1 Introduction

The eigenvalues of geometric operators have always been an active subject in the study of geometry and analysis of manifolds. Recently, there has been increasing attentions on the eigenvalue problems under various geometric flows. In [13], Perelman introduced the so-called \({\mathcal {F}}\)-functional and proved that it is nondecreasing along the Ricci flow coupled to a backward heat equation. The nondecreasing of the functional implies the monotonicity of the first eigenvalue of \(-4\Delta +R\) along the Ricci flow. As an application, he was able to rule out nontrivial steady or expanding breathers on closed manifolds. In [12], Ma obtained the monotonicity of the first eigenvalue of the Laplace–Beltrami operator on a domain with Dirichlet boundary condition along the Ricci flow. Cao [1] showed that the eigenvalues of \(-\Delta +\frac{R}{2}\) are nondecreasing under the Ricci flow on manifolds with nonnegative curvature operator. Li [8] introduced families of functionals of the type of \({\mathcal {F}}\)-functional and \({\mathcal {W}}\)-functional of Perelman and proved that they are nondecreasing under the Ricci flow. In particular, he got the monotonicity of eigenvalues of \(-4\Delta +kR\) and ruled out compact steady Ricci breathers. Later, Cao [2] also improved his own previous results and proved that the first eigenvalues of \(-\Delta +cR \left( c\ge \frac{1}{4}\right) \) are nondecreasing under the Ricci flow on manifolds without curvature assumption. Ling [10] showed a Faber–Krahn type of comparison theorem, gave a sharp bound for the first eigenvalue of the Laplace–Beltrami operator under the normalized Ricci flow, and constructed a class of monotonic quantities on closed n-dimensional manifolds [11]. Cao et al. [3] derived various monotonicity formulae and estimates for the first eigenvalue of \(-\Delta +cR\left( 0<c\le \frac{1}{2}\right) \) on closed surfaces. Moreover, Zhao got some monotonic quantities for the first eigenvalue of the Laplace–Beltrami operator under the Yamabe flow [15], and proved that the first eigenvalue of the p-Laplacian operator is increasing and the differentiable almost everywhere along powers of the mth mean curvature flow [16] and the \(H^k\)-flow [17]. Guo and his collaborators [7] derived an explicit formula for the evolution of the lowest eigenvalue of the Laplace–Beltrami operator with potential in abstract geometric flows. The first author, Xu and Zhu [4] proved the monotonicity of eigenvalues of \(-\Delta _{\phi }+cR\left( c>\frac{1}{4}\right) \) along the system of Ricci flow coupled to a heat equation. Not long ago the first author, Yang and Zhu also generalized Cao’s result [1] to the Witten–Laplacian operator and obtained evolution equations and monotonicity of eigenvalues of \(-\Delta _{\phi }+\frac{R}{2}\) along the Yamabe flow [5] and the Ricci flow [6].

In this paper, we consider an n-dimensional closed Riemannian manifold M with a time-dependent Riemannian metric g(t). (M, g(t)) is a smooth solution to the Ricci flow

$$\begin{aligned} \frac{\partial }{\partial t}g_{ij}(t)=-2R_{ij}(t), \end{aligned}$$
(1.1)

or the normalized Ricci flow

$$\begin{aligned} \frac{\partial }{\partial t}g_{ij}=-2R_{ij}+\frac{2r}{n}g_{ij}, \end{aligned}$$
(1.2)

where \(r=\dfrac{\int _M R\mathrm{d}\nu }{\int _M \mathrm{d}\nu }\) is the average scalar curvature. Let \(\nabla \) be the Levi–Civita connection on (M, g), \(\Delta \) the Laplace–Beltrami operator, \(d\nu \) the Riemannian volume measure, and \(\mathrm{d}\mu \) the weight volume measure on (M, g), i.e.,

$$\begin{aligned} \mathrm{d}\mu =e^{-\phi (x)}\mathrm{d}\nu , \end{aligned}$$

where \(\phi \in C^{\infty }(M)\). Then the Witten–Laplacian (also called symmetric diffusion operator)

$$\begin{aligned} \Delta _{\phi }=\Delta -\nabla \phi \cdot \nabla \end{aligned}$$

is a symmetric operator on \(L^2(M,\mu )\). When \(\phi \) is a constant function, the Witten–Laplacian operator is just the Laplace–Beltrami operator. As an extension of the Laplace–Beltrami operator, many classical results in Riemannian geometry asserted in terms of the Laplace–Beltrami operator have been extended to the analogous ones on the Witten–Laplacian operator. For example, we can refer to [4–6, 9, 14]. Inspired by Cao et al. [3], we study the eigenvalues of geometric operator \(-\Delta _{\phi }+cR\) under the Ricci flow and the normalized Ricci flow. The purpose of this paper is to obtain the monotonicity and some bounds for the first eigenvalue of the operator along the Ricci flow on closed surfaces under some curvature assumptions for the case \(0<c\le \frac{1}{2}\).

