Evolution for First Eigenvalue of L_T,f on an Evolving Riemannian Manifold

Abstract

In this paper, evolution formulas for the first non-zero eigenvalue of the operator $L_{T,f}$ on a weighted closed Riemannian manifold along the Ricci flow as well as along the Yamabe flow are formulated. Some monotonic quantities are also derived for the normalized Ricci flow on Bianchi classes.

1. Introduction

Let $(M^n,g)$ be a closed Riemannian manifold of dimension n with weighted volume measure $e^{-f\left(x\right)}d\nu$, where $f\in C^2\left(M\right)$ and $d\nu$ be the Riemannian volume measure on $(M^n,g)$. The triple $(M^n,g,e^{-f}d\nu )$ is a smooth metric measure space. One can see from the analysis that there are many similarities between Riemannian manifolds and weighted Riemannian manifolds. In this space, one has the ability to change the measure of the space without hampering the underlying geometric structure. This ability happens to be a powerful tool for analysis. Study of such spaces is an interesting area of research in mathematics.

A smooth 1-parameter family of metrics $g\left(t\right)$ on a Riemannian manifold M is called Hamilton’s Ricci flow if it satisfies

$$\frac{\partial }{\partial t}{g}_{ij}=-2R_{ij},$$

(1)

with $g\left(0\right)=g_0$, where $R_{ij}$ is the Ricci tensor and normalized Ricci flow is defined by

$$\frac{\partial }{\partial t}{g}_{ij}=-2R_{ij}+\frac{2}{n}r{g}_{ij},$$

(2)

where $r=\frac{{\int }_{M}Rd\nu }{{\int }_{M}d\nu }$ is the average of the scalar curvature of the metric $g\left(t\right)$. For any smooth metric $g_0$ on a closed Riemannian manifold M, the existence and uniqueness of the solution $g\left(t\right)$ to the Ricci flow equation on the time interval $[0,T_0)$ for sufficient $T_0$ was proved by Hamilton [1].

A smooth 1-parameter family of smooth metrics $g\left(t\right)$ on a Riemannian manifold is called a Yamabe flow if it satisfies

$$\frac{\partial }{\partial t}{g}_{ij}=-R{g}_{ij},$$

(3)

with $g\left(0\right)=g_0$, where R is the scalar curvature of M and normalized Yamabe flow is defined by

$$\frac{\partial }{\partial t}{g}_{ij}=-(R-r)g_{ij},$$

(4)

with $g\left(0\right)=g_0$, where $r=\frac{{\int }_{M}Rd\nu }{{\int }_{M}d\nu }$ is the average of the scalar curvature of the Riemannian metric $g\left(t\right)$. The Yamabe flow has been studied in [2]. In case of Riemannian manifolds with time dependent metrics, the potential function f depends on time t unless otherwise stated. There are many papers studied special flows; see [3].

The topic of evolution of the eigenvalues of geometric operators has been widely studied in recent years. Many mathematicians investigated properties of eigenvalues of geometric operators such as Laplace operator, p-Laplace operator, weighted Laplace operator, etc. along various geometric flows. Perelman, in [4], showed that the first eigenvalue of the geometric operator $-4\mathrm{\Delta }+R$ is non-decreasing along the Ricci flow. Later, Cao [5] showed the similar result for the operator $-\mathrm{\Delta }+\frac{R}{2}$ on a manifold with non-negative curvature operator. Zeng et al. [6] extended the geometric operator $-4\mathrm{\Delta }+R$ to the operator $-\mathrm{\Delta }+cR$ on closed Riemannian manifolds and studied the monotonicity of eigenvalues of the operator $-\mathrm{\Delta }+cR$ along the Ricci–Bourguignon flow. Azami, in [7], showed that the first eigenvalue of the Witten–Laplace operator is monotonic along the Ricci–Bourguignon flow with some assumptions and in [8], studied the evolution of the first eigenvalue of the weighted p-Laplacian along the Yamabe flow. Different geometric operators along different geometric flows on Riemannian manifolds are also studied in [9,10,11,12].

2. Preliminaries

We use the identification of a $(1,1)$-tensor T with its associated $(0,2)$-tensor by

$$g(T\left(X\right),Y)=\tilde{T}(X,Y),\mathrm{for}\mathrm{all}X,Y\in \chi \left(M\right).$$

For simplification, we slightly abuse the notation $\tilde{T}$ and denote it by T. Thus, the above equation becomes

$$g\left(T\right(X),Y)=T(X,Y),\mathrm{for}\mathrm{all}X,Y\in \chi \left(M\right).$$

(5)

In particular, the metric tensor g is identified with the identity $Id$ in $\chi \left(M\right)$. The divergence of the $(1,1)$-tensor T on $(M,g)$ is given by

$$\left(divT\right)\left(v\right)=tr(w⟼\left({\nabla }_{w}T\right)\left(v\right)),\forall v,w\in T_p\left(M\right).$$

(6)

For any smooth vector field X on M, the weighted version of the divergence is given by

$$di{v}_{f}X=div\left(X\right)-\langle \nabla f,X\rangle .$$

(7)

Then, the weighted version of the divergence theorem for any smooth vector field X on M is given by

$${\int }_{M}di{v}_{f}X{e}^{-f}d\nu =0.$$

(8)

The integration by parts formula gives

$${\int }_{M}udi{v}_{f}X{e}^{-f}d\nu =-{\int }_M\langle \nabla u,X\rangle e^{-f}d\nu ,$$

(9)

for any smooth function u and any vector field X on M. Let $L_{T,f}$ be the second order differential operator defined for any smooth function u on M given by

$$L_{T,f}u=-div(T\nabla u)+\langle \nabla f,T\nabla u\rangle .$$

(10)

We consider the following eigenvalue problem

$$L_{T,f}u=\lambda u\mathrm{in}M.$$

(11)

The eigenvalues of the operator $L_{T,f}$ are discrete and increasing. The evolution of the first non-zero eigenvalue of the above operator is given by the following equation:

$$\lambda \left(t\right)=\frac{{\int }_{M}T(\nabla u,\nabla u)e^{-f}d\nu }{{\int }_M{u}^2{e}^{-f}d\nu }.$$

(12)

For simplicity, we consider the normalized condition, i.e., ${\int }_M{u}^2{e}^{-f}d\nu =1$. Then, the evolution of the first non-zero eigenvalue of the operator $L_{T,f}$ is given by

$$\lambda \left(t\right)={\int }_{M}T(\nabla u,\nabla u)e^{-f}d\nu .$$

(13)

The $C^1$-differentiability of the eigenvalue $\lambda \left(t\right)$ and its corresponding normalized eigenfunctions are assumed to be considered.

3. Evolution along Ricci flow

In this section, the evolution formula for the first non-zero eigenvalue of the operator $L_{T,f}$ is formulated along the Ricci flow as well as along the normalized Ricci flow. Before going to our main results, we recall that, under the Ricci flow,

$$\frac{\partial }{\partial t}d\nu =-Rd\nu ,$$

and thus we have

$$\frac{\partial }{\partial t}\left(e^{-f}d\nu \right)=-(f_t+R)e^{-f}d\nu .$$

(14)

Theorem 1.

Let $(M^n,g\left(t\right))$ be a solution to the Ricci flow on a closed Riemannian manifold $(M^n,g_0)$. If $\lambda \left(t\right)$ denotes the evolution of the first non-zero eigenvalue of the operator $L_{T,f}$ along the Ricci flow, then

$$\begin{array}{cc}\hfill \frac{d\lambda }{dt}=& {\int }_M\frac{\partial }{\partial t}\left(T^{ij}\right)u_i{u}_j{e}^{-f}d\nu +\lambda {\int }_M(R+f_t)u^2{e}^{-f}d\nu -{\int }_{M}T(\nabla u,\nabla u)R{e}^{-f}d\nu \hfill \\ \hfill & -{\int }_{M}T(\nabla u,\nabla u)f_t{e}^{-f}d\nu ,\hfill \end{array}$$

(15)

where u is the normalized eigenfunction corresponding to the eigenvalue λ.

Proof. 

