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eISSN 0454-8124
pISSN 1225-6951

### Article

KYUNGPOOK Math. J. 2019; 59(2): 341-352

Published online June 23, 2019

### Evolution of the First Eigenvalue of Weighted p-Laplacian along the Yamabe Flow

Shahroud Azami

Department of Mathematics, Faculty of Sciences, Imam Khomeini International University, Qazvin, Iran
e-mail : azami@sci.ikiu.ac.ir

Received: July 14, 2018; Accepted: January 21, 2019

### Abstract

Let M be an n-dimensional closed Riemannian manifold with metric g, = eφ(x) be the weighted measure and Δp,φ be the weighted p-Laplacian. In this article we will study the evolution and monotonicity for the first nonzero eigenvalue problem of the weighted p-Laplace operator acting on the space of functions along the Yamabe flow on closed Riemannian manifolds. We find the first variation formula of it along the Yamabe flow. We obtain various monotonic quantities and give an example.

Keywords: Laplace, Yamabe ﬂow, eigenvalue.

### 1. Introduction

Let (M, g) be a Riemannian manifold, be the Riemannian volume measure on (M, g) and = eφ(x) is the weighted volume measure. Then triple (M, g, dμ) is a smooth metric measure space. Such spaces have been used more widely in the work of mathematicians, for instance, Perelman used it in [10]. Let M be an n-dimensional closed Riemannian manifold with metric g.

The geometric flows as the Yamabe flow have been a topic of active research interest in both mathematics and physics. A geometric flow is an evolution of a geometric structure under a differential equation related to a functional on a manifold, usually associated with some curvature. Asmooth one-parameter family of Riemannian metrics g(t) on M with scalar curvature R = Rg(t) is a called unnormalized Yamabe flow if

$ddtg(t)=-Rg(t)g(t), g(0)=g0.$

Also, g(t) is a solution of normalized Yamabe flow if

$ddtg(t)=(-Rg(t)-rg(t))g(t), g(0)=g0,$

where rg(t) denotes the mean value of the scalar of the metric g(t), i.e.

$rg(t)=∫MRgdvg(t)Vol(M).$

Let W1, p(M) be the Sobolev space and f : M → ℝ, fW1, p(M). For p ∈ [1, +∞) and any smooth function f on M, we define the weighed p-Laplacian on M by

$Δp,φf=eφdiv(e-φ∣∇f∣p-2∇f)=Δpf-∣∇f∣p-2∇φ.∇f,$

where the p-Laplacian Δpf defined as

$Δpf=div(∣∇f∣p-2∇f)= ∣∇f∣p-2Δf+(p-2)∣∇f∣p-4(Hessf)(∇f,∇f).$

The Witten-Laplacian is defined by Δφ = Δ − ∇ φ. ∇, which is a symmetric diffusion operator on L2(M, μ) and is self-adjoint. The weighted p-Laplacian is generalization of p-Laplacian and the Witten-Laplace operators, for instance, when φ is a constant function, the weighted p-Laplace operator is just the p-Laplace operator and when p = 2, the weighted p-Laplace operator is the Witten-Laplace operator.

We say Λ is an eigenvalue of the weighted p-Laplacian Δp,φ at time t ∈ [0, T) whenever for some fW1, p(M),

$-Δp,φf=Λ∣f∣p-2f,$

or equivalently

$-∫MfΔp,φfdμ=Λ∫M∣f∣pdμ,$

where = eφ(x) and is the Riemannian volume measure. We have

$∫M∣∇f∣pdμ=Λ∫M∣f∣pdμ,$

where f(x, t) called eigenfunction corresponding to eigenvalue Λ(t). The first nonzero eigenvalue λ(t) = λ(M, g(t), ) is characterised as follows

$λ(t)=inf0≠f∈W01,p(M){∫M∣∇f∣pdμ:∫M∣f∣pdμ=1},$

where $W01,p(M)$ is the completion of $C0∞(M)$ with respect Sobolev norm

$‖f‖W1,p=(∫M∣f∣pdμ+∫M∣∇f∣pdμ)1p.$

The eigenvalue problem for weighted p-Laplacian has been extensively studied in the literature [11, 12].

The problem of evolution and monotonicity of the eigenvalue of geometric operator is a topic problem. Recently many mathematicians investigate properties of Laplace, p-Laplace, Witten-Laplace and etc geometric operators, under various geometric flows. For first time Perelman in [10] showed that the first eigenvalue of the geometric operator −4Δ + R is nondecreasing along the Ricci flow, where R is scalar curvature.