The rest of this paper is organized as follows. In Sect. 2, we will derive the evolution equations of eigenvalues under the Ricci flow and the normalized Ricci flow. In Sect. 3, we consider the first eigenvalue of the geometric operator \(-\Delta _{\phi }+cR\left( 0<c\le \frac{1}{2}\right) \) on closed surfaces. We will show that the first eigenvalue is nondecreasing along the Ricci flow with some curvature conditions. In Sect. 4, we will obtain some monotonicity formulae and lower bounds for the first eigenvalue on closed surfaces.

Throughout this paper, we use Einstein convention, i.e., repeated index implies summation.

2 Evolution Equations of Eigenvalues

In this section, we establish the evolution of eigenvalues of geometric operator \(-\Delta _{\phi }+cR\) under the Ricci flow and the normalized Ricci flow, respectively.

Let (M, g(t)) be a closed Riemannian manifold, and \((M, g(t)), t\in [0,T)\) be a smooth solution to the Ricci flow Eq. (1.1). Let \(\lambda \) be an eigenvalue of the operator \(-\Delta _{\phi }+cR\) at time \(t_0\) where \(0 \le t_0 < T\), and f the corresponding eigenfunction, i.e.,

$$\begin{aligned} -\Delta _{\phi }f+cRf=\lambda f, \end{aligned}$$

with the normalization

$$\begin{aligned} \int _M f^2\mathrm{d}\mu =1. \end{aligned}$$
(2.1)

Assume that f(x, t) is a \(C^1\)-family of smooth functions on M, and satisfies the normalization condition for any t. We need to use the following functional

$$\begin{aligned} \lambda (f,t)=\int _M\left( -f\Delta _{\phi }f+cRf^2\right) \mathrm{d}\mu =\int _M \left( -\Delta _{\phi }f+cRf\right) f\mathrm{d}\mu , \end{aligned}$$

where f satisfies the equality (2.1). At time t, if f is the eigenfunction of \(\lambda \), then

$$\begin{aligned} \lambda (f,t)=\lambda (t). \end{aligned}$$

Let us first recall the evolution equation of the above functional under the general geometric flow, which had been proved in [4].

Lemma 2.1

Suppose that \(\lambda \) is an eigenvalue of the operator \(-\Delta _{\phi }+cR\), f is the eigenfunction of \(\lambda \) at time \(t_0\), and the metric g(t) evolves by

$$\begin{aligned} \frac{\partial }{\partial t}g_{ij}=v_{ij}, \end{aligned}$$

where \(v_{ij}\) is a symmetric two-tensor. Then we have

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t}\lambda (f,t)|_{t=t_0}= & {} \int _M\left( v_{ij}f_{ij}-v_{ij}\phi _if_j+c\frac{\partial R}{\partial t}f\right) f\mathrm{d}\mu \nonumber \\&+\int _M\left( v_{ij,i}-\frac{1}{2}V_{j}\right) f_jf\mathrm{d}\mu , \end{aligned}$$
(2.2)

where \(V={\text {Tr}} (v)\).

Remark 2.2

In fact, Lemma 2.1 can also give us the evolution of eigenvalues. By the eigenvalue perturbation theory, we may assume that there is a \(C^1\)-family of smooth eigenvalues and eigenfunctions. Therefore, we have

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t}\lambda (t)=\frac{\mathrm{d}}{\mathrm{d}t}\lambda (f,t) \end{aligned}$$
(2.3)

for any time t, when f is the eigenfunction of \(\lambda \) at time t. In particular, we further assume that the first eigenvalue and first eigenfunction are smooth in the time along the Ricci flow (for example, see [3]).

Now we can calculate the evolution equation of eigenvalues of the geometric operator under the Ricci flow. In Lemma 2.1, when the symmetric two-tensor \(v_{ij}=-2R_{ij}\), we get the following result.

