]), where $d\upsilon$ is the volume form of $(M^{m},g)$. That is, $\forall\ u,v\in C^{\infty}(M^{m})$,

[TABLE]

Lemma 2.1**.**

Let $g$ be a generalized $\tau$-quasi Ricci-harmonic metric defined in Definition 1.1. Then one can get

[TABLE]

Proof.

The equation (2.1) is direct consequence of the first equation in (1.1).

We use the second contracted Bianchi identity

[TABLE]

as well as the fact that

[TABLE]

and

[TABLE]

to deduce

[TABLE]

Using the first equation in (1.1), the Ricci identity

[TABLE]

and

[TABLE]

we have

[TABLE]

here we have used the second equation in (1.1) in the last equality. Using the equation (2.1) yields

[TABLE]

Substituting (2.7) and remembering that $\nabla|\nabla f|^{2}=2\nabla_{\nabla f}\nabla f$ we use (2.6) to write

[TABLE]

We now use (2.8) to write

[TABLE]

From the first equation of (1.1), we have

[TABLE]

Insertting (2.7) and (2.10) into (2.9) leads to (2.2), which finishes the second statement of the lemma.

Initially by using (2.2) to compute the divergence of $\nabla R_{\phi}$ we obtain

[TABLE]

By (2.5) we have

[TABLE]

By (2.2) we have

[TABLE]

Plugging (2.13) into (2.12) leads to

[TABLE]

Insertting (2.14) into (2.11) leads to

[TABLE]

Note that

[TABLE]

Plugging (2.16) into (2.15), we arrive at (2.3). ∎

Lemma 2.2**.**

If $g$ is a generalized $\tau$-quasi Ricci-harmonic metric defined in Definition 1.1 and $\lambda=F(f)$, where $F(t)$ is a smooth function, then there exists a constant $\zeta$, so that

[TABLE]

where

[TABLE]

Proof.

Using (2.2) and the first equation of (1.1) we can write

[TABLE]

By the definition of $G$ and the fact that $\nabla\lambda=F^{{}^{\prime}}(f)\nabla f$, (2.18) can be rewritten as

[TABLE]

Therefore,

[TABLE]

is constant, and (2.17) follows. ∎

Lemma 2.3**.**

Let $g$ be a $(\tau,\rho)$-quasi Ricci-harmonic metric defined in Definition 1.2. Then one can get

[TABLE]

Proof.

Taking $\lambda=\rho R+\mu$ in the equations (2.1) and (2.3), we obtain the formulas (2.19) and (2.20). ∎

Lemma 2.4**.**

([26]) Let $g$ be a $\tau$-quasi Ricci-harmonic metric defined in Definition 1.3. Then one can get

[TABLE]

[TABLE]

and

[TABLE]

Moreover, there exists a constant $\varrho$ such that

[TABLE]

and

[TABLE]

Proof.

Since $\lambda$ is constant, then from Lemmas 2.1 and 2.2 we conclude the proof of the lemma. ∎

3. Proof of Theorem 1.1

In this section, we prove Theorem 1.1. Firstly, we introduce an integral formula for a compact generalized $\tau$-quasi Ricci-harmonic metric.

Lemma 3.1**.**

Let $(M^{m},g)$ be an $m$-dimensional compact manifold with a metric $g$ satisfying generalized $\tau$-quasi Ricci-harmonic metric equation (1.1) with $m\geq 3$. Then, we have

[TABLE]

Remark 3.1*.*

If $\phi:(M,g)\to(\mathbb{R},dr^{2})$ is a constant function in (3.1), we obtain the formula (3.1) in [17].

Proof.

We have from (2.1)

[TABLE]

In particular, (2.3) can be written as

[TABLE]

Integrating the above formula yields

[TABLE]

We complete the proof of Lemma 3.1.∎

Now we prove Theorem 1.1.

