The proof of Theorem 3 is similar to the proof of Theorem 16 in [3]. Let(M,g) be a compact Einstein manifold of dimensionnwith the property that the curvature tensor of(M,g)lies in the interior ofCfor all points p∈M. If(M,g)is Ricci flat, then the curvature tensor of(M,g)vanishes identically. Hence, it suffices to con-sider the case that(M,g)has positive Einstein constant. After rescaling the metric if necessary, we may assume that Ric= (n−1)g. As in [3], we define an algebraic curvature tensorSby
Si jkl=Ri jkl−κ(gikgjl−gilgjk), (1) whereκ is a positive constant. Letκ be the largest real number with the property thatSlies in the coneCfor all pointsp∈M. Since the curvature tensorRlies in the interior of the coneCfor all points p∈M, we conclude thatκ >0. On the other hand, the curvature tensorShas nonnegative scalar curvature. From this, we deduce thatκ≤1.
Proposition 1The tensor S satisfies
∆S+Q(S) =2(n−1)S+2(n−1)κ(κ−1)I, where Ii jkl=gikgjl−gilgjk.
Proof The curvature tensor of(M,g)satisfies
∆R+Q(R) =2(n−1)R (2)
(see [3], Proposition 3). Using (1), we compute
Q(S)i jkl=Q(R)i jkl+2(n−1)κ2(gikgjl−gilgjk)
−2κ(Ricikgjl−Ricilgjk−Ricjkgil+Ricjlgik).
Since Ric= (n−1)g, it follows that
Q(S) =Q(R) +2(n−1)κ(κ−2)I. (3) Combining (2) and (3), we obtain
∆S+Q(S) =2(n−1)R+2(n−1)κ(κ−2)I.
SinceR=S+κI, the assertion follows.
In the following, we denote byTSCthe tangent cone toCatS.
Proposition 2At each point p∈M, we have∆S∈TSC and Q(S)∈TSC.
Proof It follows from the definition ofκ that S lies in the coneC for all points p∈M. Hence, the maximum principle implies that∆S∈TSC. Moreover, since the coneCis invariant under the Hamilton ODE, we haveQ(S)∈TSC.
Proposition 3Suppose thatκ<1. Then S lies in the interior of the cone C for all points p∈M.
Proof Let us fix a pointp∈M. By Proposition 2, we have∆S∈TSCandQ(S)∈TSC.
Furthermore, we have−S∈TSCsinceCis a cone. Putting these facts together, we obtain
∆S+Q(S)−2(n−1)S∈TSC.
Using Proposition 1, we conclude that
2(n−1)κ(κ−1)I∈TSC.
Since 0<κ<1, it follows that−2I∈TSC. On the other hand,Ilies in the interior of the tangent coneTSC. Hence, the sum −2I+I=−I lies in the interior of the tangent coneTSC. By Proposition 5.4 in [4], there exists a real numberε>0 such thatS−εI∈C. Therefore,Slies in the interior of the coneC, as claimed.
Proposition 4The algebraic curvature tensor S defined in (1) vanishes identically.
Proof By definition ofκ, there exists a pointp0∈Msuch thatS∈∂Catp0. Hence, it follows from Proposition 3 thatκ=1. Consequently, the Ricci tensor ofSvanishes identically. SinceS∈Cfor all pointsp∈M, we conclude thatSvanishes identically.
SinceSvanishes identically, the manifold (M,g)has constant sectional curva-ture. This completes the proof of Theorem 3.
References
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Xiaodong Cao and Zhou Zhang
AbstractLet(M,g(t))be a solution to the Ricci flow on a closed Riemannian man-ifold. In this paper, we prove differential Harnack inequalities for positive solutions of nonlinear parabolic equations of the type
∂
∂tf=∆f−flnf+R f.
We also comment on an earlier result of the first author on positive solutions of the conjugate heat equation under the Ricci flow.
Keywordsdifferential Harnack inequality, nonlinear parabolic equation, Ricci flow, Ricci soliton.
