Evolution of a geometric constant along the Ricci flow

R E S E A R C H

Open Access

Evolution of a geometric constant along

the Ricci flow

Guangyue Huang

*

_{and Zhi Li}

*_{Correspondence:} [email protected]

Henan Engineering Laboratory for Big Data Statistical Analysis and Optimal Control, College of Mathematics and Information Science, Henan Normal University, Xinxiang, Henan 453007, People’s Republic of China

Abstract

In this paper, we establish the first variation formula of the lowest constant

λ

b a(g) along the Ricci flow and the normalized Ricci flow, such that to the following nonlinear equation there exist positive solutions:

–u+aulogu+bRu=

λ

b_au

with_Mu2_{dv= 1, whereais a real constant. In particular, the results proved in this} paper generalize partial results in Cao (Proc. Am. Math. Soc. 136:4075-4078, 2008) and Li (Math. Ann. 338:927-946, 2007).

MSC: 58C40; 53C44

Keywords: Ricci flow; normalized Ricci flow; conjugate heat equation

1 Introduction

Let (M,g) be ann-dimensional compact Riemannian manifold. In [], Perelman intro-duced the functional

F(g,f) =

M

|∇f|+Re–fdv (.)

and proved that theF-functional is nondecreasing under the Ricci flow coupled to a back-ward heat-type equation

∂

∂tgij= –Rij,

ft= –f +|∇f|–R,

(.)

whereRis the scalar curvature depending on the metricg. More precisely, they proved that under the system (.),

d dtF= 

M|

Rij+fij|e–fdv≥. (.)

If we define

λ(g) =inf

f F(g,f), (.)

where the infimum is taken over all smooth functionsf which satisfy

M

e–fdv= , (.)

then the nondecreasing of theF-functional implies the nondecreasing ofλ(g). In partic-ular,λ(g) defined in (.) is the lowest eigenvalue of the operator

–+R. (.)

In [], Cao considered the eigenvalues of the operator –+R_ on manifolds with non-negative curvature operator and showed that the eigenvalues are nondecreasing along the Ricci flow. Using the same technique, Li [] also obtained the same monotonicity of the first eigenvalue of the operator –+ R_ by removing the assumption on a nonnegative curvature operator.

Later, Cao [] proved the first eigenvalues of the operator –+bRwith the constant

b≥/ are nondecreasing along the Ricci flow. That is, they assumeu=u(x,t) is the cor-responding positive eigenfunction ofλ(t):

(–+bR)u=λbu (.)

with_Mu_{dv= , then}

d dtλ

b₌ 

M

|Rij+fij|e–fdv+

b–  M

|Rij|e–fdv≥ (.)

by lettingf= –logu. Multiplying both sides of (.) withuand integrating onM, we see that the first eigenvalue given in (.) satisfies

λ(t) =infF˜b(g,u), (.)

where

˜

Fb_{(g,u) =}

M

|∇u|+bRudv. (.)

In particular,

˜

Fb_{(g,u) =}  F

b_{(g,f), (.)}

where

Fc_{(g,f) =}

M

|∇f|+cRe–fdv

In this paper, we consider the monotonicity along the Ricci flow of lowest constantλb a(g) such that to the following nonlinear equation there exist positive solutions:

–u+aulogu+bRu=λb_au (.) with

M

udv= , (.)

whereais a real constant. In particular, (.) can be seen a special case of (.) when

a= . For the lowest constantλb_a(g) such that to the nonlinear equation (.) there exist positive solutions, we prove the following.

Theorem . Let g(t),t∈[,T)be a solution to the Ricci flow

∂

∂tgij= –Rij (.) on a compact Riemannian manifold M.Then for b≥_,the lowest constantλb

a(g)such that

to the nonlinear equation(.)with(.)there exist positive solutions satisfies d

dt

λb_a(t) +na 

 t

= 

M

Rij+fij+

a

gij

e–fdv+

b–  M

|Rij|e–fdv

≥, (.)

where f = –logu.

For the normalized Ricci flow, we can obtain the following.

Theorem . Let g(t),t∈[,T)be a solution to the normalized Ricci flow

∂ ∂tgij= –

Rij–

r ngij

(.)

on a compact Riemannian manifold M,where r= (_MR dv)/(_Mdv)is the average scalar curvature.Then the lowest constantλb

a(g)such that to the nonlinear equation(.)with (.)there exist positive solutions satisfies

d dt

λb_a+na



 t

+r

nλ

b₌ 

M

Rij+fij+

a

gij

e–fdv

+

b–   M|

Rij|e–fdv, (.)

where f = –logu andλbis the lowest eigenvalue of(.).

