Sobolev-type inequalities and heat kernel bounds along the geometric flow

Abstract

This paper is concerned with Sobolev-type inequalities and upper bound for the fundamental solution to the heat-type equation defined on compact manifold whose metric evolves by the generalized geometric flow. It turns out that the pointwise estimates obtained in this paper depend on the constants in the uniform Sobolev inequalities for the flow or the best constants in the euclidean Sobolev embedding. We give various illustrations to show that our results are valid in many contexts of geometric flow, where we may not need explicit curvature constraint. Moreover, our approach here also demonstrates equivalence of Sobolev inequalities, log-Sobolev inequalities, ultracontractive estimates and heat kernel upper bounds.

1 Introduction

Let $M$ be an $n$-dimensional compact Riemannian manifold endowed with metric $g(x,t)$ evolving by the geometric flow in the interval $0 \le t \le T$, $T < T_\epsilon$, where $T_\epsilon$ is the time where there is (possibly) a blow-up of the curvature, so we do not need to deal with singularities. Let $u(x,t)$ be a positive solution to a heat-type equation on $M \times [0, T]$, we consider the following coupled system

$$\begin{aligned} {\left\{ \begin{array}{ll} \displaystyle \partial _t g_{ij}(x, t) = - 2 h_{ij}(x,t), &{} (x,t) \in M \times [0,T]\\ \displaystyle (- \partial _t - \Delta _g + \mathcal {H}(x,t)) u(x,t) = 0, &{} (x, t) \in M \times [0, T], \end{array}\right. } \end{aligned}$$

(1.1)

where $h_{ij}$ is a general time-dependent symmetric $(0, 2)$-tensor, $\mathcal {H} = g^{ij}h_{ij}$, the metric trace of $2$-tensor $h_{ij}$, is a $C^\infty$-function $\mathcal {H}: M \times [0, T] \rightarrow \mathbf {R}$ and $\Delta _g$ is the usual Laplace–Beltrami operator acting on functions in space with respect to metric $g(t)$ in time. The first in the system (1.1) is the generalized geometric flow equation, that is, the flow of metric by tensor with respect to abstract time. In practice, the geometric flow deforms and smoothens out irregularities in the metric to give a “nicer” form and thus, provides useful geometric and topological information on the manifold. For example if $h_{ij} = R_{ij}$, the Ricci curvature tensor, we then have the famous Hamilton Ricci flow [18], which has proven to be of fundamental importance in the global analysis on manifolds. The second equation is a heat-type-conjugate equation on $M$ whose positive solution is a smooth function, at least $C^2$ in $x$ and $C^1$ in $t$, $u(x, t) = u \in C^{2, 1}(M \times [0,T])$. In this paper we study the behaviour of the fundamental (minimal positive) solution to the associated heat-type equation along the geometric flow. Let $U, V : M \times [0,T) \rightarrow (0, \infty )$ satisfy

$$\begin{aligned} \square U = (\partial _t - \Delta _g) U = 0 \quad and \quad \square ^* V = ( - \partial _t - \Delta _g + \mathcal {H} ) V = 0 \end{aligned}$$

with

$$\begin{aligned} \int _0^T \int _M \square U V d \mu _g dt = \int _0^T \int _M U\square ^*V d \mu _g dt. \end{aligned}$$

(1.2)

We say $U$ and $V$ are respectively solutions to the heat equation and heat-type-conjugate equation. An application of this is to solve geometric flow forward in time and solve the heat-type-conjugate equation backward in time. A very good example is the conjugate heat equation $\square ^* u = (\Delta _g - \partial _t + g^{ij} R_{ij})u(x,t) = 0$ (i.e., adjoint to the heat operator $\square = (\Delta _g - \partial _t)$, where $g^{ij} R_{ij} = R$, the scalar curvature), which was introduced in a fundamental paper [25], there Perelman obtained Li–Yau Harnack estimates for the minimal positive solution to this equation among many results. The author also studied this in [1] under both forward and backward in time Ricci flow. The case $h_{ij} = - R_{ij}$ (resp. $R_{ij}$) is precisely when the manifold is being evolved with respect to forward (resp. backward) Ricci flow. In fact, one of the motivations to study this subject arises from the question; is there any merit or demerit of flowing Riemannian manifold by the Ricci flow? The readers who are interested more in this question can find any of the following books [9–11] and [30]. See also our short note [2] on Ricci flow on a closed manifold with positive Euler characteristics and [3] for further details.

Coupling geometric flow to the heat equation can be associated with some physical interpretation in terms of heat conduction process. Precisely, the manifold $M$ with initial metric $g(x, 0)$ 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 geometric flow and simultaneously allow the heat to diffuse on $M$, then, the solution $u(x, t)$ will represent the space-time temperature on $M$. Moreover, if $u(x, t)$ approaches $\delta$-function at the initial time, we know that $u(x,t) >0$, this gives another physical interpretation that temperature is always positive, whence we can consider the potential $f = \log u$ as an entropy or unit mass of heat supplied and the local production entropy is given by $|\nabla f|^2 = \frac{ |\nabla u|^2}{u^2}$. Suffice to say that heat kernel governs the evolution of temperature on a manifold with certain amount of heat energy prescribed at the initial time.

