GRADIENT ESTIMATES FOR HEAT-TYPE EQUATIONS ON MANIFOLDS EVOLVING BY THE RICCI FLOW

In this paper, certain localized and global gradient estimates for all positive solutions to the geometric heat equation coupled to the Ricci flow either forward or backward in time are proved. As a by product, we obtain various Li-Yau type differential Harnack estimates. We also discuss the case when the diffusion operator is perturbed with the curvature operator (precisely, when the Laplacian is replaced with ”∆−R(x, t)”, R being the scalar operator). This is well generalised to the case of an adjoint heat equation under the Ricci flow. AMS Subject Classification: 35K05, 53C25, 53C44


Introduction
Let M be an n-dimensional compact (or noncompact without boundary) manifold on which a one parameter family of Riemannian metrics g(t), t ∈ [0, T ) is defined.We say (M, g(t)) is a solution to the Ricci flow if it is evolving by the following nonlinear weakly parabolic partial differential equation ∂ ∂t g(x, t) = −2Rc(x, t), (x, t) ∈ M × [0, T ] (1.1) with g(x, 0) = g(0), where Rc is the Ricci curvature.By the positive solution to the heat equation on the manifold, we mean a smooth function at least C 2 in x and C 1 in t, u ∈ C 2,1 (M × [0, T ]) which satisfies the following equation where the symbol ∆ is the Laplace-Beltrami operator acting on functions in space with respect to metric g(t) in time.We couple the Ricci flow to the equation of the form (1.2) (either forward, backward, or perturbed with curvature operator), and obtain several gradient estimate on the logarithm of the positive solutions.We start with localized version for gradient estimate when Laplacian operator in (1.2) is replaced with ∆ − R(x, t), where R is the scalar curvature of the Ricci flow g(t).We show how the local results imply the the global ones.As an application of our gradient estimates, we derive several Harnack estimates.In recent years, there have been increasing efforts towards the study of the Ricci flow coupled to heat-type equation.This actually arose from R. Hamilton's work [15], where he conceived the idea of investigating Ricci flow coupled to harmonic maps heat flow.He combined this with his previous results [13,14] to study the formation of singularities in the Ricci flow.In [14], he proved a Harnack estimate on Riemannian manifolds with nonnegative positive curvature operator.Hamilton also proved Harnack estimates for surfaces [12] whose positive scalar curvature under the Ricci flow satisfies the heat equation with soliton potential.B. Chow [7] completed the proof of Harnack estimate for surfaces of positive scalar curvature in general.Thus, Harnack estimates for the Ricci flow on surfaces give a control on curvature growth, while in higher dimension, one uses the Harnack estimates to classify the ancient solutions of nonnegative curvature operators.The importance of gradient estimates as well as those of Harnack inequalities can not be overemphasised in the fields of Differential geometry and Analysis among their numerous applications.Differential Harnack inequalities are used to study the behaviours of solutions to the heat equation in space-time.Li and Yau's paper [17] can be said to mark the beginning of rigorous applications of these concepts, they derived gradient estimates for positive solutions to the heat operator defined on closed manifold with bounded Ricci curvature, from which they obtained Harnack inequalities.These inequalities were in turn used to establish various lower and upper bounds on the heat kernel.They also studied manifold satisfying Dirichlet and Neumann conditions.G. Perelman [19] recorded a breakthrough in the fields of Topology and Geometry by the application of differential inequality, where he obtained gradient estimate for the fundamental solution to the conjugate heat equation on compact manifold evolving by the Ricci flow.Perelman's results are one of important ingredients in the proof of Poincaré conjecture.C. Guenther [11] also proved gradient estimates for positive solutions to the heat equation under the Ricci flow by adapting the methodology of Bakry and Qian [1] to time dependent metric case.As an application of her results, she got a Harnack-type inequality and obtain a lower bound for fundamental solutions.She also studied existence and basic properties of these solutions.S. Kuang and Q. Zhang [16] established a gradient estimate that holds for all positive solutions of the conjugate heat equation defined on a closed manifold whose metric is evolving by the Ricci flow.X. Cao [4] also used Perelman's approach to establish a differential inequality for all positive solutions to the conjugate heat equation also under the Ricci flow with nonnegative condition on the scalar curvature.Immediate consequences of their result in [16] and [5] are Harnack-type inequalities.Qi.Zhang [21] derived local gradient estimates for positive solutions to the heat equation coupled to the backward in time Ricci flow with assumption of lower bound on the Ricci curvature, his gradient estimate is used to prove a Gaussian bound for the conjugate heat equation.In [2], Bǎileşteanu, Cao, and Pulemotov applied Zhang's method to prove both space-only and space-time gradient estimates for heat equation coupled to forward in time Ricci flow.They also study manifolds with nonempty convex boundary evolving under the Ricci flow (see also [18] for a related result.Ecker, Knopf, Ni and Topping [10] have a local result on gradient estimate in relation to ean value theorem and monotonicity for heat kernel.