In particular, whenn= , we haveRij=R_gijand the normalized Ricci flow (.) becomes ∂

∂tgij= –(R–r)gij. Hence, d

Theorem . Let g(t),t∈[,T)be a solution to the normalized Ricci flow(.)on a com-pact surface M_{.Then for b≥} 

,the lowest constantλ b

a(g)such that to the nonlinear

equa-tion(.)with(.)there exist positive solutions satisfies d

dt

λb_a+a



t+r

t



λb(s)ds

= 

M

Rij+fij+

a

gij

e–fdv

+

b–   M

|Rij|e–fdv

≥, (.)

where f = –logu andλb_{is the lowest eigenvalue of(.).}

Remark . In particular, whena= , our estimate (.) reduces to Theorem . of Cao

in [] and the estimate (.) reduces to the Corollary . of Cao in [], respectively.

On the other hand, under the transformationf = –logu= –logvwithu_{=v, equation} (.) becomes

_∂

∂tgij= –Rij,

vt= –v+Rv.

(.)

In particular, the second equation in (.) is exactly the conjugate heat equation intro-duced by Perelman. In [], Cao and Zhang obtained differential Harnack inequalities for positive solutions of the nonlinear parabolic equation of the typevt =v–vlogv+Rv. Extending the second equation in (.) to the following nonlinear version:

vt= –v+avlogv+Sv, (.)

Guo and Ishida [, ] studied Harnack inequalities for positive solutions of equation (.) on a compact Riemannian manifold with a family of g(t) evolving by a geometric flow

∂

∂tgij= –Sij, whereSijis a family of smooth symmetric two-tensor andS=g ij_S

ij. Clearly, there is a one-to-one relation for the following two equations:

∂

∂tv= –v+avlogv+Rv ⇐⇒

∂

∂tf= –f+|∇f|

_{+af –R (.)}

underf = –logv. Therefore, a natural problem is to consider the monotonicity of

Fcd(g,f) =

M

|∇f|_{+cR+d(f + )e}–f_{dv (.)}

under the Ricci flow coupled to a nonlinear backward heat-type equation

d

dtgij= –Rij,

ft= –f +|∇f|+af–R,

(.)

For the functionalFc_d(g,f), we derive the following monotonicity formula.

Theorem . Let g(t),t∈[,T)be a solution to the Ricci flow(.)on a compact Rieman-nian manifold M.Then all functionalsFc_d(g,f)defined by(.)under the system(.)

satisfy d dtF

k

nak

 (g,f) = 

M

Rij+fij–

a

fgij

e–fdv+ (k– )

M

Rij–

a

fgij

e–fdv

+na  kF

_{(g,f) +aF}_{(g,f). (.)}

In particular,if R(t)≥for all t and a≥,k≥,then_dtdFknak

 (g,f)≥.

Remark . Choosinga=  in (.), we obtain Theorem . of Li in [].

2 Proof of Theorems 1.1 and 1.2

Proof of Theorems. Letube a positive solution to the following nonlinear elliptic equa-tion:

–u+aulogu+bRu=λb_au. (.)

Multiplying both sides of (.) withuand integrating onM, we have

λb_a=

M

|∇u|+aulogu+bRudv. (.)

If the metricg(t) evolves by (.), we have _∂∂_tdv= –R dv. It follows from (.) that

d dtλ

b a=

M

Rijuiuj+ (ut)iui+ auutlogu+auut+bRtu+ bRuut

dv

–

M

|∇u|+aulogu+bRuR dv. (.)

Applying



M

Rijuiujdv=

M

–R,iuiu– Rijuiju

dv (.)

and

–

M

|∇u|R dv=

M

Ru+R,iui

u dv (.)

into (.) yields

d dtλ

b a=

M

–Rijuiju+bRtu+auut

+ ut(–u+aulogu+bRu) –Ru(–u+aulogu+bRu)

=

M

–Rijuiju+bRtu+

a



u_t

dv+λ

M

udv

t

=

M

–Rijuiju+bRtu+

a

Ru 

dv, (.)

where the last equality used

M

u_t–Rudv=  (.)

from (.). NoticingRt=R+ |Rij|for the Ricci flow, hence from (.) we have

d dtλ

b a=

M

–Rijuiju+bu

R+ |Rij|

+a Ru



dv

=

M

–Rijuiju+bR

u+ b|Rij|u+

a

Ru 

dv. (.)

Taking a transformationf = –logu, which is equivalent tou_=e–f_{, then}

uij=

– f

ij₊  f

i_fj

e–f_{. (.)}

Thus, (.) can be written as

d dtλ

b a=

M

Rijfij–  Rijf

i_fj_{–bRf+bR|∇f|}_{+ b|R} ij|+

a

R

e–fdv. (.)

