A Ricci-type flow on globally null manifolds and its gradient estimates

Locally, a screen integrable globally null manifold $M$ splits through a Riemannian leaf $M'$ of its screen distribution and a null curve $\mathcal{C}$ tangent to its radical distribution. The leaf $M'$ carries a lot of geometric information about $M$ and, in fact, forms a basis for the study of expanding and non-expanding horizons in black hole theory. In the present paper, we introduce a degenerate Ricci-type flow in $M'$ via the intrinsic Ricci tensor of $M$. Several new gradient estimates regarding the flow are proved.


INTRODUCTION
Let M be a compact m-dimensional Riemannian manifold on which a one parameter family of Riemannian metrics g(t), t ∈ [0, T ], T < T ǫ , where T ǫ is the time where there is (possibly) a blow-up of the curvature is defined. We say (M, g(t)) is a solution to the Ricci flow if it is evolving by the following non-linear weakly parabolic partial differential equation [1] ∂ t g ij (x, t) = −2Ric ij (x, t), (x, t) ∈ M × [0, T ], (1.1) with g ij (x, 0) = g ij (x), where Ric ij (x, t) is the Ricci curvature tensor of the evolving metric g ij (x, t). This evolution system was initially introduced by Hamilton in [17]. The evolution equation for the metric tensor implies the evolution equation for the curvature tensor R in the form ∂ t R = ∆R + Q, where ∆ denotes the Laplacian operator on M and Q is a quadratic expression of the curvatures. In particular, the scalar curvature R satisfies ∂ t R = ∆ R + 2| Ric| 2 , so by the maximum principle its minimum is non-decreasing along the flow. By developing a maximum principle for tensors, Hamilton [17], proved that Ricci flow preserves the positivity of the Ricci tensor in dimension three and of the curvature operator in all dimensions; moreover, the eigenvalues of the Ricci tensor in dimension three and of the curvature operator in dimension four are getting pinched point-wisely as the curvature is getting large. In [25], Perelman used Ricci flow and its surgery to prove Poincare Conjecture. In the papers [1] and [8], the authors used Ricci flow coupled to heat-like equations to study some gradient estimates. For more information about the classical Ricci flow, see the papers [1,2,3,7,17,25,26] and references therein. When the underlying manifold M is null (sometimes called degenerate or lightlike), one may not define, in the usual way, the Ricci flow associated to the degenerate metric g on M . In fact, it is well-known in [10] that, in general, there is no Ricci tensor on M via the null metric g. When M is embedded into a semi-Riemannian manifold M as a null hypersurface, special classes of M do exist with an induced symmetric Ricci tensors. For instance, in [13] the authors shows that null hypersurfaces C[M ] 0 of genus zero exhibits an induced Ricci tensor. As the Ricci flow is an intrinsic geometric flow, one needs not to know much about the ambient space in which M is embedded. In [16], Kupeli studies null manifolds using a factor bundle approach and proved the existence of many geometric objects on M by taking the assumption that it is stationary, i.e., the normal bundle of M is a killing distribution, which secures a Levi-Civita connection on M . In fact, stationary M is the same as a globally null manifold studied by Duggal in [11]. A screen integrable globally null manifold M is locally a product manifold of a leaf M ′ of its screen distribution and a null curve C tangent to the normal bundle of M . The leaves M ′ are fundamental in studying expanding and non-expanding black hole horizons in mathematical physics, see for instance [16] and [24] and references therein. As the intrinsic Ricci tensor of M is symmetric on M ′ , we introduce a degenerate Ricci flow-type flow on M using such a Ricci tensor and investigate its properties in terms of the associated gradient estimates.
