Harnack Estimates for Conjugate Heat Kernel on Evolving Manifolds
aa r X i v : . [ m a t h . DG ] A ug HARNACK ESTIMATES FOR CONJUGATE HEATKERNEL ON EVOLVING MANIFOLDS
XIAODONG CAO, HONGXIN GUO, AND HUNG TRAN
Abstract.
In this article we derive Harnack estimates for con-jugate heat kernel in an abstract geometric flow. Our calculationinvolves a correction term D . When D is nonnegative, we are ableto obtain a Harnack inequality. Our abstract formulation providesa unified framework for some known results, in particular includ-ing corresponding results of Ni [8], Perelman [10] and Tran [12] asspecial cases. Moreover it leads to new results in the setting ofRicci-Harmonic flow and mean curvature flow in Lorentzian man-ifolds with nonnegative sectional curvature. Contents
1. Introduction 11.1. Main Results 22. Preliminaries 32.1. Evolution Equations 32.2. Asymptotic Behavior and Reduced Geometry 42.3. Entropy Formulas 63. Estimates on the Heat Kernel 83.1. A Gradient Estimate 83.2. L ∞ Bound 103.3. Proofs of Main Results 12References 151.
Introduction
Assume that M is an n -dimensional closed manifold endowed with aone-parameter family of Riemannian metrics g ( t ), t ∈ [0 , T ], evolvingby(1.1) ∂g ( t, x ) ∂t = − α ( t, x ) . Date : August 21, 2018.2010
Mathematics Subject Classification.
Primary 53C44.
Here α ( t, x ) is a one-parameter family of smooth symmetric 2-tensorson M . In particular, when α = Rc, Eq.(1.1) is R. Hamilton’s Ricciflow. Let S( t, x ) + g ij α ij be the trace of α with respect to the time-dependent metric g ( t ).In [7], R. M¨uller studied reduced volumes for the abstract flow (1.1)and defined the following quantity for tensor α and vector V , D α ( V ) + ∂ S ∂t − ∆S − | α | + 2 (Rc − α ) ( V, V ) + h α ) − ∇ S , V i , (1.2)where Div is the divergence operator defined by Div( α ) k = g ij ∇ i α jk (in local coordinates). Under the assumption that D α ≥
0, M¨ullerobtained monotonicity of the reduced volumes [7]. Most recently, in[6], the authors proved monotonicity for the entropy and the lowesteigenvalue. In [5], a Harnack inequality for positive solutions of theconjugate heat equation and heat equation with potential has beenproved.The main purpose of this article is to derive Harnack inequalities fora conjugate heat kernel in the abstract setting with D α ≥ . Main Results.
We consider (
M, g ( t )), 0 ≤ t ≤ T , to be a solutionof (1.1) and τ + T − t , (cid:3) ∗ + − ∂∂t − ∆ + S = ∂∂τ − ∆ + S . A function u = (4 πτ ) − n/ e − f is a solution to the conjugate heat equa-tion if, (cid:3) ∗ u = 0 . (1.3)We also denote H ( x, t ; y, T ) = (4 π ( T − t )) − n/ e − h = (4 πτ ) − n/ e − h to be a heat kernel. That is, based at a fixed ( x, t ), H is the fundamentalsolution of heat equation (cid:3) H = 0, and similarly for fixed ( y, T ) andconjugate heat equation (cid:3) ∗ H = 0. Our first result is computational. Theorem 1.1.
Let v = (cid:0) τ (cid:0) f − |∇ f | + S (cid:1) + f − n (cid:1) u, ARNACK ESTIMATES ON EVOLVING MANIFOLDS 3 then we have (cid:3) ∗ v = − τ u (cid:12)(cid:12)(cid:12) α + ∇∇ f − g τ (cid:12)(cid:12)(cid:12) − τ u D α ( ∇ f ) . (1.4)Secondly, we obtain the following Harnack estimate. Theorem 1.2. If D α ≥ , then the following inequality holds, τ (cid:0) h − |∇ h | + S (cid:1) + h − n ≤ . (1.5) Remark 1.3.
