Differential Harnack estimates for conjugate heat equation under the Ricci flow
aa r X i v : . [ m a t h . DG ] S e p Differential Harnack estimates for conjugate heatequation under the Ricci flow
Abimbola Abolarinwa ∗† Abstract
We prove certain localized and global differential Harnack inequality for allpositive solutions to the geometric conjugate heat equation coupled to the for-ward in time Ricci flow. In this case, the diffusion operator is perturbed withthe curvature operator, precisely, the Laplace-Beltrami operator is replacedwith ”∆ − R ( x, t )”, where R is the scalar curvature of the Ricci flow, whichis well generalised to the case of nonlinear heat equation with potential. Ourestimates improve on some well known results by weakening the curvature con-straints. As a by product, we obtain some Li-Yau type differential Harnackestimate. The localized version of our estimate is very useful in extending theresults obtained to noncampact case. Keywords:
Ricci Flow, Conjugate Heat Equation, Harnack inequality, Gra-dient Estimate, Laplace-Beltrami operator, Laplacian Comparison Theorem.
Let M be an n -dimensional compact (or noncompact without boundary) manifold onwhich a one parameter family of Riemannian metrics g ( t ) , t ∈ [0 , T ) is defined. Wesay ( M, g ( t )) is a solution to the Ricci flow if it is evolving by the following nonlinearweakly parabolic partial differential equation ∂∂t g ( x, t ) = − Ric ( x, t ) , ( x, t ) ∈ M × [0 , T ) (1.1)with g ( x,
0) = g (0), where Ric is the Ricci curvature and T ≤ ∞ . By the positivesolution to the heat equation on the manifold, we mean a smooth function atleast C in x and C in t , u ∈ C , ( M × [0 , T ]) which satisfies the following equation (cid:16) ∆ − ∂∂t (cid:17) u ( x, t ) = 0 , ( x, t ) ∈ M × [0 , T ] , (1.2) ∗ Department of Mathematics, University of Sussex, Brighton, BN1 9QH, United Kingdom. † E-mail: [email protected] g ( t ) in time. We can couple the Ricci flow (1.1) to equation ofthe form (1.2), either forward, backward, or perturbed with some potential function,see the author’s papers [1, 2] for more details. In the present we consider a more gen-eralized situation where the heat equation is perturbed, in this case, the Laplacianis replaced with ∆ − R ( x, t ), where R is the scalar curvature of the Ricci flow g ( t )and we obtain some Harnack and gradient estimates on the logarithm of the positivesolutions. The author also obtained various estimates on positive solutions and fun-damental solution in [3]. Throughout, we assume, that the manifold is endowed withbounded curvature, we remark that boundedness and nonnegativity of the curvatureis preserved as long as Ricci flow exists [17].Heat equation coupled to the Ricci flow can be associated with some physical inter-pretation in terms of heat conduction process. Precisely, the manifold M with initialmetric g ( x,
0) can be thought of as having the temperature distribution u ( x,
0) at t = 0 . If we now allow the manifold to evolve under the Ricci flow and simultaneouslyallow the heat to diffuse on M , then, the solution u ( x, t ) will represent the space-timetemperature on M . Moreover, if u ( x, t ) approaches δ -function at the initial time, weknow that u ( x, t ) >
0, this gives another physical interpretation that temperature isalways positive, whence we can consider the potential f = log u as an entropy or unitmass of heat supplied and the local production entropy is given by |∇ f | = |∇ u | u .Harnack inequalities are indeed very powerful tools in geometric analysis. Thepaper of Li and Yau [24] paved way for the rigorous studies and many interestingapplications of Harnack inequalities. They derived gradient estimates for positivesolutions to the heat operator defined on closed manifold with bounded Ricci cur-vature and from where their Harnack inequalities follows. These inequalities werein turn used to establish various lower and upper bounds on the heat kernel. Theyalso studied manifold satisfying Dirichlet and Neumann conditions. On the otherhand, Perelman in [26] obtained differential Harnack estimate for the fundamentalsolution to the conjugate heat equation on compact manifold evolving by the Ricciflow. Perelman’s results are unprecedented as they play a key factor in the proofof Poincar´e conjecture. Meanwhile, shortly before Perelman’s paper appeared on-line, C. Guenther [19] had found gradient estimates for positive solutions to the heatequation under the Ricci flow by adapting the methodology of Bakry and Qian [5] totime dependent metric case. As an application of her results, she got a Harnack-typeinequality and obtain a lower bound for fundamental solutions. These techniqueswere first brought