January 9, 2017 Some new gradient estimates for two nonlinear parabolic equations under Ricci flow WENWANG HUIZHOU 7 1 0 Abstract. In this paper, by maximum principle and cutoff function, we in- 2 vestigate gradient estimates for positive solutions to two nonlinear parabolic n equationsunderRicciflow. TherelatedHarnackinequalitiesarededuced. An a resultaboutpositivesolutionsonclosedmanifoldsunderRicciflowisabtained. J Asapplications,gradientestimatesandHarnackinequalitiesforpositivesolu- tions to the heat equation under Ricci flow are derived. These results in the 6 papercanberegardasgeneralizingthegradientestimatesofLi-Yau,J.Y.Li, HamiltonandLi-XutotheRicciflow. Ourresultsalsoimprovetheestimates ] G ofS.P.LiuandJ.SuntothenonlinearparabolicequationunderRicciflow. D . h t a 1. Introduction m Beginning with the pioneering work of Li and Yau [14], gradient estimates are [ also known as differential Harnack inequalities, which have tremendous impact in 1 geometric analysis, as shown for example in [14, 15, 16]. Moreover, both have v very important applications in singularity analysis. In perelman’s geometrization 1 5 conjecture [22, 23] on the poincar´e conjecture, a differential Harnack inequality 6 played an important role. 1 Next, we simply introduce research progress associated with this article. 0 Let (Mn,g) be a complete Riemannian manifold. Li and Yau [14] established a . 1 famous gradient estimate for positive solutions to the following heat equation 0 7 u =∆u (1.1) t 1 : on (Mn,g), which is described as v Theorem A (Li-Yau [14]) Let (Mn,g) be a complete Riemannian manifold. i X Suppose that on the ball B , Ricci(B ) K. Then for any α>1, 2R 2R ≥− r a u2 u Cα2 α2 nα2K nα2 sup |∇ | α t +√KR + + . (1.2) u2 − u ≤ R2 α2 1 α 1 2t B2R(cid:18) (cid:19) (cid:18) − (cid:19) − In general, on a complete Riemannian manifold, if Ricci(M) k, by letting ≥ − R in (1.2), one inferred →∞ u2 u nα2k nα2 |∇ | α t + . (1.3) u2 − u ≤ 2(α 1) 2t − 2010 Mathematics Subject Classification. 58J35, 35K05,53C21. Keywordsandphrases. Gradientestimate,nonlinearparabolicequation,heatequation,Ricci flow,Harnackinequality. Correspondingauthor: WenWang,E-mail: [email protected]. ThispaperwastypesetusingAMS-LATEX. 