Using the second Bianchi identityR,i= Rij,jagain, we have

–b

M

Rfe–fdv=

M

bR,ifi–bR|∇f|

e–fdv

=

M

–bRijfij+ bRijfifj–bR|∇f|

e–fdv. (.)

Therefore, inserting (.) into (.) yields

d dtλ

b

a= ( – b)

M

Rijfije–fdv+

b–  M

Rijfifje–fdv

+ b

M

|Rij|e–fdv+

a



M

Re–fdv. (.)

Integrating by parts again, one has

M

Rijfije–fdv=

M

Rijfifje–fdv–  

M

Re–fdv (.)

and

M

Rijfije–fdv+

M

|fij|e–fdv

= 

M

|∇f|e–fdv–

M

(f)ifie–fdv–  

M

= –

M

f –

|∇f| ₊

R

e–fdv

=

b–   M

Re–fdv–a

M

|∇f|e–fdv, (.)

where the last equality in (.) was used with

λb_a=f – |∇f|

_{–af+ bR. (.)}

By virtue of (.), subtracting (.), we obtain

M

|fij|e–fdv= b

M

Re–fdv–

M

Rijfifje–fdv–a

M

|∇f|e–fdv. (.)

It follows from (.) and (.) that

d dtλ

b

a= ( – b)

M

Rijfije–fdv+

b–  M

Rijfifje–fdv

+ b

M

|Rij|e–fdv+

a



M

Re–fdv

=

M

Rijfije–fdv–  

M

Rijfifje–fdv

+ b

M

|Rij|e–fdv+

a



M

Re–fdv+b

M

Re–fdv

=

M

Rijfije–fdv+ b

M

|Rij|e–fdv+

a



M

Re–fdv

+ 

M|

fij|e–fdv+

a



M

(f)e–fdv

= 

M

Rij+fij+

a

gij

e–fdv+

b–   M

|Rij|e–fdv

–na 

 , (.)

and the desired estimate (.) is achieved.

Proof of Theorem. If the metricg(t) evolves by (.), we have _∂∂_tdv= –(R–r)dv. It follows from (.) that

d dtλ

b a=

M

Rijuiuj– r

n|∇u|

_{+ (u}

t)iui+ auutlogu+auut+bRtu + bRuut

dv–

M

|∇u|+aulogu+bRu(R–r)dv. (.)

Applying (.) and

–

M

|∇u|(R–r)dv=

M

(R–r)u+R,iui

to (.) yields d dtλ b a= M

–Rijuiju– r

n|∇u|

_+bR

tu+auut

+ ut(–u+aulogu+bRu) – (R–r)u(–u+aulogu+bRu)

dv = M

–Rijuiju– r

n|∇u|

_+bR tu+

a



u_t

dv+λ

M

udv

t = M

–Rijuiju– r

n|∇u|

_+bR tu+

a

Ru 

dv. (.)

NoticingRt=R+ |Rij|–_nrRfor the normalized Ricci flow, we obtain from (.)

d dtλ b a= M

–Rijuiju– r

n|∇u|

_+bR tu+

a Ru  dv = M

–Rijuiju– r

n|∇u|

_+bu

R+ |Rij|– r nR +a Ru  dv = M

–Rijuiju+bR

u+ b|Rij|u+

a

Ru 

dv

–r

n

M

|∇u|+bRudv. (.)

Using (.), then (.) can be written as

d dtλ b a= M

Rijfij–  Rijf

i_fj_{–bRf+bR|∇f|}_{+ b|R} ij|+

a

R

e–fdv

–r

n

M

 |∇f|

_+bR

e–fdv. (.)

By virtue of a similar computation, we can obtain

d dtλ b a=   M

Rij+fij+

a

gij

e–fdv+

b–   M

|Rij|e–fdv

–na   – r n M 

f +bR

e–fdv, (.)

which gives

d dt

λb_a+na



 t

+r

nλ

b₌ 

M

Rij+fij+

a

gij

e–fdv

+

b–   M

|Rij|e–fdv. (.)

3 Proof of Theorem 1.4

Under the following coupled system (.), by a direct computation, we have the following:

∂ ∂t

e–fdv= –(ft+R)e–fdv=

f–|∇f|–afe–fdv

= –e–fdv–afe–fdv, (.)

∂ ∂t|∇f|

_{= R}ij_f

ifj+ fi(ft)i = Rijfifj+ fi

–f+|∇f|+af –R_i

= Rijfifj– fi(f)i+ fijfifj+ a|∇f|– Rifi. (.)

