Evolution of a geometric constant along the Ricci flow

Abstract

In this paper, we establish the first variation formula of the lowest constant $\lambda_{a}^{b}(g)$ along the Ricci flow and the normalized Ricci flow, such that to the following nonlinear equation there exist positive solutions:

$$-\Delta u+au\log u+bRu=\lambda_{a}^{b} u $$

with $\int_{M}u^{2}\,dv=1$, where a is a real constant. In particular, the results proved in this paper generalize partial results in Cao (Proc. Am. Math. Soc. 136:4075-4078, 2008) and Li (Math. Ann. 338:927-946, 2007).

Introduction

Let $(M, g)$ be an n-dimensional compact Riemannian manifold. In [3], Perelman introduced the functional

$$ \mathcal{F}(g,f)= \int _{M}\bigl(|\nabla f|^{2}+R\bigr)e^{-f} \,dv $$

(1.1)

and proved that the $\mathcal{F}$-functional is nondecreasing under the Ricci flow coupled to a backward heat-type equation

$$ \left \{ \textstyle\begin{array}{@{}l} \frac{\partial}{\partial t}g_{ij}=-2R_{ij},\\ f_{t}=-\Delta f+|\nabla f|^{2}-R, \end{array}\displaystyle \right . $$

(1.2)

where R is the scalar curvature depending on the metric g. More precisely, they proved that under the system (1.2),

$$ \frac{d}{d t}\mathcal{F}=2 \int _{M}|R_{ij}+f_{ij}|^{2}e^{-f} \,dv\geq0. $$

(1.3)

If we define

$$ \lambda(g)=\inf_{f}\mathcal{F}(g,f), $$

(1.4)

where the infimum is taken over all smooth functions f which satisfy

$$ \int _{M}e^{-f}\,dv=1, $$

(1.5)

then the nondecreasing of the $\mathcal{F}$-functional implies the nondecreasing of $\lambda(g)$. In particular, $\lambda(g)$ defined in (1.4) is the lowest eigenvalue of the operator

$$ -4\Delta+R. $$

(1.6)

In [4], Cao considered the eigenvalues of the operator $-\Delta+\frac{R}{2}$ on manifolds with nonnegative curvature operator and showed that the eigenvalues are nondecreasing along the Ricci flow. Using the same technique, Li [2] also obtained the same monotonicity of the first eigenvalue of the operator $-\Delta+\frac{R}{2}$ by removing the assumption on a nonnegative curvature operator.

Later, Cao [1] proved the first eigenvalues of the operator $-\Delta+bR$ with the constant $b\geq1/4$ are nondecreasing along the Ricci flow. That is, they assume $u=u(x, t)$ is the corresponding positive eigenfunction of $\lambda(t)$:

$$ (-\Delta+bR)u=\lambda^{b} u $$

(1.7)

with $\int_{M}u^{2}\,dv=1$, then

$$ \frac{d}{d t}\lambda^{b}=\frac{1}{2} \int _{M} |R_{ij}+f_{ij}|^{2}e^{-f} \,dv+ \biggl(2b-\frac{1}{2} \biggr) \int _{M} |R_{ij}|^{2}e^{-f}\,dv \geq0 $$

(1.8)

by letting $f=-2\log u$. Multiplying both sides of (1.7) with u and integrating on M, we see that the first eigenvalue given in (1.7) satisfies

$$ \lambda(t)=\inf\tilde{\mathcal{F}}^{b}(g,u), $$

(1.9)

where

$$ \tilde{\mathcal{F}}^{b}(g,u)= \int _{M}\bigl(|\nabla u|^{2}+bRu^{2} \bigr)\,dv. $$

(1.10)

In particular,

$$ \tilde{\mathcal{F}}^{b}(g,u)=\frac{1}{4} \mathcal{F}^{4b}(g,f), $$

(1.11)

where

$$\mathcal{F}^{c}(g,f)= \int _{M}\bigl(|\nabla f|^{2}+cR \bigr)e^{-f}\,dv $$

if we let $f=-2\log u$. It is easy to see from (1.11) that the nondecreasing of the $\tilde{\mathcal{F}}^{b}$-functional is equivalent to the nondecreasing of $\lambda(t)$.

In this paper, we consider the monotonicity along the Ricci flow of lowest constant $\lambda_{a}^{b}(g)$ such that to the following nonlinear equation there exist positive solutions:

$$ -\Delta u+au\log u+bRu=\lambda_{a}^{b} u $$

(1.12)

with

$$ \int _{M}u^{2}\,dv=1, $$

(1.13)

where a is a real constant. In particular, (1.7) can be seen a special case of (1.12) when $a=0$. For the lowest constant $\lambda_{a}^{b}(g)$ such that to the nonlinear equation (1.12) there exist positive solutions, we prove the following.