into the study of Ricci flow by R. Hamilton, see [21] for instance.As useful as Harnack inequalities are, they have also been discovered in other geo-metric flows; See the following- H-D. Cao [7] for heat equation on K¨ahler manifolds,B. Chow [12] for Gaussian curvature flow and [13] for Yamabe flow, also B. Chowand R. Hamilton [15], and R. Hamilton [22] on mean curvature flow. The followingreferences among many others are found relevant [4, 9, 19, 29]. See also the followingmonographs [27] on Gradient estimates and [14, 16, 17, 25] for theory and applicationof Ricci flow.Recently, [6] and [23] have extended results in [29] to heat equation and its con-jugate respectively. We remark that our results are similar to those of [23] but withdifferent approach, the application to heat conduction that we have in mind has2reatly motivated our approach. The detail descriptions of our results are presentedin Sections 2 and 3 (localized version), while we collect some elements of the Ricciflow used in our calculation as an appendix in the last section. Let (cid:3) := ∂ t − ∆ be the heat operator acting on functions u : M × [0 , T ] → R , where M × [0 , T ] is endowed with the volume form dµ ( x ) dt . The conjugate to the heatoperator Γ is defined by (cid:3) ∗ = − ∂ t − ∆ x + R, (2.1)where R is the scalar curvature. We remark that for any solution g ( t ) , t ∈ [0 , T ] tothe Ricci flow and smooth functions u, v : M × [0 , T ] → R , the following identityholds Z T Z M ( (cid:3) u ) vdµ ( x ) dt = Z T Z M u ( (cid:3) ∗ v ) dµ ( x ) dt. (2.2)By direct application of integration by parts with the fact that the functions u and v are C with compact support (or if M is compact) and using evolution of dµ underthe Ricci flow the last identity can be shown easily. In a special case u ≡
1, we have ddt Z M vdµ = − Z M (cid:3) ∗ dµ. Proposition 2.1.
Let u = (4 πτ ) − n e − f be a positive solution to the conjugate heatequation. The evolution equation ∂f∂t = − ∆ f + |∇ f | − R + n τ (2.3) is equivalent to the following evolution (cid:3) ∗ u = 0 . (2.4) Proof. (cid:3) ∗ u = ( − ∂ t − ∆ x + R )(4 πτ ) − n e − f . By direct calculation, it follows that ∂ t [(4 πτ ) − n e − f ] = ( n τ − ∂ t f )(4 πτ ) − n e − f ∆[(4 πτ ) − n e − f ] = ( − ∆ f + |∇ f | )(4 πτ ) − n e − f . Then (cid:3) ∗ u = (cid:16) − n τ + ∂ t f + ∆ f − |∇ f | + R (cid:17) u = 0 , where we have made use of ∂ t τ = − u >
0, the claimed is then proved.3et (
M, g ( t )) , t ∈ [0 , T ] be a solution of the Ricci flow on a closed manifold. Here T > u be a positivesolution to the conjugate heat equation, then we have the following coupled system. ∂g ij ∂t = − R ij − ∂u∂t − ∆ g ( t ) u + R g ( t ) u = 0 , (2.5)which we refer to as Perelman’s conjugate heat equation coupled to the Ricci flow.We will prove differential Harnack and gradient estimates for all positive solutions ofthe conjugate heat equation in the above system. A differential Harnack estimate ofLi-Yau type yields a space-time gradient estimate for a positive solution to a heat-typeequation, which when integrated compares the solution at different points in spaceand time. We will later apply the maximum principle to obtain a localized version ofthe estimates. The main result of this subsection is contained in Theorem (2.2) and as an applicationwe arrived at Theorem (2.5) which gives the corresponding Li-Yau type gradientestimate for all positive solution to the conjugate heat equation in the system (2.5).
Theorem 2.2.
Let u ∈ C , ( M × [0 , T ]) be a positive solution to the conjugate heatequation (cid:3) ∗ u = ( − ∂ t − ∆ + R ) u = 0 and the metric g ( t ) evolve by the Ricci flow inthe interval [0 , T ) on a closed manifold M with nonnegative scalar curvature. Supposefurther that u = (4 πτ ) − n e − f , where τ = T − t , then for all points ( x, t ) ∈ ( M × [0 , T ]) ,we have the Harnack quantity P = 2∆ f − |∇ f | + R − nτ ≤ . (2.6) Then P evolves as ∂∂t P = − ∆ P + 2 h∇ f, ∇ P i + 2 (cid:12)(cid:12)(cid:12) R ij + ∇ i ∇ j f − τ g ij (cid:12)(cid:12)(cid:12) + 2 τ P + 2 τ |∇ f | + 4 nτ + 2 τ R. (2.7) for all t > . Moreover P ≤ for all t ∈ [0 , T ] . Note that u = (4 πτ ) − n e − f implies ln u = − f − n ln(4 πτ ) and we can write (2.6)as |∇ u | u − u t u − R − nτ ≤ , (2.8)which is similar to the celebrated Li-Yau [24] gradient estimate for the heat equationon manifold with nonnegative Ricci curvature.We need the usual routine computations as in the following;4 emma 2.3. Let ( g, f ) solve the system (2.5) above. Suppose further that u =(4 πτ ) − n e − f with τ = T − t . Then we have ( ∂∂t + ∆)∆ f = 2 R ij ∇ i ∇ j f + ∆ |∇ f | − ∆ R and ( ∂∂t + ∆) |∇ f | = 4 R ij ∇ i f ∇ j f + 2 h∇ f, ∇|∇ f | i + 2 |∇∇ f | − h∇ f, |∇ R | i . Proof.