The theory of null submanifolds of a semi-Riemannian manifold is one of the most important topics of differential geometry. More precisely, null hypersurfaces appear in general relativity as models of different types of black hole horizons [10,13,24]. The study of non-degenerate submanifolds of semi-Riemannian manifolds has many similarities with the Riemannian submanifolds. However, in case the induced metric on the submanifold is degenerate, the study becomes more difficult and is strikingly different from the study of nondegenerate submanifolds [10]. Some of the pioneering works on null geometry is due to Duggal-Bejancu [10], Duggal-Sahin [13] and Kupeli [16]. Such works motivated many other researchers to invest in the study of null submanifolds, for example, [7,10,12,16,19,20,21,22,23] and many more references therein. The rest of the paper is organized as follows. In Section 2, we review the basics on globally null manifolds. In Section 3, we define a degenerate Ricci-type flow evolution on a globally null manifold and give some examples. In Section 4 -5, we develop several gradient estimates for the degenerate Ricci flow-type.

GLOBALLY NULL MANIFOLDS
We recall the basic concepts on globally null manifolds (see [11] for more details and references therein). Let (M, g) be a real m-dimensional smooth manifold where g is a symmetric tensor field of type (0, 2). We assume that M is paracompact. For x ∈ M , the radical or null space of T x M is subspace, denoted by Rad T x M , defined by (see [15] for more details) The dimension, say r, of Rad T x M is called nullity degree of g. Rad T x M is called the radical distribution of rank r on M . Clearly, g is degenerate or nondegenerate on M if and only if r > 0 or r = 0, respectively. We say that (M, g) is a null manifold if 0 < r ≤ m.
In this paper, we assume that 0 < r < m. Consider a complementary distribution S(T M ) to Rad T M in T M . We call S(T M ) a screen distribution on M , and its existence is secured by the paracompactness of M . It is easy to see that S(T M ) is semi-Riemannian. Therefore, we have the following decomposition (2. 2) The associated quadratic form of g is a mapping h : T x M −→ R given by h(X) = g(X, X), for any X ∈ T x M . In general, h is of type (p, q, r), where p+q +r = m, where q is the index of g on T x M . We use the following range of indices: I, J ∈ {1, . . . , q}, A, B ∈ {q+1, . . . , q+p}, α, β ∈ {1, . . . , r} and a, b ∈ {r+1, . . . , m}, i, j ∈ {1, . . . , m}. Throughout the paper we consider Γ(Ξ) to be a set of smooth sections of the vector bundle Ξ.
Using a well-known result from linear algebra, we have the following canonical form for h (with respect to a local basis of where ω 1 , . . . , ω p+q are linearly independent local differential 1forms on M . With respect to a local coordinates system (x i ), the above relation leads to Then it follows from the Frobenius theorem that leaves of Rad T M determine a foliation on M of dimension r, that is, M is a disjoint union of connected subsets {L t } and each point x ∈ M , M has a coordinate system (U , x i ), where i ∈ {1, . . . , m} and L t ∩ U is locally given by the equation x a = c a , a ∈ {r + 1, . . . , m} for real constants c a , and (x α ), α ∈ {1, . . . , r}, are local coordinates of a leaf L of Rad T M . Consider another coordinate system (U , x α ) on M . The transformation of coordinates on M , endowed with an integrable distribution, has the following special form. 0 = dx a = ∂x a ∂x b dx b + ∂x a ∂x α dx α = ∂x a ∂x α dx α , which imply ∂x a ∂x α = 0, ∀a ∈ {r + 1, . . . , m} and α ∈ {1, . . . , r}. Hence the transformation of coordinates on M is given by As g is degenerate on T M , by using (2.1) and the canonical form for h we obtain g αβ = g αa = g aα = 0. Thus, the matrix of g with respect to the natural frame {∂ i } becomes . holds for any other system of coordinate adapted to the foliation induced by Rad T M . We, therefore, suppose that (2.4) holds. Also, one can show that the screen distribution S(T M ) is invariant with respect to the transformations in (2.3). Next, we assume that r = 1. Thus, the 1-dimensional nullity distribution Rad T M is integrable. Using the basic formula where L is the Liederivative operator. The following result was established in [11]. Let C be a null curve in an m-dimensional null manifold (M, g), with m > 1 and locally given by where ⊥ means the orthogonal direct sum. It follows that S(T M ) is the transversal vector bundle of C in T M . Suppose S(T M ) is Riemannian and m = 3, we obtain the following differential equations where h and {k 1 , k 2 , k 3 } are smooth functions on U and {W 1 , W 2 } is an orthonormal basis of Γ(S(T M )) U . We call F = d dt = E, W 1 , W 2 a Frénet frame on M along C with respect to the screen distribution S(T M ). The functions {k 1 , k 2 , k 3 } and the differential equations (2.6) are called curvature functions of C and Frénet equations for F , respectively. The result can be generalized for higher dimensions. It is important to mention that for any m, the first Frénet equation for F remains the same. Now we show that it is always possible to find a parameter on C such that h = 0, using the same screen distribution S(T M ). Consider another coordinate neighborhood U * , and its Frénet frame F * , with U ∩ U * = φ. Then, d dt * = dt dt * d dt . Writing the first Frénet equation in (2.6) for both F and F * and using above transformation, we obtain d 2 t dt * 2 + h dt dt * 2 = h * dt dt * . Consider the differential equation d 2 t dt * 2 − h * dt dt * = 0, whose general solution comes from It follows that a solution of (2.7), with a = 0, may be taken as a special parameter on C such that h = 0. Denote such a parameter by p = t−b a , where we call t the general parameter as given in (2.7), and p a distinguished parameter of C. Then, the first Frénet equation is given by ∇ d dp d dp = 0 and, therefore, C is a null geodesic of M , with respect to the distinguished parameter p. Since the first Frénet equation is the same for any m, the following holds. In view of the above theorem, the author in [11] defined globally null manifolds as follows.
A null manifold (M, g) is said to be a globally null manifold if it admits a single global null vector field and a complete Riemannian hypersurface. As a consequence, the following characterization holds on globally null manifolds.
). Let (M, g) be a globally null manifold. Then, the following assertions are equivalent: As an example, we have the following.
It is well known that Minkowski spacetime is globally hyperbolic. Take two null coordinates u = t + r and v = t − r (u > v). Then, we have The absence of the terms du 2 and dv 2 in (2.8) implies that the two hypersurfaces {v = constant}, {u = constant} are null. Denote one of these null hypersurfaces by (M, g), where g is the induced degenerate metric tensor of g. A leaf of the 2dimensional screen distribution S(T M ) is topologically a 2-sphere with complete Riemannian metric dΩ 2 = r 2 (dθ 2 + sin 2 θdφ 2 ) which is the intersection of the two hypersurfaces. Since by definition a spacetime admits a global timelike vector field, it follows that both its null hypersurfaces admit a single global null vector field. Thus, there exists a pair of globally null manifolds, as null hypersurfaces of a Minkowski spacetime.
Remark 2.5. Globally null manifolds were also studied by Kupeli in [16], under the name of stationary singular semi-Riemannian manifolds.

A DEGENERATE RICCI-TYPE FLOW
Let (M, g) be a null manifold and S(T M ) be its screen distribution. A globally null manifold (M, g) is said to be a screen integrable globally null manifold if its screen distribution S(T M ) is integrable.
Let (M, g) be a screen integrable globally null manifold and (M ′ , g ′ ) be a leaf of S(T M ) such that g ′ = g| M ′ , immersed in M as a non-degenerate submanifold. Denote by P the projection morphism of T M into S(T M ). Let R denote the curvature tensor of M with respect to the metric connection ∇ on M . If R ′ is the curvature tensor of M ′ with respect to the connection ∇ ′ on M ′ , then, in view of [16, p. 48], we have Moreover, the associated curvaturelike tensor of type (0,4) is given by Let (M, g) be a screen integrable global null submanifold. Then, the Ricci ten- the gradient of f and div(·) denotes the divergence operator on M ′ . For more details of curvature properties on globally null manifolds, see Kupeli [16]. In view of the above background, we define a degenerate Ricci-type flow on (M, g) as follows.
In harmonic local coordinates around a point x ∈ M , the Ricci tensor at x is Thus, a degenerate Ricci-type flow resembles a heat flow evolution.