Theorem 1.1

Let $g(t)$, $t\in[0,T)$ be a solution to the Ricci flow

$$ \frac{\partial}{\partial t}g_{ij}=-2R_{ij} $$

(1.14)

on a compact Riemannian manifold M. Then for $b\geq\frac{1}{4}$, the lowest constant $\lambda_{a}^{b}(g)$ such that to the nonlinear equation (1.12) with (1.13) there exist positive solutions satisfies

$$\begin{aligned} \frac{d}{d t} \biggl(\lambda_{a}^{b}(t)+ \frac{na^{2}}{8}t \biggr)=&\frac{1}{2} \int _{M} \biggl|R_{ij}+f_{ij}+ \frac{a}{2}g_{ij}\biggr|^{2}e^{-f}\,dv+ \biggl(2b- \frac{1}{2} \biggr) \int _{M} |R_{ij}|^{2}e^{-f}\,dv \\ \geq&0, \end{aligned}$$

(1.15)

where $f=-2\log u$.

For the normalized Ricci flow, we can obtain the following.

Theorem 1.2

Let $g(t)$, $t\in[0,T)$ be a solution to the normalized Ricci flow

$$ \frac{\partial}{\partial t}g_{ij}=-2 \biggl(R_{ij}- \frac{r}{n}g_{ij} \biggr) $$

(1.16)

on a compact Riemannian manifold M, where $r=(\int_{M}R \,dv)/(\int_{M} \,dv)$ is the average scalar curvature. Then the lowest constant $\lambda_{a}^{b}(g)$ such that to the nonlinear equation (1.12) with (1.13) there exist positive solutions satisfies

$$\begin{aligned} \frac{d}{d t} \biggl(\lambda_{a}^{b}+ \frac{na^{2}}{8}t \biggr)+\frac{2r}{n}\lambda^{b} ={}& \frac{1}{2} \int _{M} \biggl|R_{ij}+f_{ij}+ \frac{a}{2}g_{ij}\biggr|^{2}e^{-f}\,dv \\ &{}+ \biggl(2b-\frac{1}{2} \biggr) \int _{M} |R_{ij}|^{2}e^{-f}\,dv, \end{aligned}$$

(1.17)

where $f=-2\log u$ and $\lambda^{b}$ is the lowest eigenvalue of (1.7).

In particular, when $n=2$, we have $R_{ij}=\frac{R}{2}g_{ij}$ and the normalized Ricci flow (1.16) becomes $\frac{\partial}{\partial t}g_{ij}=-(R-r)g_{ij}$. Hence, $\frac{d}{d t}r=0$, which implies that r is a constant (or see p.455 in [5] for an alternative proof). Then from the estimate (1.17), we obtain the following.

Theorem 1.3

Let $g(t)$, $t\in[0,T)$ be a solution to the normalized Ricci flow (1.16) on a compact surface $M^{2}$. Then for $b\geq\frac{1}{4}$, the lowest constant $\lambda_{a}^{b}(g)$ such that to the nonlinear equation (1.12) with (1.13) there exist positive solutions satisfies

$$\begin{aligned} \frac{d}{d t} \biggl(\lambda_{a}^{b}+ \frac{a^{2}}{4}t+r \int _{0}^{t}\lambda^{b}(s)\,ds \biggr) ={}&\frac{1}{2} \int _{M} \biggl|R_{ij}+f_{ij}+ \frac{a}{2}g_{ij}\biggr|^{2}e^{-f}\,dv \\ &{}+ \biggl(2b-\frac{1}{2} \biggr) \int _{M} |R_{ij}|^{2}e^{-f}\,dv \\ \geq{}&0, \end{aligned}$$

(1.18)

where $f=-2\log u$ and $\lambda^{b}$ is the lowest eigenvalue of (1.7).

Remark 1.1

In particular, when $a=0$, our estimate (1.15) reduces to Theorem 1.5 of Cao in [1] and the estimate (1.18) reduces to the Corollary 2.4 of Cao in [1], respectively.