Differentiating (13) with respect to t, we obtain

$$\begin{array}{cc}\hfill \frac{d}{dt}\lambda =& {\int }_M\frac{\partial }{\partial t}\left(T(\nabla u,\nabla u)\right)e^{-f}d\nu +{\int }_{M}T(\nabla u,\nabla u)\frac{\partial }{\partial t}\left(e^{-f}d\nu \right)\hfill \\ \hfill =& {\int }_M\frac{\partial }{\partial t}\left(T^{ij}\right)u_i{u}_j{e}^{-f}d\nu +2{\int }_M{T}^{ij}{\left(u_t\right)}_i{u}_j{e}^{-f}d\nu -{\int }_{M}T(\nabla u,\nabla u)(f_t+R)e^{-f}d\nu \hfill \\ \hfill =& {\int }_M\frac{\partial }{\partial t}\left(T^{ij}\right)u_i{u}_j{e}^{-f}d\nu -2{\int }_{M}div(T\nabla u{e}^{-f})u_{t}d\nu -{\int }_{M}T(\nabla u,\nabla u)R{e}^{-f}d\nu \hfill \\ \hfill & -{\int }_{M}T(\nabla u,\nabla u)f_t{e}^{-f}d\nu .\hfill \end{array}$$

(16)

Differentiating the normalized condition ${\int }_M{u}^2{e}^{-f}d\nu =1$ with respect to t, we have

$$\begin{array}{c}\hfill {\int }_{M}u{u}_t{e}^{-f}d\nu =\frac{1}{2}{\int }_M{u}^2(R+f_t)e^{-f}d\nu .\end{array}$$

(17)

Now,

$$\begin{array}{cc}\hfill {\int }_{M}div(T\nabla u{e}^{-f})u_{t}d\nu & ={\int }_M(div(T\nabla u)-\langle \nabla f,T\nabla u\rangle )u_t{e}^{-f}d\nu \hfill \\ \hfill & =-{\int }_M{L}_{T,f}u{u}_t{e}^{-f}d\nu \hfill \\ \hfill & =-\lambda {\int }_{M}u{u}_t{e}^{-f}d\nu =-\frac{\lambda }{2}{\int }_M{u}^2(R+f_t)e^{-f}d\nu .\hfill \end{array}$$

(18)

Finally, using (18) in (16), we have our result (15). □

Remark 1.

If we take $T=Id$, then we have $di{v}_f(T\nabla u)=\mathrm{\Delta }u-\langle \nabla u,\nabla f\rangle ={\mathrm{\Delta }}_{f}u$, i.e., weighted Laplacian. Thus, the eigenvalue $\lambda \left(t\right)$ along the Ricci flow satisfied the following equation:

$$\begin{array}{cc}\hfill \frac{d\lambda }{dt}=& {\int }_M\frac{\partial }{\partial t}\left(g^{ij}\right)u_i{u}_j{e}^{-f}d\nu +\lambda {\int }_M(R+f_t)u^2{e}^{-f}d\nu -{\int }_M{|\nabla u|}^{2}R{e}^{-f}d\nu \hfill \\ \hfill & -{\int }_M{|\nabla u|}^2{f}_t{e}^{-f}d\nu .\hfill \end{array}$$

(19)

Again along the Ricci flow, we have $\frac{\partial }{\partial t}\left(g^{ij}\right)=2R^{ij}$. Thus, from (19), we obtain

$$\begin{array}{cc}\hfill \frac{d\lambda }{dt}=& 2{\int }_{M}Ric(\nabla u,\nabla u)e^{-f}d\nu +\lambda {\int }_M(R+f_t)u^2{e}^{-f}d\nu -{\int }_M{|\nabla u|}^{2}R{e}^{-f}d\nu \hfill \\ \hfill & -{\int }_M{|\nabla u|}^2{f}_t{e}^{-f}d\nu .\hfill \end{array}$$

(20)

Theorem 2.

Let $(M^n,g\left(t\right))$ be a solution to the Ricci flow on a closed homogeneous Riemannian manifold $(M^n,g_0)$. If $\lambda \left(t\right)$ denotes the evolution of the first non-zero eigenvalue of the operator $L_{T,f}$ then along the Ricci flow

$$\begin{array}{cc}\hfill \frac{d\lambda }{dt}=& {\int }_M\frac{\partial }{\partial t}\left(T^{ij}\right)u_i{u}_j{e}^{-f}d\nu +\lambda {\int }_M{f}_t{u}^2{e}^{-f}d\nu -{\int }_{M}T(\nabla u,\nabla u)f_t{e}^{-f}d\nu ,\hfill \end{array}$$

where u is the normalized eigenfunction corresponding to the eigenvalue λ.

Proof. 

For a homogeneous Riemannian manifold evolving along the Ricci flow, the scalar curvature R remains constant. Then, from (15), we have

$$\begin{array}{cc}\hfill \frac{d\lambda }{dt}=& {\int }_M\frac{\partial }{\partial t}\left(T^{ij}\right)u_i{u}_j{e}^{-f}d\nu +\lambda R{\int }_M{u}^2{e}^{-f}d\nu -R{\int }_{M}T(\nabla u,\nabla u)e^{-f}d\nu \hfill \\ \hfill & +\lambda {\int }_M{f}_t{u}^2{e}^{-f}d\nu -{\int }_{M}T(\nabla u,\nabla u)f_t{e}^{-f}d\nu \hfill \\ \hfill =& {\int }_M\frac{\partial }{\partial t}\left(T^{ij}\right)u_i{u}_j{e}^{-f}d\nu +\lambda {\int }_M{f}_t{u}^2{e}^{-f}d\nu -{\int }_{M}T(\nabla u,\nabla u)f_t{e}^{-f}d\nu .\hfill \end{array}$$

This completes the proof. □

Corollary 1.

Let $(M^n,g\left(t\right))$ be a solution to the Ricci flow on a closed Riemannian manifold $(M^n,g_0)$, $f_t=0$, $\frac{\partial T^{ij}}{\partial t}\ge (R+\alpha )T^{ij}$ and $R\ge -\alpha$ for some constant α. If $\lambda \left(t\right)$ denotes the evolution of the first non-zero eigenvalue of the operator $L_{T,f}$ along the Ricci flow, then $\lambda \left(t\right)$ is a non-decreasing function:

Proof. 

Applying the conditions $f_t=0$, $\frac{\partial T^{ij}}{\partial t}\ge (R+\alpha )T^{ij}$ and $R\ge -\alpha$ in (15) we have

$$\begin{array}{cc}\hfill \frac{d\lambda }{dt}\ge & {\int }_M(R+\alpha )T(\nabla u,\nabla u)e^{-f}d\nu +\lambda {\int }_{M}R{u}^2{e}^{-f}d\nu -{\int }_{M}T(\nabla u,\nabla u)R{e}^{-f}d\nu \hfill \\ \hfill \ge & \alpha {\int }_{M}T(\nabla u,\nabla u)e^{-f}d\nu -\alpha {\int }_{M}T(\nabla u,\nabla u)e^{-f}d\nu =0.\hfill \end{array}$$

This shows that $\lambda \left(t\right)$ is a non-decreasing function with respect to time variable t under the Ricci flow. □

Corollary 2.

Let $(M^3,g\left(t\right)),t\in [0,T_0)$ be a solution to the Ricci flow on a closed Riemannian manifold whose Ricci curvature is positive initially, $f_t=0$ and $\frac{\partial T^{ij}}{\partial t}\ge \alpha R^{ij}$ for some constant α such that $0<\alpha <1$. Then, there exists $\overline{t}\in [0,T_0)$ depending on $g_0$ such that, for each $t\in [\overline{t},T_0)$, the first eigenvalue $\lambda \left(t\right)$ of $L_{T,f}$ is increasing:

Proof. 