1 2 W.WANG In 1991, Li [15] generalized Li and Yau’s estimates to the nonlinear parabolic equation ∂ ∆ u(x,t)+h(x,t)uα(x,t)=0 (1.4) − ∂t (cid:18) (cid:19) on (Mn,g). In 1993, Hamilton in [8] generalized the constant α of Li and Yau’s result to the function α(t) = e2Kt. In 2006, Sun [27] also obtained a gradient estimate of different coefficient. In 2011, Li and Xu in [17] further promoted Li and Yau’s result, and found two new functions α(t). Recently, first author and Zhang in [28] further generalized Li and Xu’s results to the nonlinear parabolic equation (1.4). Related results can be found in [5, 11, 32]. In this paper, we investigate the two nonlinear parabolic equations ∂ u(x,t)=∆u(x,t)+h(x,t)ul(x,t) (1.5) t and ∂ u(x,t)=∆u(x,t)+au(x,t)logu(x,t) (1.6) t under Ricci flow, where the function h(x,t) 0 is defined on Mn [0,T],which is ≥ × C2 in the first variable and C1 in the second variable, T is a positive constant and l,a R, respectively. ∈ Recently,thereareanumberofstudiesonRicciflowonmanifoldsbyR.Hamilton [9,10]andothers,becausetheRicciflowisapowerfultoolinanalyzingthestructure of manifolds. Assume Mn is an n-dimensionalmanifold without boundary, and let (Mn,g(t))t∈[0,T] be an n-dimensional complete manifold with metric g(t) evolving by the Ricci flow ∂g(t) = 2R , (x,t) Mn [0,T]. (1.7) ∂t − ij ∈ × In 2008,Kuangand Zhang [11] proveda gradientestimate for positive solutions to the conjugate heat equationunder Ricci flow on a closedmanifold. In 2009,Liu [18] derived a gradient estimate for positive solutions to the heat equation under Ricci flow. Afterwards, Sun[26] generalized Liu’s results to general geometric flow. In 2010, Bailesteanu, Cao and Pulemotov [1] established some gradient estimates for positive solutions to the heat equation under Ricci flow. In 2016, Li and Zhu [19] generalized J. Y. Li’s [15] estimates under Ricci flow. Recently, Cao and Zhu [3] derived some Aronson and B´enilan estimates for porous medium equation u =∆um, m>1 t under Ricci flow. Li, Bai and Zhang [13] studied fast diffusion equation u =∆um, 0<m<1 t undertheRicciflow. ZhaoandFang[31]generalizedYang’sresult[30]totheRicci flow. Firstly,weintroducethreeC1functionsα(t),ϕ(t)andγ(t):(0,+ ) (0,+ ). ∞ → ∞ SupposethatthreeC1 functionsα(t),ϕ(t)andγ(t)satisfythefollowingconditions: (C1) α(t)>1, ϕ(t) and γ(t). NEW GRADIENT ESTIMATES 3 (C2) α(t) and ϕ(t) satisfy the following system 2ϕ 2ϕ 1 2αK ( α′) , n − ≥ n − α  2ϕ  ϕ2n −α′ >0, +αϕ′ 0. (C3) γ(t) satisfies  n ≥ γ′ 2ϕ 1 ( α′) 0. γ − n − α ≤ (C4)γ(t)isnon-decreasing,andα(t)isalsonon-decreasingorisboundeduniformly. This paper is organized as follows: We prove gradient estimates for the equa- tion (1.5) in Section 2 and gradient estimates for the equation (1.6) in Section 3. We derive related Harnack inequalities in Section 4. As special case, we deduce gradient estimates and Harnack inequality to the heat equation in section 5. De- tailed calculationof some specific functions α(t), ϕ(t) and γ(t) are givenin section 6. 2. Gradient estimates for the equation (1.5) In this section, we will derivesome new gradientestimates for positive solutions to equation (1.5) under the Ricci flow. 2.1. Main results. We state our results as follows. Theorem 2.1. Let (Mn,g(t))t∈[0,T] be a complete solution to the Ricci flow (1.7). Assume that Ric(x,t) K for some K > 0 and all t [0,T]. Suppose that there | | ≤ ∈ exist three functions α(t), ϕ(t) and γ(t) satisfy conditions (C1), (C2), (C3) and (C4). Given x Mn and R > 0, let u be a positive solution of the equation (1.5) in 0 ∈ the cube B := (x,t)d(x,x ,t) 2R,0 t T . Let h(x,t) be a function 2R,T 0 { | ≤ ≤ ≤ } defined on Mn [0,T] which is C1 in t and C2 in x, satisfying h2 δ h and 2 × |∇ | ≤ ∆h δ on B for some positive constants δ and δ . 3 2R,T 2 3 (1≥)−l 1. If γα4 C for some constant C , then ≤ α−1 ≤ 1 1 u2 u |∇ | α t +αh(x,t)ul−1 u2 − u Cα2 1 + √K +K + Cn2α4 +n23α2K ≤ R2 R ! R2γ +α nu1δ3+nα2u1δ1+ 2−lα23 nu1δ2+αϕ. 2 r If γ C for spome constant C , then p α−1 ≤ 2 2 u2 u |∇ | α t +αh(x,t)ul−1 u2 − u Cα2 1 + √K +K + Cn2α4 +n23α2K ≤ R2 R ! R2γ 4 W.WANG +α nu1δ3+nα2u1δ1+ 2−lα23 nu1δ2+αϕ, 2 r where C is a positivepconstant depending only on n apnd set u := max ul−1, δ := max h(x,t). 1 1 B2R,T B2R,T (2) l>1. If γα4 C for some constant C , then α−1 ≤ 1 1 u2 u |∇ | α t +αh(x,t)ul−1 u2 − u ≤Cα2 R12 + √RK +K!+ CRn22γα4 +n32α2K+nα2(l−1)δ1u2 +α n(lαl−11)u2δ2 +α23 n(l−1)δ1ϕ+α32 nδ3u2+αϕ. r − If γ C for some constant Cp, then p α−1 ≤ 2 2 u2 u |∇ | α t +αh(x,t)ul−1 u2 − u ≤Cα2 R12 + √RK +K!+ CRn22γα4 +n32α2K+nα2(l−1)δ1u2 +α n(lαl−11)u2δ2 +α23 n(l−1)δ1ϕ+α32 nδ3u2+αϕ, r − where C is a positive constant depenpding only on n andpset u := max ul−1, δ := max h(x,t). 2 1 B2R,T B2R,T Let us list some examples to illustrate the Theorem 2.1 holds for different cir- cumstances and see appendix in section 6 for detailed calculation process. Corollary 2.1. Suppose that (Mn,g(t))t∈[0,T] satisfies the hypotheses of Theorem 2.1. Then the following special estimates are valid. 