Then Cao [4] and Zeng and et’al [14] extended the geometric operator −4Δ+R to the operator −Δ+cR on closed Riemannian manifolds, and studied the monotonicity of eigenvalues of the operator −Δ+cR along the Ricci flow and the Ricci-Bourguignon flow, respectively.

Author in [1] studied the evolution for the first eigenvalue of p-Laplacian along the Yamabe flow and in [2] shown that the first eigenvalue of Witten-Laplace operator −Δφ is monotonic along the Ricci-Bourguignon flow with some assumptions.

In [9], I have been studied the evolution for the first eigenvalue of geometric operator $-Δφ+R2$ under the Yamabe flow. For the other recent research in this direction, see [5, 6, 13].

Motivated by the above works, in this paper we will investigate the evolution of the nonzero first eigenvalue of the weighted p-Laplace operator whose metric satisfying the Yamabe flow (1.1) and φ evolves by $∂φ∂t=Δφ$.

### 2. Preliminaries

In this section, we will first introduce a smooth function where at time t0 is the first nonzero eigenvalue of the weighted p-Laplace operator Δp,φ then we will find the formula for the evolution of the first non-zero eigenvalue of the weighted p-Laplace operator along the evolution equation system (1.1) on a connected, smooth, closed oriented Riemannnian n-manifold. Let M be a closed oriented Riemannian n-manifold and (M, g(t), φ(t)) be a smooth solution of the evolution equation system (1.1) for t ∈ [0, T).

The first non-zero eigenvalue of weighted p-Laplacian is nonlinear and its corresponding eigenfunction are not known to be C1-differentiable. For this reason, we apply techniques of Cao [3] and Wu [13] and assume that at time t0, f0 = f(t0) is the eigenfunction for the first nonzero eigenvalue λ(t0) of Δp,φ. Then we get

$∫M∣f(t0)∣pdμg(t0)=1.$

We consider the following smooth function

$h(t):=f0[det(gij(t0))det(gij(t))]12(p-1),$

along the Yamabe flow g(t). We assume that

$f(t)=h(t)(∫M∣h(t)∣pdμ)1p,$

where f(t) is smooth function under the Yamabe flow, satisfied in ∫M |f|p= 1 and at time t0, f is the eigenfunction for λ of Δp,φ. Therefore if ∫M |f|p= 1 and

$λ(t,f(t))=-∫MfΔp,φfdμ,$

then λ(t0, f(t0)) = λ(t0).

### 3. Variation of λ(t)

In this section, we first recall evolution of some geometric structure along the Yamabe flow and then we give some useful evolution formulas for λ(t) under the Yamabe flow. From [7] we have

### Lemma 3.1

Under the Yamabe flowequation (1.1), we get

$∂∂tgij=Rgij,∂∂t(dν)=-n2Rdν,∂∂t(dμ)=-(φt+n2R)dμ,∂∂t(Γijk)=-12(∇jRδik+∇iRδjk-∇kRgij),∂∂tR=(n-1)ΔR+R2,$

and along the normalized Yamabe flow we have

$∂∂tgij=(R-r)gij,∂∂t(dν)=-n2(R-r)dν,∂∂t(dμ)=-(φt+n2(R-r))dμ,∂∂tR=(n-1)ΔR+R(R-r).$

### Lemma 3.2

Let (M, g(t)), t ∈ [0, T) be a solution to the Yamabe flow (1.1) on a closed oriented Riemannain manifold. Let fC(M) be a smooth function on (M, g(t)). Then we have the following evolutions:

$∂∂t∣∇f∣2=∣∇f∣2+2gij∇if∇jft,$$∂∂t∣∇f∣p-2=(p-2)∣∇f∣p-4{R2∣∇f∣2+gij∇if∇jft},$$∂∂t(Δf)=Δft+RΔf+2-n2∇kR∇kf,$$∂∂t(Δpf)=RΔpf+gij∇i(Zt∇jf)+gij∇i(Z∇jft) -n-22Zgij∇iR∇jf,$$∂∂t(Δp,φf)=gij∇i(Zt∇jf)+gij∇i(Z∇jft)-n-22Zgij∇iR∇jf +RΔp,φf-Zt∇φ.∇f-Z∇φt.∇f-Z∇φ.∇ft,$

where Z := |∇f|p−2and$ft=∂f∂t$.