From (15), we have

$$\begin{array}{cc}\hfill \frac{d\lambda }{dt}\ge & \alpha {\int }_{M}RT(\nabla u,\nabla u)e^{-f}d\nu +\lambda {\int }_{M}R{u}^2{e}^{-f}d\nu -{\int }_{M}RT(\nabla u,\nabla u)e^{-f}d\nu \hfill \\ \hfill =& (\alpha -1){\int }_{M}RT(\nabla u,\nabla u)e^{-f}d\nu +\lambda {\int }_{M}R{u}^2{e}^{-f}d\nu .\hfill \end{array}$$

As proved in [1], for any $\eta >0$, we can obtain $T_{\eta }\in [0,T_0)$ such that for $t\in [T_{\eta },T_0)$, $R\ge (1-\eta )R_{max}$. Then, there exists a $\overline{t}\in [0,T_0)$ such that, for $t\in [\overline{t},T_0)$, $R\ge (1-\alpha )R_{max}$. Hence,

$$\frac{d\lambda }{dt}\ge (\alpha -1)R_{max}\lambda +\lambda R_{min}=\lambda (R_{min}+(\alpha -1)R_{max})\ge 0.$$

Therefore, the proposition follows. □

Corollary 3.

Let $(M^n,g\left(t\right)),t\in [0,T_0)$ be a solution to the Ricci flow on a closed Riemannian manifold $(M^n,g_0)$, $\frac{\partial T^{ij}}{\partial t}\ge \alpha (R+f_t)T^{ij}$ and $R+f_t\ge \beta$ for some constant $\alpha ,\beta$ such that $\alpha \ge 1$. Then, the quantity $\lambda \left(t\right)e^{-\alpha \beta t}$ is increasing along the Ricci flow.

Proof. 

From (15), we have

$$\begin{array}{cc}\hfill \frac{d\lambda }{dt}\ge & (\alpha -1){\int }_M(R+f_t)T(\nabla u,\nabla u)e^{-f}d\nu +\lambda {\int }_M(R+f_t)T(\nabla u,\nabla u)e^{-f}d\nu \hfill \\ \hfill \ge & (\alpha -1)\beta \lambda +\lambda \beta =\alpha \beta \lambda ,\hfill \end{array}$$

which implies that $\frac{1}{\lambda }\frac{d\lambda }{dt}\ge \alpha \beta$. Integrating this inequality with respect to t on $[t_1,t_2]\subseteq [0,T_0)$, we obtain

$$ln\frac{\lambda \left(t_2\right)}{\lambda \left(t_1\right)}\ge \alpha \beta (t_2-t_1).$$

Hence,

$$\frac{\lambda \left(t_2\right)}{\lambda \left(t_1\right)}\ge e^{\alpha \beta (t_2-t_1)};$$

equivalently, we have

$$\lambda \left(t_2\right)e^{-\alpha \beta t_2}\ge \lambda \left(t_1\right)e^{-\alpha \beta t_1}.$$

Thus, the quantity $\lambda \left(t\right)e^{-\alpha \beta t}$ is increasing along the Ricci flow. □

Corollary 4.

Let $(M^n,g\left(t\right)),t\in [0,T_0)$ be a solution to the Ricci flow on a closed homogeneous Riemannian manifold $(M^n,g_0)$, $\frac{\partial T^{ij}}{\partial t}\ge (\alpha +f_t)T^{ij}$ and $f_t\ge \beta$ for some positive constants $\alpha ,\beta$. Then, the quantity $\lambda \left(t\right)e^{-(\alpha +\beta )t}$ is non-decreasing along the Ricci flow.

Proof. 

Using the assumptions, from Theorem 2, we have

$$\begin{array}{cc}\hfill \frac{d\lambda }{dt}\ge & {\int }_M(\alpha +f_t)T(\nabla u,\nabla u)e^{-f}d\nu +\lambda \beta {\int }_M{u}^2{e}^{-f}d\nu -{\int }_{M}T(\nabla u,\nabla u)f_t{e}^{-f}d\nu .\hfill \end{array}$$

Thus,

$$\frac{d\lambda }{dt}\ge (\alpha +\beta )\lambda .$$

Integrating this inequality with respect to t on $[t_1,t_2]\subseteq [0,T_0)$, we obtain

$$ln\frac{\lambda \left(t_2\right)}{\lambda \left(t_1\right)}\ge (\alpha +\beta )(t_2-t_1),$$

hence

$$\frac{\lambda \left(t_2\right)}{\lambda \left(t_1\right)}\ge e^{(\alpha +\beta )(t_2-t_1)}.$$

This implies that

$$\lambda \left(t_2\right)e^{-(\alpha +\beta )t_2}\ge \lambda \left(t_1\right)e^{-(\alpha +\beta )t_1}.$$

Thus, the quantity $\lambda \left(t\right)e^{-(\alpha +\beta )t}$ is increasing along the Ricci flow. □

Let us now consider a smooth function $h:M\to \mathbb{R}$, such that $T=hId$, i.e., $T^{ij}=h{g}^{ij}$. Then, two cases arises (i) h is independent of time t and (ii) h is dependent on time t. First, we consider the case that h is independent of time t, then we have

$$\frac{\partial }{\partial t}\left(T^{ij}\right)=\frac{\partial }{\partial t}\left(h{g}^{ij}\right)=2h{R}^{ij},$$

(21)

using the fact that under the Ricci flow $\frac{\partial }{\partial t}\left(g^{ij}\right)=2R^{ij}$.

Theorem 3.

Let $(M^n,g\left(t\right))$ be a solution to the Ricci flow on a closed Riemannian manifold $(M^n,g_0)$. If $\lambda \left(t\right)$ denotes the evolution of the first non-zero eigenvalue of the operator $L_{T,f}$ with $T=hId$, then

$$\begin{array}{cc}\hfill \frac{d\lambda }{dt}=& 2{\int }_{M}hRic(\nabla u,\nabla u)e^{-f}d\nu -{\int }_{M}h{|\nabla u|}^2(R+f_t)e^{-f}d\nu \hfill \\ \hfill & +\lambda {\int }_M(R+f_t)u^2{e}^{-f}d\nu ,\hfill \end{array}$$

(22)

where h is independent of time t and u is normalized eigenfunction corresponding to the eigenvalue λ.

Proof. 

From Theorem 1, we obtain

$$\begin{array}{cc}\hfill \frac{d\lambda }{dt}=& {\int }_M\frac{\partial }{\partial t}\left(T^{ij}\right)u_i{u}_j{e}^{-f}d\nu +\lambda {\int }_M(R+f_t)u^2{e}^{-f}d\nu -{\int }_{M}T(\nabla u,\nabla u)R{e}^{-f}d\nu \hfill \\ \hfill & -{\int }_{M}T(\nabla u,\nabla u)f_t{e}^{-f}d\nu \hfill \\ \hfill =& 2{\int }_{M}h{R}^{ij}{u}_i{u}_j{e}^{-f}d\nu +\lambda {\int }_M(R+f_t)u^2{e}^{-f}d\nu -{\int }_{M}h{|\nabla u|}^{2}R{e}^{-f}d\nu \hfill \\ \hfill & -{\int }_{M}h{|\nabla u|}^2{f}_t{e}^{-f}d\nu .\hfill \end{array}$$

This completes the proof. □

Corollary 5.

Let $(M^n,g\left(t\right))$ be a solution to the Ricci flow on a closed homogeneous Riemannian manifold $(M^n,g_0)$. If $\lambda \left(t\right)$ denotes the evolution of the first non-zero eigenvalue of the operator $L_{T,f}$ with $T=hId$, then

$$\begin{array}{c}\hfill \frac{d\lambda }{dt}=2{\int }_{M}hRic(\nabla u,\nabla u)e^{-f}d\nu +\lambda {\int }_M{u}^2{f}_t{e}^{-f}d\nu -{\int }_{M}h{|\nabla u|}^2{f}_t{e}^{-f}d\nu ,\end{array}$$

(23)

where h is independent of time t and u is normalized eigenfunction corresponding to the eigenvalue λ.

Now, we consider the second case $T=hId$ where h depends on time t. Thus, we obtain

$$\frac{\partial }{\partial t}\left(T^{ij}\right)=\frac{\partial }{\partial t}\left(h{g}^{ij}\right)=h_t{g}^{ij}+2h{R}^{ij},$$

(24)

using $\frac{\partial }{\partial t}\left(g^{ij}\right)=2R{g}^{ij}$, where $\frac{\partial }{\partial t}\left(h\right)=h_t$.

Theorem 4.