Theorem 2.3

Let \(g(t),t\in [0,T)\), be a solution to the Ricci flow (1.1) on a closed manifold \(M^n\). Assume that there is a \(C^1\)-family of smooth functions \(f(x,t)>0\), which satisfy

$$\begin{aligned} -\Delta _{\phi }f(x,t)+cRf(x,t)=\lambda (t) f(x,t), \end{aligned}$$

and the normalization

$$\begin{aligned} \int _M f(x,t)^2\mathrm{d}\mu =1. \end{aligned}$$

Then the eigenvalue \(\lambda (t)\) satisfies

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t}\lambda (t)&=(4c-2)\int _MR_{ij}(f_{ij}-\phi _if_j)f\mathrm{d}\mu +2c\int _MR_{ij}f_if_j\mathrm{d}\mu +2c\int _M|Rc|^2f^2\mathrm{d}\mu \nonumber \\&\quad \,+2c\int _MR_{ij}(f\phi _i-f_i)(f\phi _j-f_j)\mathrm{d}\mu -2c\int _MR_{ij}\phi _{ij}f^2\mathrm{d}\mu . \end{aligned}$$
(2.4)

Proof

The proof also follows from a direct computation. Note that the evolution of scalar curvature is

$$\begin{aligned} \frac{\partial R}{\partial t}=\Delta R+2|Rc|^2 \end{aligned}$$

and

$$\begin{aligned} {\mathbf {div}} Rc=\frac{1}{2}\nabla R. \end{aligned}$$

Using (2.3) and plugging \(v_{ij}=-2R_{ij}\) into the equality (2.2), we have

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t}\lambda (t)=\int _M\left( -2R_{ij}f_{ij}f+2R_{ij}\phi _if_jf+c\Delta Rf^2+2c|Rc|^2f^2\right) \mathrm{d}\mu . \end{aligned}$$
(2.5)

From integration by parts and \(\frac{1}{2}\Delta R={\mathbf {div}}({\mathbf {div}} Rc)\), we get

$$\begin{aligned}&\frac{1}{2}\int _M\Delta Rf^2\mathrm{d}\mu \nonumber \\&\quad =\int _M\left( 2R_{ij}f_if_j+2R_{ij}f_{ij}f-4R_{ij}\phi _if_jf+R_{ij}\phi _i\phi _jf^2-R_{ij}\phi _{ij}f^2\right) \mathrm{d}\mu .\qquad \quad \end{aligned}$$
(2.6)

Substituting (2.6) into (2.5), we have

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t}\lambda (t)&=(4c-2)\int _MR_{ij}f_{ij}f\mathrm{d}\mu +4c\int _MR_{ij}f_if_j\mathrm{d}\mu +2c\int _M|Rc|^2f^2\mathrm{d}\mu \\&\quad \,+(2-8c)\int _MR_{ij}\phi _if_jf\mathrm{d}\mu +2c\int _MR_{ij}\phi _i\phi _jf^2\mathrm{d}\mu -2c\int _MR_{ij}\phi _{ij}f^2\mathrm{d}\mu \\&=(4c-2)\int _MR_{ij}(f_{ij}-\phi _if_j)f\mathrm{d}\mu +2c\int _MR_{ij}f_if_j\mathrm{d}\mu +2c\int _M|Rc|^2f^2\mathrm{d}\mu \\&\quad \,+2c\int _MR_{ij}(f\phi _i-f_i)(f\phi _j-f_j)\mathrm{d}\mu -2c\int _MR_{ij}\phi _{ij}f^2\mathrm{d}\mu . \end{aligned}$$

\(\square \)

By Lemma 2.1 we can also get the evolution equation of eigenvalues of the geometric operator under the normalized Ricci flow. We have the following theorem.