Proof

By derivative respect to variable time t in local coordinates we have

$∂∂t∣∇f∣2=∂∂t(gij∇if∇jf)=∂gij∂t∇if∇jf+2gij∇if∇jft =R∣∇f∣2+2gij∇if∇jft$

which is (3.1). For prove (3.2) by using (3.1) we get

$∂∂t∣∇f∣p-2=∂∂t(∣∇f∣2)p-22=p-22(∣∇f∣2)p-42∂∂t(∣∇f∣2) =p-22∣∇f∣p-4{R∣∇f∣2+2gij∇if∇jft} =(p-2)∣∇f∣p-4{R2∣∇f∣2+gij∇if∇jft}$

which is exactly (3.2). Now Lemma 3.1 and 2∇iRij = ∇jR result

$∂∂t(Δf)=∂∂t[gij(∂2f∂xi∂xj-Γijk∂f∂xk)] =∂gij∂t(∂2f∂xi∂xj-Γijk∂f∂xk)+gij(∂2ft∂xi∂xj-Γijk∂ft∂xk)-gij∂∂t(Γijk)∂f∂xk =RΔf+Δft+12gij(∇jRδik+∇iRδjk-∇kRgij)∇kf =Δft+RΔf+2-n2∇kR∇kf.$

For prove (3.4), let Z = |∇f|p−2. We obtain

$∂∂t(Δpf)=∂∂t(div(∣∇f∣p-2∇f))=∂∂t(gij∇i(Z∇jf)) =∂∂t(gij∇iZ∇jf+gijZ∇i∇jf) =∂gij∂t∇iZ∇jf+gij∇iZt∇jf+gij∇iZ∇jft+ZtΔf+Z∂∂t(Δf) =Rgij∇iZ∇jf+gij∇iZt∇jf+gij∇iZ∇jft+ZtΔf +Z{Δft+RΔf+2-n2∇kR∇kf} =RΔpf+gij∇i(Zt∇jf)+gij∇i(Z∇jft)-n-22Zgij∇iR∇jf.$

Taking derivative with respect to time t of both sides of equation Δp,φf = Δpf − |∇f|p−2φ.f and (3.4) imply that

$∂∂t(Δp,φf)=∂∂t(Δpf)-Z∂gij∂t∇iφ∇jf-Ztgij∇iφ∇jf-Zgij∇iφt∇jf -Zgij∇iφ∇jft =RΔpf+gij∇i(Zt∇jf)+gij∇i(Z∇jft)-n-22Zgij∇iR∇jf -ZRgij∇iφ∇jf-Ztgij∇iφ∇jf-Zgij∇iφt∇jf-Zgij∇iφ∇jft,$

it results in (3.5).

### Proposition 3.3

Let (M, g(t)), t ∈ [0, T) be a solution of the Yamabe flow (1.1) on the smooth closed Riemannain manifold (Mn, g0) and$∂φ∂t=Δφ$. If λ(t) denotes the evolution the first non-zero eigenvalue of the weighted p-Laplacian Δp,φ corresponding to the eigenfunction f(x, t) under the flow (1.1), then

$∂∂tλ(t,f(t))∣t=t0=n2λ(t0)∫MR∣f∣pdμ+p-n2∫MR∣∇f∣pdμ +λ(t0)∫M(Δφ)∣f∣pdμ-∫M(Δφ)∣∇f∣pdμ.$
Proof

Let f(t) be a smooth function where f(t0) is the corresponding eigenfunction to λ(t0) = λ(t0, f(t0)). Function λ(t, f(t)) is smooth and taking derivative λ(t, f(t)) = − λM fΔp,φf dμ with respect to t, we get

$∂∂tλ(t,f(t))∣t=t0=-∂∂t∫MfΔp,φf dμ.$

Now, by condition ∫M |f|p= 1 and taking the time derivative of it, we can write

$0=∂∂t∫M∣f∣p-2f2dμ=∫M(p-1)∣f∣p-2f ftdμ+∫M∣f∣p-2f∂∂t(fdμ),$

hence

$∫M∣f∣p-2f [(p-1)ftdμ+∂∂t(fdμ)]=0.$

On the other hand, using (3.5), we obtain

$∂∂t∫MfΔp,φf dμ=∫M∂∂t(Δp,φf)f dμ+∫MΔp,φf∂∂t(f dμ) =∫MRΔp,φf f dμ+∫Mgij∇i(Zt∇jf)f dμ +∫Mgij∇i(Z∇jft)f dμ-n-22∫MZ∇R.∇f f dμ -∫MZt∇φ.∇f f dμ-∫MZ∇φt.∇f f dμ -∫MZ∇φ.∇ftf dμ-∫Mλ∣f∣p-2f∂∂t(fdμ).$