Let $(M^n,g\left(t\right))$ be a solution to the Ricci flow on a closed Riemannian manifold $(M^n,g_0)$. If $\lambda \left(t\right)$ denotes the evolution of the first non-zero eigenvalue of the operator $L_{T,f}$ with $T=hId$, then

$$\begin{array}{cc}\hfill \frac{d\lambda }{dt}=& {\int }_M{h}_t{|\nabla u|}^2{e}^{-f}d\nu +2{\int }_{M}hRic(\nabla u,\nabla u)e^{-f}d\nu +\lambda {\int }_M{u}^2(R+f_t)e^{-f}d\nu \hfill \\ \hfill & -{\int }_{M}h{|\nabla u|}^2(R+f_t)e^{-f}d\nu ,\hfill \end{array}$$

(25)

where h depends on time t and u is normalized eigenfunction corresponding to the eigenvalue λ.

Proof. 

From Theorem 1, we obtain

$$\begin{array}{cc}\hfill \frac{d\lambda }{dt}=& {\int }_M\frac{\partial }{\partial t}\left(T^{ij}\right)u_i{u}_j{e}^{-f}d\nu +\lambda {\int }_M(R+f_t)u^2{e}^{-f}d\nu -{\int }_{M}T(\nabla u,\nabla u)R{e}^{-f}d\nu \hfill \\ \hfill & -{\int }_{M}T(\nabla u,\nabla u)f_t{e}^{-f}d\nu \hfill \\ \hfill =& {\int }_M(h_t{g}^{ij}+2h{R}^{ij})u_i{u}_j{e}^{-f}d\nu +\lambda {\int }_M(R+f_t)u^2{e}^{-f}d\nu -{\int }_{M}h{|\nabla u|}^{2}R{e}^{-f}d\nu \hfill \\ \hfill & -{\int }_{M}h{|\nabla u|}^2{f}_t{e}^{-f}d\nu .\hfill \end{array}$$

This completes the proof. □

Theorem 5.

Let $(M^n,g\left(t\right))$ be a solution to the normalized Ricci flow on a closed Riemannian manifold $(M^n,g_0)$. If $\lambda \left(t\right)$ denotes the evolution of the first non-zero eigenvalue of the operator $L_{T,f}$ along the normalized Ricci flow, then

$$\begin{array}{cc}\hfill \frac{d\lambda }{dt}=& {\int }_M\frac{\partial }{\partial t}\left(T^{ij}\right)u_i{u}_j{e}^{-f}d\nu +\lambda {\int }_M(R+f_t)u^2{e}^{-f}d\nu -{\int }_{M}T(\nabla u,\nabla u)f_t{e}^{-f}d\nu \hfill \\ \hfill & -{\int }_{M}T(\nabla u,\nabla u)R{e}^{-f}d\nu ,\hfill \end{array}$$

(26)

where u is the normalized eigenfunction corresponding to the eigenvalue λ.

Proof. 

Differentiating (13) with respect to time t, we obtain

$$\begin{array}{cc}\hfill \frac{d}{dt}\lambda =& {\int }_M\frac{\partial }{\partial t}\left(T^{ij}\right)u_i{u}_j{e}^{-f}d\nu +2{\int }_M{T}^{ij}{\left(u_t\right)}_i{u}_j{e}^{-f}d\nu -{\int }_{M}T(\nabla u,\nabla u)f_t{e}^{-f}d\nu \hfill \\ \hfill & -{\int }_{M}T(\nabla u,\nabla u)R{e}^{-f}d\nu +r{\int }_{M}T(\nabla u,\nabla u)e^{-f}d\nu \hfill \end{array}$$

(27)

$$\begin{array}{cc}\hfill =& {\int }_M\frac{\partial }{\partial t}\left(T^{ij}\right)u_i{u}_j{e}^{-f}d\nu -2{\int }_{M}div(T\nabla u{e}^{-f})u_{t}d\nu -{\int }_{M}T(\nabla u,\nabla u)R{e}^{-f}d\nu \hfill \\ \hfill & -{\int }_{M}T(\nabla u,\nabla u)f_t{e}^{-f}d\nu +r\lambda ,\hfill \end{array}$$

(28)

where in (27) we have used the fact that

$$\frac{d}{dt}\left(e^{-f}d\nu \right)=-\{f_t+(R-r)\}{e}^{-f}d\nu .$$

(29)

Now, differentiating the normalized condition ${\int }_M{u}^2{e}^{-f}d\nu =1$ with respect to time t, we have

$${\int }_{M}u{u}_t{e}^{-f}d\nu =\frac{1}{2}{\int }_M\{f_t+(R-r)\}{u}^2{e}^{-f}d\nu .$$

(30)

Thus,

$$\begin{array}{cc}\hfill -{\int }_{M}div(T\nabla u{e}^{-f})u_{t}d\nu =& {\int }_M(-div(T\nabla u)+\langle T\nabla u,\nabla f\rangle )u_t{e}^{-f}d\nu \hfill \\ \hfill =& {\int }_M\lambda u{u}_t{e}^{-f}d\nu \hfill \\ \hfill =& \frac{\lambda }{2}{\int }_M\{f_t+(R-r)\}{u}^2{e}^{-f}d\nu .\hfill \end{array}$$

(31)

Thus, using (31) in (28), we have our result. □

4. Eigenvalue Bounds on Bianchi Classes

Locally homogeneous 3-manifold are divided into nine classes. The first group includes $H\left(3\right)$, $H\left(2\right)\times {\mathbb{R}}^1$, $SO\left(3\right)\times {\mathbb{R}}^1$ and the second group which is also known as Bianchi classes includes ${\mathbb{R}}^3$, Heisenberg, $E(1,1)$, $E\left(2\right)$, $SU\left(2\right)$ and $SL(2,\mathbb{R})$. In [13], Milnor showed that there exists a frame ${\left\{X_i\right\}}_{i=1}^3$ such that the metric and the Ricci tensors are diagonalized, and this property is preserved along the Ricci flow. Let ${\left\{{\theta }_i\right\}}_{i=1}^3$ be the dual frame to the Milnor frame. Then, the metric $g\left(t\right)$ can be written as

$$g\left(t\right)=A\left(t\right){\theta }_1\otimes {\theta }_1+B\left(t\right){\theta }_2\otimes {\theta }_2+C\left(t\right){\theta }_3\otimes {\theta }_3,$$

and the Ricci flow becomes a system of ODE in three variables $\left\{A\right(t),B(t),C(t\left)\right\}$. Let $A_0,B_0,C_0$ be the initial value of $A,B,C$ respectively. Recently, many authors studied evolution of the first non-zero eigenvalues of different geometric operators along Ricci flow on Bianchi classes [14,15,16,17,18]. If we assume that $T=hId$ where h is independent of time t and

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & \frac{\partial g_{ij}}{\partial t}=-2R_{ij}+\frac{2}{3}r{g}_{ij},\hfill \\ \hfill & \frac{\partial f}{\partial t}=R,\hfill \end{array}\end{array}$$

(32)

then, from (26), we obtain

$$\begin{array}{cc}\hfill \frac{d\lambda }{dt}=& 2{\int }_{M}h{R}^{ij}{u}_i{u}_j{e}^{-f}d\nu -\frac{2}{3}r{\int }_{M}h{|\nabla u|}^2{e}^{-f}d\nu +2\lambda {\int }_{M}R{u}^2{e}^{-f}d\nu \hfill \\ \hfill & -2{\int }_{M}Rh{|\nabla u|}^2{e}^{-f}d\nu .\hfill \end{array}$$

Thus, on Bianchi classes, we have

$$\frac{d\lambda }{dt}=2{\int }_{M}h{R}^{ij}{u}_i{u}_j{e}^{-f}d\nu -\frac{2}{3}R\lambda .$$

(33)

In addition, if we suppose that $T=hId$ where h is independent of time t and

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & \frac{\partial g_{ij}}{\partial t}=-2R_{ij}+\frac{2}{3}r{g}_{ij},\hfill \\ \hfill & \frac{\partial f}{\partial t}=-R,\hfill \end{array}\end{array}$$

(34)

then, on Bianchi classes, we have

$$\frac{d\lambda }{dt}=2{\int }_{M}h{R}^{ij}{u}_i{u}_j{e}^{-f}d\nu -\frac{2}{3}R\lambda .$$

(35)

Thus, we see that the evolution equations for the eigenvalue $\lambda \left(t\right)$ under the systems (32) and (34) are the same and given by

$$\frac{d\lambda }{dt}=2{\int }_{M}h{R}^{ij}{u}_i{u}_j{e}^{-f}d\nu -\frac{2}{3}R\lambda .$$

(36)

Now, using (36), we investigate the bounds of the first non-zero eigenvalue $\lambda \left(t\right)$ of the operator $L_{T,f}$ on Bianchi classes such as Heisenberg, $E(1,1)$, $E\left(2\right)$, $SU\left(2\right)$, and $SL(2,\mathbb{R})$ under the systems (32) and (34).