For the Ricci flow, where α = Rc , one has D = 0 ; (1.5) has been proved by G. Perelman [10] . On a static Riemannianmanifold where α = 0 one has D = Rc , and (1.5) has been provedby L. Ni in [8] for static manifolds with nonnegative Ricci curvature.Another special case of (1.5) was recently proved by the third authorfor the extended Ricci flow in [12] . (1.5) is new for M¨uller’s Ricci-Harmonic flow and mean curvature flow in a Lorentzian manifold withnonnegative sectional curvature. The detailed calculations of D can befound in [7] . Acknowledgement . X. Cao was partially supported by a grantfrom the Simons Foundation (
Preliminaries
Evolution Equations.
In this section, we collect several evolu-tion equations and prove Theorem 1.1.For the Laplace-Beltrami operator ∆ with respect to g ( t ) we have, (cid:18) ∂∂t ∆ (cid:19) f = 2 h α, ∇∇ f i + h α ) − ∇ S , ∇ f i , (2.6)where f is any smooth function on M . This formula can be found instandard textbooks, for instance [3].Now we assume u is a solution to the conjugate heat equation. Theoperator − (cid:3) ∗ acting on the term u log u produces, − (cid:3) ∗ u log u = u |∇ log u | + u S . (2.7)The same operator acts once more and we have, − (cid:3) ∗ (cid:0) u |∇ log u | + u S (cid:1) = 2 uα ( ∇ log u, ∇ log u ) + 4 h∇ S , ∇ u i + u ∂ S ∂t + 2 u |∇∇ log u | XIAODONG CAO, HONGXIN GUO, AND HUNG TRAN + 2 u Rc( ∇ log u, ∇ log u ) + u ∆S= 2 u |∇∇ log u − α | + 4 u h α, ∇∇ log u i + 2 uα ( ∇ log u, ∇ log u )(2.8) − u | α | + 4 h∇ S , ∇ u i + u ∂ S ∂t + 2 u Rc( ∇ log u, ∇ log u ) + u ∆S . Notice that, − (cid:3) ∗ (∆ u ) =2 h α, ∇∇ u i + h α ) − ∇ S , ∇ u i + 2 h∇ S , ∇ u i + u ∆S=2 u h α, ∇∇ log u i + 2 uα ( ∇ log u, ∇ log u )(2.9) + h α ) − ∇ S , ∇ u i + 2 h∇ S , ∇ u i + u ∆S . Thus, by (1.2), we have, for V = −∇ log u ,(2.10) − (cid:3) ∗ (cid:0) u |∇ log u | + u S − u (cid:1) = 2 u |∇∇ log u − α | + u D α ( V ) . Moreover, − (cid:3) ∗ (cid:16) τ (cid:0) u |∇ log u | + u S − u (cid:1) − u log u − nu τ (cid:17) (2.11) = 2 τ u | α − ∇∇ log u − g τ | + τ u D α ( −∇ log u ) . In the calculation above, if we add a normalization term c n u to the lefthand side, we get the same on the right hand side since (cid:3) ∗ u = 0 . Thus,we have the following result.
Lemma 2.1. (cid:3) ∗ h τ (cid:0) u |∇ log u | + u S − u (cid:1) − u log u − nu πτ ) − nu i (2.12)= − τ u (cid:12)(cid:12)(cid:12) α − ∇∇ log u − g τ (cid:12)(cid:12)(cid:12) − τ u D α ( −∇ log u ) . Theorem 1.1 follows by realizing that f + − log u − n log(4 πτ ) . Asymptotic Behavior and Reduced Geometry.
Let’s recallthe asymptotic behavior of the heat kernel as t → T . Theorem 2.2. [1, Theorem 24.21]
For τ = T − t , H ( x, t ; y, T ) ∼ e − d T ( x,y )4 τ (4 πτ ) n/ Σ ∞ j =0 τ j u j ( x, y, τ ) . More precisely, there exist t > and a sequence u j ∈ C ∞ ( M × M × [0 , t ]) such that, H ( x, t ; y, T ) − e − d T ( x,y )4 τ (4 πτ ) n/ Σ kj =0 τ j u j ( x, y, T − l ) = w k ( x, y, τ ) , ARNACK ESTIMATES ON EVOLVING MANIFOLDS 5 with u ( x, x,
0) = 1 ,w k ( x, y, τ ) = O ( τ k +1 − n ) , as τ → uniformly for all x, y ∈ M . Then following [7], we can define reduced length and distance.