Preliminaries and Notation
We remark that our manifold is endowed with Riemannian metric ds 2 = g = g ij dx i dx j , where {x i }, 1 ≤ i ≤ n is a local coordinate system and n is the dimension of the manifold.The operator ∆ is the Laplace-Beltrami operator on (M, g) which is defined by and ∇ = g ij ∂ i is the gradient operator, where |g| = determinant of g and g ij = (g ij ) −1 , inverse metric.A natural function that will be defined on M is the distance function from a given point, namely, let p ∈ M and define d(x, p) for all x ∈ M where dist(•, •) is the geodesic distance.Note that d(x, p) is only Lipschitz continuous, i.e., everywhere continuous except on the cut locus of p and on the point where x and p coincide.It is then easy to see that 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 and where C 1 , C 2 are absolute constants, such that ϕ(x, t) = ψ d(x, p, t) ρ and ϕ We will apply maximum principle and invoke Calabi's trick to assume everywhere smoothness of ϕ(x) since ψ(s) is in general Lipschitz.We need Laplacian comparison theorem to do some calculation on ϕ(x, t).Here is the statement of the theorem; Let M be a complete n-dimensional Riemannian manifold whose Ricci curvature is bounded from below by Rc ≥ (n − 1)k for some constant k ∈ R. Then the Laplacian of the distance function satisfies (1.4) Throughout, we will impose boundedness condition on the Ricci curvature of the metric and note that when the metric evolves by the Ricci flow, boundedness and sign assumptions are preserved as long as the flow exists, so also the metrics are uniformly equivalent, precisely, if −K 1 g ≤ Rc ≤ K 2 g, where g(t), t ∈ [0, T ] is a Ricci flow, then e −k 1 T g(0) ≤ g(t) ≤ e k 2 T g(0). (1.5) See [8] and [9] for details on the theory of the Ricci flow.Most of our calculations are done in local coordinates where {x i } is fixed in a neighbourhood of every point x ∈ M. We switch between the Rc and R ij for the Ricci curvature notation, and any time R ij is used we assumed to be on local coordinates at x.We also define some classical identities on function f ∈ C ∞ at point x in a local normal coordinate system, namely, Bochner-Weitzenböck identity and Ricci identity ij We make use of the above identities and we try as much as possible to be explicit at any point where they are used.The rest of this paper is planned as follows; In section two we discuss gradient estimate on the operator of the form (∂ t − ∆ + R) under both backward and forward in time Ricci flow, where we only require boundedness of scalar curvature.In section three, we discuss local space-time gradient estimates for heat equation on manifold evolving by the Ricci flow, while as an application of these results we obtain various Li-Yau type Harnack inequality in section four.

The Conjugate Heat Equation under Ricci Flow
In this section, we discuss the localized version of gradient estimate on the heat equation perturbed with curvature operator under both forward and backward Ricci flow.The estimate under backward action of Ricci flow is related to the local monotonicity for heat kernel and mean value theorem of Ecker, Knopf, Ni and Topping in [10, Section 3].Here, they worked in general geometric flow, we follow their approach.We also remark that our result can be easily generalized to Perelman's conjugate heat equation and entropy monotonicity [19] and we can use it to prove a global Gaussian estimate as done in [21, Section 5].