Heisenberg: Under a given metric $g_0$, there exists a Milnor frame ${\left\{X_i\right\}}_{i=1}^3$ such that

$$[X_2,X_3]=X_1,[X_3,X_1]=0,[X_1,X_2]=0.$$

Taking the normalization condition $A_0{B}_0{C}_0=1$, we have

$$\begin{array}{cc}\hfill & R_{11}=\frac{1}{2}{A}^3,R_{22}=-\frac{1}{2}{A}^{2}B,\hfill \\ \hfill & R_{33}=-\frac{1}{2}{A}^{2}C,R=-\frac{1}{2}{A}^2.\hfill \end{array}$$

(37)

Theorem 6.

Let $\lambda \left(t\right)$ be the first eigenvalue of the operator $L_{T,f}$ with $T=hId$ for some positive time-independent function h, on a three-dimensional Heisenberg weighted Riemannian manifold $(H^3,g_0)$ under (32) or (34) and also $B_0\ge C_0$. Then, $\lambda \left(t\right)e^{-{\int }_{\tau }^t\frac{4}{3}{A}^{2}ds}$ is non-increasing and $\lambda \left(t\right)e^{{\int }_{\tau }^t\frac{2}{3}{A}^{2}ds}$ is non-decreasing for some τ satisfying $t\ge \tau$.

Proof. 

Given $B_0\ge C_0$. Using (37) in (36), we obtain

$$-A^2\lambda \left(t\right)-\frac{2}{3}R\lambda \left(t\right)\le \frac{d}{dt}\lambda \left(t\right)\le A^2\lambda \left(t\right)-\frac{2}{3}R\lambda \left(t\right),$$

(38)

equivalently,

$$-\frac{2}{3}{A}^2\lambda \left(t\right)\le \frac{d}{dt}\lambda \left(t\right)\le \frac{4}{3}{A}^2\lambda \left(t\right),$$

this shows that $\lambda \left(t\right)e^{-{\int }_{\tau }^t\frac{4}{3}{A}^{2}ds}$ is non-increasing and $\lambda \left(t\right)e^{{\int }_{\tau }^t\frac{2}{3}{A}^{2}ds}$ is non-decreasing. □

E(2): In this case, there exists a Milnor frame for a given metric $g_0$ such that

$$[X_2,X_3]=X_1,[X_3,X_1]=X_2,[X_1,X_2]=0.$$

Under the normalization condition $A_0{B}_0{C}_0=1$, we have

$$\begin{array}{cc}\hfill & R_{11}=\frac{1}{2}A(A^2-B^2),R_{22}=\frac{1}{2}B(B^2-A^2),\hfill \\ \hfill & R_{33}=-\frac{1}{2}C{(A-B)}^2,R=-\frac{1}{2}{(A-B)}^2.\hfill \end{array}$$

(39)

Theorem 7.

Let $\lambda \left(t\right)$ be the first eigenvalue of the operator $L_{T,f}$ with $T=hId$ for some positive time-independent function h, on three-dimensional homogeneous weighted manifold $(E\left(2\right),g_0)$ under (32) or (34) and $A_0\ge B_0$. Then, $\lambda \left(t\right)e^{-{\int }_{\tau }^t\frac{4}{3}(A^2-B^2)ds}$ is non-increasing and $\lambda \left(t\right)e^{{\int }_{\tau }^t(A^2-B^2)ds}$ is non-decreasing for some time τ such that $t\ge \tau$.

Proof. 

Since $A_0\ge B_0$, by using (39), from (36), we obtain

$$-(A^2-B^2)\lambda \left(t\right)\le \frac{d}{dt}\lambda \left(t\right)\le \frac{4}{3}(A^2-B^2)\lambda \left(t\right).$$

(40)

This shows that $\lambda \left(t\right)e^{-{\int }_{\tau }^t\frac{4}{3}(A^2-B^2)ds}$ is non-increasing and $\lambda \left(t\right)e^{{\int }_{\tau }^t(A^2-B^2)ds}$ is non-decreasing. □

E(1,1): For a given metric $g_0$, there exists a Milnor frame such that

$$[X_1,X_2]=0,[X_2,X_3]=-X_1,[X_3,X_1]=X_2.$$

Under the normalization condition $A_0{B}_0{C}_0=1$, we have

$$\begin{array}{cc}\hfill & R_{11}=\frac{1}{2}A(A^2-C^2),R_{22}=-\frac{1}{2}B{(A+C)}^2,\hfill \\ \hfill & R_{33}=\frac{1}{2}C(C^2-A^2),R=-\frac{1}{2}{(A+C)}^2.\hfill \end{array}$$

(41)

Theorem 8.

Let $\lambda \left(t\right)$ be the first eigenvalue of the operator $L_{T,f}$ with $T=hId$ for some positive time-independent function h, on three-dimensional homogeneous weighted manifold $(E(1,1),g_0)$ under (32) or (34). Then, there is a time $\tau \le t$ such that

(1) if $A_0=C_0$, then $\lambda \left(t\right)e^{{\int }_{\tau }^t{(A+C)}^{2}ds}$ is non-decreasing and $\lambda \left(t\right)e^{-{\int }_{\tau }^t\frac{4}{3}{(A+C)}^{2}ds}$ is non-increasing.

(2) If $A_0>C_0$, then $\lambda \left(t\right)e^{{\int }_{\tau }^t\frac{2}{3}{(A+C)}^{2}ds}$ is non-decreasing and $\lambda \left(t\right)e^{-{\int }_{\tau }^t\frac{4}{3}{(A+C)}^2}ds$ is non-increasing.

Proof. 

(1) We have by simple calculation that

$$-{(A+C)}^2\lambda \left(t\right)\le \frac{d\lambda \left(t\right)}{dt}\le \frac{4}{3}{(A+C)}^2\lambda \left(t\right)$$

(42)

Taking integration from $\tau$ to t we have that $\lambda \left(t\right)e^{{\int }_{\tau }^t{(A+C)}^{2}ds}$ is non-decreasing and

$\lambda \left(t\right)e^{-{\int }_{\tau }^t\frac{4}{3}{(A+C)}^{2}ds}$ is non-increasing.

(2) Using the condition $A_0>C_0$, we have that

$$-\frac{2}{3}{(A+C)}^2\lambda \left(t\right)\le \frac{d\lambda \left(t\right)}{dt}\le \frac{4}{3}{(A+C)}^2\lambda \left(t\right).$$

(43)

Then, using the similar method, we obtain that $\lambda \left(t\right)e^{{\int }_{\tau }^t\frac{2}{3}{(A+C)}^{2}ds}$ is non-decreasing and $\lambda \left(t\right)e^{-{\int }_{\tau }^t\frac{4}{3}{(A+C)}^{2}ds}$ is non-increasing. □

SU(2): Similarly for a given metric $g_0$, there is a Milnor frame such that

$$[X_2,X_3]=X_1,[X_3,X_1]=X_2,[X_1,X_2]=X_3.$$

Under the normalization condition $A_0{B}_0{C}_0=1$, we have

$$\begin{array}{cc}\hfill & R_{11}=\frac{1}{2}A[A^2-{(B-C)}^2],R_{22}=\frac{1}{2}B[B^2-{(A-C)}^2],\hfill \\ \hfill & R_{33}=\frac{1}{2}C[C^2-{(A-B)}^2],\hfill \end{array}$$

(44)

and

$$R=\frac{1}{2}[A^2-{(B-C)}^2]+\frac{1}{2}[B^2-{(A-C)}^2]+\frac{1}{2}[C^2-{(A-B)}^2].$$

(45)

Theorem 9.