Thus, we have

d dt

M

e–fdv= –a

M

fe–fdv, (.)

d dt

M

Re–fdv=

M

R+ |Rij|–afR

e–fdv–

M

Re–fdv

=

M

|Rij|–afR

e–fdv, (.)

d dt

M

fe–fdv=

M

(af –R)e–fdv–

M

fe–fdv–

M

afe–fdv

=

M

af –af– (R+f)e–fdv (.)

and

d dt

M

|∇f|e–fdv

=

M

Rijfifj– fi(f)i+ fijfifj+ a|∇f|– Rifi

e–fdv

–

M

e–f|∇f|dv–

M

af|∇f|e–fdv

=

M

–f_ij– fi(f)i+ fijfifj+ a|∇f|– Rifi

e–fdv

–

M

af|∇f|e–fdv. (.)

By virtue of the Bochner formula with respect to thef-Laplacian, we have 

f|∇u| _=u

ij+ui(fu)i+

Rij+fijuiuj, ∀u, and hence

 =

M

f_ij+fi(ff)i+

Rij+fijfifj

e–fdv

=

M

f_ij+fi(f)i+Rijfifj–fijfifj

Therefore, (.) becomes

d dt

M

|∇f|e–fdv=

M

f_ij+ Rijfifj+ a|∇f|– Rifi

e–fdv–

M

af|∇f|e–fdv

=

M

f_ij+ Rijfij+ a|∇f|

e–fdv–

M

a(f + )(f)e–fdv. (.)

Therefore, from (.) and (.), we obtain

d dt

M

R+|∇f|e–fdv= 

M

Rij+fij–

a

fgij

e–fdv

–na 



M

fe–fdv+a

M

|∇f|e–fdv. (.)

Noticing (.) tells us that

–a

M

fe–fdv= d

dt

M

(f+ )e–fdv

+

M

(R+f)e–fdv. (.)

Thus, (.) can be written as

d dt

M

R+|∇f|+na  (f + )

e–fdv

= 

M

Rij+fij–

a

fgij

e–fdv+na 

M

R+|∇f|e–fdv+a

M

|∇f|e–fdv. (.)

Since (.) holds, we have

d dt

M

Re–fdv= 

M

Rij–

a

fgij

e–fdv–na 



M

fe–fdv, (.)

which gives d dt M

R+na  (f+ )

e–fdv= 

M

Rij–

a

fgij

e–fdv

+na 

M

R+|∇f|e–fdv. (.)

Therefore, we have

d dt

M

|∇f|+k

R+na  (f+ )

e–fdv

= 

M

Rij+fij–

a

fgij

e–fdv+ (k– )

M

Rij–

a

fgij

e–fdv

+na  k

M

R+|∇f|e–fdv+a

M|∇

f|e–fdv (.)

4 Conclusions

We establish the first variation formula of the lowest constantλb

a(g) along the Ricci flow and the normalized Ricci flow, such that to the following nonlinear equation there exist positive solutions:

–u+aulogu+bRu=λb_au (.) with _Mu_{dv= , where ais a real constant. Equation (.) can be seen as a nonlinear} version of eigenvalue problem of the operator –u+bR. In particular, whena= , our estimate (.) in Theorem . reduces to Theorem . of Cao in [] and the estimate (.) in Theorem . reduces to the Corollary . of Cao in [], respectively.

On the other hand, we obtained the first variation formula (.) of the functional

Fcd(g,f) =

M

|∇f|+cR+d(f + )e–fdv

under the Ricci flow coupled to a nonlinear backward heat-type equation

d

dtgij= –Rij,

ft= –f +|∇f|+af–R,

where c,d are two real constants. In particular, whena=  in (.), we obtain Theo-rem . of Li in [].

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

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

Acknowledgements

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

Received: 14 November 2015 Accepted: 4 February 2016

References

1. Cao, X-D: First eigenvalues of geometric operators under the Ricci flow. Proc. Am. Math. Soc.136, 4075-4078 (2008) 2. Li, J-F: Eigenvalues and energy functionals with monotonicity formulae under Ricci flow. Math. Ann.338, 927-946

(2007)

3. Perelman, G: The entropy formula for the Ricci flow and its geometric applications. arXiv:math.DG/0211159 4. Cao, X-D: Eigenvalues of (–+R

2) on manifolds with nonnegative curvature operator. Math. Ann.337, 435-441 (2007) 5. Cao, X-D, Hou, S, Ling, J: Estimate and monotonicity of the first eigenvalue under the Ricci flow. Math. Ann.354,

451-463 (2012)

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

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

8. Guo, H, Ishida, M: Harnack estimates for nonlinear heat equations with potentials in geometric flows. Manuscr. Math.