By direct calculation and Proposition 2.1 ∂∂t (∆ f ) = ∂∂t ( g ij ∂ i ∂ j f ) = ∂∂t ( g ij ) ∂ i ∂ j f + g ij ∂ i ∂ j ∂∂t f = 2 R ij ∂ i ∂ j f + ∆( − ∆ f + |∇ f | − R + n τ )= 2 R ij ∇ i ∇ j f − ∆(∆ f ) + ∆ |∇ f | − ∆ R then, (cid:16) ∂∂t + ∆ (cid:17) ∆ f = 2 R ij ∇ i ∇ j f − ∆(∆ f ) + ∆ |∇ f | − ∆ R + ∆(∆ f )= 2 R ij ∇ i ∇ j f + ∆ |∇ f | − ∆ R Part 1 is proved. ∂∂t |∇ f | = 2 R ij ∂ i f ∂ j f + 2 g ij ∂ i f ∂ j ∂∂t f = 2 R ij ∂ i f ∂ j f + 2 h∇ f, ∇ ( − ∆ f + |∇ f | − R + n τ ) i = 2 R ij ∇ i f ∇ j f + 2 h∇ f, ∇|∇ f | i − h∇ f, ∇ ∆ f i − h∇ f, ∇ R i then, (cid:16) ∂∂t + ∆ (cid:17) |∇ f | = 2 R ij ∂ i f ∂ j f + 2 h∇ f, ∇|∇ f | i − h∇ f, ∇ ∆ f i − h∇ f, ∇ R i + ∆ |∇ f | . Using the Bochner identity∆ |∇ f | = 2 |∇∇ f | + 2 h∇ f, ∇ ∆ f i + 2 Rc ( ∇ f, ∇ f )we obtain the identity in part (2). Proof.
Proof of Theorem 2.2. Since P = 2∆ f − |∇ f | + R − nτ and by directcomputation and using Lemma 2.3, we have5 ∂∂t + ∆ (cid:17) P = 2 (cid:16) ∂∂t + ∆ (cid:17) ∆ f − (cid:16) ∂∂t + ∆ (cid:17) |∇ f | + (cid:16) ∂∂t + ∆ (cid:17) R − ∂∂t (cid:16) nτ (cid:17) = 4 R ij ∇ i ∇ j f + 2∆ |∇ f | − R − Rc ( ∇ f, ∇ f ) − h∇ f, ∇|∇ f | i− |∇∇ f | + 2 h∇ f, ∇ R i + 2∆ R + 2 | Rc | + 2 nτ = 4 R ij ∇ i ∇ j f + 2 | Rc | + 2 nτ − h∇ f, ∇|∇ f | i + 2 h∇ f, ∇ R i + 2∆ |∇ f | − Rc ( ∇ f, ∇ f ) − |∇∇ f | = 4 R ij ∇ i ∇ j f + 2 | Rc | + 2 nτ − h∇ f, ∇|∇ f | i + 2 h∇ f, ∇ R i + ∆ |∇ f | − Rc ( ∇ f, ∇ f ) + 2 h∇ f, ∇ ∆ f i = 4 R ij ∇ i ∇ j f + 2 | Rc | + 2 nτ + 2 |∇∇ f | − h∇ f, ∇|∇ f | i + 2 h∇ f, ∇ R i + 4 h∇ f, ∇ ∆ f i = 4 R ij ∇ i ∇ j f + 2 | Rc | + 2 nτ + 2 |∇∇ f | + 2 h∇ f, ∇ P i = 2 | R ij + ∇ i ∇ j f | + 2 nτ + 2 h∇ f, ∇ P i . By direct computation we notice that (cid:12)(cid:12)(cid:12) R ij + ∇ i ∇ j f − τ g ij (cid:12)(cid:12)(cid:12) = | R ij + ∇ i ∇ j f | − τ ( R + ∆ f ) + nτ , which implies2 | R ij + ∇ i ∇ j f | + 2 nτ = 2 (cid:12)(cid:12)(cid:12) R ij + ∇ i ∇ j f − τ g ij (cid:12)(cid:12)(cid:12) + 4 τ ( R + ∆ f ) . Also 4 τ ( R + ∆ f ) = 2 τ ( R + 2∆ f ) + 2 τ R = 2 τ P + 2 τ |∇ f | + 4 nτ + 2 τ R. Therefore, by putting these together we have (cid:16) ∂∂t + ∆ (cid:17) P = 2 h∇ f, ∇ P i + 2 (cid:12)(cid:12)(cid:12) R ij + ∇ i ∇ j f − τ g ij (cid:12)(cid:12)(cid:12) + 2 τ P + 2 τ |∇ f | + 4 nτ + 2 τ R, which proves the evolution equation for P .To prove that P ≤ t ∈ [0 , T ], we know that for small τ , P ( τ ) < W -entropy monotonicity R ij + ∇ i ∇ j f − τ g ij ≥ g ( t ) is a shrinking gradient soliton. So our conclusionwill follow from a theorem in [10, Theorem 4].For completeness we show this; by Cauchy-Schwarz inequality and the fact that R = g ij R ij and P i,j g ij = n , we have | R ij + ∇ i ∇ j f − τ g ij | ≥ n ( R + ∆ f − nτ ) and by definition of P P + R + |∇ f | = 2( R + ∆ f − nτ ) . Hence 2 (cid:12)(cid:12)(cid:12) R ij + ∇ i ∇ j f − τ (cid:12)(cid:12)(cid:12) ≥ n ( P + R + |∇ f | ) . Putting the last identity into the evolution equation for P yields ∂P∂t ≥ − ∆ P + 2 h∇ P, ∇ f i + 12 n ( P + R + |∇ f | ) + 2 τ ( P + R + |∇ f | ) + 4 nτ = − ∆ P + 2 h∇ P, ∇ f i + 12 n ( P + R + |∇ f | + 2 nτ ) + 2 nτ . This implies that ∂P∂τ ≤ ∆ P − h∇ P, ∇ f i − n ( P + R + |∇ f | + 2 nτ ) − nτ . Then ∂P∂τ ≤ ∆ P − h∇ P, ∇ f i . (2.9)Applying the maximum principle to the evolution equation (2.9) yields clearly that P ≤ τ , hence, for all t ∈ [0 , T ) . The result here is an improvement on Kuang and Zhang’s [23] since it holds withno assumption on the curvature. This result can also be compared with those of [8, 9]where they define a general Harnack quantity for conjugate heat equation and deriveits evolution under the Ricci flow.
The aim of this subsection is to state and proof the Li-Yau type pointwise Harnackestimate corresponding to the Harnack inequality proved in the last subsection. Weintroduce some notations. Given x , x ∈ M and t , t ∈ [0 , T ] satisfying t < t Θ( x , t ; x , t ) = inf γ Z t t (cid:12)(cid:12)(cid:12) ddt γ ( t ) (cid:12)(cid:12)(cid:12) dt, where the infimum is taken over all the smooth path γ : [ t , x ] → M connecting x and x . The norm | . | depends on t . We now present a lemma which is crucial to theproof of our main result in this subsection.7 emma 2.4. Let ( M, g ( t )) be a complete solution to the Ricci flow. Let u : M × [0 , T ] → R be a smooth positive solution to the heat equation (1.1). Define f = log u and assumed that − ∂f∂t ≤ α (cid:16) βt − |∇ f | (cid:17) , ( x, t ) ∈ M × [0 , T ] for some α, β > . Then, the inequality u ( x , t ) ≤ u ( x , t ) (cid:16) t t (cid:17) αβ exp (cid:16) α x , t ; x , t ) (cid:17) (2.10) holds for all ( x , t ) and ( x , t ) such that t < t . Proof.
Obtain the time differential of a function f depending on the path γ as follows ddt f ( γ ( t ) , t ) = ∇ f ( γ ( t ) , t ) ddt γ ( t ) − ∂∂s ( γ ( t ) , s ) (cid:12)(cid:12)(cid:12) s = t ≤ (cid:12)(cid:12)(cid:12) ∇ f (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ddt γ ( t ) (cid:12)(cid:12)(cid:12) + 1 α (cid:16) βt − |∇ f | (cid:17) ≤ α (cid:12)(cid:12)(cid:12) ddt γ ( t ) (cid:12)(cid:12)(cid:12) + βαt . The last inequality was obtained by the application of completing the square methodin form of a quadratic inequality satisfying ax − bx ≥ − b a , ( a, b > t to t , we have f ( x , t ) − f ( x , t ) = Z t t ddt f ( γ ( t ) , t ) dt ≤ α Z t t (cid:12)(cid:12)(cid:12) ddt γ ( t ) (cid:12)(cid:12)(cid:12) dt + βα log t (cid:12)(cid:12)(cid:12) t t . The required estimate ( 2.10) follows immediately after exponentiation.We have the following as an immediate consequence of the above theorem
Corollary 2.5. (Harnack Estimates). Let u ∈ C , ( M × [0 , T )) be a positive solutionto the conjugate heat equation Γ ∗ u = 0 and g ( t ) , t ∈ [0 , T ) evolve by the Ricci flowon a closed manifold M with nonnegative scalar curvature R . Then for any points ( x , t ) and ( x , t ) in M × (0 , T ) such that < t ≤ t < T , the following estimateholds u ( x , t ) u ( x , t ) ≤ (cid:16) τ τ (cid:17) n exp h Z | γ ′ ( s ) | τ − τ ) ds + ( τ − τ )2 R i , (2.11) where τ i = T − t i , i = 1 , and γ : [0 , is a geodesic curve connecting points x and x in M. Proof.
