KYUNGPOOK Math. J. 2019; 59(2): 341-352  
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; Published online: June 23, 2019.
© Kyungpook Mathematical Journal. All rights reserved.

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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-φfp-2f)=Δpf-fp-2φ.f,

where the p-Laplacian Δpf defined as

Δpf=div(fp-2f)=fp-2Δf+(p-2)fp-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=Λfp-2f,

or equivalently

-MfΔp,φfdμ=ΛMfpdμ,

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

Mfpdμ=ΛMfpdμ,

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)=inf0fW01,p(M){Mfpdμ:Mfpdμ=1},

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

fW1,p=(Mfpdμ+Mfpdμ)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 φt=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

Mf(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)(Mh(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:

tf2=f2+2gijifjft,tfp-2=(p-2)fp-4{R2f2+gijifjft},t(Δf)=Δft+RΔf+2-n2kRkf,t(Δpf)=RΔpf+giji(Ztjf)+giji(Zjft)-n-22ZgijiRjf,t(Δp,φf)=giji(Ztjf)+giji(Zjft)-n-22ZgijiRjf+RΔp,φf-Ztφ.f-Zφt.f-Zφ.ft,

where Z := |∇f|p−2andft=ft.

Proof

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

tf2=t(gijifjf)=gijtifjf+2gijifjft=Rf2+2gijifjft

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

tfp-2=t(f2)p-22=p-22(f2)p-42t(f2)=p-22fp-4{Rf2+2gijifjft}=(p-2)fp-4{R2f2+gijifjft}

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

t(Δf)=t[gij(2fxixj-Γijkfxk)]=gijt(2fxixj-Γijkfxk)+gij(2ftxixj-Γijkftxk)-gijt(Γijk)fxk=RΔf+Δft+12gij(jRδik+iRδjk-kRgij)kf=Δft+RΔf+2-n2kRkf.

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

t(Δpf)=t(div(fp-2f))=t(giji(Zjf))=t(gijiZjf+gijZijf)=gijtiZjf+gijiZtjf+gijiZjft+ZtΔf+Zt(Δf)=RgijiZjf+gijiZtjf+gijiZjft+ZtΔf+Z{Δft+RΔf+2-n2kRkf}=RΔpf+giji(Ztjf)+giji(Zjft)-n-22ZgijiRjf.

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)-Zgijtiφjf-Ztgijiφjf-Zgijiφtjf-Zgijiφjft=RΔpf+giji(Ztjf)+giji(Zjft)-n-22ZgijiRjf-ZRgijiφjf-Ztgijiφjf-Zgijiφtjf-Zgijiφjft,

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) andφt=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)MRfpdμ+p-n2MRfpdμ+λ(t0)M(Δφ)fpdμ-M(Δφ)fpdμ.
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=-tMfΔp,φfdμ.

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

0=tMfp-2f2dμ=M(p-1)fp-2fftdμ+Mfp-2ft(fdμ),

hence

Mfp-2f[(p-1)ftdμ+t(fdμ)]=0.

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

tMfΔp,φfdμ=Mt(Δp,φf)fdμ+MΔp,φft(fdμ)=MRΔp,φffdμ+Mgiji(Ztjf)fdμ+Mgiji(Zjft)fdμ-n-22MZR.ffdμ-MZtφ.ffdμ-MZφt.ffdμ-MZφ.ftfdμ-Mλfp-2ft(fdμ).

By the application of integration by parts we have

Mgiji(Ztjf)fdμ=-MZtf2dμ+MZtf.φfdμ,

and

Mgiji(Zjft)fdμ=-MZft.fdμ+MZft.φfdμ.

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

tMfΔp,φfdμ=-MλRfpdμ-n-22MZR.ffdμ-MZtf2dμ-MZft.fdμ-MZφt.ffdμ-Mλfp-2ft(fdμ).

On the other hand

Zt=t(fp-2)=(p-2)fp-4{R2f2+gijifjft}.

Therefore replacing this into (3.13), we have

-tλ(t,f(t))t=t0=-λ(t0)MRfpdμ-p-22MRfpdμ-n-22MZR.ffdμ-(p-1)MZft.fdμ-MZφt.ffdμ-λ(t0)Mfp-2ft(fdμ).

Also

MZft.fdμ=-MftΔp,φfdμ=Mλfp-2fftdμ.

Then we arrive at

-tλ(t,f(t))t=t0=-λ(t0)MRfpdμ-p-22MRfpdμ-n-22MZR.ffdμ-MZφt.fdμ-λ(t0)Mfp-2f((p-1)ftdμ+t(fdμ)).

Hence (3.9) results in

-tλ(t,f(t))t=t0=-λ(t0)MRfpdμ-p-22MRfpdμ-n-22MZR.ffdμ-MZφt.fdμ.

By integration by parts we get

MZφt.ffdμ=Mλfp(Δφ)dμ-M(Δφ)fpdμ

and

MZR.ffdμ=MλRfpdμ-MRfpdμ.

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)MRfpdμ+p-n2MRfpdμ.

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) andφt=Δφ. Letp-n2R<Δφand 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+22is nondecreasing along the flow (1.1) forT1Rmin(0).

Proof

According to (3.6) of Proposition 3.3, we have

tλ(t,f(t))t=t0n2λ(t0)MRfpdμ+p-n2MRfpdμ+p-n2λ(t0)MRfpdμ-M(Δφ)fpdμp2λ(t0)MRfpdμ

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

Rt=(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=t0p2λ(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) andφt=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)MRfpdμ+p-n2MRfpdμ+λ(t0)M(Δφ)fpdμ-M(Δφ)fpdμ.
Proof

In the normalized case, we have

t(Δp,φf)=giji(Ztjf)+giji(Zjft)-n-22ZgijiRj+(R-r)Δp,φf-Ztφ.f-Zφt.f-Zφ.ft,

and

Zt=t(fp-2)=(p-2)fp-4{R-r2f2+gijifjft},

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)MRfpdμ+p-n2MRfpdμ.

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

  • Ifφt=Δφthentλ(t,f(t))t=t0=-pr2λ(t0)+λ(t0)MRfpdμ+p-22MRfpdμ+λ(t0)M(Δφ)fpdμ-M(Δφ)fpdμ.

  • If φ is independent of t thentλ(t,f(t))t=t0=-pr2λ(t0)+λ(t0)MRfpdμ+p-22MRfpdμ.

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+ctRc,

  • 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

gt=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+1Mfpdμ+p-n2an-nat0+1Mfpdμ,

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).

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