Theorem 2.4

Let \(g(t),t\in [0,T)\), be a solution to the normalized Ricci flow (1.2) on a closed manifold \(M^n\). Assume that there is a \(C^1\)-family of smooth functions \(f(x,t)>0\), which satisfy

$$\begin{aligned} -\Delta _{\phi }f(x,t)+cRf(x,t)=\lambda (t) f(x,t), \end{aligned}$$
(2.7)

and the normalization

$$\begin{aligned} \int _M f(x,t)^2\mathrm{d}\mu =1. \end{aligned}$$

Then the eigenvalue \(\lambda (t)\) satisfies

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t}\lambda (t)&=-\frac{2r\lambda }{n}+(1-2c)\lambda \int _MRf^2\mathrm{d}\mu +c(2c-1)\int _MR^2f^2\mathrm{d}\mu \nonumber \\&\quad \,+(2c-1)\int _M R|\nabla f|^2\mathrm{d}\mu \nonumber \\&\quad \,+2\int _MR_{ij}f_if_j\mathrm{d}\mu +c\int _MR(-2ff_i\phi _i-f^2\Delta \phi +f^2|\nabla \phi |^2)\mathrm{d}\mu \nonumber \\&\quad \,+2c\int _M|Rc|^2f^2\mathrm{d}\mu . \end{aligned}$$
(2.8)

Proof

We note that the evolution of scalar curvature under the normalized Ricci flow is

$$\begin{aligned} \frac{\partial R}{\partial t}=\Delta R+2|Rc|^2-\frac{2r}{n}R \end{aligned}$$

and

$$\begin{aligned} v_{ij}=-2R_{ij}+\frac{2r}{n}g_{ij}. \end{aligned}$$

By Lemma 2.1 it is easy to get the extra term \(-\frac{2r\lambda }{n}\) than (2.5), i.e., we have

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t}\lambda (t)=-\frac{2r\lambda }{n}+\int _M\left( -2R_{ij}f_{ij}f+2R_{ij}\phi _if_jf+c\Delta Rf^2+2c|Rc|^2f^2\right) \mathrm{d}\mu .\nonumber \\ \end{aligned}$$
(2.9)

By the contracted second Bianchi identity \( \nabla _iR_{ij} = \frac{1}{2}\nabla _j R\) and integration by parts, it follows that

$$\begin{aligned} -2\int _MR_{ij}f_{ij}f\mathrm{d}\mu = \,&2\int _MR_{ij}f_if_j\mathrm{d}\mu -2\int _MR_{ij}\phi _if_jf\mathrm{d}\mu \nonumber \\&-\int _M R|\nabla f|^2\mathrm{d}\mu -\int _M Rf\Delta _{\phi }f\mathrm{d}\mu . \end{aligned}$$
(2.10)

Integrating by parts again, we derive that

$$\begin{aligned} \int _Mc\Delta R f^2\mathrm{d}\mu= & {} c\int _MR(2f\Delta f+2|\nabla f|^2-4ff_i\phi _i-f^2\Delta \phi +f^2|\nabla \phi |^2)\mathrm{d}\mu \nonumber \\= & {} c\int _MR(2f\Delta _{\phi } f+2|\nabla f|^2-2ff_i\phi _i-f^2\Delta \phi +f^2|\nabla \phi |^2)\mathrm{d}\mu .\nonumber \\ \end{aligned}$$
(2.11)

Combining (2.9)-(2.11), we have

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t}\lambda (t)&=-\frac{2r\lambda }{n}+(2c-1)\int _MRf\Delta _{\phi } f\mathrm{d}\mu +(2c-1)\int _M R|\nabla f|^2\mathrm{d}\mu \\&\quad \,+2\int _MR_{ij}f_if_j\mathrm{d}\mu \\&\quad \,+c\int _MR(-2ff_i\phi _i-f^2\Delta \phi +f^2|\nabla \phi |^2)\mathrm{d}\mu +2c\int _M|Rc|^2f^2\mathrm{d}\mu \\&=-\frac{2r\lambda }{n}+(1-2c)\lambda \int _MRf^2\mathrm{d}\mu +c(2c-1)\int _MR^2f^2\mathrm{d}\mu \nonumber \\&\quad \,+(2c-1)\int _M R|\nabla f|^2\mathrm{d}\mu \\&\quad \,+2\int _MR_{ij}f_if_j\mathrm{d}\mu +c\int _MR(-2ff_i\phi _i-f^2\Delta \phi +f^2|\nabla \phi |^2)\mathrm{d}\mu \\&\quad \,+2c\int _M|Rc|^2f^2\mathrm{d}\mu . \end{aligned}$$

In the above formula the last equality holds because of (2.7). We complete the proof of the theorem. \(\square \)

Remark 2.5

In Theorem 2.4 if we let \(\phi \) be a constant function on M, our theorem reduce to Cao et al.’s Lemma 3.1 in [3]. So our result is an extension version of Cao, Hou, and Ling’s.