1. Li-Yau type: αn nKα2 α(t)=constant, ϕ(t)= + ,γ(t)=tθ with 0<θ 2. t α 1 ≤ − If l 1, then ≤ u2 u |∇ | α t +αh(x,t)ul−1 u2 − u 1 α2 1 Cα2 (1+√KR)+ +K +αϕ ≤ R2 α 1R2 (cid:20) − (cid:21) +n23α2K+α nu1δ3+nα2u1δ1+ 2−lα23 nu1δ2. 2 r If l>1, then p p u2 u |∇ | α t +αh(x,t)ul−1 u2 − u 1 α2 1 Cα2 (1+√KR)+ +K +αϕ ≤ R2 α 1R2 (cid:20) − (cid:21) NEW GRADIENT ESTIMATES 5 +n23α2K+nα2(l−1)δ1u2+α n(lαl−11)u2δ2 r − 3 3 +α2 n(l 1)δ1ϕ+α2 nδ3u2. − 2. Hamilton type: p p n α(t)=e2Kt, ϕ(t)= e4Kt, γ(t)=te2Kt. t If l 1, then ≤ u2 u |∇ | α t +αh(x,t)ul−1 u2 − u 1 Cα4 Cα2 (1+√KR)+K + +αϕ ≤ R2 R2te2Kt (cid:20) (cid:21) +n32α2K+α nu1δ3+nα2u1δ1+ 2−lα nu1δ2. 2 r If l>1, then p p u2 u |∇ | α t +αh(x,t)ul−1 u2 − u 1 Cα4 Cα2 (1+√KR)+K + +αϕ ≤ R2 R2te2Kt (cid:20) (cid:21) +n23α2K+nα2(l−1)δ1u2+α n(lαl−11)u2δ2 r − 3 3 +α2 n(l 1)δ1ϕ+α2 nδ3u2.. − 3. Li-Xu type: p p sinh(Kt)cosh(Kt) Kt α(t)=1+ − , ϕ(t)=2nK[1+coth(Kt)], sinh2(Kt) γ(t)=tanh(Kt). If l 1, then ≤ u2 u |∇ | α t +αh(x,t)ul−1 u2 − u 1 C C (1+√KR)+K + +αϕ ≤ R2 R2tanh(Kt) (cid:20) (cid:21) +n32α2K+α nu1δ3+nα2u1δ1+ 2−lα nu1δ2. 2 r If l>1, then p p u2 u |∇ | α t +αh(x,t)ul−1 u2 − u 1 C Cα2 (1+√KR)+K + +αϕ ≤ R2 R2tanh(Kt) (cid:20) (cid:21) +n23α2K+nα2(l−1)δ1u2+α n(lαl−11)u2δ2 r − 3 3 +α2 n(l 1)δ1ϕ+α2 nδ3u2, − where α(t) is bounded upniformly. p 6 W.WANG 4. Linear Li-Xu type: n 1 α(t)=1+2Kt,ϕ(t)= +nK(1+2Kt+µKt),γ(t)=Kt with µ . t ≥ 4 If l 1, then ≤ u2 u |∇ | α t +αh(x,t)ul−1 u2 − u 1 Cα4 Cα2 (1+√KR)+K + +αϕ ≤ R2 R2Kt (cid:20) (cid:21) +n32α2K+α nu1δ3+nα2u1δ1+ 2−lα nu1δ2. 2 r If l>1, then p p u2 u |∇ | α t +αh(x,t)ul−1 u2 − u 1 Cα4 Cα2 (1+√KR)+K + +αϕ ≤ R2 R2Kt (cid:20) (cid:21) +n23α2K+nα2(l−1)δ1u2+α n(lαl−11)u2δ2 r − 3 3 +α2 n(l 1)δ1ϕ+α2 nδ3u2. − Remark 2.1. Theabovepresultscanberegaprdasgeneralizingthegradientestimates of Li-Yau [14], J. Y. Li [15], Hamilton [8] and Li-Xu [17] to the Ricci flow. Our resultsalsogeneralizetheestimatesofS.P.Liu[18]andJ.Sun[26]tothenonlinear parabolic equation under the Ricci flow. The local estimates in Theorem 2.1 imply global estimates. Corollary 2.2. Let (Mn,g(t))t∈[0,T] be a complete solution to the Ricci flow (1.7). Assumethat Ric(x,t) K for someK >0andall (x,t) Mn [0,T]. Letu(x,t) | |≤ ∈ × be a positive solution to equation (1.5) on Mn [0,T]. Let h(x,t) be a function × defined on Mn [0,T] which is C1 in t and C2 in x, satisfying h2 δ h and 2 × |∇ | ≤ ∆h δ on Mn [0,T] for some positive constants δ and δ . 