Let $\lambda \left(t\right)$ be the first eigenvalue of the operator $L_{T,f}$ where $T=hId$ for some positive time-independent function h, on three-dimensional homogeneous weighted manifold $(SU\left(2\right),g_0)$ under (32) or (34). Then, there exists $t\ge \tau$ such that the following results hold:

(1) if $A_0=B_0\ge C_0$, then $\lambda \left(t\right)e^{-{\int }_{\tau }^t\frac{2}{3}[A^2-{(B-C)}^2]ds}$ is non-increasing and $\lambda \left(t\right)e^{-{\int }_{\tau }^{t}2C^{2}ds}$ is non-decreasing.

(2) If $A_0>B_0\ge C_0$, then $\lambda \left(t\right)e^{-{\int }_{\tau }^t((C^2-{(A-C)}^2)-(A^2-{(B-C)}^2))ds}$ is non-decreasing and

$\lambda \left(t\right)e^{-{\int }_{\tau }^t((A^2-{(B-C)}^2)-(C^2-{(A-B)}^2))ds}$ is non-increasing.

Proof. 

(1) Under the assumption $A_0=B_0\ge C_0$, it is easy to calculate that

$$2C^2\lambda \left(t\right)\le \frac{d\lambda \left(t\right)}{dt}\le \frac{2}{3}[A^2-{(B-C)}^2]\lambda \left(t\right).$$

(46)

Taking integration from $\tau$ to t we obtain $\lambda \left(t\right)e^{-{\int }_{\tau }^t\frac{2}{3}[A^2-{(B-C)}^2]ds}$ is non-increasing and

$\lambda \left(t\right)e^{-{\int }_{\tau }^{t}2C^{2}ds}$ is non-decreasing for $t\ge \tau$.

(2) Using the given condition that $A_0>B_0\ge C_0$ and (44), from (36), we have

$$\frac{d\lambda \left(t\right)}{dt}\ge ((C^2-{(A-C)}^2)-(A^2-{(B-C)}^2))\lambda \left(t\right)$$

(47)

and

$$\frac{d\lambda \left(t\right)}{dt}\le ((A^2-{(B-C)}^2)-(C^2-{(A-B)}^2))\lambda \left(t\right).$$

(48)

Taking integration from $\tau$ to t, we obtain the theorem. □

SL(2,$\mathbb{R}$): In this case for a given metric $g_0$, there is a Milnor frame such that

$$[X_2,X_3]=-X_1,[X_3,X_1]=X_2,[X_1,X_2]=X_3.$$

Under the normalization condition $A_0{B}_0{C}_0=1$, we have

$$\begin{array}{cc}\hfill & R_{11}=\frac{1}{2}A[A^2-{(B-C)}^2],R_{22}=\frac{1}{2}B[B^2-{(A+C)}^2],\hfill \\ \hfill & R_{33}=\frac{1}{2}C[C^2-{(A+B)}^2],\hfill \end{array}$$

(49)

and

$$R=\frac{1}{2}[A^2-{(B-C)}^2]+\frac{1}{2}[B^2-{(A+C)}^2]+\frac{1}{2}[C^2-{(A+B)}^2].$$

(50)

Theorem 10.

Let $\lambda \left(t\right)$ be the first eigenvalue of the operator $L_{T,f}$ where $T=hId$ for some positive time-independent function h, on a three-dimensional weighted Riemannian manifold $(SL(2,\mathbb{R}),g_0)$ under (32) or (34). Then, there exists a time τ such that

(1) if $A>B=C$, then $\lambda \left(t\right)e^{-{\int }_{\tau }^t\frac{2}{3}[C^2-{(A+B)}^2]ds}$ is non-decreasing and $\lambda \left(t\right)e^{{\int }_{\tau }^t\frac{5}{3}[C^2-{(A+B)}^2]ds}$ is non-increasing for $t\ge \tau$.

(2) If $A\le B-C$, then $\lambda \left(t\right)e^{{\int }_{\tau }^t[\frac{1}{3}{B}^2+{(A+B)}^2]ds}$ is non-decreasing and $\lambda \left(t\right)e^{-{\int }_{\tau }^t\frac{5}{3}{(A+B)}^{2}ds}$ is non-increasing for $t\ge \tau$.

Proof. 

(1) Using the condition $A>B=C$ and (49), from (36), we have the following

$$\frac{2}{3}[C^2-{(A+B)}^2]\lambda \left(t\right)\le \frac{d\lambda \left(t\right)}{dt}\le \frac{5}{3}[{(A+B)}^2-C^2]\lambda \left(t\right).$$

(51)

Taking the integration of the above on the interval $[\tau ,t]$, we have that $\lambda \left(t\right)e^{-{\int }_{\tau }^t\frac{2}{3}[C^2-{(A+B)}^2]ds}$ is non-decreasing and $\lambda \left(t\right)e^{{\int }_{\tau }^t\frac{5}{3}[C^2-{(A+B)}^2]ds}$ is non-increasing.

(2) Using the condition $A\le B-C$ and (49), from (36), we have the following

$$-[\frac{1}{3}{B}^2+{(A+B)}^2]\lambda \left(t\right)\le \frac{d\lambda \left(t\right)}{dt}\le \frac{5}{3}{(A+B)}^2\lambda \left(t\right).$$

(52)

Thus, $\lambda \left(t\right)e^{{\int }_{\tau }^t[\frac{1}{3}{B}^2+{(A+B)}^2]ds}$ is non-decreasing and $\lambda \left(t\right)e^{-{\int }_{\tau }^t\frac{5}{3}{(A+B)}^{2}ds}$ is non-increasing. □

5. Evolution along Yamabe Flow

In this section, we establish the evolution formula for the first non-zero eigenvalue of the operator $L_{T,f}$ along the Yamabe flow on M. Along the Yamabe flow, we have the following evolution equations:

$$\begin{array}{cc}\hfill & \left(i\right)\frac{\partial }{\partial t}\left(e^{-f}d\nu \right)=-(f_t+\frac{n}{2}R)e^{-f}d\nu ,\hfill \end{array}$$

(53)

$$\begin{array}{cc}\hfill & \left(ii\right)\frac{\partial }{\partial t}\left(g^{ij}\right)=-g^{ik}{g}^{jl}\frac{\partial }{\partial t}\left(g_{kl}\right)=R{g}^{ij}.\hfill \end{array}$$

(54)

Theorem 11.

Let $(M^n,g\left(t\right))$ be a solution to the Yamabe flow on a closed Riemannian manifold $(M^n,g_0)$. If $\lambda \left(t\right)$ denotes the evolution of the first eigenvalue of the operator $L_{T,f}$ along the Yamabe flow, then

$$\begin{array}{cc}\hfill \frac{d\lambda }{dt}=& {\int }_M\frac{\partial }{\partial t}\left(T^{ij}\right)u_i{u}_j{e}^{-f}d\nu +\lambda {\int }_M{u}^2{f}_t{e}^{-f}d\nu +\frac{n}{2}\lambda {\int }_{M}R{u}^2{e}^{-f}d\nu \hfill \\ \hfill & -{\int }_{M}T(\nabla u,\nabla u)f_t{e}^{-f}d\nu -\frac{n}{2}{\int }_{M}T(\nabla u,\nabla u)R{e}^{-f}d\nu ,\hfill \end{array}$$

(55)

where u is the normalized eigenfunction corresponding to the eigenvalue λ.

Proof. 