3 Monotonicity of the First Eigenvalues

In this section, we consider the first eigenvalues of geometric operators \(-\Delta _{\phi }+cR\left( 0<c\le \frac{1}{2}\right) \) on closed surfaces with nonnegative scalar curvature. We will obtain evolution equations and monotonicity of the first eigenvalues along the Ricci flow using the results in Sect. 2.

Let \(\lambda (t)\) be the first eigenvalue of the operator \(-\Delta _{\phi }+cR\), and \(f(x,t)>0\) be the corresponding eigenfunction, which satisfies

$$\begin{aligned} -\Delta _{\phi }f(x,t)+cRf(x,t)=\lambda (t) f(x,t), \end{aligned}$$
(3.1)

and the normalization

$$\begin{aligned} \int _M f(x,t)^2\mathrm{d}\mu =1. \end{aligned}$$

From Theorem 2.3 in Sect. 2, we can easily get the evolution equation of the first eigenvalue under the Ricci flow.

Theorem 3.1

Let \(g(t),t\in [0,T)\), be a solution to the Ricci flow (1.1) on a closed surface \(M^2\). Assume that \(\lambda (t)\) and f(x, t) are defined as above. Then we have

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t}\lambda (t)=&(1-2c)\lambda \int _MRf^2\mathrm{d}\mu +2c^2\int _MR^2f^2\mathrm{d}\mu +c\int _MR|\nabla f|^2\mathrm{d}\mu \nonumber \\&+c\int _MR|f\nabla \phi -\nabla f|^2\mathrm{d}\mu -c\int _MR\Delta \phi f^2\mathrm{d}\mu . \end{aligned}$$
(3.2)

Proof

When M is a 2-dimensional manifold, the Ricci curvature satisfies \(R_{ij}=\frac{1}{2}Rg_{ij}\). Plugging it into (2.4), we have

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t}\lambda (t)= \,&(2c-1)\int _MR\Delta _{\phi }ff\mathrm{d}\mu +c\int _MR|\nabla f|^2\mathrm{d}\mu +c\int _MR^2f^2\mathrm{d}\mu \\&+c\int _MRg_{ij}(f\phi _i-f_i)(f\phi _j-f_j)\mathrm{d}\mu -c\int _MR\Delta \phi f^2\mathrm{d}\mu \\ = \,&(1-2c)\lambda \int _MRf^2\mathrm{d}\mu +2c^2\int _MR^2f^2\mathrm{d}\mu +c\int _MR|\nabla f|^2\mathrm{d}\mu \\&+c\int _MR|f\nabla \phi -\nabla f|^2\mathrm{d}\mu -c\int _MR\Delta \phi f^2\mathrm{d}\mu . \end{aligned}$$

The last equality follows from (3.1). We complete the proof of the theorem. \(\square \)

It is obvious now that the first eigenvalue of the geometric operator is monotonic along the Ricci flow as a consequence of Theorem 3.1.

Corollary 3.2

Let \(g(t),t\in [0,T)\), be a solution to the Ricci flow (1.1) on a closed surface \(M^2\) with nonnegative scalar curvature. Assume that \(0<c\le \frac{1}{2}\) and \(R\ge \frac{1}{2c}\Delta \phi ,\forall t\in [0,T)\), then the first eigenvalue of the operator \(-\Delta _{\phi }+cR\) is nondecreasing under the Ricci flow.

Remark 3.3

For \(c=\frac{1}{2}\) the same result was obtained by the first author, Yang and Zhu in [5, 6]. Moreover, for \(c>\frac{1}{4}\) the first author, Xu and Zhu proved a similar monotonicity along the system of Ricci flow coupled to a heat equation in [4].

4 Some Monotonic Quantities and Estimates

In the last section, we derive some monotonic quantities and lower bounds for the first eigenvalues on closed surfaces using the previous results.

Now let us come to the normalized Ricci flow. By Theorem 2.4 we can get the evolution equation of the first eigenvalue under the normalized Ricci flow.