Definition 2.3.
Given τ ( t ) = T − t , we define the L -length of a curve γ : [ τ , τ ] N , [ τ , τ ] ⊂ [0 , T ] by, (2.13) L ( γ ) := Z τ τ √ τ (S( γ ( τ )) + | ˙ γ ( τ ) | ) dτ. For a fixed point y ∈ N and τ = 0 , the backward reduced distance isdefined as, (2.14) ℓ ( x, τ ) := inf γ ∈ Γ { τ L ( γ ) } , where Γ = { γ : [0 , τ ] M, γ (0) = y, γ ( τ ) = x } .The backward reduced volume is defined as (2.15) V ( τ ) := Z M (4 πτ ) − n/ e − ℓ ( y,τ ) dµ τ ( y ) . The next result, mainly from [7], relates the reduced distance definedin (2.14) with the distance at time T.
Lemma 2.4.
Let L ( x, τ ) = 4 τ ℓ ( x, τ ) then we have the followings: a. Assume that there exists k , k ≥ such that − k g ( t ) ≤ α ( t ) ≤ k g ( t ) for t ∈ [0 , T ] , then L is smooth amost everywhere and a localLipschitz function on N × [0 , T ] . Furthermore, e − k τ d T ( x, y ) − k n τ ≤ L ( x, τ ) ≤ e k τ d T ( x, y ) + 4 k n τ . b. If D α ≥ , then (cid:3) ∗ (cid:16) e − L ( x,τ )4 τ (4 πτ ) n/ (cid:17) ≤ . c. For the same point y in the definition of reduced distance and H ( x, t ; y, T ) = (4 πτ ) − n/ e − h , then h ( x, t ; y, T ) ≤ ℓ ( x, T − t ) .Proof. Parts a. and b. follow from [7, Lemmas 4.1, 5.15] respectively.For part c. we provide a brief argument here (for more details, see[2, Lemma 16.49]).We first observe that part a. implies lim τ → L ( x, τ ) = d T ( y, x ) and,lim τ → e − Lw ( x,τ )4 τ (4 πτ ) n/ = δ y ( x ) , XIAODONG CAO, HONGXIN GUO, AND HUNG TRAN since Riemannian manifolds locally look like Euclidean. It then followsfrom part b. and maximum principle that, H ( x, t ; y, T ) ≥ e − L ( x,τ )4 τ (4 πτ ) n/ = e − L ( x,T − t )4 τ (4 π ( T − t )) n/ . Hence, h ( x, t ; y, T ) ≤ L ( x, τ )4 τ = ℓ ( x, τ ) = ℓ ( x, T − l ) . (cid:3) Entropy Formulas.
In this subsection, we define several func-tionals and collect their properties.
Definition 2.5.
Along flow (1.1), for h satisfying R M (4 πτ ) − n/ e − h dµ =1 , we define (2.16) W α ( g, τ, h ) + Z M (cid:16) τ ( |∇ h | + S) + ( h − n ) (cid:17) (4 πτ ) − n/ e − h dµ. Associated functionals are defined as follows: µ α ( g, τ ) = inf f W α ( g, h, τ ) , (2.17) υ α ( g ) = inf τ> µ α ( g, τ ) . (2.18) Remark 2.6.
Since α is a (2 , -tensor, S scales like the inverse of themetric. Thus, these functionals satisfy diffeomorphism invariance andthe following scaling rules: W α ( g, τ, h ) = W α ( cg, cτ, h ) ,µ α ( g, τ ) = µ α ( cg, cτ ) ,υ α ( g, u ) = υ α ( cg ) . Next, we collect some useful results.
Lemma 2.7.
On a closed Riemannian manifold ( M, g ( t )) , t ∈ [0 , T ] ,evolved by (1.1), with D α ≥ . Let τ = T − t , the following holds: a. W α ( g, τ, h ) is non-decreasing in time t (non-increasing in τ ). b. There exists a smooth minimizer h τ for W α ( g, τ, . ) which satisfies τ (2 △ h τ − |∇ h τ | + S) + h τ − n = µ α ( g, τ ) . In particular, µ α ( g, τ ) is finite. c. µ α ( g, τ ) is non-decreasing in time t. d. lim τ → + µ α ( g, τ ) = 0 . ARNACK ESTIMATES ON EVOLVING MANIFOLDS 7
Proof.