Let γ : [0 ,
1] be a minimizing geodesic connecting points x and x in M suchthat γ (0) = x and γ (1) = x with | γ ′ ( s ) | being the length of the vector γ ′ ( s ) at time τ ( s ) = (1 − s ) τ + sτ , ≤ τ ≤ τ ≤ T. Define η ( s ) = ln u ( γ ( s ) , (1 − s ) τ + sτ ).Clearly, η (0) = ln u ( x , t ) and η (1) = ln u ( x , t ) . η ( s ), we obtainln u ( x , t ) − ln u ( x , t ) = Z (cid:16) ∂∂s ln u ( γ ( s ) , (1 − s ) τ + sτ ) (cid:17) ds i.e., ln (cid:16) u ( x , t ) u ( x , t ) (cid:17) = ln u ( γ ( t ) , t ) (cid:12)(cid:12)(cid:12) . By direct computation, we have on the path γ ( s ) that ∂∂s η ( s ) = dds ln u = ∇ ln u · γ ′ ( s ) + ∂∂t ln u = ∇ uu · γ ′ ( s ) − u t ( τ − τ ) u = ( τ − τ ) (cid:16) ∇ uu · γ ′ ( s ) τ − τ − u t u (cid:17) . From Theorem 2.2, we have |∇ u | u − u t u ≤ R + 2 nτ , which implies − u t u ≤
12 ( R + 2 nτ ) − |∇ u | u . By this, we have dds ln u ≤ ( τ − τ ) (cid:16) ∇ uu · γ ′ ( s )( τ − τ ) − |∇ u | u + 12 ( R + 2 nτ ) (cid:17) = − ( τ − τ )2 (cid:16) ∇ uu − γ ′ ( s )( τ − τ ) (cid:17) + ( τ − τ )2 | γ ′ ( s ) | ( τ − τ ) + ( τ − τ )2 (cid:16) R + 2 nτ (cid:17) ≤ | γ ′ ( s ) | τ − τ ) + ( τ − τ )2 (cid:16) R + 2 nτ (cid:17) . Now integrating with respect to s , from 0 to 1, we haveln u (cid:12)(cid:12)(cid:12) ≤ Z | γ ′ ( s ) | τ − τ ) + ( τ − τ )2 Z Rds + ln (cid:16) τ τ (cid:17) n , (2.12)exponentiating both sides, we get u ( x , t ) u ( x , t ) ≤ (cid:16) τ τ (cid:17) n exp h Z | γ ′ ( s ) | τ − τ ) ds + ( τ − τ )2 R i . Main Result III. (Localising the Harnack andGradient Estimates)
We establish a localised form of the Harnack and gradient estimates obtained in thelast subsection. The main idea is the application of the Maximum principle on somesmooth cut-off function. It was also the basic idea used by Li and Yau in [24], thistype of approach has since become tradition. It has been systematically developedover the years since the paper of Cheng and Yau [11], see also [27, 28], however ourcomputation is more involved as the metric is also evolving.A natural function that will be defined on M is the distance function from a givenpoint, namely, let p ∈ M and define d ( x, p ) for all x ∈ M, where dist ( · , · ) is thegeodesic distance. Note that d ( x, p ) is only Lipschitz continuous, i.e., everywherecontinuously differentiable except on the cut locus of p and on the point where x and p coincide. It is then easy to see that |∇ d | = g ij ∂ i d ∂ j d = 1 on M \ {{ p } ∪ cut ( p ) } . Let d ( x, y, t ) be the geodesic distance between x and y with respect to the metric g ( t ), we define a smooth cut-off function ϕ ( x, t ) with support in the geodesic cube Q ρ,T := { ( x, t ) ∈ M × (0 , T ] : d ( x, p, t ) ≤ ρ } , for any C -function ψ ( s ) on [0 , + ∞ ) with ψ ( s ) = (cid:26) , s ∈ [0 , , , s ∈ [2 , + ∞ )and ψ ′ ( s ) ≤ , | ψ ′ | ψ ≤ C and | ψ ′′ ( s ) | ≤ C , where C , C are absolute constants depending only on the dimension of the manifold,such that ϕ ( x, t ) = ψ (cid:16) d ( x, p, t ) ρ (cid:17) and ϕ (cid:12)(cid:12)(cid:12) Q ρ,T = 1 . We will apply the maximum principle and invoke Calabi’s trick [ ? ] to assume every-where smoothness of ϕ ( x, t ) since ψ ( s ) is in general Lipschitz. We need Laplaciancomparison theorem to do some calculation on ϕ ( x, t ). Here is the statement of thetheorem; Let M be a complete n -dimensional Riemannian manifold whose Ricci cur-vature is bounded from below by Rc ≥ ( n − k for some constant k ∈ R . Then theLaplacian of the distance function satisfies∆ d ( x, p ) ≤ ( n − √ k cot( √ kρ ) , k > n − ρ − , k = 0( n − p | k | coth( p | k | ρ ) , k < . (3.1)10or detail of the Laplacian comparison theorem see [17, Theorem 1.128] or the book[27]. We are now set to prove the localized version of the gradient estimate for thesystem (2.5). Theorem 3.1.