On the other hand, under the transformation $f=-2\log u=-\log v$ with $u^{2}=v$, equation (1.2) becomes

$$ \left \{ \textstyle\begin{array}{@{}l} \frac{\partial}{\partial t}g_{ij}=-2R_{ij},\\ v_{t}=-\Delta v+Rv. \end{array}\displaystyle \right . $$

(1.19)

In particular, the second equation in (1.19) is exactly the conjugate heat equation introduced by Perelman. In [6], Cao and Zhang obtained differential Harnack inequalities for positive solutions of the nonlinear parabolic equation of the type $v_{t}=\Delta v-v\log v+Rv$. Extending the second equation in (1.19) to the following nonlinear version:

$$ v_{t}=-\Delta v+a v\log v+Sv, $$

(1.20)

Guo and Ishida [7, 8] studied Harnack inequalities for positive solutions of equation (1.20) on a compact Riemannian manifold with a family of $g(t)$ evolving by a geometric flow $\frac{\partial}{\partial t}g_{ij}=-2S_{ij}$, where $S_{ij}$ is a family of smooth symmetric two-tensor and $S=g^{ij}S_{ij}$. Clearly, there is a one-to-one relation for the following two equations:

$$ \frac{\partial}{\partial t}v=-\Delta v+a v\log v+Rv\quad\Longleftrightarrow \quad\frac{\partial}{\partial t}f=-\Delta f+|\nabla f|^{2}+af-R $$

(1.21)

under $f=-\log v$. Therefore, a natural problem is to consider the monotonicity of

$$ \overline{\mathcal {F}}^{c}_{d}(g,f)= \int _{M}\bigl[|\nabla f|^{2}+cR+d(f+1) \bigr]e^{-f}\,dv $$

(1.22)

under the Ricci flow coupled to a nonlinear backward heat-type equation

$$ \left \{ \textstyle\begin{array}{@{}l} \frac{d}{d t}g_{ij}=-2R_{ij},\\ f_{t}=-\Delta f+|\nabla f|^{2}+af-R, \end{array}\displaystyle \right . $$

(1.23)

where c, d are two real constants.

For the functional $\overline{\mathcal{F}}^{c}_{d}(g,f)$, we derive the following monotonicity formula.

Theorem 1.4

Let $g(t)$, $t\in[0,T)$ be a solution to the Ricci flow (1.14) on a compact Riemannian manifold M. Then all functionals $\overline{\mathcal{F}}^{c}_{d}(g,f)$ defined by (1.22) under the system (1.23) satisfy

$$\begin{aligned} \frac{d}{d t}\overline{\mathcal{F}}_{\frac{nak}{8}}^{k}(g,f) ={}&2 \int _{M}\biggl|R_{ij}+f_{ij}- \frac{a}{4}fg_{ij}\biggr|^{2}e^{-f}\,dv +2(k-1) \int _{M}\biggl|R_{ij}-\frac{a}{4}fg_{ij}\biggr|^{2}e^{-f} \,dv \\ &{}+\frac{na}{8}k\mathcal{F}^{1}(g,f)+a\mathcal{F}^{0}(g,f). \end{aligned}$$

(1.24)

In particular, if $R(t)\geq0$ for all t and $a\geq0$, $k\geq1$, then $\frac{d}{d t}\overline{\mathcal{F}}_{\frac{nak}{8}}^{k}(g,f)\geq0$.

Remark 1.2

Choosing $a=0$ in (1.24), we obtain Theorem 4.2 of Li in [2].

Proof of Theorems 1.1 and 1.2

Proof of Theorems 1.1

Let u be a positive solution to the following nonlinear elliptic equation:

$$ -\Delta u+au\log u+bRu=\lambda_{a}^{b} u. $$

(2.1)

Multiplying both sides of (2.1) with u and integrating on M, we have

$$ \lambda_{a}^{b}= \int _{M} \bigl(|\nabla u|^{2}+au^{2}\log u+bRu^{2}\bigr)\,dv. $$

(2.2)

If the metric $g(t)$ evolves by (1.14), we have $\frac{\partial}{\partial t}\,dv=-R \,dv$. It follows from (2.2) that

$$\begin{aligned} \frac{d}{d t}\lambda_{a}^{b}={}& \int _{M} \bigl(2R_{ij}u^{i}u^{j}+2(u_{t})^{i}u_{i}+2auu_{t} \log u+auu_{t}+bR_{t}u^{2}+2bRuu_{t}\bigr) \,dv \\ &{}- \int _{M} \bigl(|\nabla u|^{2}+au^{2}\log u+bRu^{2}\bigr)R \,dv. \end{aligned}$$

(2.3)

Applying

$$ 2 \int _{M}R_{ij}u^{i}u^{j} \,dv= \int _{M}\bigl(-R_{,i}u^{i}u-2R_{ij}u^{ij}u \bigr)\,dv $$