Proof of Theorem1.1. Since $m\geq 3$ and the condition

[TABLE]

then from Lemma 3.1 we obtain

[TABLE]

which gives $Ric_{\phi}-\frac{R_{\phi}}{m}g=0$ and $\tau_{g}\phi=0$, completing the proof of the theorem.

4. Proof of Theorem 1.2

In this section, we study rigid properties of a special generalized $\tau$-quasi Ricci-harmonic metric with $\lambda=\rho R_{\phi}+\mu$, where $\rho,\mu$ are two real constants. First, we will use the following important lemma.

Lemma 4.1**.**

Let $(M^{m},g)$ be an $m$-dimensional compact manifold with a metric $g$ satisfying $(\tau,\rho)$-quasi Ricci-harmonic metric equation (1.2). If the generalized scalar curvature $R_{\phi}$ is constant, then $(M^{m},g)$ is harmonic-Einstein.

Proof.

Integrating (2.19) with respect to $e^{-\frac{f}{\tau}}d\upsilon$ gives

[TABLE]

Hence $(m\rho-1)R_{\phi}+m\mu=0$ and then

[TABLE]

Integrating (4.1) with respect to $d\upsilon$ gives

[TABLE]

which gives $f$ is a constant and $(M^{m},g)$ is harmonic-Einstein. ∎

Now we prove Theorem 1.2.

Proof of Theorem1.2. (1) If $\rho=\frac{1}{m}$, then (2.19) shows

[TABLE]

which gives that $\mu=0$ and $f$ is constant and $(M^{m},g)$ is harmonic-Einstein.

Now we consider the case $\rho>\frac{1}{m}$. Integrating (2.19) on $M$ with respect to the measure $e^{-\frac{f}{\tau}}d\upsilon$ leads to

[TABLE]

Integrating by parts and using the first equation in (1.2), we have

[TABLE]

Hence

[TABLE]

Note that

[TABLE]

We then get

[TABLE]

where we have used (4.2). Since $1-m\rho<0$, then (4.3) implies

[TABLE]

which gives

[TABLE]

We use (4.4) and the Cauchy-Schwartz inequality

[TABLE]

to obtain the relation

[TABLE]

We then use (4.2) to obtain

[TABLE]

which gives $R_{\phi}$ is constant and $(M^{m},g)$ is harmonic-Einstein.

(2) Applying (2.20) to a point of maximal of $R_{\phi}$, we have

[TABLE]

If $\mu>0$, (4.5) implies

[TABLE]

If $\mu\leq 0$, (4.5) gives

[TABLE]

Then if $R_{\phi}$ is not a constant, from (4.2) we have

[TABLE]

which shows

[TABLE]

This contradicts with (4.6). Hence, $R_{\phi}$ is a constant and $(M^{m},g)$ is harmonic-Einstein.

(3) Applying (2.20) to a point of minimum of $R_{\phi}$, we have

[TABLE]

If $\mu>0$, (4.7) implies

[TABLE]

If $\mu\leq 0$, (4.7) gives

[TABLE]

Then if $R_{\phi}$ is not a constant, from (4.2) we have

[TABLE]

which shows

[TABLE]

This contradicts with (4.8). Hence, $R_{\phi}$ is a constant and $(M^{m},g)$ is harmonic-Einstein.

(4) This result follows from (2) and (3).

We complete the proof of the theorem.

5. Proof of Theorems 1.3 and 1.4

In this section, using estimates for the generalized Ricci curvature and the generalized scalar curvature, we shall give two gap theorems for compact $\tau$-quasi Ricci-harmonic metrics by showing some necessary and sufficient conditions for the metrics to be harmonic-Einstein. The following lemma plays crucial roles in this section:

Lemma 5.1**.**

Let $(M^{m},g)$ be an $m$-dimensional compact manifold with a metric $g$ satisfying $\tau$-quasi Ricci-harmonic metric equation (1.3) with $\lambda>0$ and $\tau>1$. Then

[TABLE]

[TABLE]

and

[TABLE]

Proof.