Let u ∈ C , ( M × [0 , T ]) be a positive solution to the conjugate heatequation Γ ∗ u = ( − ∂ t − ∆ + R ) u = 0 defined in geodesic cube Q ρ,T and the metric g ( t ) evolves by the Ricci flow in the interval [0 , T ] on a closed manifold M with boundedRicci curvature, say Rc ≥ − Kg , for some constant K > . Suppose further that u = (4 πτ ) − n e − f , where τ = T − t , then for all points in Q ρ,T we have the followingestimate |∇ u | u − u t u − R ≤ n − δn ( τ + C ρ + √ Kρ + Kρ + 1 T !) , (3.2) where C is an absolute constant depending only on the dimension of the manifold and δ such that δ < n . Proof.
Recall the evolution equation for the differential Harnack quantity P = 2∆ f − |∇ f | + R − nτ ,∂∂t P ≥ − ∆ P + 2 h∇ f, ∇ P i + 2 | R ij + ∇ i ∇ j f − τ g ij | + 2 τ P + 4 nτ + 2 τ |∇ f | , using the non negativity of the scalar curvature Multiplying the quantity P by tϕ ,since ϕ is time-dependent we have at any point where ϕ = 0 that1 τ ∂∂t ( τ ϕP ) = ϕ ∂P∂t + ∂ϕ∂t P − ϕPτ ≥ ϕ (cid:16) − ∆ P + 2 h∇ f, ∇ P i + 2 τ P + 4 nτ + 2 τ |∇ f | (cid:17) + 2 ϕ | R ij + ∇ i ∇ j f − τ g ij | + ∂ϕ∂t P − ϕPτ = − ∆( ϕP ) + 2 ∇ ϕ ∇ P + 2 h∇ f, ∇ P i ϕ + P (∆ + ∂ t ) ϕ + 4 nτ ϕ + ϕPτ + 2 τ ϕ |∇ f | + 2 ϕ | R ij + ∇ i ∇ j f − τ g ij | . The last equality is due to derivative test on ( ϕP ) at the minimum point as obtainedin the condition (3.5) below. The approach is to estimate ∂∂t ( τ ϕP ) at the pointwhere minimum (or maximum) value for ( τ ϕP ) is attained and do some analysisat the minimum (or maximum) point. We know that the support of ( τ ϕP )( x, t ) iscontained in Q ρ × [0 , T ] since Supp ( ϕ ) ⊂ Q ρ,T := { ( x, t ) ∈ M × (0 , T ] : d ( x, p, t ) ≤ ρ } . Now let ( x , t ) be a point in Q ρ,T at which ( τ ϕP ) attains its minimum value. Atthis point, we have to assume that P is positive since if P ≤
0, we have the sameestimate and ( τ ϕP )( x , t ) ≤ τ ϕP )( x, t ) ≤ x ∈ M such that thedistance d ( x, x , t ) ≤ ρ and the theorem will follow trivially.11ote that at the minimum point ( x , t ) we have by the derivative test that (0 ≤ ϕ ≤ ∇ ( τ ϕP )( x , t ) = 0 , ∂∂t ( τ ϕP )( x , t ) ≤ and ∆( τ ϕP )( x , t ) ≥ . (3.3)We shall obtain a lower bound for τ ϕP at this minimum point. Therefore0 ≥ − ∆( ϕP ) + 2 ∇ ϕ ∇ P + 2 h∇ f, ∇ P i ϕ + P (∆ + ∂ t ) ϕ + ϕPτ + 4 nτ ϕ + 2 τ ϕ |∇ f | + 2 ϕ | R ij + ∇ i ∇ j f − τ g ij | . (3.4)By the argument in (3.3) and product rule we have ∇ ( ϕP )( x , t ) − P ∇ ϕ ( x , t ) = ϕ ∇ P ( x , t )which means ϕ ∇ P can always be replaced by − P ∇ ϕ . Similarly, − ϕ ∆ P = − ∆( ϕP ) + P ∆ ϕ + 2 ∇ ϕ ∇ P, (3.5)which we have already used before the last inequality. Notice that by direct calculationusing product rule ∇ ϕ ∇ P = ∇ ϕϕ · ∇ ( ϕP ) − |∇ ϕ | ϕ P and 2 h∇ f, ∇ P i ϕ = h∇ f, ∇ ( ϕP ) i − h∇ f, ∇ ϕ i P. Putting the last two equations into (3.4) we have0 ≥ − ∆( ϕP ) + 2 ∇ ϕϕ · ∇ ( ϕP ) − |∇ ϕ | ϕ P + 2 h∇ f, ∇ ( ϕP ) i − h∇ f, ∇ ϕ i P + P (∆ + ∂ t ) ϕ + ϕPτ + 4 nτ ϕ + 2 τ ϕ |∇ f | + 2 ϕ | R ij + ∇ i ∇ j f − τ g ij | . By using the argument in (3.3)0 ≥ − |∇ ϕ | ϕ P − h∇ f, ∇ ϕ i P + P (∆ + ∂ t ) ϕ + ϕPτ + 4 nτ ϕ + 2 τ ϕ |∇ f | + 2 ϕ | R ij + ∇ i ∇ j f − τ g ij | . (3.6)Observe that for any δ > |∇ f ||∇ ϕ | P = 2 ϕ |∇ f | |∇ ϕ | ϕ P ≤ δϕ |∇ f | P + δ − |∇ ϕ | ϕ P |∇ f ||∇ ϕ | P ≤ δϕ |∇ f | P + δϕP + δ − |∇ ϕ | ϕ P (3.7)12nd also that | R ij + ∇ i ∇ j f − τ g ij | ≥ n (cid:16) R + ∆ f − nτ (cid:17) . It is equally clear that P = 2∆ f − |∇ f | + R − nτ = 2 (cid:16) R + ∆ f − nτ (cid:17) − |∇ f | − R, which implies ( P + |∇ f | + R ) = 2 (cid:16) ∆ f + R − nτ (cid:17) . Therefore 2 ϕ | R ij + ∇ i ∇ j f − τ g ij | ≥ ϕ n (cid:16) P + |∇ f | + R (cid:17) . Notice also that( P + |∇ f | + R ) ( y, s ) = ( P + |∇ f | + R + − R − ) ( y, s ) ≥