(2.4)

and

$$ - \int _{M}|\nabla u|^{2}R \,dv= \int _{M}\bigl(R\Delta u+R_{,i}u^{i} \bigr)u \,dv $$

(2.5)

into (2.3) yields

$$\begin{aligned} \frac{d}{d t}\lambda_{a}^{b}={}& \int _{M} \bigl[-2R_{ij}u^{ij}u+bR_{t}u^{2}+auu_{t} \\ &{}+2u_{t}(-\Delta u+au\log u+bRu)-Ru(-\Delta u+au\log u+bRu)\bigr] \,dv \\ ={}& \int _{M} \biggl[-2R_{ij}u^{ij}u+bR_{t}u^{2}+ \frac{a}{2}\bigl(u^{2}\bigr)_{t}\biggr]\,dv+\lambda \biggl( \int _{M}u^{2}\,dv \biggr)_{t} \\ ={}& \int _{M} \biggl[-2R_{ij}u^{ij}u+bR_{t}u^{2}+ \frac{a}{2}Ru^{2}\biggr]\,dv, \end{aligned}$$

(2.6)

where the last equality used

$$ \int _{M}\bigl[\bigl(u^{2}\bigr)_{t}-Ru^{2} \bigr]\,dv=0 $$

(2.7)

from (1.13). Noticing $R_{t}=\Delta R+2|R_{ij}|^{2}$ for the Ricci flow, hence from (2.6) we have

$$\begin{aligned} \frac{d}{d t}\lambda_{a}^{b}&= \int _{M} \biggl[-2R_{ij}u^{ij}u+bu^{2} \bigl(\Delta R+2|R_{ij}|^{2}\bigr)+\frac{a}{2}Ru^{2} \biggr]\,dv \\ &= \int _{M} \biggl[-2R_{ij}u^{ij}u+bR\Delta \bigl(u^{2}\bigr)+2b|R_{ij}|^{2}u^{2}+ \frac{a}{2}Ru^{2}\biggr]\,dv. \end{aligned}$$

(2.8)

Taking a transformation $f=-2\log u$, which is equivalent to $u^{2}=e^{-f}$, then

$$ u^{ij}=\biggl(-\frac{1}{2}f^{ij}+ \frac {1}{4}f^{i}f^{j}\biggr)e^{-\frac{f}{2}}. $$

(2.9)

Thus, (2.8) can be written as

$$ \frac{d}{d t}\lambda_{a}^{b}= \int _{M} \biggl[R_{ij}f^{ij}- \frac{1}{2}R_{ij}f^{i}f^{j}-bR\Delta f+bR| \nabla f|^{2}+2b|R_{ij}|^{2}+\frac{a}{2}R \biggr]e^{-f}\,dv. $$

(2.10)

Using the second Bianchi identity $R_{,i}=2R_{ij,}{}^{j}$ again, we have

$$\begin{aligned} -b \int _{M}R\Delta f e^{-f}\,dv&= \int _{M}\bigl(bR_{,i}f^{i}-bR|\nabla f|^{2}\bigr) e^{-f}\,dv \\ &= \int _{M} \bigl(-2bR_{ij}f^{ij}+2bR_{ij}f^{i}f^{j}-bR| \nabla f|^{2} \bigr) e^{-f}\,dv. \end{aligned}$$

(2.11)

Therefore, inserting (2.11) into (2.10) yields

$$\begin{aligned} \frac{d}{d t}\lambda_{a}^{b}={}&(1-2b) \int _{M} R_{ij}f^{ij}e^{-f}\,dv+ \biggl(2b-\frac{1}{2} \biggr) \int _{M} R_{ij}f^{i}f^{j}e^{-f} \,dv \\ &{}+2b \int _{M} |R_{ij}|^{2}e^{-f}\,dv+ \frac{a}{2} \int _{M} Re^{-f}\,dv. \end{aligned}$$

(2.12)

Integrating by parts again, one has

$$ \int _{M} R_{ij}f^{ij}e^{-f}\,dv= \int _{M} R_{ij}f^{i}f^{j}e^{-f} \,dv-\frac{1}{2} \int _{M} R\Delta e^{-f}\,dv $$