Mathematics Subject Classification (2010)53C44.
1 Introduction
Let(M,g(t)),t∈[0,T), be a solution to the Ricci flow on a closed manifoldM.
In the first part of this paper, we deal with positive solutions of nonlinear parabolic equations onM. We establish Li-Yau type differential Harnack inequalities for such Xiaodong Cao
Department of Mathematics, Cornell University, Ithaca, NY 14853-4201, USA, e-mail:
cao@math.cornell.edu Zhou Zhang
Department of Mathematics, University of Michigan, at Ann Arbor, MI 48109, USA, e-mail:
zhangou@umich.edu
W. Ebeling et al. (eds.), Complex and Differential Geometry, Springer Proceedings
in Mathematics 8, DOI 10.1007/978-3-642-20300-8_5, © Springer-Verlag Berlin Heidelberg 2011 87
equations
positive solutions. More precisely,g(t)evolves under the Ricci flow
∂g(t)
∂t =−2Rc, (1)
whereRcdenotes the Ricci curvature ofg(t). We first assume that the initial metric g(0)has nonnegative curvature operator, which implies that for all timet∈[0,T), g(t)has nonnegative curvature operator (for example, in the case that dimension is 4, see [7]). Consider a positive function f(x,t)defined onM×[0,T), which solves the following nonlinear parabolic equation,
∂f
∂t =4f−flnf+R f, (2) where the symbol4stands for the Laplacian of the evolving metricg(t)andRis the scalar curvature ofg(t). For simplicity, we omitg(t)in the above notations. All geometry operators are with respect to the evolving metricg(t).
Differential Harnack inequalities were originated by P. Li and S.-T. Yau in [12]
for positive solutions of the heat equation (therefore also known as Li-Yau type Har-nack estimates). The technique was then brought into the study of geometric evolu-tion equaevolu-tion by R. Hamilton (for example, see [8]) and has ever since been playing an important role in the study of geometric flows. Applications include estimates on the heat kernel; curvature growth control; understanding the ancient solutions for geometric flows; proving noncollapsing result in the Ricci flow ([17]); etc. See [16]
for a recent survey on this subject by L. Ni.
Using the maximum principle, one can see that the solution for (2) remains pos-itive along the flow. It exists as long as the solution for (1) exists. The study of the Ricci flow coupled with a heat-type (or backward heat-type) equation started from R. Hamilton’s paper [9]. Recently, there has been some interesting study on this topic. In [17], G. Perelman proved a differential Harnack inequality for the funda-mental solution of the conjugate heat equation under the Ricci flow. In [2], the first author proved a differential Harnack inequality for general positive solutions of the conjugate heat equation, which was also proved independently by S. Kuang and Q.
S. Zhang in [11]. The study has also been pursued in [3, 6, 15, 20]. Various estimates are obtained recently by M. Bailesteanu, A. Pulemotov and the first author in [1], and by S. Liu in [13]. For nonlinear parabolic equations under the Ricci flow, local gradient estimates for positive solutions of equation
∂
∂tf=∆f+a flnf+b f,
wherea andb are constants, have been studied by Y. Yang in [19]. For general evolving metrics a similar estimate has been obtained by A. Chau, L.-F. Tam and C. Yu in [4], by S.-Y. Hsu in [10], and by J. Sun in [18]. In [14], L. Ma proved a gradient estimate for the elliptic equation
∆f+a flnf+b f=0.
In (2), if one defines
u(x,t) =−lnf(x,t),
then the functionu=u(x,t)satisfies the following evolution equation
∂u
∂t =∆u− |∇u|2−R−u. (3) The computation from (2) to (3) is standard, which also gives the explicit relation between these two equations.
Our motivation to study (2) under the Ricci flow comes from the geometric in-terpretation of (3), which arises from the study of expanding Ricci solitons. Recall that given a gradient expanding Ricci soliton(M,g)satisfying
Ri j+∇i∇jw=−1 4gi j, wherewis called soliton potential function, we have
R(g) +∆gw=−n 4.