12 ( P + |∇ f | + R + ) ( y, s ) − ( R − ) ( y, s ) ≥
12 ( P + |∇ f | ) ( y, s ) − ( R − ) ( y, s ) ≥
12 ( P + |∇ f | )( y, s ) − ( sup Q ρ,T R − ) ≥
12 ( P + |∇ f | )( y, s ) − n K , where we have applied some inequalities, namely; 2( a − b ) ≥ a − b and ( a + b ) ≥ a + b with a, b ≥ R ij ≥ − K, which implies R = − nK = ⇒ R − ≤ nK and R = − R − . Hence2 ϕ | R ij + ∇ i ∇ j f − τ g ij | ≥ ϕ n P + ϕ n |∇ f | . (3.8)Whereever P <
0, we then obtain from (3.6) - (3.8) that0 ≥ (cid:16) n − δ (cid:17) ϕP + ( ( δ − − |∇ ϕ | ϕ + (∆ + ∂ t ) ϕ + ϕτ ) P − (cid:16) δ − n (cid:17) ϕ |∇ f | + 2 τ ϕ |∇ f | + 4 nτ ϕ, using the inequality of the form m |∇ f | − n |∇ f | ≥ − n m and multiplying by ϕ again( ϕ = 0), we have a quadratic polynomial in ( ϕP ) which we use to bound ( ϕP ) in thefollowing (cid:16) n − δ (cid:17) ( ϕP ) + ( ( δ − − |∇ ϕ | ϕ + (∆ + ∂ t ) ϕ + ϕτ ) ( ϕP ) − nτ (cid:16) − nδ − (cid:17) ϕ ≤ . . (3.9)13ote that if there is a number x ∈ R satisfying inequality px + qx + r ≤ , when p > , q > r <
0, then q − pr > − q − p q − pr p ≤ x ≤ − q + p q − pr p , which clearly implies − q − √− prp ≤ x ≤ q + √− prp . Now, choosing δ such that δ < n and denoting Z = ( δ − − |∇ ϕ | ϕ + (∆ + ∂ t ) ϕ, we obtain τ ϕP ≥ − n − δn ( τ Z + ϕ + 4 ϕ √ δn ) . Moreover, since τ ≤ τ ≤ T and 0 ≤ ϕ ≤
1, we have τ P ≥ − n − δn n τ Z + 1 + C o , where C depends on n and δ . It remains to estimate Z via appropriate choice ofa cut function ϕ : M × [0 , T ] → [0 ,
1] such that ∂∂t ϕ, ∆ ϕ and |∇ ϕ | ϕ have appropriateupper bounds. The main difficulty with this kind of approach lies in the fact thatfor any cut-off function, one gets different kind of estimates and therefore the cut-offfunction in use must be chosen so related to the result one is looking for.Define a C -function 0 ≤ ψ ≤
1, on [0 , ∞ ) satisfying ψ ′ ( s ) ≤ , | ψ ′ | ψ ≤ C and | ψ ′′ ( s ) | ≤ C and define ϕ by ϕ ( x, t ) = ψ (cid:16) d ( x, x , t ) ρ (cid:17) and we have the following after some computations |∇ ϕ | ϕ = | ψ ′ | · |∇ d | ρ ϕ ≤ C ρ , and by the Laplacian comparison Theorem (3.1) we have∆ ϕ = ψ ′ ∆ dρ + ψ ′′ |∇ d | ρ ≤ C ρ √ K + C ρ Next is to estimate time derivative of ϕ : consider a fixed smooth path γ : [ a, b ] → M whose length at time t is given by d ( γ ) = R ba | γ ′ ( t ) | g ( t ) dr , where r is the arc length.Differentiating we get ∂∂t ( d ( γ )) = 12 Z ba (cid:12)(cid:12)(cid:12) γ ′ ( t ) (cid:12)(cid:12)(cid:12) − g ( t ) ∂g∂t (cid:16) γ ′ ( t ) , γ ′ ( t ) (cid:17) dr = Z γ Rc ( ξ, ξ ) dr, ξ is the unit tangent vector to the path γ . For detail see [16, Lemma 3.11].Now ∂∂t ϕ = ψ ′ (cid:16) dρ (cid:17) ρ ddt ( d ( x, p, t )) = ψ ′ (cid:16) dρ (cid:17) ρ Z γ Rc ( ξ ( s ) , ξ ( s )) ds ≤ √ C ρ ψ K. Therefore Z ≤ C ′ ρ + C ρ √ K + √ C ρ K + C ρ , where