Part a. follows from [6, Theorem 5.2].Part b. is deducted from the regularity theory for elliptic equationsbased on Sobolev spaces. The details can be found in [1, Proposition17.24]. Replacing R by S, the argument works exactly the same.Part c. is an immediate consequence of the monotonicity formula(part a. ) and the existence of a minimizer realizing the µ α functional(part b. ).The proof of part d. is mostly identical to that of [11, Prop 3.2] (also[1, Prop 17.19, 17.20]), but it is subtle so we give a brief argument here.Assume that the flow exists for τ ∈ [0 , τ ]. The idea is to constructcut-off functions reflecting the local geometry which looks like Eu-clidean. Then it is shown that the limit of W α functional on thesefunctions is 0 if a certain parameter approaches 0. Thus, by the mono-tonicity of µ α and L. Gross’s logarithmic-Sobolev inequality on an Eu-clidean space [4], the result then follows.The construction of cut-off functions follows [11, Prop 3.2]. Let τ = τ − ǫ for small ǫ . Using normal coordinates at a point p on( M, g ( τ )), we define a cut-off function f = ( | x | ǫ if d ( x, p ) = | x | < ρ, ρ ǫ elsewhere , where ρ is a positive number smaller than the injectivity radius (whichexists since M is closed). Then by the choice of our coordinate, dµ ( τ ) = 1 + O ( | x | ) , | x | << , and let e − C + Z M (4 πǫ ) − n/ e f , then C → ǫ → f = f + C then, u + (4 πǫ ) − n/ e − f ;1 = Z M (4 πǫ ) − n/ e − f dµ ( τ ); |∇ f | = |∇ f | = |∇ | x | ǫ | = | x | ǫ , for | x | < ρ. We solve f backward using equation ∂∂t f = − S − ∆ f + |∇ f | + n τ . XIAODONG CAO, HONGXIN GUO, AND HUNG TRAN
The solution clearly depends on the choice of ǫ . Now using (2.16), wecalculate, W ( g ( τ ) , τ − τ , f ( τ )) = Z | x | <ρ (cid:16) ǫ (S + | x | ǫ ) + | x | ǫ + C − n (cid:17) udµ + Z d ( x,p ) ≥ ρ ( ǫ S + ρ ǫ + C − n ) udµ = Z | x | <ρ ( | x | ǫ − n ) udµ + ǫ Z M S udµ + C Z M udµ + Z d ( x,p ) ≥ ρ ( ρ ǫ − n ) udµ = I + II + III + IV.
By a change of variable, as ǫ →
0, we have, II + III → IV = Z d ( x,p ) ≥ ρ ( ρ ǫ − n ) e − ρ ǫ − C → I = e − C Z | y |≤ ρ √ ǫ ( | y | − n )(2 π ) − n/ e −| y | / (1 + O ( ǫ | y | ) dy → Z R n ( | y | − n )(2 π ) − n/ e −| y | / dy = 0 . Thus, by part a. and b , µ α ( g ( t ) , τ − t ) ≤ t ≤ τ . The proofthat the limit is actually 0 when τ → + follows from a rather standardblow-up argument whose details can be found in either [11, Prop 3.2]or [1, Prop 17.20]. (cid:3) Estimates on the Heat Kernel
In this section, we obtain several estimates on the heat kernel usingmaximum principle and the monotone framework. Particularly, wederive a gradient estimate and an upper bound for positive solutionsof the conjugate heat equation. Then we prove our main result.3.1.
A Gradient Estimate.
We first establish a space-only gradientestimate. Recall that, (cid:3) ∗ = ∂∂τ − ∆ + S . ARNACK ESTIMATES ON EVOLVING MANIFOLDS 9
Lemma 3.1.
Assume there exist k , k , k , k > such that the follow-ings hold on N × [0 , T ] , Rc( g ( t )) ≥ − k g ( t ) ,α ≥ − k g ( t ) , |∇ S | ≤ k , | S | ≤ k . Let q be any positive solution to the conjugate heat equation on M × [0 , T ] , i.e., (cid:3) ∗ q = 0 , and τ = T − t . If q < Q for some constant Q then there exist C , C depending on k , k , k , k and n, such that for < τ ≤ min { , T } , we have (3.19) τ |∇ q | q ≤ (1 + C τ )(ln Qq + C τ ) . Proof.