Differentiating (13) with respect to t and using integration by parts, we obtain

$$\begin{array}{cc}\hfill \frac{d}{dt}\lambda =& {\int }_M\frac{\partial }{\partial t}\left(T^{ij}{u}_i{u}_j\right)e^{-f}d\nu -{\int }_{M}T(\nabla u,\nabla u)(f_t+\frac{n}{2}R)e^{-f}d\nu \hfill \\ \hfill =& {\int }_M\frac{\partial }{\partial t}\left(T^{ij}\right)u_i{u}_j{e}^{-f}d\nu -2{\int }_{M}div(T\nabla u{e}^{-f})u_{t}d\nu -{\int }_M{T}^{ij}{u}_i{u}_j{f}_t{e}^{-f}d\nu \hfill \\ \hfill & -\frac{n}{2}{\int }_M{T}^{ij}{u}_i{u}_{j}R{e}^{-f}d\nu .\hfill \end{array}$$

(56)

Differentiating the normalization condition ${\int }_M{u}^2{e}^{-f}d\nu =1$ with respect to t, we have

$${\int }_{M}u{u}_t{e}^{-f}d\nu =\frac{1}{2}{\int }_M{u}^2(f_t+\frac{n}{2}R)e^{-f}d\nu .$$

(57)

Using (57), the second term on the RHS of (56) yields

$${\int }_{M}div(T\nabla u{e}^{-f})u_{t}d\nu =-\frac{\lambda }{2}{\int }_M{u}^2(f_t+\frac{n}{2}R)e^{-f}d\nu .$$

(58)

Hence, the proof is complete. □

Remark 2.

If we take $T=Id$, then the operator $-L_{T,f}$ reduces to the weighted Laplacian operator. Then, the result (55) reduces to

$$\begin{array}{cc}\hfill \frac{d}{dt}\lambda =& {\int }_M\frac{\partial }{\partial t}\left(g^{ij}\right)u_i{u}_j{e}^{-f}d\nu +\lambda {\int }_M{u}^2{f}_t{e}^{-f}d\nu +\frac{n}{2}\lambda {\int }_{M}R{u}^2{e}^{-f}d\nu \hfill \\ \hfill & -{\int }_M{|\nabla u|}^2{f}_t{e}^{-f}d\nu -\frac{n}{2}{\int }_M{|\nabla u|}^{2}R{e}^{-f}d\nu \hfill \\ \hfill =& (1-\frac{n}{2}){\int }_{M}R{|\nabla u|}^2{e}^{-f}d\nu +\lambda {\int }_M{u}^2{f}_t{e}^{-f}d\nu +\frac{n}{2}\lambda {\int }_{M}R{u}^2{e}^{-f}d\nu \hfill \\ \hfill & -{\int }_M{|\nabla u|}^2{f}_t{e}^{-f}d\nu .\hfill \end{array}$$

(59)

Corollary 6.

Let $(M^n,g\left(t\right))$ be a solution to the Yamabe flow on a closed Riemannian manifold $(M^n,g_0)$, $f_t=0$, $\frac{\partial T^{ij}}{\partial t}\ge (\alpha +\frac{n}{2}R)T^{ij}$ and $R\ge \alpha$ for some positive constant α. If $\lambda \left(t\right)$ denotes the evolution of the first non-zero eigenvalue of the operator $L_{T,f}$ along the Yamabe flow, then $\lambda \left(t\right)$ is a non-decreasing function.

Proof. 

Using the assumptions, we have

$$\begin{array}{cc}\hfill \frac{d\lambda }{dt}\ge & {\int }_M(\alpha +\frac{n}{2}R)T(\nabla u,\nabla u)e^{-f}d\nu +\frac{n}{2}\lambda {\int }_{M}R{u}^2{e}^{-f}d\nu -\frac{n}{2}{\int }_{M}RT(\nabla u,\nabla u)e^{-f}d\nu \hfill \\ \hfill \ge & \alpha {\int }_{M}T(\nabla u,\nabla u)e^{-f}d\nu +\frac{n}{2}\alpha \lambda .\hfill \end{array}$$

Then, $\lambda \left(t\right)$ is a non-decreasing function with respect to time variable t under the Yamabe flow. □

Theorem 12.

Let $(M^n,g\left(t\right))$ be a solution to the Yamabe flow on a closed homogeneous Riemannian manifold $(M^n,g_0)$. If $\lambda \left(t\right)$ denotes the evolution of the first eigenvalue of the operator $L_{T,f}$, then along the Yamabe flow

$$\begin{array}{cc}\hfill \frac{d\lambda }{dt}=& {\int }_M\frac{\partial }{\partial t}\left(T^{ij}\right)u_i{u}_j{e}^{-f}d\nu +\lambda {\int }_M{u}^2{f}_t{e}^{-f}d\nu -{\int }_{M}T(\nabla u,\nabla u)f_t{e}^{-f}d\nu ,\hfill \end{array}$$

where u is the normalized eigenfunction corresponding to the eigenvalue λ.

Proof. 

As the scalar curvature remains constant for a homogeneous Riemannian manifold, we obtain from Theorem 11

$$\begin{array}{cc}\hfill \frac{d\lambda }{dt}=& {\int }_M\frac{\partial }{\partial t}\left(T^{ij}\right)u_i{u}_j{e}^{-f}d\nu +\lambda {\int }_M{u}^2{f}_t{e}^{-f}d\nu +\frac{n}{2}\lambda {\int }_{M}R{u}^2{e}^{-f}d\nu \hfill \\ \hfill & -{\int }_{M}T(\nabla u,\nabla u)f_t{e}^{-f}d\nu -\frac{n}{2}{\int }_{M}T(\nabla u,\nabla u)R{e}^{-f}d\nu \hfill \\ \hfill =& {\int }_M\frac{\partial }{\partial t}\left(T^{ij}\right)u_i{u}_j{e}^{-f}d\nu +\lambda {\int }_M{u}^2{f}_t{e}^{-f}d\nu +\frac{n}{2}\lambda R{\int }_M{u}^2{e}^{-f}d\nu \hfill \\ \hfill & -{\int }_{M}T(\nabla u,\nabla u)f_t{e}^{-f}d\nu -\frac{n}{2}R{\int }_{M}T(\nabla u,\nabla u)e^{-f}d\nu .\hfill \end{array}$$

Hence the theorem. □

Corollary 7.

Let $(M^n,g\left(t\right)),t\in [0,{\tilde{T}}_0)$ for sufficient ${\tilde{T}}_0$, be a solution to the Yamabe flow on a closed homogeneous Riemannian manifold $(M^n,g_0)$, $\frac{\partial T^{ij}}{\partial t}\ge (\alpha +f_t)T^{ij}$ and $f_t\ge \beta$ for some positive constants $\alpha ,\beta$. Then, the quantity $\lambda \left(t\right)e^{-(\alpha +\beta )t}$ is increasing along the Yamabe flow.

Proof. 

The proof is similar as proof of Corollary 4. □

As in Section 3, we also consider $T=hId$ here, where $h:M\to \mathbb{R}$ is a smooth map. Then, two cases arise: (i) h is independent of time t and (ii) h depends on time t. If the smooth function h is independent of time t, then

$$\frac{\partial }{\partial t}\left(T^{ij}\right)=\frac{\partial }{\partial t}\left(h{g}^{ij}\right)=hR{g}^{ij}.$$

(60)

Thus, we have the following theorem.

Theorem 13.

Let $(M^n,g\left(t\right))$ be a solution to the Yamabe flow on a closed Riemannian manifold $(M^n,g_0)$. If $\lambda \left(t\right)$ denotes the evolution of the first non-zero eigenvalue of the operator $L_{T,f}$ with $T=hId$, then

$$\begin{array}{cc}\hfill \frac{d\lambda }{dt}=& (1-\frac{n}{2}){\int }_M{hR|\nabla u|}^2{e}^{-f}d\nu +\lambda {\int }_M{u}^2{f}_t{e}^{-f}d\nu -{\int }_{M}h{|\nabla u|}^2{f}_t{e}^{-f}d\nu \hfill \\ \hfill & +\frac{n}{2}\lambda {\int }_{M}R{u}^2{e}^{-f}d\nu ,\hfill \end{array}$$

(61)

where h is independent of time t and u is normalized eigenfunction corresponding to the eigenvalue λ.

Proof. 

