# 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].

## References

1. Cao, X-D: First eigenvalues of geometric operators under the Ricci flow. Proc. Am. Math. Soc. 136, 4075-4078 (2008)

2. Li, J-F: Eigenvalues and energy functionals with monotonicity formulae under Ricci flow. Math. Ann. 338, 927-946 (2007)

3. Perelman, G: The entropy formula for the Ricci flow and its geometric applications. arXiv:math.DG/0211159

4. Cao, X-D: Eigenvalues of $$(-\Delta+\frac{R}{2})$$ on manifolds with nonnegative curvature operator. Math. Ann. 337, 435-441 (2007)

5. Cao, X-D, Hou, S, Ling, J: Estimate and monotonicity of the first eigenvalue under the Ricci flow. Math. Ann. 354, 451-463 (2012)

6. Cao, X-D, Zhang, Z: Differential Harnack estimates for parabolic equations. In: Complex and Differential Geometry. Springer Proc. Math., vol. 8, pp. 87-98 (2011)

7. Guo, H, Ishida, M: Harnack estimates for nonlinear backward heat equations in geometric flows. J. Funct. Anal. 267, 2638-2662 (2014)

8. Guo, H, Ishida, M: Harnack estimates for nonlinear heat equations with potentials in geometric flows. Manuscr. Math. 148, 471-484 (2015)

## Acknowledgements

The research of the first author is supported by NSFC (No. 11371018, 11171091), Henan Provincial Core Teacher (No. 2013GGJS-057) and IRTSTHN (14IRTSTHN023). The research of the second author is partially supported by NSFC (No. 11401179) and Henan Provincial Education department (No. 14B110017).

## Author information

Authors

### Corresponding author

Correspondence to Guangyue Huang.

### Competing interests

The authors declare that they have no competing interests.

### Authors’ contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

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Reprints and Permissions

Huang, G., Li, Z. Evolution of a geometric constant along the Ricci flow. J Inequal Appl 2016, 53 (2016). https://doi.org/10.1186/s13660-016-1003-6

• Accepted:

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• DOI: https://doi.org/10.1186/s13660-016-1003-6

• 58C40
• 53C44

### Keywords

• Ricci flow
• normalized Ricci flow
• conjugate heat equation