We consider the conjugate heat equation coupled to the backward and forward Ricci flow respectively as follows    (2.2) Suppose u = u(x, t) solves the conjugate heat equation and satisfies 0 < u ≤ A in the geodesic cube Q 2ρ ⊂ M as defined by then we have The boundedness assumption may be weakened in the case of the forward Ricci flow (2.2) as we can state our result as follows; By Bochner-Weitzenböck identity (1.6), we have Next we compute Hence Rearranging Note that we can simplify some terms in the last equation further.The first term becomes Using the Ricci identity on the next three terms we have since the Ricci identity (1.7) implies then the second to the last three terms gives Putting all these together, we get then using the curvature conditions we are left with the following inequality (2.9) We now apply a cut-off function in order to derive the desired estimate.Now, let ψ(s) be a smooth cut function defined on [0, ∞) such that 0 ≤ ψ(s) ≤ 1, with for some constants C 1 , C 2 > 0 depending on the dimension of the manifold only.Define a distance function d(p, x) between points p and x such that It is easily seen that ϕ(x, t) has its support in the closure of Q 2ρ,T .We note that ϕ(x, t) is smooth at (y, s) ∈ M × [0, T ] whenever point y does not either coincide with p or fall in the cut locus of p, with respect to the metric g(y, s).In what follows, we consider the function ϕw supported in Q 2ρ,T × [0, ∞), though, this is in general Lipchitz continuous.For the purpose of the application of the maximum principle, we may therefore, without loss of generality assume that ϕ(x, t) with support in Q 2ρ,T is C 2 at the maxima.The assumption is made possible by a standard argument called Calabi's trick (due to Calabi 1958 [3]).This approach is used in [6], see also [10,20,21].Therefore we can calculate and by the Laplacian comparison theorem we have Let (x 0 , t 0 ) be a point in Q 2ρ,T at which F = ϕw attains its maximum value.
At this point we have to assume that F is positive, since, if F = 0, implies ϕw(x 0 , t 0 ) = 0 =⇒ ϕw(x, t) = 0 =⇒ w(x, t) = 0 for all x ∈ M such that the distance d(x, x 0 , t) < 2ρ, this yields ∇u(x, t) = 0 and the theorem will follow trivially at (x, t).The approach here is to estimate (∂ t − ∆)(tF ) and do some analysis on the result at the maximum point.The argument is as follows Note that at the maximum point (x 0 , t 0 ), we have by derivative test that ∇F (x 0 , t 0 ) = 0, ∂ ∂t F (x 0 , t 0 ) ≥ 0 and ∆F (x 0 , t 0 ) ≤ 0. (2.16) Taking tF on M × [0, T ], we have (∂ t − ∆)(tF ) ≥ 0 whenever (tF ) achieves its maximum.Similarly, by this argument we have ∇(ϕw)(x 0 , t 0 ) − w∇ϕ(x 0 , t 0 ) = ϕ∇w(x 0 , t 0 ), which means ϕ∇w can always be replaced by −w∇ϕ.By (2.9), ( 2.14) and ( 2.16) Taking 0 ≤ ϕ ≤ 1 and noticing that 1 1−f ≤ 1, then the last inequality becomes Using the following relation as noticed in [10]; and by the Young's inequlity Notice also that by bounds given in (2.10) and (2.11) we have Putting these together and dividing through by (1 − f ), while noticing that Therefore we have From here we can conclude that (2.17) By standard scaling argument we can assume A ≡ 1, hence the proof is completed.
Proof.(of Theorem 2. 2) The proof of Theorem 2.2 is similar to that of Theorem 2.1, the disparity between the estimates (2.4) and (2.5) arises in some calculation which we briefly point out here.
Similarly, set f = log u and w Notice that g(t) evolves by the Ricci flow (2.2), where the inverse metric evolves as ∂ t (g ij ) = 2R ij and then Using the Ricci identity, the second braced terms vanish and we are left with following inequality as in (2.9) We then proceed as in the the rest of the proof of theorem (2.1).