(2.13)

and

$$\begin{aligned} & \int _{M} R_{ij}f^{ij}e^{-f}\,dv+ \int _{M} |f_{ij}|^{2}e^{-f}\,dv \\ &\quad=\frac{1}{2} \int _{M} \Delta|\nabla f|^{2}e^{-f}\,dv- \int _{M}(\Delta f)_{i}f^{i} e^{-f}\,dv-\frac{1}{2} \int _{M}R\Delta e^{-f}\,dv \\ &\quad=- \int _{M} \biggl[\Delta f-\frac{1}{2}|\nabla f|^{2}+\frac{1}{2}R\biggr]\Delta e^{-f}\,dv \\ &\quad= \biggl(2b-\frac{1}{2} \biggr) \int _{M}R\Delta e^{-f}\,dv-a \int _{M}|\nabla f|^{2}e^{-f}\,dv, \end{aligned}$$

(2.14)

where the last equality in (2.14) was used with

$$ 2\lambda_{a}^{b}=\Delta f-\frac{1}{2}| \nabla f|^{2}-af+2bR. $$

(2.15)

By virtue of (2.14), subtracting (2.13), we obtain

$$ \int _{M} |f_{ij}|^{2}e^{-f} \,dv=2b \int _{M}R\Delta e^{-f}\,dv- \int _{M} R_{ij}f^{i}f^{j}e^{-f} \,dv-a \int _{M}|\nabla f|^{2}e^{-f}\,dv. $$

(2.16)

It follows from (2.13) and (2.14) that

$$\begin{aligned} \frac{d}{d t}\lambda_{a}^{b}={}&(1-2b) \int _{M} R_{ij}f^{ij}e^{-f}\,dv+ \biggl(2b-\frac{1}{2} \biggr) \int _{M} R_{ij}f^{i}f^{j}e^{-f} \,dv \\ &{}+2b \int _{M} |R_{ij}|^{2}e^{-f}\,dv+ \frac{a}{2} \int _{M} Re^{-f}\,dv \\ ={}& \int _{M} R_{ij}f^{ij}e^{-f}\,dv- \frac{1}{2} \int _{M} R_{ij}f^{i}f^{j}e^{-f} \,dv \\ &{}+2b \int _{M} |R_{ij}|^{2}e^{-f}\,dv+ \frac{a}{2} \int _{M} Re^{-f}\,dv+b \int _{M}R\Delta e^{-f}\,dv \\ ={}& \int _{M} R_{ij}f^{ij}e^{-f} \,dv+2b \int _{M} |R_{ij}|^{2}e^{-f}\,dv+ \frac{a}{2} \int _{M} Re^{-f}\,dv \\ &{}+\frac{1}{2} \int _{M} |f_{ij}|^{2}e^{-f}\,dv+ \frac{a}{2} \int _{M}(\Delta f)e^{-f}\,dv \\ ={}&\frac{1}{2} \int _{M} \biggl|R_{ij}+f_{ij}+ \frac{a}{2}g_{ij}\biggr|^{2}e^{-f}\,dv+ \biggl(2b- \frac{1}{2} \biggr) \int _{M} |R_{ij}|^{2}e^{-f}\,dv \\ &{}-\frac{na^{2}}{8}, \end{aligned}$$

(2.17)

and the desired estimate (1.15) is achieved. □

Proof of Theorem 1.2

If the metric $g(t)$ evolves by (1.16), we have $\frac{\partial}{\partial t}\,dv=-(R-r)\,dv$. It follows from (2.2) that

$$\begin{aligned} \frac{d}{d t}\lambda_{a}^{b}={}& \int _{M} \biggl(2R_{ij}u^{i}u^{j}- \frac{2r}{n}|\nabla u|^{2}+2(u_{t})^{i}u_{i}+2auu_{t} \log u+auu_{t}+bR_{t}u^{2} \\ &{}+2bRuu_{t}\biggr)\,dv- \int _{M} \bigl(|\nabla u|^{2}+au^{2}\log u+bRu^{2}\bigr) (R-r)\,dv. \end{aligned}$$

(2.18)

Applying (2.4) and

$$ - \int _{M}|\nabla u|^{2}(R-r)\,dv= \int _{M}\bigl[(R-r)\Delta u+R_{,i}u^{i} \bigr]u \,dv $$