We compute that( − ∂∂t − △ ) |∇ q | q =S |∇ q | q + 1 q ( − ∂∂t − △ ) |∇ q | + 2 |∇ q | ∇ q ∇ ln q − ∇|∇ q | ∇ q , q ( − ∂∂t − △ ) |∇ q | = 1 q h − α + Rc)( ∇ q, ∇ q ) − ∇ q ∇ (S q ) − |∇ q | i , |∇ q | ∇ q ∇ ln q = − |∇ q | q , − ∇|∇ q | ∇ q =4 ∇ q ( ∇ q, ∇ q ) q . Thus( − ∂∂t − △ ) |∇ q | q = − q |∇ q − dq ⊗ dqq | + S |∇ q | q + − α + Rc)( ∇ q, ∇ q ) − ∇ q ∇ q − q ∇ q ∇ S q ≤ [2( k + k ) + nk ] |∇ q | q + 2 |∇ q ||∇ S |≤ [2 k + (2 + n ) k + 1] |∇ q | q + k q. Furthermore, we have( − ∂∂t − △ )( q ln Qq ) = − S q ln Qq + S q + |∇ q | q ≥ |∇ q | q − nk q − k q ln Qq .
Let Φ = a ( τ ) |∇ q | q − b ( τ ) q ln Qq − c ( τ ) q, then( − ∂∂t − △ )Φ ≤ |∇ q | q (cid:16) a ′ ( τ ) + a ( τ )(2 k + (2 + n ) k + 1) − b ( τ ) (cid:17) + q ln Qq (cid:16) k b ( τ ) − b ′ ( τ ) (cid:17) + q (cid:16) k a ( τ ) − c ′ ( τ ) + nk b ( τ ) + c ( τ ) k (cid:17) . We can now choose a, b and c appropriately such that ( − ∂ t − △ )Φ ≤ a = τ k + (2 + n ) k + 1) τ ,b = e k τ ,c = ( e k k τ nk + k ) τ, for k = 1 + k nk . Then by maximum principle, noticing that Φ ≤ τ = 0, we arrive at a |∇ q | q ≤ b ( τ ) q ln Qq + cq. The result then follows from simple algebra. (cid:3) L ∞ Bound.
Second, we shall derive an upper bound for positiveconjugate heat solutions. Our main statement says that any normalizedsolution can not blow up too fast.
Lemma 3.2.
Let q be any normalized positive solution to the conjugateheat equation on M × [0 , T ] , i.e., (cid:3) ∗ q = 0 with R qdµ g (0) = 1 . Let τ = T − t , then there exists a constant C depending on the geometry of g ( t ) t ∈ [0 ,T ] , such that (3.20) q ( y, τ ) ≤ Cτ n/ . Proof.
The proof is modeled after [9, Lemma 2.2] (also see [2, Lemma16.47]). As the solution and the flow is well defined in M × [0 , T ], thereexists y , τ such that(3.21) sup M × [0 , min { ,T } ] τ n/ q ( y, τ ) = τ n/ q ( y , τ ) . ARNACK ESTIMATES ON EVOLVING MANIFOLDS 11
In particular,sup M × [ τ / ,τ ] q ( y, τ ) ≤ τ n/ τ n/ q ( y , τ ) ≤ n/ q ( y , τ ) := Q. Applying Lemma 3.1 to q ( y, τ ) on M × [ τ / , τ ] we obtain, τ |∇ q | q ( y, τ ) ≤ (1 + C τ Qq ( y, τ ) ) + C τ . Let G ( y, τ ) := log( Qq ( y,τ ) ) + C τ , then the inequality above can berewritten as |∇√ G | = | ∇ GG / | = | G / ∇ qq | = 14 G |∇ q | q ≤ C τ τ . Therefore, with B τ ( y, r ) denoting the ball of radius r measured by g ( τ )around the point y, we havesup B τ ( y , r τ C τ ) √ G ( y, τ ) ≤ √ G ( y , τ ) + 1 √ r n C τ √ . Writing the above inequality in terms of q ( y, τ ) yields, q ( y, τ ) ≥ q ( y , τ ) exp (cid:8) − − n − √ r n C (cid:9) := C q ( y , τ ) . Now we observe that there exists a constant C depending on the ge-ometry of ( M, g ( τ )), such thatVol g ( τ ) (cid:16) B τ ( y , r τ C τ (cid:17) ≥ C τ n/ . Therefore we have, q ( y , τ ) ≤ C C τ n/ Z M q ( y, τ ) dµ τ ( y ) := C τ n/ Z M q ( y, τ ) dµ τ ( y ) . By our choice of y , τ and the fact that R M q ( y, τ ) dµ τ ( y ) remains con-stant along the flow, the statement follows. (cid:3) Remark 3.3.