This can be seen as an improvement since the curvature assumption is weakened, we only require boundedness of scalar curvature.Global estimates can be obtained from the two gradient estimates, simply by sending ρ to ∞.For instance, when u is globally defined and g(t) has nonnegative curvature, the estimate simply read as (2.20)

Gradient Estimates on Forward Heat equation
Let M be an n-dimensional complete Riemannian manifold without boundary, we discuss space-time gradient estimates for positive solutions of the heat equation along the Ricci flow In general, our estimate is a local one as we obtain it in the interior of geodesic cube.We will show how this local estimate can lead to achieving a global one.We first proof a lemma which is very crucial to our derivation.Define the geodesic cube Lemma 3.1.Let (M, g(t)) be a complete solution to the Ricci flow in some time interval [0, T ].Suppose that −k 1 g ≤ Rc ≤ k 2 g for some positive constants k 1 and k 2 and for all t ∈ [0, T ].For any smooth positive solution u ∈ C 2,1 (M × [0, T ]) to the heat equation in the geodesic cube Q 2ρ,T , it holds that where f = log u, G = t(|∇f | 2 − α∂ t f ) and α ≥ 1 are given such that 1 p + 1 q = 1 α for any real numbers p, q > 0.
Proof.We have Working in local coordinates system at any point x ∈ M and using Einstein summation convection where repeated indices are summed up.We have by Bochner-Weitzenböck's identity (1.6) ∆|∇f By the hypothesis of the lemma that g(x, t) evolves by the Ricci flow we have With the above computations we obtain the following at an arbitrary point On the other hand Noticing that f = log u implies the evolution Now we can choose any two real numbers p, q > 0 such that 1 p + 1 q = 1 α , so that we can write where we have used completing the square method to arrive at the last inequality.Also by Cauchy-Schwarz inequality holds at an arbitrary point (x, t) ∈ Q 2ρ,T , therefore we have f 2 ij ≥ 1 n (∆f ) 2 .We can also write the boundedness condition on the Ricci curvature as Hence the result.Our calculation is valid in the cube Q 2ρ,T .
In the next, we state and prove a result for local gradient estimate (spacetime) for the positive solutions to the heat equation in the geodesic cube Q 2ρ,T of bounded Ricci curvature manifold evolving by the Ricci flow.Theorem 3.2.Let (M, g(t)), t ∈ [0, T ] be a complete solution of the Ricci flow ( 3.1) such that the Ricci curvature is bounded in Q 2ρ,T , i.e., −k 1 g(x, t) ≤ Rc(x, t) ≤ k 2 g(x, t) for some positive constants k 1 and k 2 , with (x, t) ∈ Q 2ρ,T ⊂ (M × [0, T ]).Suppose a smooth positive function u ∈ C 2,1 (M × [0, t]) solves the heat equation (3.2 ) in the cube Q 2ρ,T .Then, for any given α > 1 with where C is an arbitrary constant depending only on the dimension of the manifold.
Proof.Note that as before we let f = log u and The approach is also by using cut-off function and estimating (∆ − ∂ t )(tϕG) at the point where the maximum value for (ϕG) is attained as we did in Theorem 2.1.The argument follows; (3.8) Suppose (ϕG) attains its maximum value at (x 0 , t 0 ) ∈ M × [0, T ], for t 0 > 0. Since (ϕG)(x, 0) = 0 for all x ∈ M , we have by derivative test that where the function (ϕG) is being considered with support on Q 2ρ × [0, T ] and we have assumed that (ϕG)(x 0 , t 0 ) > 0 for t 0 > 0. By ( 3.9) we notice that Using (3.8)-(3.9)and the last lemma, we have Noticing also that ϕ∇G can be replaced be −G∇ϕ by the condition ∇(ϕG) = 0. Therefore As we have noted earlier, Calabi's trick and Laplacian comparison theorem allows us to do the following calculation on the cut-off function depending on the geodesic distance, since we know that cut locus does not intersect with the geodesic cube where we have taken C 3 to be the maximum of C 1 , C 2 , so our computation becomes Multiplying through by 0 ≤ ϕ ≤ 1 again we have Using a standard argument from Li and Yau [17], see also Schoen and Yau [20], we let y = ϕ|∇f | 2 and Z = ϕf t to have ϕ 2 |∇f | 2 = ϕy ≤ y and y Noticing that and using the inequality of the form ax 2 − bx ≥ − b 2 4a , (a, b > 0), we have (3.13) Putting together (3.11) -(3.13), the second and the third terms in the right hand side of (3.10) are further computed as follows, using t(y − αz) = ϕG; Hence by (3.10) where C 4 depends on n.We see that the left hand side of the last inequality is a quadratic polynomial in (ϕG), then using the quadratic formula and elementary inequality of the form √ pq.