3 2 3 ≥− × If l 1 and for (x,t) Mn (0,T], then ≤ ∈ × u2 u |∇ | α t +αh(x,t)ul−1 u2 − u 2 l αϕ+Cα αK + u δ +αu δ + − u δ , 1 3 1 1 1 2 ≤ " r 2 # p p where where C is a positive constant depending only on n and set u := max ul−1, δ := max h(x,t). 1 1 Mn×[0,T] Mn×[0,T] If l >1 and for (x,t) Mn (0,T], then ∈ × u2 u |∇ | α t +αh(x,t)ul−1 αϕ u2 − u ≤ ≤Cα αK+(l−1)αu2δ1+ (lα−l 1)1u2δ2 +α21 (l−1)δ1ϕ+α12 u2δ3 , h r − p p i NEW GRADIENT ESTIMATES 7 where where C is a positive constant depending only on n and set u := max ul−1, δ := max h(x,t). 1 1 Mn×[0,T] Mn×[0,T] We can derive a gradient estimate for an any positive solution to the following nonlinearparabolicequationundertheRicciflowonaclosedmanifoldwithoutany curvature conditions. The method of the proof is inspired by Hamilton [10], Shi [23] and Liu [18]. Theorem 2.2. Let (Mn,g(x,t))t∈[0,T] be a solution to the Ricci flow (1.7) on a closed manifold. If u is a positive solution to equation ∂ u=∆u+h(t)ul−1, t where h(t) is a C1 function and h(t) 0. Then for l 1, we have ≤ ≥ 1 u(x,t)2 max u2(x,0) u2(x,t) for (x,t) Mn [0,T]. (2.1) |∇ | ≤ 2t x∈Mn − ∈ × (cid:18) (cid:19) 2.2. Auxilliary lemma. To prove main results, we need a lemma. Let f =lnu. Then f =∆f + f 2+hul−1. (2.2) t |∇ | Let F = f 2 αf +αhul−1 αϕ, where α=α(t)>1 and ϕ=ϕ(t)>0. t |∇ | − − Lemma 2.1. Suppose that (Mn,g(t))t∈[0,T] satisfies the hypotheses of Theorem 2.1. We also assume that α(t)>1 and ϕ(t)>0 satisfy the following system 2ϕ 2ϕ 1 2αK ( α′) , n − ≥ n − α  2ϕ  ϕ2n −α′ >0, (2.3) +αϕ′ 0, and α(t) is non-decreasing.Then n ≥ ϕ 2ϕ 1 (∆ ∂ )F f + g 2+( α′) F α2n2K2 2 f F − t ≥ | ij n ij| n − α − − ∇ ∇ +2c(α 1)(l 1)ul−1 f 2+α(l 1)2hul−1 f 2 − − |∇ | − |∇ | +α(l 1)hul−1∆f +αul−1∆h+2(α 1)ul−1 f h. (2.4) − − ∇ ·∇ Proof. By directly computing, we have ∆F = ∆ f 2 α∆(f )+α∆(hul−1) t |∇ | − = 2f 2+2f f +2R f f α∆(f )+αh∆(ul−1) ij j iij ij i j t | | − +αul−1∆h+2α h ul−1 ∇ ∇ = 2 f 2+αR f +2f f +2R f f α(∆f) ij ij ij j iij ij i j t | | − +(cid:16)αh∆(ul−1)+αu(cid:17)l−1∆h+2α h ul−1, ∇ ∇ where we have used Bochner’s formula and n ∆(f )=(∆f) 2 R f . t t ij ij − i,j=1 X 8 W.WANG Applying Young’s inequality α 1 R f R f R 2+ f 2, ij ij ≤| ij|| ij|≤ 2| ij| 2α| ij| we conclude for R K, ij | |≤ ∆F f 2 α2 R 2+2f f +2R f f α(∆f) ij ij j iij ij i j t ≥ | | − | | − +αh∆(uXl−1)+αul−1∆h++2α h ul−1 ∇ ∇ f 2 α2n2K2+2f f +2R f f α(∆f) ij j iij ij i j t ≥ | | − − +αh∆(ul−1)+αul−1∆h+2α h ul−1. (2.5) ∇ ∇ On the other hand, we infer ∂ F = ( f 2) αf α′f +α′hul−1+αh(ul−1) t t tt t t |∇ | − − +αul−1h αϕ′ α′ϕ t − − = 2 f (f )+2R f f αf α′f +α′hul−1+αul−1h t ij i j tt t t ∇ ∇ − − +αh(ul−1) αϕ′ α′ϕ. (2.6) t − − We follow from (2.5) and (2.6), (∆ ∂ )F f 2 α2n2K2+2 f (∆f) α(∆f) +αh∆(ul−1) t ij t − ≥ | | − ∇ ∇ − +αul−1∆h+2α h ul−1 2 f (f )+αf +α′f t tt t ∇ ∇ − ∇ ∇ α′hul−1 αh(ul−1) αul−1h +αϕ′+α′ϕ t t − − − = f 2 α2n2K2+2 f (∆f) α(f f 2 hul−1) ij t t | | − ∇ ∇ − −|∇ | − +αh∆(ul−1)+αul−1∆h+2α h ul−1 2 f (f )+αf t tt ∇ ∇ − ∇ ∇ +α′f α′hul−1 αh(ul−1) αul−1h +αϕ′+α′ϕ t t t − − − = f 2 α2n2K2+2 f (∆f)+α( f 2) +αh∆(ul−1) ij t | | − ∇ ∇ |∇ | +αul−1∆h+2α h ul−1 2 f (f )+α′f t t ∇ ∇ − ∇ ∇ α′hul−1+αϕ′+α′ϕ. − By using the formula ( f 2) =2 f (f )+2Ric( f, f), t t |∇ | ∇ ·∇ ∇ ∇ we obtain (∆ ∂ )F f 2 α2n2K2+2 f (∆f)+2α f (f ) t ij t − ≥ | | − ∇ ∇ ∇ ∇ +2αR f f +αh∆(ul−1)+αul−1∆h+2α h ul−1 ij i j ∇ ∇ 2 f (f )+α′f α′hul−1+αϕ′+α′ϕ t t − ∇ ∇ − = f 2+2αR f f α2n2K2 2 f F ij ij i j | | − − ∇ ∇ +2(α 1) f (hul−1)+αh∆(ul−1)+αul−1∆h − ∇ ∇ +2α h ul−1+α′f α′hul−1+αϕ′+α′ϕ. (2.7) t ∇ ∇ − Applying the following two equations (ul−1) = (l 1)ul−1 f, ∇ − ∇ ∆(ul−1) = (l 1)2ul−1 f 2+(l 1)ul−1∆f, − |∇ | − to (2.7), we have (∆ ∂ )F f 2+2αR f f α2n2K2+2 f F +αul−1∆h t ij ij i j − ≥ | | − ∇ ∇ NEW GRADIENT ESTIMATES 9 +2h(α 1)(l 1)ul−1 f 2+2 (α 1)+α(l 1) ul−1 f h − − |∇ | − − ∇ ·∇ +hα(l 1)2ul−1 f 2+hα(l (cid:2)1)ul−1∆f (cid:3) − |∇ | − +α′f α′cul−1+αϕ′+α′ϕ. (2.8) t − Further applying unit matrix (δij)n×n and (2.8), we derive ϕ (∆ ∂ )F f + δ 2 2αK f 2 α2n2K2+2 f F − t ≥ | ij n ij| − |∇ | − ∇ ∇ +2h(α 1)(l 1)ul−1 f 2+2 (α 1)+α(l 1) ul−1 f h − − |∇ | − − ∇ ·∇ +hα(l 1)2ul−1 f 2+hα(l (cid:2)1)ul−1∆f +αul−1∆(cid:3)h − |∇ | − ϕ2 ϕ +α′f α′cul−1+αϕ′+α′ϕ 2 ∆f. t− − n − n Applying (2.2), we have ϕ 2ϕ 2ϕ (∆ ∂ )F f + δ 2+( 2αK) f 2 ( α′)f − t ≥ | ij n ij| n − |∇ | − n − t 2ϕ 2ϕ αϕ +( α′)cul−1 ( α′) α2n2K2 2 f F n − − n − α − − ∇ ∇ +2h(α 1)(l 1)ul−1 f 2+2 (α 1)+α(l 1) ul−1 f h − − |∇ | − − ∇ ·∇ +hα(l 1)2ul−1 f 2+hα(l (cid:2)1)ul−1∆f +αul−1∆(cid:3)h − |∇ | − ϕ2 2ϕ αϕ +αϕ′+α′ϕ +( α′) . (2.9) − n n − α Therefore, (2.4) is derived from (2.3) and (2.9). The proof is complete. (cid:3) 2.3. Proof of Theorem 2.1 and 2.2. In this section, we will prove the Theorem 2.1 