It is interesting to note that in [13] , a Harnack inequalityis used to obtain an off-diagonal bound, while here the argument goesthe opposite direction.
Proofs of Main Results.
Finally, we are ready to finish ourproof of the main theorem.
Lemma 3.4.
Let H ( x, t ; y, T ) = (4 πτ ) − n/ e − h be a heat kernel and Φ be any positive solution to the heat equation. Then we have Z M hH Φ dµ ≤ n y, T ) , i.e, Z M ( h − n H Φ dµ ≤ . Proof.
By Lemma 2.4 we havelim sup τ → Z M hH Φ dµ ≤ lim sup τ → Z M ℓ ( x, τ ) H Φ dµ ( x ) ≤ lim sup τ → Z M d T ( x, y )4 τ H Φ dµ ( x ) . Using Theorem 2.2, it follows that,lim τ → Z M d T ( x, y )4 τ H Φ dµ ( x ) = lim τ → Z M d T ( x, y )4 τ e − d T ( x,y )4 τ (4 πτ ) n/ Φ dµ ( x ) . Either by differentiating twice under the integral sign or using thesefollowing identities on Euclidean spaces Z ∞−∞ e − a x d x = r πa and Z ∞−∞ x e − a x d x = 12 a r πa , we obtain that Z R n | x | e − a | x | dx = n ( Z ∞−∞ x e − a x d x ) (cid:16) Z ∞−∞ e − a x d x (cid:17) n − = n a ( πa ) n/ . Therefore, lim τ → d T ( x, y )4 τ e − d T ( x,y )4 τ (4 πτ ) n/ = n δ y ( x ) , hence lim τ → Z d T ( x, y )4 τ e − d T ( x,y )4 τ (4 πτ ) n/ Φ dµ N ( x ) = n y, T ) . Thus the result follows. (cid:3)
The following result implies that the equality actually holds.
Proposition 3.5.
Let H ( x, t ; y, T ) = (4 πτ ) − n/ e − h be a heat kerneland Φ be any positive solution to the heat equation. Then for v = h ( T − t )(2 △ h − |∇ h | + S ) + h − n i H,ρ Φ ( t ) = Z M v Φ dµ, ARNACK ESTIMATES ON EVOLVING MANIFOLDS 13 we have lim t → T ρ Φ ( t ) = 0 . Proof.