(2.19)

to (2.18) yields

$$\begin{aligned} \frac{d}{d t}\lambda_{a}^{b}={}& \int _{M} \biggl[-2R_{ij}u^{ij}u- \frac{2r}{n}|\nabla u|^{2}+bR_{t}u^{2}+auu_{t} \\ &{}+2u_{t}(-\Delta u+au\log u+bRu)-(R-r)u(-\Delta u+au\log u+bRu) \biggr]\,dv \\ ={}& \int _{M} \biggl[-2R_{ij}u^{ij}u- \frac{2r}{n}|\nabla u|^{2}+bR_{t}u^{2}+ \frac{a}{2}\bigl(u^{2}\bigr)_{t}\biggr]\,dv +\lambda \biggl( \int _{M}u^{2}\,dv \biggr)_{t} \\ ={}& \int _{M} \biggl[-2R_{ij}u^{ij}u- \frac{2r}{n}|\nabla u|^{2}+bR_{t}u^{2}+ \frac{a}{2}Ru^{2}\biggr]\,dv. \end{aligned}$$

(2.20)

Noticing $R_{t}=\Delta R+2|R_{ij}|^{2}-\frac{2r}{n}R$ for the normalized Ricci flow, we obtain from (2.20)

$$\begin{aligned} \frac{d}{d t}\lambda_{a}^{b}={}& \int _{M} \biggl[-2R_{ij}u^{ij}u- \frac{2r}{n}|\nabla u|^{2}+bR_{t}u^{2}+ \frac{a}{2}Ru^{2}\biggr]\,dv \\ ={}& \int _{M} \biggl[-2R_{ij}u^{ij}u- \frac{2r}{n}|\nabla u|^{2}+bu^{2}\biggl(\Delta R+2|R_{ij}|^{2}-\frac{2r}{n}R\biggr)+ \frac{a}{2}Ru^{2}\biggr]\,dv \\ ={}& \int _{M} \biggl[-2R_{ij}u^{ij}u+bR\Delta \bigl(u^{2}\bigr)+2b|R_{ij}|^{2}u^{2}+ \frac{a}{2}Ru^{2}\biggr]\,dv \\ &{}-\frac{2r}{n} \int _{M}\bigl(|\nabla u|^{2}+bRu^{2} \bigr)\,dv. \end{aligned}$$

(2.21)

Using (2.9), then (2.21) can be written as

$$\begin{aligned} \frac{d}{d t}\lambda_{a}^{b}={}& \int _{M} \biggl[R_{ij}f^{ij}- \frac{1}{2}R_{ij}f^{i}f^{j}-bR\Delta f+bR| \nabla f|^{2}+2b|R_{ij}|^{2}+\frac{a}{2}R \biggr]e^{-f}\,dv \\ &{}-\frac{2r}{n} \int _{M}\biggl(\frac{1}{4}|\nabla f|^{2}+bR \biggr)e^{-f}\,dv. \end{aligned}$$

(2.22)

By virtue of a similar computation, we can obtain

$$\begin{aligned} \frac{d}{d t}\lambda_{a}^{b}={}& \frac{1}{2} \int _{M} \biggl|R_{ij}+f_{ij}+ \frac{a}{2}g_{ij}\biggr|^{2}e^{-f}\,dv+ \biggl(2b- \frac{1}{2} \biggr) \int _{M} |R_{ij}|^{2}e^{-f}\,dv \\ &{}-\frac{na^{2}}{8}-\frac{2r}{n} \int _{M}\biggl(\frac{1}{4}\Delta f+bR \biggr)e^{-f}\,dv, \end{aligned}$$

(2.23)

which gives

$$\begin{aligned} \frac{d}{d t} \biggl(\lambda_{a}^{b}+ \frac{na^{2}}{8}t \biggr)+\frac{2r}{n}\lambda^{b}={}& \frac {1}{2} \int _{M} \biggl|R_{ij}+f_{ij}+ \frac{a}{2}g_{ij}\biggr|^{2}e^{-f}\,dv \\ &{}+ \biggl(2b-\frac{1}{2} \biggr) \int _{M} |R_{ij}|^{2}e^{-f}\,dv. \end{aligned}$$

(2.24)

Then the desired estimate (1.17) is attained. □

Proof of Theorem 1.4

Under the following coupled system (1.23), by a direct computation, we have the following:

$$\begin{aligned}& \begin{aligned}[b] \frac{\partial}{\partial t}\bigl(e^{-f}\,dv \bigr)&=-(f_{t}+R)e^{-f}\,dv=\bigl[\Delta f-|\nabla f|^{2}-af\bigr]e^{-f}\,dv \\ &=-\bigl(\Delta e^{-f}\bigr)\,dv-afe^{-f}\,dv, \end{aligned} \end{aligned}$$