Next, we will give some examples of degenerate Ricci flows. is flat, we notice that M is flat as well (see [10] for more details). Consequently, the intrinsic curvature R vanishes and each leaf M ′ is flat. Thus, the flat metric on M ′ has zero Ricci curvature, so it does not evolve at all under the degenerate Ricci-type flow.
where R 0 is the initial radius of the sphere. Notice that the degenerate Ricci-type flow of (M, g) will become singular at t = (1/2)R 2 0 . At this time, the leaf M ′ has collapsed to a point.
The existence and uniqueness of degenerate Ricci-type flows can be established in the same way as in the classical Ricci flow, as well as the associated geometric evolution equations for the induced objects associated to curvature (see [2, p. 90] for more details).
As each leaf M ′ is Riemannian, we can define the distance function on M ′ in a natural way. For a point p ∈ M ′ , define d(x, p) for all x ∈ M ′ , where d(·, ·) is the geodesic distance. It is important to note that d is only Lipschitz continuous, that is, everywhere continuous except at the cut locus of p and on the point where x coincides with p. One can easily see that |∇d| = g ′ab ∂ a d∂ b d = 1 on M ′ /{{p} ∪ cut(p)}. Let d(x, y, t) be the geodesic distance between x and y with respect to the Riemannaian metric g ′ (t), we define a smooth cut-off function ϕ(x, t) with support in the geodesic cube for any C 2 -function ψ on [0, ∞) with ψ(s) = 1, for s ∈ [0, 1] and ψ(s) = 0, for s ∈ [2, +∞] (see [1] for more details). Furthermore, ψ ′ (s) ≤ 0, ψ ′′ (s) ≥ −c 1 and |ψ ′ | 2 ψ ≤ c 2 , where c 1 , c 2 are constants such that ϕ(x, t) = ψ(d(x, p, t)/ρ) and ϕ| Q 2ρ,T = 1.
As in [1], let M ′ be a complete n-dimensional leaf of the integrable secreen distribution S(T M ) of null manifold (M, g) whose Ricci curvature is bounded from below by Ric ′ ≥ (n − 1)k, for some constant k ∈ R. Then the Laplacian of the distance function satisfies Throughout, we will impose boundedness condition on the Ricci curvature of the metric and note that when the metric evolves by the degenerate Ricci-type flow, boundedness and sign assumptions are preserved as long as the flow exists, so also the metrics are uniformly equivalent,

SOME GRADIENT ESTIMATES
In this section, we discuss the localized version of gradient estimate on the heat equation perturbed with curvature operator under both forward and backward degenerate Ricci-type flow. The estimate under backward action of degenerate Riccitype flow is related to the local monotonicity for heat kernel and mean value theorem of Ecker, Knopf, Ni and Topping in [8]. They worked in general geometric flow, we follow their approach.
Let (M, g) be a screen integrable globally null manifold and (M ′ , g ′ = g| M ′ ) be a leaf of the screen distribution S(T M ). We consider the conjugate heat equation coupled to the backward and forward degenerate degenerate Ricci-type flow, respectively, as follows: and (4.1) Throughout this paper, we denote by · the norm on M ′ with respect to g ′ . 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 We set ω i := ∂ i ω, ω ij := ∂ i ∂ j ω and so on, for some smooth function ω on M ′ , where 1 ≤ i, j ≤ n. Then we have Proof. Let us define f = ln u A . It is easy to see that Then, a straightforward calculation using (4.1) gives the evolution equation of f as ∂ t f = ∆f + ∇f 2 − Scal ′ . Next, we compute the evolution equation of φ. To that end, differentiating φ with respect to t and using the evolution equation of f , gives On the other hand, the term ∂ t ( ∇f 2 ) is given by in which we have used (4.1) and the evolution equation of f . Applying (4.4) to (4.3) and using the evolution equation of f , gives In view of Bochner-Weitzenböck identity (see [8] for details), we have ∆( Next, we compute ∆φ. To that end, a direct calculation gives Applying the definition of ∆ and relation (4.7), we get from which we deduce that Rearranging the terms in (4.8), gives We can simplify some terms in the braces of (4.9) further. The first term becomes Using the Ricci identity on the next three terms we have Similarly, we have 2(1 − f ) −2 g ′ (∇f, ∇ ∇f 2 ) = 2g ′ (∇f, ∇φ), and then the second to the last three terms gives Putting all these together in (4.9), we get Then, using the curvature conditions we are left with the following inequality Next, we apply a cut-off function in order to derive the desired estimate. To that end, let ψ be a smooth cut function defined on [0, ∞) such that 0 ≤ ψ(s) ≤ 1, with ψ ′ (s) ≤ 0, ψ ′′ (s) ≥ c 1 and |ψ ′ | 2 /ψ ≤ −c 2 , for some constants c 1 , c 2 > 0, depending on the dimension of the manifold only. Let us define a distance function d(p, x) between the point p and x such that ϕ(x, t) = ϕ(d(p, x, t)) = ψ d g ′ (t) (p, x)/ρ , for smooth function ϕ : M ′ × (0, T ] −→ R. 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 ϕφ supported in Q 2ρ,T × [0, ∞) is C 2 at the maxima, in which such assumption is supported by a standard argument by Calabi known as Calabi's trick (see [5] for more details). This approach is used in [6], see also [8,26,28]. Therefore, we obtain (4.11) and by the Laplacian Comparison Theorem (see [1] for more details), we have Let (x 0 , t 0 ) be a point in Q 2ρ,T at which F = ϕφ attains its maximum value. At this point we have to assume that F is positive, since F = 0 implies that ϕφ(x 0 , t 0 ) = 0 and hence, φ(x, t) = 0, for all x ∈ M . Then, we 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 analyses on the result at the maximum point. To that end, we have (4.13) 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. (4.14) Taking tF on M ′ × (0, T ], we have (∂ t − ∆)(tF ) ≥ 0, whenever (tF ) achieves its maximum. Similarly, by this argument, we have ∇(ϕφ)(x 0 , t 0 ) − φ∇ϕ(x 0 , t 0 ) = ϕ∇φ(x 0 , t 0 ), which means ϕ∇φ can always be replaced by −φ∇ϕ. By (4.10), (4.13) and (4.14), one has Taking 0 ≤ ϕ ≤ 1 and noticing that 1 1−f ≤ 1, then the last inequality becomes Applying the following relation 2ρ 3 F 1 2 ≤ ρ 3 F + ρ 3 (see [8] for details) and by the Young's inequality, we derive Notice also that by bounds given in (4.11) and (4.12), we have Putting these together and dividing through by (1−f ), while noticing that 1 Therefore, we have Then, From here we can conclude that at (x 0 , t 0 ). Therefore, which completes the proof.
The boundedness assumption may be weakened in the case of the forward degenerate Ricci-type flow. Therefore, we have the following.
Proof. The proof of this theorem is similar to that of Theorem 4.1. The disparity between the estimate inequalities in Theorem 4.1 and Theorem 4.2 arises in some calculation which we briefly point out here. Similary, set f = ln u A and φ = ∇ln (1−f ) 2 . Notice that g ′ (t) evolves by the forward degenerate Ricci flow, where the inverse metric evolves as ∂ t (g ′ij ) = 2Ric ′ ij and then hence, the counterpart of (4.9) is Now, using the Ricci identity, some braced terms vanish. Then, we can conclude that Using the curvature conditions, we are left with the following inequality as in (4.10) The rest of the proof follows as in the one of Theorem 4.1.

GRADIENT ESTIMATES ON FORWARD HEAT EQUATION
Let (M, g) be a screen integrable globally null manifold and (M ′ , g ′ ) be an n-dimensional complete Riemannian integral manifold of the screen distribution S(T M ) without boundary. Next, we discuss space-time gradient estimates for positive solutions of the forward heat equation along the degenerate Ricci flow. More precisely, we consider As mentioned in [1], in general, our degenerate version of our estimate is local one too we also obtain it in the interior of geodesic cube. We will show how this local estimate can lead to achieving a global one. We start by proving the following lemma, which is useful to this section. Let us define the geodesic cube where f = ln 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.