and 2.2. Proof of Theorem 2.1. Let G=γ(t)F and γ(t)>0 be non-decreasing. Then (∆ ∂ )G=γ(∆ ∂ )F γ′F t t − − − ϕ 2ϕ 1 γ f + δ 2+( α′) G γα2n2K2 2 f G ≥ | ij n ij| n − α − − ∇ ∇ +2hγ(α 1)(l 1)ul−1 f 2+2(lα 1)γul−1 f h − − |∇ | − ∇ ·∇ +hγα(l 1)2ul−1 f 2+hγα(l 1)ul−1∆f +αγul−1∆h γ′F − |∇ | − − ϕ 2ϕ 1 γ′ =γ f + g 2+ ( α′) G γα2n2K2 2 f G | ij n ij| n − α − γ − − ∇ ∇ +2hγ(α 1)(l h1)ul−1 f 2+2 (cid:3)(α 1)+α(l 1) γul−1 f h − − |∇ | − − ∇ ·∇ +γα(l 1)2hul−1 f 2+γα(l (cid:2)1)hul−1∆f +αγul−(cid:3)1∆h. (2.10) − |∇ | − Now let ϕ(r) be a C2 function on [0, ) such that ∞ 1 if r [0,1], ϕ(r)= ∈ (0 if r [2, ), ∈ ∞ and ϕ′(r) 0 ϕ(r) 1, ϕ′(r) 0, ϕ′′(r) 0, | | C, ≤ ≤ ≤ ≤ ϕ(r) ≤ 10 W.WANG where C is an absolute constant. Let define by d(x,x ,t) ρ(x,t) φ(x,t)=ϕ(d(x,x ,t))=ϕ 0 =ϕ , 0 R R (cid:18) (cid:19) (cid:18) (cid:19) whereρ(x,t)=d(x,x ,t). Byusingmaximumprinciple,theargumentofCalabi[2] 0 allows us to suppose that the function φ(x,t) with support in B , is C2 at the 2R,T maximum point. By utilizing the Laplacian theorem, we deduce that φ2 C C |∇ | , ∆φ (1+√KR), (2.11) φ ≤ R2 − ≤ R2 For any 0 T T, let H = φG and (x ,t ) be the point in B at which ≤ 1 ≤ 1 1 2R,T1 H attains its maximum value. We can suppose that the value is positive, because otherwise the proof is trivial. Then at the point (x ,t ), we infer 1 1 0= (φG)=G φ+φ G, ∇ ∇ ∇ ∆(φG) 0,  (2.12) ∂ (φG)≤0.  t ≥ BytheevolutionformulaofthegeodesiclengthundertheRicciflow[6],wecalculate ρ 1 dρ ρ φ G= Gφ′ =Gφ′ Ric(S,S)ds t − R R dt R (cid:16) (cid:17) (cid:16) (cid:17)Zγt1 ρ 1 ρ Gφ′ Kρ Gφ′ K G√CK, ≤ R R ≤ R 2 ≤ (cid:16) (cid:17) (cid:16) (cid:17) whereγ isthegeodesicconnectingxandx underthemetricg(t ), S isthe unite t1 0 1 tangent vector to γ , and ds is the element of the arc length. t1 All the followingcomputationsareatthe point(x ,t ). Itis notdifficultto find 1 1 that ϕ 1 ϕ 2 f + δ 2 trf + δ | ij n ij| ≥ n | ij n ij| 1(cid:16) (cid:17) = ∆f +ϕ n 1(cid:16) 1 (cid:17)1 2 = F (α 1) f 2 n − α − α − |∇ | 1h G 2 i = +(α 1) f 2 . (2.13) α2n γ − |∇ | h i and ∆f = f f 2 cul−1 t −|∇ | − F α 1 = − f 2 ϕ<0. (2.14) −α − α |∇ | − To obtain main results, two cases will be shown. Case 1 l 1. ≤ From (2.14), we have ∆f 0. Then by substituting it into (2.10), we obtain ≤ (∆ ∂ )G=γ(∆ ∂ )F γ′F t t − − − ϕ 2ϕ 1 γ′ γ f + δ 2+ ( α′) G γα2n2K2 2 f G ≥ | ij n ij| n − α − γ − − ∇ ∇ +2hγ(α 1)(l h1)ul−1 f 2+αγ(cid:3)ul−1∆h − − |∇ |