Integrating by parts yields that ρ Φ ( t ) = Z M h τ (2 △ h − |∇ h | + S ) + h − n i H Φ dµ = − Z M τ ∇ h ∇ ( H Φ) dµ − Z M τ |∇ h | H Φ dµ + Z M ( τ S + h − n ) H Φ dµ = Z M τ |∇ h | H Φ dµ − τ Z M ∇ Φ ∇ hHdµ + Z M ( τ S + h − n ) H Φ dµ = Z M τ |∇ h | H Φ dµ − τ Z M H △ Φ dµ + Z M ( τ S + h − n ) H Φ dµ = Z M τ |∇ h | H Φ dµ + Z M hH Φ dµ − τ Z M H △ Φ dµ + Z M ( τ S − n ) H Φ dµ. Notice that, except the first two terms, the rest approaches − n Φ( y, T )as τ →
0. For the first term, using Lemmas 3.2 and 3.1 for any space-( τ ) time point on M × [ τ , τ ] we arrive at τ Z M |∇ h | H Φ dµ ≤ (2 + C τ ) Z M (ln ( C Hτ n/ ) + C τ ) H Φ dµ ≤ (2 + C τ ) Z M (ln C + h + n π ) + C τ ) H Φ dµ, with C , C defined as in Lemma 3.1, while C is a constant dependingon the geometry of g ( t ), τ ≤ T − t ≤ τ . As τ →
0, ln C + n ln(4 π )is bounded from above by another constant C also depending on thegeometry of g ( t ), t ∈ [0 , T ]. Consequently, by Lemma 3.4, which claimsthe finiteness of R M hH Φ dµ ,lim τ → ( Z M τ |∇ h | dµ + Z M hH Φ dµ ) ≤ Z M hH Φ dµ + 2 ln C Φ( x, T ) ≤ ( 3 n C )Φ( x, T ) . Thus we have lim t → T ρ Φ ( t ) ≤ C Φ( x, T ) . Since Φ is a positive solution satisfying ∂ t Φ = △ Φ, applying Theorem1.1 yields that,(3.22) ∂ t ρ Φ ( t ) = ∂ t Z v Φ dµ = Z ( (cid:3) Φ v − Φ (cid:3) ∗ v ) dµ ≥ . The above implies that there exists β , such thatlim t → T ρ Φ ( t ) = β. Hence lim τ → ( ρ Φ ( T − τ ) − ρ Φ ( T − τ . By the above equation (3.22), Theorem 1.1, and the mean-value theo-rem, there exists a sequence τ i →
0, such thatlim τ i → τ i Z M (cid:16) | α + Hess h − g τ i | + 12 D α ( ∇ h ) (cid:17) H Φ dµ = 0 . Now standard inequalities yield that, h Z M τ i (S + △ h − n τ i ) H Φ dµ i ≤ h Z M τ i ( S + △ h − n τ i ) H Φ dµ ih Z M H Φ dµ i ≤ n h Z M τ i |S + Hess h − g τ i | H Φ dµ ih Z m H Φ dµ i . Since lim τ i → Z M H Φ dµ = Φ( y, T ) < ∞ , and D α ( ∇ h ) ≥
0, we derive thatlim τ i → Z M τ i ( S + △ h − n τ i ) H Φ dµ = 0 . Therefore, by Lemma 3.4,lim t → T ρ Φ ( t ) = lim τ i → Z M h τ i (2 △ h − |∇ h | + S ) + h − n i H Φ dµ = lim τ i → Z M h τ i ( △ h − |∇ h | ) + h − n i H Φ dµ = lim τ i → h Z M − τ i H △ Φ dµ + Z M ( h − n H Φ dµ i = lim τ i → Z M ( h − n H Φ dµ ≤ . So β ≤
0. To show that equality holds, we proceed by contradiction.Without loss of generality, we may assume Φ( y, T ) = 1. Let H Φ =(4 πτ ) − n/ e − ˜ h (that is, ˜ h = h − ln Φ), then integrating by parts yields,(3.23) ρ Φ ( t ) = W α ( g, τ, ˜ h ) + Z M (cid:16) τ ( |∇ Φ | Φ ) − Φ ln Φ (cid:17)
Hdµ.
ARNACK ESTIMATES ON EVOLVING MANIFOLDS 15
By the choice of Φ the last term converges to 0 as τ →
0. So iflim t → T ρ Φ ( t ) = β < τ → µ α ( g, τ ) < β = 0. (cid:3) Now Theorem 1.2 follows immediately.
Proof. (Theorem 1.2) Recall from inequality (3.22) ∂ t Z M v Φ dµ = Z M ( (cid:3) Φ v − Φ (cid:3) ∗ v ) dµ ≥ . By Proposition 3.5, lim t → T Z M v Φ dµ = 0 . Since Φ is arbitrary, v ≤ (cid:3) References [1] Bennett Chow, Sun-Chin Chu, David Glickenstein, Christine Guenther, JamesIsenberg, Tom Ivey, Dan Knopf, Peng Lu, Feng Luo, and Lei Ni.
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Department of Mathematics, Cornell University, Ithaca, NY 14853
E-mail address : [email protected] School of mathematics and information science, Wenzhou Univer-sity, Wenzhou, Zhejiang 325035, China.
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