From Theorem 11, we find

$$\begin{array}{cc}\hfill \frac{d\lambda }{dt}=& {\int }_M\frac{\partial }{\partial t}\left(h{g}^{ij}\right)u_i{u}_j{e}^{-f}d\nu +\lambda {\int }_M{u}^2{f}_t{e}^{-f}d\nu +\frac{n}{2}\lambda {\int }_{M}R{u}^2{e}^{-f}d\nu \hfill \\ \hfill & -{\int }_{M}hg(\nabla u,\nabla u)f_t{e}^{-f}d\nu -\frac{n}{2}{\int }_{M}hg(\nabla u,\nabla u)R{e}^{-f}d\nu \hfill \\ \hfill =& {\int }_{M}hR{|\nabla u|}^2{e}^{-f}d\nu +\lambda {\int }_M{u}^2{f}_t{e}^{-f}d\nu +\frac{n}{2}\lambda {\int }_{M}R{u}^2{e}^{-f}d\nu \hfill \\ \hfill & -{\int }_M{h|\nabla u|}^2{f}_t{e}^{-f}d\nu -\frac{n}{2}{\int }_{M}h{|\nabla u|}^{2}R{e}^{-f}d\nu .\hfill \end{array}$$

(62)

Hence, the proof is complete. □

Next, if we consider case that h depends on the time t, then we obtain

$$\frac{\partial }{\partial t}\left(T^{ij}\right)=\frac{\partial }{\partial t}\left(h{g}^{ij}\right)=h_t{g}^{ij}+h{R}^{ij},$$

(63)

using (54).

Theorem 14.

Let $(M^n,g\left(t\right))$ be a solution of the Yamabe flow on a closed Riemannian manifold $(M^n,g_0)$. If $\lambda \left(t\right)$ denotes the evolution of the first non-zero eigenvalue of the operator $L_{T,f}$ with $T=hId$, then

$$\begin{array}{cc}\hfill \frac{d\lambda }{dt}=& (1-\frac{n}{2}){\int }_M{hR|\nabla u|}^2{e}^{-f}d\nu +\lambda {\int }_M{u}^2{f}_t{e}^{-f}d\nu -{\int }_{M}h{|\nabla u|}^2{f}_t{e}^{-f}d\nu \hfill \\ \hfill & +{\int }_M{h}_t{|\nabla u|}^2{e}^{-f}d\nu +\frac{n}{2}\lambda {\int }_{M}R{u}^2{e}^{-f}d\nu ,\hfill \end{array}$$

(64)

where h is dependent of time t and u is normalized eigenfunction corresponding to the eigenvalue λ.

Proof. 

From Theorem 11, we find that

$$\begin{array}{cc}\hfill \frac{d\lambda }{dt}=& {\int }_M(h_t{g}^{ij}+hR{g}^{ij})u_i{u}_j{e}^{-f}d\nu +\frac{n}{2}\lambda {\int }_{M}R{u}^2{e}^{-f}d\nu +\lambda {\int }_M{u}^2{f}_t{e}^{-f}d\nu \hfill \\ \hfill & -\frac{n}{2}{\int }_M{Rh|\nabla u|}^2{e}^{-f}d\nu -{\int }_{M}h{|\nabla u|}^2{f}_t{e}^{-f}d\nu \hfill \\ \hfill =& {\int }_M{h}_t{|\nabla u|}^2{e}^{-f}d\nu +{\int }_{M}hR{|\nabla u|}^2{e}^{-f}d\nu +\frac{n}{2}\lambda {\int }_{M}R{u}^2{e}^{-f}d\nu \hfill \\ \hfill & +\lambda {\int }_M{u}^2{f}_t{e}^{-f}d\nu -\frac{n}{2}{\int }_M{Rh|\nabla u|}^2{e}^{-f}d\nu -{\int }_{M}h{|\nabla u|}^2{f}_t{e}^{-f}d\nu .\hfill \end{array}$$

(65)

This completes the theorem. □

Theorem 15.

Let $(M^n,g\left(t\right))$ be a solution to the normalized Yamabe flow on a closed Riemannian manifold $(M^n,g_0)$. If $\lambda \left(t\right)$ denotes the evolution of the first non-zero eigenvalue of the operator $L_{T,f}$ then along the normalized Yamabe flow

$$\begin{array}{cc}\hfill \frac{d\lambda }{dt}=& {\int }_M\frac{\partial }{\partial t}\left(T^{ij}\right)u_i{u}_j{e}^{-f}d\nu +\lambda {\int }_M{u}^2{f}_t{e}^{-f}d\nu +\frac{n}{2}\lambda {\int }_{M}R{u}^2{e}^{-f}d\nu \hfill \\ \hfill & -{\int }_{M}T(\nabla u,\nabla u)f_t{e}^{-f}d\nu -\frac{n}{2}{\int }_{M}T(\nabla u,\nabla u)R{e}^{-f}d\nu ,\hfill \end{array}$$

(66)

where u is the normalized eigenfunction corresponding to the eigenvalue λ.

Proof. 

It is known that, along the normalized Yamabe flow,

$$\frac{\partial }{\partial t}\left(e^{-f}d\nu \right)=-\{f_t+\frac{n}{2}(R-r)\}{e}^{-f}d\nu .$$

(67)

Now, differentiating (13) with respect to time t and using integration by parts, we obtain

$$\begin{array}{cc}\hfill \frac{d}{dt}\lambda =& {\int }_M\frac{\partial }{\partial t}\left(T^{ij}\right)u_i{u}_j{e}^{-f}d\nu +2{\int }_M{T}^{ij}{\left(u_t\right)}_i{u}_j{e}^{-f}d\nu -{\int }_{M}T(\nabla u,\nabla u)f_t{e}^{-f}d\nu \hfill \\ \hfill & -\frac{n}{2}{\int }_{M}T(\nabla u,\nabla u)e^{-f}d\nu +\frac{n}{2}r{\int }_{M}T(\nabla u,\nabla u)e^{-f}d\nu \hfill \\ \hfill =& {\int }_M\frac{\partial }{\partial t}\left(T^{ij}\right)u_i{u}_j{e}^{-f}d\nu -2{\int }_{M}div(T\nabla u{e}^{-f})u_{t}d\nu -{\int }_{M}T(\nabla u,\nabla u)f_t{e}^{-f}d\nu \hfill \\ \hfill & -\frac{n}{2}{\int }_{M}T(\nabla u,\nabla u)e^{-f}d\nu +\frac{n}{2}r\lambda \hfill \end{array}$$

(68)

Differentiating the normalized condition with respect to t, we have

$$2{\int }_{M}u{u}_t{e}^{-f}d\nu ={\int }_M{u}^2\{f_t+\frac{n}{2}(R-r)\}{e}^{-f}d\nu .$$

(69)

Thus, we obtain

$$-2{\int }_{M}div(T\nabla u{e}^{-f})u_{t}d\nu =\lambda {\int }_M{u}^2\{f_t+\frac{n}{2}(R-r)\}{e}^{-f}d\nu .$$

(70)

This completes our theorem. □

6. Conclusions

There are so many geometric flows in the literature along with Ricci flow. In mathematics as well as in physics, geometric flows have great significance. In particular, Ricci flow arises in String theory as the one loop approximation to the renormalization group flow of sigma models. In cosmology, it also plays an important role; see [19,20]. Mass in two dimension as well as in higher dimension under Ricci flow have been studied in [20]. The Ricci flow may lead to better understanding of quasilocal mass. From this point of view, it is mentioned that the Ricci flows are very much connected to the theoretical physics.

The eigenvalues of an operator contain various geometric properties of the underlying space. The geometry of the Laplace operator is an important topic in recent research. The Laplace operator has been used for the diffusion equation for heat and fluid flow and quantum physics. Many mathematicians generalized such operators and studied them in different contexts. In this paper, we obtain the evolution formula for the first non-zero eigenvalue of a divergence type operator which is an extension of weighted Laplace operator on a weighted Riemannian manifold under the Ricci flow as well as along the Yamabe flow. Thereafter, some monotonic formulas are constructed. In addition, for a special case of the (1,1)-tensor T, on Bianchi classes, some bounds of the the first non-zero eigenvalue are derived under the normalized Ricci flow. To develop this area more in the future, one can consider the techniques of the Singularity theory and Submanifold theory presented in [21,22,23,24,25,26,27,28], and it may find some new and interesting results.