(3.1)

$$\begin{aligned}& \begin{aligned}[b] \frac{\partial}{\partial t}|\nabla f|^{2}&=2R^{ij}f_{i}f_{j}+2f^{i}(f_{t})_{i} \\ &=2R^{ij}f_{i}f_{j}+2f^{i}\bigl(- \Delta f+|\nabla f|^{2}+af-R\bigr)_{i} \\ &=2R^{ij}f_{i}f_{j}-2f^{i}(\Delta f)_{i}+4f^{ij}f_{i}f_{j}+2a|\nabla f|^{2}-2R_{i}f^{i}. \end{aligned} \end{aligned}$$

(3.2)

Thus, we have

$$\begin{aligned}& \frac{d}{d t} \int _{M}e^{-f}\,dv=-a \int _{M}f e^{-f}\,dv, \end{aligned}$$

(3.3)

$$\begin{aligned}& \begin{aligned}[b] \frac{d}{d t} \int _{M}Re^{-f}\,dv&= \int _{M}\bigl[\Delta R+2|R_{ij}|^{2}-afR \bigr]e^{-f}\,dv- \int _{M}R\bigl(\Delta e^{-f}\bigr)\,dv \\ &= \int _{M}\bigl[2|R_{ij}|^{2}-afR \bigr]e^{-f}\,dv, \end{aligned} \end{aligned}$$

(3.4)

$$\begin{aligned}& \begin{aligned}[b] \frac{d}{d t} \int _{M}fe^{-f}\,dv&= \int _{M}(af-R)e^{-f}\,dv- \int _{M} f\bigl(\Delta e^{-f}\bigr)\,dv- \int _{M} af^{2}e^{-f}\,dv \\ &= \int _{M}\bigl[af-af^{2}-(R+\Delta f) \bigr]e^{-f}\,dv \end{aligned} \end{aligned}$$

(3.5)

and

$$\begin{aligned} &\frac{d}{d t} \int _{M}|\nabla f|^{2}e^{-f}\,dv \\ &\quad= \int _{M}\bigl[2R^{ij}f_{i}f_{j}-2f^{i}( \Delta f)_{i}+4f^{ij}f_{i}f_{j}+2a| \nabla f|^{2}-2R_{i}f^{i}\bigr]e^{-f}\,dv \\ &\qquad{}- \int _{M} \bigl(\Delta e^{-f}\bigr)|\nabla f|^{2}\,dv- \int _{M} af|\nabla f|^{2}e^{-f}\,dv \\ &\quad= \int _{M}\bigl[-2f_{ij}^{2}-4f^{i}( \Delta f)_{i}+4f^{ij}f_{i}f_{j}+2a| \nabla f|^{2}-2R_{i}f^{i}\bigr]e^{-f}\,dv \\ &\qquad{}- \int _{M} af|\nabla f|^{2}e^{-f}\,dv. \end{aligned}$$

(3.6)

By virtue of the Bochner formula with respect to the f-Laplacian, we have

$$\frac{1}{2}\Delta_{f}|\nabla u|^{2}=u_{ij}^{2}+u_{i}( \Delta_{f}u)_{i}+\bigl(R^{ij}+f^{ij} \bigr)u_{i}u_{j}, \quad \forall u, $$

and hence

$$\begin{aligned} 0&= \int _{M}\bigl[f_{ij}^{2}+f_{i}( \Delta _{f}f)_{i}+\bigl(R^{ij}+f^{ij} \bigr)f_{i}f_{j}\bigr]e^{-f}\,dv \\ &= \int _{M}\bigl[f_{ij}^{2}+f_{i}( \Delta f)_{i}+R^{ij}f_{i}f_{j}-f^{ij}f_{i}f_{j} \bigr]e^{-f}\,dv. \end{aligned}$$

(3.7)

Therefore, (3.6) becomes

$$\begin{aligned} \frac{d}{d t} \int _{M}|\nabla f|^{2}e^{-f}\,dv&= \int _{M}\bigl[2f_{ij}^{2}+4R^{ij}f_{i}f_{j}+2a| \nabla f|^{2}-2R_{i}f^{i}\bigr]e^{-f}\,dv - \int _{M} af|\nabla f|^{2}e^{-f}\,dv \\ &= \int _{M}\bigl[2f_{ij}^{2}+4R^{ij}f_{ij}+2a| \nabla f|^{2}\bigr]e^{-f}\,dv - \int _{M} a(f+1) (\Delta f)e^{-f}\,dv. \end{aligned}$$

(3.8)