By the application of integration by parts we have

$∫Mgij∇i(Zt∇jf)f dμ=-∫MZt∣∇f∣2dμ+∫MZt∇f.∇φf dμ,$

and

$∫Mgij∇i(Z∇jft)f dμ=-∫MZ∇ft.∇f dμ+∫MZ∇ft.∇φf dμ.$

Now, plugging (3.11) and (3.12) into (3.10), we get

$∂∂t∫MfΔp,φf dμ=-∫MλR∣f∣pdμ-n-22∫MZ∇R.∇ffdμ -∫MZt∣∇f∣2dμ-∫MZ∇ft.∇f dμ-∫MZ∇φt.∇f f dμ -∫Mλ∣f∣p-2f∂∂t(f dμ).$

On the other hand

$Zt=∂∂t(∣∇f∣p-2)=(p-2)∣∇f∣p-4{R2∣∇f∣2+gij∇if∇jft}.$

Therefore replacing this into (3.13), we have

$-∂∂tλ(t,f(t))∣t=t0=-λ(t0)∫MR∣f∣pdμ-p-22∫MR∣∇f∣pdμ -n-22∫MZ∇R.∇f fdμ-(p-1)∫MZ∇ft.∇f dμ -∫MZ∇φt.∇f f dμ-λ(t0)∫M∣f∣p-2f∂∂t(fdμ).$

Also

$∫MZ∇ft.∇f dμ=-∫MftΔp,φf dμ=∫Mλ∣f∣p-2f ft dμ.$

Then we arrive at

$-∂∂tλ(t,f(t))∣t=t0=-λ(t0)∫MR∣f∣pdμ-p-22∫MR∣∇f∣pdμ -n-22∫MZ∇R.∇f fdμ-∫MZ∇φt.∇f dμ -λ(t0)∫M∣f∣p-2f ((p-1)ftdμ+∂∂t(fdμ)).$

Hence (3.9) results in

$-∂∂tλ(t,f(t))∣t=t0=-λ(t0)∫MR∣f∣pdμ-p-22∫MR∣∇f∣pdμ -n-22∫MZ∇R.∇f fdμ-∫MZ∇φt.∇f dμ.$

By integration by parts we get

$∫MZ∇φt.∇f f dμ=∫Mλ∣f∣p(Δφ) dμ-∫M(Δφ)∣∇f∣pdμ$

and

$∫MZ∇R.∇f f dμ=∫MλR∣f∣pdμ-∫MR∣∇f∣pdμ.$

Pluggin (3.19) and (3.20) into (3.18) imply that (3.6).

### Corollary 3.4

Let (M, g(t)), t ∈ [0, T) be a solution of the Yamabe flow (1.1) on the smooth closed Riemannain manifold (Mn, g0). If λ(t) denotes the evolution the first non-zero eigenvalue of the weighted p-Laplacian Δp,φ corresponding to the eigenfunction f(x, t) under the flow (1.1) where φ is independent of t, then

$∂∂tλ(t,f(t))∣t=t0=n2λ(t0)∫MR∣f∣pdμ+p-n2∫MR∣∇f∣pdμ.$

### Theorem 3.5

Let (M, g(t)), t ∈ [0, T) be a solution of the Yamabe flow (1.1) on the smooth closed Riemannain manifold (Mn, g0) and$∂φ∂t=Δφ$. Let$p-n2R<Δφ$and Rmin(0) ≥ 0 along the Yamabe flow (1.1). Suppose that λ(t) denotes the evolution the first non-zero eigenvalue of the weighted p-Laplacian Δp,φ then the quantity $λ(t)(1-Rmin(0)t)p+22$is nondecreasing along the flow (1.1) for$T≤1Rmin(0)$.