C ′ depends on n and δ. Hence ϕP ≥ − n − δn ( τ + C ρ + √ Kρ + Kρ + 1 τ !) , where C = max { C , C , C } . The required estimate follows since both minimum andmaximum points for ( ϕP ) are contained in the cube Q ρ,T . Our main results in Section 2 hold for all positive solutions and calculations are donewithout recourse to reduced length, therefore, they can be seen as improvement onPerelman’s which only works for the fundamental solution via his reduced distancefunction. After a simple modification and ǫ regularisation method we can get acorresponding result for heat kernel-type function, namely, if the function u ( x, t ) isdefined on M × (0 , T ] instead of M × [0 , T ], it suffices to replace u ( x, t ) and g ( x, t )with u ( x, t + ǫ ) and g ( x, t + ǫ ) for a sufficiently small ǫ >
0, do similar analysis andlater send ǫ to 0. The local estimate in Section 3 is desirable to extend our result tothe case the manifold is noncompact, for example, in the local monotonicity formulaand mean value theorem considered in [18] a local version is needed. The estimatesobtained here are used to prove on diagonal and gaussian-type upper bound for heatkernel under a mild assumption on curvature and a technical lemma involving thebest constant in the Sobolev embedding. This will be announced in a forthcomingpaper. Appendix
Elements of the Ricci Flow
Given an n -dimensional Riemannaian manifold M endowed with metric g . In localcoordinate { x i } , ≤ i ≤ n , we can write the metric in component form ds = g = g ij dx i dx j . f defined on M , then, the Laplace-Beltrami operatoracting on f is defined by∆ f = 1 p | g | n X i,j ∂∂x i (cid:16)p | g | g ij ∂∂x j f (cid:17) = g ij (cid:16) ∂ i ∂ j f − Γ kij ∂ k f (cid:17) , where ( g ij ) = ( g ij ) − is the metric inverse, | g | is the matrix determinant of ( g ij )and Γ kij are the Christoffel’s symbols. The degenerate parabolic partial differentialequation ∂∂t g ij = − R ij is the Ricci flow on ( M, g ( t )), where R ij is the component of the Ricci curvaturetensor and g ( t ) is a one-parameter family of Riemannian metrics. The degeneracyof the pde is due to the group of differomorphism invariance, but we are sure of theexistence of solution at least for short a time (Hamilton [20]).Interestingly all geometric quantities associated with M also evolve along the Ricciflow, in the present study we have made use of the following evolutions metric inverse : ∂∂t g ij = 2 R ij volume element : ∂∂t dµ = − RdµScalar curvature : ∂∂t R = ∆ R + 2 | R ij | Laplacian : ∂∂t ∆ g ( t ) = 2 R ij · ∇ i ∇ j Christof f el ′ s symbols : ∂∂t Γ kij = − g kl (cid:16) ∂ i R jl + ∂ j R il − ∂ l R ij (cid:17) , where R ij is the Ricci curvature ( R = g ij R ij ) and P ij g ij = n . The metric is boundedunder the Ricci flow (Cf. [14, 16, 17], for more on this and detail of geometric andanalytical aspect of the Ricci flow). Acknowledgment
The author wishes to acknowledge his PhD thesis advisor Prof. Ali Taheri for con-stant encouragement. He also thanks the anonymous reviewers for their valuablecomments. His research is supported by TETFund of Federal Government of Nigeriaand University of Sussex, United Kingdom.
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