Therefore, from (3.4) and (3.8), we obtain

$$\begin{aligned} \frac{d}{d t} \int _{M}\bigl(R+|\nabla f|^{2}\bigr)e^{-f} \,dv={}&2 \int _{M}\biggl|R_{ij}+f_{ij}- \frac {a}{4}fg_{ij}\biggr|^{2}e^{-f}\,dv \\ &{}-\frac{na^{2}}{8} \int _{M}f^{2}e^{-f}\,dv+a \int _{M}|\nabla f|^{2}e^{-f}\,dv. \end{aligned}$$

(3.9)

Noticing (3.5) tells us that

$$ -a \int _{M}f^{2}e^{-f}\,dv=\frac{d}{d t} \biggl( \int _{M}(f+1)e^{-f}\,dv \biggr)+ \int _{M}(R+\Delta f)e^{-f}\,dv. $$

(3.10)

Thus, (3.9) can be written as

$$\begin{aligned} &\frac{d}{d t} \int _{M}\biggl[R+|\nabla f|^{2}+ \frac{na}{8}(f+1)\biggr]e^{-f}\,dv \\ &\quad=2 \int _{M}\biggl|R_{ij}+f_{ij}- \frac{a}{4}fg_{ij}\biggr|^{2}e^{-f}\,dv +\frac{na}{8} \int _{M}\bigl(R+|\nabla f|^{2}\bigr)e^{-f} \,dv+a \int _{M}|\nabla f|^{2}e^{-f}\,dv. \end{aligned}$$

(3.11)

Since (3.4) holds, we have

$$ \frac{d}{d t} \int _{M}Re^{-f}\,dv=2 \int _{M}\biggl|R_{ij}-\frac {a}{4}fg_{ij}\biggr|^{2}e^{-f} \,dv -\frac{na^{2}}{8} \int _{M}f^{2}e^{-f}\,dv, $$

(3.12)

which gives

$$\begin{aligned} \frac{d}{d t} \int _{M}\biggl[R+\frac{na}{8}(f+1)\biggr]e^{-f} \,dv={}&2 \int _{M}\biggl|R_{ij}-\frac{a}{4}fg_{ij}\biggr|^{2}e^{-f} \,dv \\ &{}+\frac{na}{8} \int _{M}\bigl(R+|\nabla f|^{2}\bigr)e^{-f} \,dv. \end{aligned}$$

(3.13)

Therefore, we have

$$\begin{aligned} &\frac{d}{d t} \int _{M}\biggl\{ |\nabla f|^{2}+k\biggl[R+ \frac{na}{8}(f+1)\biggr]\biggr\} e^{-f}\,dv \\ &\quad=2 \int _{M}\biggl|R_{ij}+f_{ij}- \frac{a}{4}fg_{ij}\biggr|^{2}e^{-f}\,dv +2(k-1) \int _{M}\biggl|R_{ij}-\frac{a}{4}fg_{ij}\biggr|^{2}e^{-f} \,dv \\ &\qquad{}+\frac{na}{8}k \int _{M}\bigl(R+|\nabla f|^{2}\bigr)e^{-f} \,dv+a \int _{M}|\nabla f|^{2}e^{-f}\,dv \end{aligned}$$

(3.14)

and the desired estimate (1.24) is obtained.

Conclusions

We establish the first variation formula of the lowest constant $\lambda_{a}^{b}(g)$ along the Ricci flow and the normalized Ricci flow, such that to the following nonlinear equation there exist positive solutions:

$$ -\Delta u+au\log u+bRu=\lambda_{a}^{b} u $$

(4.1)

with $\int_{M}u^{2}\,dv=1$, where a is a real constant. Equation (4.1) can be seen as a nonlinear version of eigenvalue problem of the operator $-\Delta u+bR$. In particular, when $a=0$, our estimate (1.15) in Theorem 1.1 reduces to Theorem 1.5 of Cao in [1] and the estimate (1.18) in Theorem 1.3 reduces to the Corollary 2.4 of Cao in [1], respectively.

On the other hand, we obtained the first variation formula (1.24) of the functional

$$\overline{\mathcal{F}}^{c}_{d}(g,f)= \int _{M}\bigl[|\nabla f|^{2}+cR+d(f+1) \bigr]e^{-f}\,dv $$

under the Ricci flow coupled to a nonlinear backward heat-type equation

$$\left \{ \textstyle\begin{array}{@{}l} \frac{d}{d t}g_{ij}=-2R_{ij},\\ f_{t}=-\Delta f+|\nabla f|^{2}+af-R, \end{array}\displaystyle \right . $$

where $c,d$ are two real constants. In particular, when $a=0$ in (1.24), we obtain Theorem 4.2 of Li in [2].