First, by (2.21) and (2.22), we have

[TABLE]

which proves (5.1). Secondly, by the first equation of (1.3), we have

[TABLE]

where in the third equality we have used the fact that

[TABLE]

Hence we arrive at (5.2). Finally, in order to prove (5.3), recall from (2.24) that

[TABLE]

for some real constant $\varrho$. By the compactness of manifold $M$, there exists some global maximum point $p\in M$ of the function $\varrho e^{\frac{2}{\tau}f}$. Then, for any point $x\in M$, we have

[TABLE]

which implies that $R_{\phi}$ achieves its maximal value $(R_{\phi})_{max}$ at $p$ and

[TABLE]

holds for all $x\in M$, and we obtain (5.3). ∎

5.1. Proof of Theorems 1.3

It was proved in [26] that when $\tau>1$,

[TABLE]

Therefore, by (5.2) we have

[TABLE]

where in the third inequality we have used (5.3). Let

[TABLE]

Integrating (2.21) on $M$ with respect to the measure $e^{-\frac{f}{\tau}}d\upsilon$ leads to

[TABLE]

Note that $R_{\phi}=m\lambda-\Delta_{\tau,f}(f)$, which gives

[TABLE]

Integrating the above equation on $M$ with respect to the measure $e^{-\frac{f}{\tau}}d\upsilon$ leads to

[TABLE]

By (5.6), (5.7) and (5.8), we have

[TABLE]

which gives

[TABLE]

Hence, by (1.8) in the theorem, the equality in (5.9) must be achieved. This shows that the equality in (5.3) must also attain. Therefore, by (2.24) we have

[TABLE]

Hence $f$ is a constant and $(M^{m},g)$ is harmonic-Einstein.

5.2. Proof of Theorems 1.4

Firstly, we establish the following lemma, which plays crucial roles in the proof of Theorems 1.4.

Lemma 5.2**.**

Let $(M^{m},g)$ be an $m$-dimensional compact manifold with a metric $g$ satisfying $\tau$-quasi Ricci-harmonic metric equation (1.3) with $\lambda>0$ and $\tau>64m$. If

[TABLE]

for some $\delta$ satisfying $0<\delta<\frac{1}{6}$. Then

[TABLE]

and

[TABLE]

Proof.

Since $Ric_{\phi}\geq(1-\delta)\lambda g$ for some $\delta$ satisfying $0<\delta<\frac{1}{6}$, the generalized scalar curvature satisfies

[TABLE]

and it follows from Myers theorem that

[TABLE]

Then, by (2.24) and (5.13), we have

[TABLE]

(5.15) can be rewritten as

[TABLE]

By the compactness of manifold $M$, there exists some global minimum point $q\in M$ of the potential function $f$. Due to (2.25), the maximum principle shows that

[TABLE]

which gives

[TABLE]

Therefore, for any point $x\in M$,

[TABLE]

Then, it follows from the above that

[TABLE]

Since $\tau>64m$ we know that

[TABLE]

and

[TABLE]

We deduce from (5.16) that (see [29])

[TABLE]

By (2.24), we have

[TABLE]

Hence, we arrive at (5.11).

Now, we prove (5.12). For simplicity, we put

[TABLE]

respectively. By (5.13) and (5.11) we know that

[TABLE]

Then, we have

[TABLE]

Due to (2.21) we have

[TABLE]

We then get that

[TABLE]

Due to (5.1) and (5.13) we have

[TABLE]

Therefore, by (5.17) and (5.18) we have

[TABLE]

By (5.10), we get that

[TABLE]

By the fact that $\tau>64m$ and $0<\delta<\frac{1}{6}$, it is easy to verify that

[TABLE]

Hence, we arrive at (5.12). ∎

Now, we may finish the proof of Theorem 1.4. We here assume that $(M^{m},g)$ is not harmonic-Einstein and deduce a contradiction. If $(M^{m},g)$ is not harmonic-Einstein, by Lemma 5.2, we know that

[TABLE]

which contradicts (1.10). This completes the proof of Theorem 1.4.