Let M be an n-dimensional compact (or noncompact without boundary) manifold with bounded Ricci curvature, using previous Lemma 3.1 and local gradient Theorem 3.2, we now present global estimates for the positive solutions to the heat equation when the metric evolves by the Ricci flow.Theorem 3.3.(Global Estimates) Let (M, g(t)) be a complete manifold and g(t) solves the Ricci flow (3.1) such that its Ricci curvature is bounded for all (x, t) ∈ M × [0, T ].Suppose u = u(x, t) > 0 is any positive solution to the heat equation (3.2), then we have for −k 1 g(x, t) ≤ Rc(x, t) ≤ k 2 g(x, t) and α > 1 with Moreover, if we assume M has nonnegative Ricci curvature in the case of compact manifold, i.e., for 0 ≤ Rc(x, t) ≤ kg(x, t), we have for all (x, t) ∈ M × [0, T ] and α ≥ 1 with 1 p + 1 q = 1 α .The extreme case when α = 1 is the cheapest to prove where we choose p = q = 2α and the result immediately yields Proof.Set f = log u and allow G to remain as before, calculating at the maximum point (x 0 , t 0 ) ∈ M × (0, T ], we will show that a new function satisfies the inequality It suffices to prove that G ≤ αnp for any (x, t) ∈ M × (0, T ] and the inequality (4.4) will follow at once.We now show this by contradiction.Suppose G > αnp and that G has its maximum at the point (x 0 , t 0 ), then we know that ∇ G(x 0 , t 0 ) = 0, ∆ G(x 0 , t 0 ) ≤ 0, ∂ ∂t G(x 0 , t 0 ) ≥ 0, and (∆ − ∂ t ) G(x 0 , t 0 ) ≤ 0 then by Lemma 3.1, we have 0 Following the calculation in Theorem 3.2 we obtain after sending ρ to ∞.Consequently, we have the following inequality resulting into a quadratic inequality, and since from (3.18) Using the quadratic formula, we have which obviously implies that G ≤ αnp, a contradiction by the assumption (3.19).By definition of G, we therefore have The desired estimate follows since t 0 was arbitrarily chosen.

Harnack Inequalities and Estimates
As an application of our gradient estimate in the last section, we obtain some Harnack estimates for all positive solutions to the heat equation under the Ricci flow.We choose to follow the traditional approach of integrating Harnack quantity along a geodesic path connecting two points and exponentiating the result.Given where the infimum is taken over all the smooth path γ : [t 1 , t 2 ] → M connecting x 1 and x 2 .The norm |.| depends on t.We now present a lemma whose aim is to give an insight into how the approach goes.(See [2] and [20]).
for some A, B, C > 0.Then, the inequality holds for all (x 1 , t 1 ) and (x 2 , t 2 ) such that t 1 < t 2 .
Proof.Obtain the time differential of a function f depending on the path γ as follows (t ∈ [t 1 , t 2 ]), The last inequality was obtained by the application of completing the square method in form of a quadratic inequality satisfying ax 2 − bx ≥ − b 2 4a , (a, b > 0).Then integrating over the path from t 1 to t 2 , we have The required estimate (4.1) follows immediately after exponentiation.
Theorem 4.3.Let (M, g(t)) be a compact Riemannian manifold evolving by the Ricci flow.Let u(x, t) be a positive solution to the heat equation for t ∈ (0, T ], if the Ricci curvature is nonnegatively bounded by 0 ≤ Rc ≤ Kg, then for 0 < t 1 < t 2 ≤ T , we have for any α > 1 with 1 p + 1 q = 1 α u(x where C(n) is some constant depending only on n.
Integrating ( ∂ ∂s log u) along η(s), we obtain log u(x 1 , t 1 ) − log u(x 2 , t 2 ) = where C(n) is some constant depending only on n.

Lemma 4 . 1 .
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) and h be a C 2 -function on M × [0, T ].Define f = − log u and assumed that