Proof

According to (3.6) of Proposition 3.3, we have

$∂∂tλ(t,f(t))∣t=t0≥n2λ(t0)∫MR∣f∣pdμ+p-n2∫MR∣∇f∣pdμ +p-n2λ(t0)∫MR∣f∣pdμ-∫M(Δφ)∣∇f∣pdμ ≥p2λ(t0)∫MR∣f∣pdμ$

On the other hand, the scalar curvature under the Yamabe flow evolves by

$∂R∂t=(n-1)ΔR+R2$

The solution of the corresponding ODE y′ = y2 with initial value y(0) = Rmin(0) is

$σ(t)=Rmin(0)1-Rmin(0)t$

on [0, T′) where $T′=min{T,1Rmin(0)}$. Using the maximum principle to (3.23), we get Rg(t)σ(t). Therefore (3.22) becomes

$ddtλ(t,f(t))∣t=t0≥p2λ(t0)σ(t0),$

and this implies that in any sufficiently small neighborhood of t0 such as I0, we get

$ddtλ(t,f(t))≥p2λ(t,f(t))σ(t).$

Integrating the last inequality with respect to t on [t1, t0] ⊂ I0, we have

$lnλ(t0,f(t0))λ(t1,f(t1))>ln(1-Rmin(0)t11-Rmin(0)t0)p2.$

Since λ(t0, f(t0)) = λ(t0) and λ(t1, f(t1)) ≥ λ(t1) we conclude that

$lnλ(t0)λ(t1)>ln(1-Rmin(0)t11-Rmin(0)t0)p2,$

that is, the quantity $λ(t)(1-Rmin(0)t)p2$ is strictly increasing in any sufficiently small neighborhood of t0. Since t0 is arbitrary, then $λ(t)(1-Rmin(0)t)p2$ is strictly increasing along the Yamabe flow on [0, T′).

### Proposition 3.6

Let (M, g(t)), t ∈ [0, T) be a solution of the normalized Yamabe flow (1.2) on the smooth closed Riemannain manifold (Mn, g0) and$∂φ∂t=Δφ$. If λ(t) denotes the evolution the first non-zero eigenvalue of the weighted p-Laplacian Δp,φ corresponding to the eigenfunction f(x, t) under the flow (1.2), then

$∂∂tλ(t,f(t))∣t=t0=-pr2λ(t0)+n2λ(t0)∫MR∣f∣pdμ+p-n2∫MR∣∇f∣pdμ +λ(t0)∫M(Δφ)∣f∣pdμ-∫M(Δφ)∣∇f∣pdμ.$
Proof

In the normalized case, we have

$∂∂t(Δp,φf)=gij∇i(Zt∇jf)+gij∇i(Z∇jft)-n-22Zgij∇iR∇j +(R-r)Δp,φf-Zt∇φ.∇f-Z∇φt.∇f-Z∇φ.∇ft,$

and

$Zt=∂∂t(∣∇f∣p-2)=(p-2)∣∇f∣p-4{R-r2∣∇f∣2+gij∇if∇jft},$

where Z = |∇f|p−2. The subsequent process is similar to the proof of Proposition 3.3, direct computation implies that (3.24).

### Corollary 3.7

Let (M, g(t)), t ∈ [0, T) be a solution of the Yamabe flow (1.1) on the smooth closed Riemannain manifold (Mn, g0). If λ(t) denotes the evolution the first non-zero eigenvalue of the weighted p-Laplacian Δp,φ corresponding to the eigenfunction f(x, t) under the flow (1.1) where φ is independent of t, then

$∂∂tλ(t,f(t))∣t=t0=-pr2λ(t0)+n2λ(t0)∫MR∣f∣pdμ+p-n2∫MR∣∇f∣pdμ.$

### 3.1. Variation of λ(t) on a Surface

Now, we write Proposition 3.6 and Corollary 3.7 in some remarkable particular cases.

Corollary 3.1.1

Let (M2, g(t)), t ∈ [0, T) be a solution of the normalized Yamabe flow on a closed Riemannian surface (M2, g0). If λ(t) denotes the evolution of the first eigenvalue of the weighted p-Laplacian under the flow (1.2), then on g0, we have

• If$∂φ∂t=Δφ$then$∂∂tλ(t,f(t))∣t=t0=-pr2λ(t0)+λ(t0)∫MR∣f∣pdμ+p-22∫MR∣∇f∣pdμ+λ(t0)∫M(Δφ)∣f∣pdμ-∫M(Δφ)∣∇f∣pdμ.$

• If φ is independent of t then$∂∂tλ(t,f(t))∣t=t0=-pr2λ(t0)+λ(t0)∫MR∣f∣pdμ+p-22∫MR∣∇f∣pdμ.$

Remark 3.1.2

Let (M2, g(t)), t ∈ [0, T) be a solution of the normalized Yamabe flow on a compact Riemannian surface (M2, g0) and φ be independent of t. Then from [8] for a constant c depending only on g0, we have

• If r < 0 then rcertRr + cert,

• If r = 0 then $-c1+ct≤R≤c$,

• If r > 0 then − certRr + cert.