Theorem 4.1

Let \(g(t),t\in [0,T)\), be a solution to the normalized Ricci flow (1.2) on a closed surface \(M^2\). Assume that \(\lambda (t)\) and f(x, t) are defined as above (3.1). Then we have

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t}\lambda (t)=&-r\lambda +(1-2c)\lambda \int _MRf^2\mathrm{d}\mu +2c^2\int _MR^2f^2\mathrm{d}\mu +c\int _MR|\nabla f|^2\mathrm{d}\mu \nonumber \\&+c\int _MR|f\nabla \phi -\nabla f|^2\mathrm{d}\mu -c\int _MR\Delta \phi f^2\mathrm{d}\mu . \end{aligned}$$
(4.1)

Proof

The same computation as Theorem 3.1, using \(R_{ij}=\frac{1}{2}Rg_{ij}\) and (2.8) we get

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t}\lambda (t)&=-r\lambda +(1-2c)\lambda \int _MRf^2\mathrm{d}\mu +c(2c-1)\int _MR^2f^2\mathrm{d}\mu \\&\quad \,+(2c-1)\int _M R|\nabla f|^2\mathrm{d}\mu \\&\quad \,+\int _MR|\nabla f|^2\mathrm{d}\mu +c\int _MR(-2ff_i\phi _i-f^2\Delta \phi +f^2|\nabla \phi |^2)\mathrm{d}\mu \\&\quad \,+c\int _MR^2f^2\mathrm{d}\mu \\&=-r\lambda +(1-2c)\lambda \int _MRf^2\mathrm{d}\mu +2c^2\int _MR^2f^2\mathrm{d}\mu +2c\int _M R|\nabla f|^2\mathrm{d}\mu \\&\quad \,+c\int _MR(-2ff_i\phi _i-f^2\Delta \phi +f^2|\nabla \phi |^2)\mathrm{d}\mu \\&=-r\lambda +(1-2c)\lambda \int _MRf^2\mathrm{d}\mu +2c^2\int _MR^2f^2\mathrm{d}\mu +c\int _MR|\nabla f|^2d\mu \\&\quad \,+c\int _MR|f\nabla \phi -\nabla f|^2\mathrm{d}\mu -c\int _MR\Delta \phi f^2\mathrm{d}\mu . \end{aligned}$$

\(\square \)

Therefore, from Theorem 4.1 it is easy to see that the following monotonicity holds under the normalized Ricci flow.

Corollary 4.2

Let \(g(t),t\in [0,T)\), be a solution to the normalized Ricci flow (1.2) on a closed surface \(M^2\) with nonnegative scalar curvature, and \(\lambda (t)\) be the first eigenvalue of the operator \(-\Delta _{\phi }+cR\). Assume that \(0<c\le \frac{1}{2}\) and \(R\ge \frac{1}{2c}\Delta \phi ,\forall t\in [0,T)\), then \(e^{rt}\lambda (t)\) is nondecreasing under the normalized Ricci flow. In particular, the first eigenvalue has a time-dependent lower bound

$$\begin{aligned} \lambda (t)\ge \lambda (0)e^{-rt}. \end{aligned}$$

Remark 4.3

When \(\phi \) is a constant function on the surface \(M^2\), the same result had been given in [3].

Since dimension \(n=2\), the average scalar curvature r is a constant. Now we assume that \(r\ne 0\). Notice that on a surface the scalar curvature under the normalized Ricci flow evolves by

$$\begin{aligned} \frac{\partial R}{\partial t}=\Delta R+R^2-rR. \end{aligned}$$

Let \(\rho (t)\) and \(\sigma (t)\) be two solutions to the ODE \(y'=y^2-ry\) with initial value respectively

$$\begin{aligned} \rho (0)=\max _{x\in M} R(0)\qquad {\text {and}} \qquad \sigma (0)=\min _{x\in M} R(0). \end{aligned}$$

By the maximum principle, we have

$$\begin{aligned} R(t)\le \rho (t)=\frac{r}{1+(\frac{r}{\rho (0)}-1)e^{rt}} \end{aligned}$$
(4.2)

and

$$\begin{aligned} R(t)\ge \sigma (t)=\frac{r}{1+(\frac{r}{\sigma (0)}-1)e^{rt}}. \end{aligned}$$
(4.3)

Thus, we can also get the following monotonic quantity on a closed surface from Theorem 4.1.