In sight of this, by taking covariant derivative for the soliton equation and applying the second Bianchi identity, one can see that
R(g) +|∇gw|2g+w
2 =constant.
Also notice that the Ricci soliton potential functionwcan be differed by a constant in the above equations. So by choosing this constant properly, we have
R(g) +|∇gw|2g=−w 2−n
8. One consequence of the above identities is the following
|∇gw|2g=∆gw− |∇gw|2g−R(g)−w. (4)
Recall that the Ricci flow solution for an expanding soliton isg(t) =c(t)·ϕ(t)∗g (c.f. [5]), wherec(t) =1+2t and the family of diffeomorphismϕ(t)satisfies, for anyx∈M,
∂
∂t(ϕ(t)(x)) = 1
c(t)·(∇gw) (ϕ(t)(x)).
Thus the corresponding Ricci soliton potentialϕ(t)∗wsatisfies
∂ ϕ(t)∗w
∂t (x) = 1
c(t)(∇gw)(w) (ϕ(t)(x)) =|∇ϕ(t)∗w|2(x).
Along the Ricci flow, (4) becomes
|∇ϕ∗w|2=∆ ϕ∗w− |∇ϕ∗w|2−R−ϕ∗w c(t). Hence the evolution equation for the Ricci soliton potential is
∂ ϕ(t)∗w
∂t =∆ ϕ∗w− |∇ϕ∗w|2−R−ϕ∗w
c(t). (5)
The second nonlinear parabolic equation that we investigate in this paper is
∂u
∂t =∆u− |∇u|2−R− u
1+2t. (6)
Notice that (3) and (6) are closely related and only differ by their last terms.
Our first result deals with (2) and (3).
Theorem 1Let(M,g(t)), t∈[0,T), be a solution to the Ricci flow on a closed man-ifold, and suppose that g(0)(and so g(t))has weakly positive curvature operator.
Let f be a positive solution to the heat equation(2), u=−lnf and H=24u− |∇u|2−3R−2n
t . (7)
Then for all time t∈(0,T)
H6n 4.
Remark 1The result can be generalized to the context ofMbeing non-compact. In order for the same argument to work, we need to assume that the Ricci flow solution g(t)is complete with the curvature and all the covariant derivatives being uniformly bounded and the solutionuand its derivatives up to the second order are uniformly bounded(in the space direction).
Our next result deals with (6), which is also a natural evolution equation to con-sider, by the previous motivation.
Theorem 2Let(M,g(t)), t∈[0,T), be a solution to the Ricci flow on a closed man-ifold, and suppose that g(0)(and so g(t))has weakly positive curvature operator.
Let u be a smooth solution to(6), and define
H=24u− |∇u|2−3R−2n
t . (7)
Then for all time t∈(0,T)
H60.
Remark 2Iff is a positive function such that f =e−u, thenf satisfies the following evolution equation
∂f
∂t =4f+R f− flnf 1+t2.
In [2], the first author studied the conjugate heat equation under the Ricci flow.
In particular, the following theorem was proved.
Theorem 3[2, Theorem 3.6] Let(M,g(t)), t∈[0,T], be a solution to the Ricci flow, and suppose that g(t)has nonnegative scalar curvature. Let f be a positive solution of the conjugate heat equation
∂
∂tf=−4f+R f. Set v=−lnf−n2ln(4π τ),τ=T−t and
P=24v− |∇v|2+R−2n τ . Then we have
∂
∂ τP=4P−2∇P·∇v−2|vi j+Ri j−1
τgi j|2−2
τP−2|∇v|2 τ −2R
τ. (8) Moreover, for all time t∈[0,T),
P60.
In the last section, we apply a similar trick as in the proof of Theorem 1 and obtain a slightly different result, where we no longer need to assume thatg(t)has nonnegative scalar curvature.