Therefore by using (2.2) we can obtain a lower bound and a upper bound for the first eigenvalue of the weighted p-Laplacian under the flow (1.2).

### 4. Example

In this section, we find the variational formula of the evolving spectrum of the weighted p-Laplace operator for some of Riemannian manifolds.

### Example 4.1

Let (Mn, g0) be an Einstein manifold i.e. there exists a constant a such that Ric(g0) = ag0. Assume that we have a solution to the Yamabe flow which is of the form

$g(t)=u(t)g0, u(0)=1$

where u(t) is a positive function. We compute

$∂g∂t=u′(t)g0, Ric(g(t))=Ric(g0)=ag0=au(t)g(t), Rg(t)=anu(t),$

for this to be a solution of the Yamabe flow, we require

$u′(t)g0=-Rg(t)g(t)=-ang0$

this shows that u(t) = −nat + 1. So g(t) is an Einstein metric. Using equation (3.21), we obtain the following relation

$ddtλ(t,f(t))∣t=t0=n2λ(t0)an-nat0+1∫M∣f∣pdμ+p-n2an-nat0+1∫M∣∇f∣pdμ,$

or equivalently

$ddtλ(t,f(t))∣t=t0=panλ(t0)2(-nat0+1).$

This implies that in any sufficiently small neighborhood of t0 such as I0, we get

$ddtλ(t,f(t))=panλ(t,f(t))2(-nat+1).$

Integrating the last inequality with respect to t on [t1, t0] ⊂ I0, we have

$lnλ(t0,f(t0))λ(t1),f(t1)=ln(-nat1+1-nat0+1)p2.$

Since λ(t0, f(t0)) = λ(t0) and λ(t1, f(t1)) ≥ λ(t1) we conclude that

$lnλ(t0)λ(t1)>ln(-nat1+1-nat0+1)p2,$

that is, the quantity $λ(t)(-nat+1)p2$ is strictly increasing along the flow (1.1) on [0, T).

### References

1. S. Azami. Eigenvalue variation of the p-Laplacian under the Yamabe flow. Cogent math., 3(2016) Art. ID 1236566, 10 pp.
2. S. Azami. Monotonicity of eigenvalues of Witten-Laplace operator along the Ricci-Bourguignon flow. AIMS mathematics., 2(2)(2017), 230-243.
3. X. Cao. Eigenvalues of (-Δ+R2) on mannifolds with nonegative curvature operator. Math Ann., 337(2)(2007), 435-441.
4. X. Cao. First eigenvalues of geometric operators under the Ricci flow. Proc Amer Math Soc., 136(2008), 4075-4078.
5. LFD. Cerbo. Eigenvalues of the Laplacian under the Ricci flow. Rend Mat Appl (7)., 27(2007), 183-195.
6. Q-M. Cheng, and HC. Yang. Estimates on eigenvalues of Laplacian. Math Ann., 331(2005), 445-460.
7. B. Chow. The Yamabe flow on locally conformally flat manifolds with positive Ricci curvature. Comm Pure Appl Math., 45(8)(1992), 1003-1014.
8. B. Chow, and D. Knopf. The Ricci flow: an introduction. Mathematical Surveys and Monographs, 110, AMS, 2004.
9. S. Fang, and F. Yang. First eigenvalues of geometric operators under the Yamabe flow. Bull Korean Math Soc., 53(2016), 1113-1122.
10. G. Perelman. . The entropy formula for the Ricci flow and its geometric applications., (2002):ArXiv:math/0211159.
11. LF. Wang. Eigenvalue estimate for the weighted p-Laplacian. Ann Mat Pura Appl (4)., 191(2012), 539-550.
12. LF. Wang. Gradient estimates on the weighted p-Laplace heat eqaution. J Differential equations., 264(2018), 506-524.
13. JY. Wu. First eigenvalue monotonicity for the p-Laplace operator under the Ricci flow. Acta Math Sin (Engl Ser)., 27(8)(2011), 1591-1598.
14. F. Zeng, Q. He, and B. Chen. Monotonicity of eigenvalues of geometric operators along the Ricci-Bourguignon flow. Pacific J Math., 296(1)(2018), 1-20.