Theorem 4.4

Let \(g(t),t\in [0,T)\), be a solution to the normalized Ricci flow (1.2) on a closed surface \(M^2\) with positive scalar curvature, and \(\lambda (t)\) be the first eigenvalue of the operator \(-\Delta _{\phi }+cR\). If \(\forall c>0\) and \(R\ge \frac{1}{2c}\Delta \phi , \forall t\in [0,T)\), then \(\dfrac{\rho (t)^{2c}}{\sigma (t)}e^{2crt}\lambda (t)\) is nondecreasing under the normalized Ricci flow. In particular, the first eigenvalue has a time-dependent lower bound

$$\begin{aligned} \lambda (t)\ge \lambda (0)\frac{\sigma (t)\rho (0)^{2c}}{\sigma (0)\rho (t)^{2c}}e^{-2crt}. \end{aligned}$$

Proof

The positive scalar curvature on \(M^2\) implies that \(\sigma (0)>0\). According to (4.1)–(4.3), we have

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t}\lambda (t)=&-r\lambda +(1-2c)\lambda \int _MRf^2\mathrm{d}\mu +2c^2\int _MR^2f^2\mathrm{d}\mu +c\int _MR|\nabla f|^2\mathrm{d}\mu \\&+c\int _MR|f\nabla \phi -\nabla f|^2\mathrm{d}\mu -c\int _MR\Delta \phi f^2\mathrm{d}\mu \\ \ge&-r\lambda +(1-2c)\lambda \int _M R f^2 \mathrm{d}\mu \\ \ge&-r\lambda +\lambda [\sigma (t)-2c\rho (t)]. \end{aligned}$$

This leads to

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t}\left[ \frac{\rho (t)^{2c}}{\sigma (t)}e^{2crt}\lambda (t)\right] \ge 0. \end{aligned}$$

The lower bound follows from the monotonicity directly.\(\square \)

Finally, if we change some conditions in Theorem 4.4, there is also another monotonic quantity on a closed surface.

Theorem 4.5

Let \(g(t),t\in [0,T)\), be a solution to the normalized Ricci flow (1.2) on a closed surface \(M^2\) with positive scalar curvature, and \(\lambda (t)\) be the first eigenvalue of the operator \(-\Delta _{\phi }+cR\). If \(0<c\le \frac{1}{2}\) and \(R\ge \frac{1}{c}\Delta \phi , \forall t\in [0,T)\), then \(\dfrac{e^{crt}}{\sigma (t)^{1-c}}\lambda (t)\) is nondecreasing under the normalized Ricci flow. In particular, the first eigenvalue has a time-dependent lower bound

$$\begin{aligned} \lambda (t)\ge \lambda (0)\left( \frac{\sigma (t)}{\sigma (0)}\right) ^{1-c}e^{-crt}. \end{aligned}$$

Proof

By the divergence theorem and (3.1), we have

$$\begin{aligned} \int _M|\nabla f|^2\mathrm{d}\mu =-\int _Mf\Delta _{\phi }f\mathrm{d}\mu =\lambda -c\int _MRf^2\mathrm{d}\mu . \end{aligned}$$
(4.4)

Note that the scalar curvature on \(M^2\) is positive and \(R\ge \dfrac{1}{c}\Delta \phi \). Substituting (4.3) and (4.4) into (4.1), we arrive at

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t}\lambda (t)=&-r\lambda +(1-2c)\lambda \int _MRf^2\mathrm{d}\mu +2c^2\int _MR^2f^2\mathrm{d}\mu +c\int _MR|\nabla f|^2\mathrm{d}\mu \\&+c\int _MR|f\nabla \phi -\nabla f|^2\mathrm{d}\mu -c\int _MR\Delta \phi f^2\mathrm{d}\mu \\ \ge&-r\lambda +(1-2c)\lambda \int _M R f^2 \mathrm{d}\mu +c^2\int _MR^2f^2\mathrm{d}\mu +c\int _MR|\nabla f|^2\mathrm{d}\mu \\ \ge&-r\lambda +(1-2c)\sigma (t)\lambda +c^2\int _MR^2f^2\mathrm{d}\mu +c\sigma (t)\lambda -c^2\int _M\sigma (t)Rf^2\mathrm{d}\mu \\ \ge&-r\lambda +(1-c)\sigma (t)\lambda . \end{aligned}$$

This implies that

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t}\left[ \frac{e^{crt}}{\sigma (t)^{1-c}}\lambda (t)\right] \ge 0. \end{aligned}$$

Hence, it follows that the time-dependent lower bound holds now. \(\square \)

Remark 4.6

Here we just consider the first eigenvalues on closed surfaces with positive Euler characteristic class. For the other cases the similar monotonic formulae and estimates can also be obtained when both the first and second derivatives of \(\phi \) along the normalized Ricci flow are bounded.