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 ∆ ∇f 2 = 2f 2 ij + 2f j f jji + 2Ric ′ ij f i f j . By the hypothesis of the lemma that g ′ (x, t) evolves by the degenerate Ricci flow With the above computations, we obtain the following at an arbitrary point (x, t) ∈ Q 2ρ,T .
On the other hand Therefore, we have Now choosing any two real numbers p, q > 0 such that 1 p + 1 q = 1 α , we can write where we have used completing the square method to arrive at the last inequality. Also by Cauchy-Schwarz inequality, (∆f ) 2 = g ′ij ∂ i ∂ j f 2 ≤ nf 2 ij 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 degenerate Ricci curvature as −(ρ 1 + ρ 2 )g ′ ≤ ≤ n(ρ 1 + ρ 2 ) 2 , since the Ricci curvature tensor is symmetric. Therefore, we have Hence the result. Our calculation is valid in the cube Q 2ρ,T .
Next, we state and prove a degenerate version result for local gradient estimate (space-time) for the positive solutions to the heat equation in the geodesic cube Q 2ρ,T of bounded Ricci curvature manifold evolving by the degenerate Ricci-type flow.
Proof. Let f = ln 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 4.1. The argument follows 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,T × (0, T ] and we have assumed that (ϕG)(x 0 , t 0 ) > 0, for t 0 > 0. By (5.7) we notice that (∆ − ∂ t )(ϕG) ≤ 0. Using (5.6), (5.7) and Lemma 5.1, we have where ρ = (ρ 1 + ρ 2 ). Noticing also that ϕ∇G can be replaced by −G∇ϕ, by the condition ∇(ϕG) = 0. Therefore, we have 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 maximum of c 1 , c 2 , so our computation becomes Multiplying through by ϕ such that 0 ≤ ϕ ≤ 1 again, we have Using a standard argument from Lie and Yau [18] (see also Schoen and Yau [26] and references therein), we let y = ϕ ∇f 2 and Z = ϕ∂ t f to have ϕ 2 ∇f 2 = ϕy ≤ y, and y and also using the inequality of the form and elementary inequality of the form 1 + a 2 + p 2 ≤ 1 + a + p yields Recall that we picked up ϕ(x, t) such that 0 ≤ ϕ ≤ 1 and particularly ϕ(x, t) = 1 in Q 2ρ,T and sine (x 0 , t 0 ) is a maximum point for (ϕG) in Q 2ρ,T , we have G(x, τ ) = (ϕG)(x, τ ) ≤ (ϕG)(x 0 , t 0 ). Hence for all x ∈ M such that d(x, x 0 , τ ) < ρ and τ ∈ (0, T ] was arbitrarily chosen, which implies the estimate (5.5) and completes the proof.
Let (M, g) be a screen integrable globally null manifold. Let (M ′ , g ′ ) be an n-dimensional compact (or noncompact without boundary) leaf of S(T M ) with bounded Ricci curvature. Using previous Lemma 5.1 and local gradient in Theorem 5.2, we now present global estimates for the positive solutions to the heat equation when the metric g ′ evolves by the degenerate Ricci-type flow. Theorem 5.3. Let (M, g) be a screen integrable globally null manifold and (M ′ , g ′ (t)) be a complete leaf of S(T M ) and g ′ (t) solves the degenerate Ricci-type flow equation such that its Ricci curvature is bounded for all (x, t) ∈ M ′ × (0, T ]. Let u = u(x, t) > 0 be any positive solution to the heat equation (5.2). Then, we have for −ρ 1 g ′ (x, t) ≤ Ric ′ (x, t) ≤ ρ 2 g ′ (x, t) and α > 1 with 1 p + 1 q = 1 α , for all (x, t) ∈ M ′ × (0, T ] and α ≥ 1 with 1 p + 1 q = 1 α . Now recall that (x 0 , t 0 ) is a maximum point for G on M ′ × (0, τ ] from this fact, we can get that for all x ∈ M . Therefore, the estimate holds at (x, τ ). Since the number τ ∈ (0, T ] can be chosen arbitrarily. Also, we have the following.