In this paper, we obtain Sobolev-type inequalities and some upper bounds for the fundamental solution (Heat kernel) to the heat-type equation defined on compact manifold whose metric is evolving by the geometric flow. Notice that the system (1.1) above is associated to Perelman’s monotonicity formula [25], which has been a vital tool in the analysis of the Ricci flow. Perelman proved a lower bound for the heat kernel satisfying the conjugate heat equation with application of the maximum principle and his reduced distance, an outstanding feature of the estimate is that it does not require explicit assumption on metric curvature, the information is being embedded in the reduced distance. In the present too, the bound obtained in this paper needs no explicit curvature assumptions, it rather depends on the Sobolev-type constants similar to those of Zhang–Ricci–Sobolev [29], which in turn depends on the best constant in the usual Sobolev embedding controlled by the infimum of the Ricci curvature and the injectivity radius of the underlying manifold. The motivation for this was Zhang’s result in [28], where he obtained upper bounds for conjugate heat kernel under backward Ricci flow, such bounds depend on Yamabe constant or Euclidean Sobolev embedding constant. He further showed that this type of heat kernel upper bounds are proper extension of an on-diagonal upper bound in the case of a fixed manifold, where one obtains a bound of the form

$$\begin{aligned} F(x, t; y, s) \le C(n) \max \left\{ \frac{1}{(t-s)^{\frac{n}{2}}} , 1\right\} \end{aligned}$$

(1.3)

with $C(n) >0$ depending on $n$ for all $t > s$ and $x, y \in M.$ We also give a special case $\mathcal {H}(x, t)$ is nonnegative to support the above assertion. Recently, Bǎileşteanu [6] has adopted Zhang’s approach to obtain similar estimate for the fundamental solution of the heat equation coupled to Ricci flow. Our calculation is based on the ideas of both papers [6] and [28] cited above, (see also [7]). We remark that the similarity in our results is a justification of the fact that heat diffusion on a bounded geometry with either static or evolving metric behaves like heat diffusion in Euclidean space, many a times, their estimates even coincide. A result of Cheeger and Yau [8] has revealed that the heat kernel of a complete manifold with bounded Ricci curvature can be compared with that of the space form whose curvature determines the lower bound for the manifold’s Ricci curvature.

2 Preliminaries and examples of geometric flow

2.1 Technical details

Throughout this paper, $M$ is assumed to be a compact Riemannian manifold without boundary. We denote the fundamental solution (heat kernel) to the heat-type equation by $F(x, t; y, s) \in ( M \times [0, T] \times M \times [0, T] )$ and partial differential operator with respect to time by $\partial _t$. We now give a formal definition and some important properties of heat kernel.

Definition 1

We say that $F( x, t; y, s)$ is a fundamental solution to the heat-type equation centred at $(y, \sigma )$ for $x, y \in M, s < t \in [0,T]$, if it satisfies the following system

$$\begin{aligned} {\left\{ \begin{array}{ll} \displaystyle (-\partial _t - \Delta _{(x, t)} + \mathcal {H}(x, t)) F( x, t; y, s) = 0 \\ \displaystyle \lim _{t \rightarrow s} F( x, t; y, s) = \delta _y(x) \\ \end{array}\right. } \end{aligned}$$

(2.1)

for any $x \in M,$ where $\delta _y(\cdot )$ is the dirac-delta function concentrated at some point $y$.

Thus, $F( x, t; y, s)$ is the unique minimal positive solution to the equation which from henceforth we refer to as the heat kernel.

Lemma 1

The heat kernel satisfies the following properties.

• 1. $\int _M F( x, t; y, s) d\mu (g(x, t)) = 1$

• 2. $F(x,t;y,0) = \int _M F( x, t; z, \frac{t}{2}) F( z, \frac{t}{2}; y, 0) d\mu (g(z, \frac{t}{2}))$ $($Semigroup property$)$

• 3. $F(x,t;y,s)$ is also the fundamental solution to the heat equation in $(y,s)$-variables i.e,

  $$\begin{aligned} {\left\{ \begin{array}{ll} \displaystyle (\partial _s - \Delta _{(y,s)}) F( x, t; y, s) = 0 \\ \displaystyle \lim _{s \rightarrow t}F( x, t; y, s) = \delta _x(y). \end{array}\right. } \end{aligned}$$

  (2.2)

• 4. $\int _M F( x, t; y, s) d\mu (g(y, s)) \le 1.$

Other important properties of heat kernel such as existence, uniqueness, smoothness, symmetry have been studied by many authors, Guenther in [17] and Garofalo and Lanconelli in [14] for examples.

Interestingly, when a manifold is being evolved under a geometric flow all the associated quantities also evolve along the flow. For examples, the Riemannian volume measure $d\mu$ of $(M, g)$ evolves by

$$\begin{aligned} \partial _t d\mu = - \mathcal {H} d\mu \end{aligned}$$

and $\mathcal {H}$ by

$$\begin{aligned} \partial _t \mathcal {H} = g^{ij} \partial _t h_{ij} + 2 |h_{ij}|^2 \end{aligned}$$

where $g^{ij}$ is the inverse of the metric $g_{ij}$ and $|h_{ij}|^2 = g^{ik} g^{jl} h_{ij} h_{kl}$. Denote $\beta := g^{ij} \partial _t h_{ij}$, in particular, under the Ricci flow, where $h_{ij} = R_{ij}$ and $\mathcal {H} = R$, we have $\beta = \Delta R$. Here in this paper we will assume that

$$\begin{aligned} \beta - \Delta \mathcal {H} \ge 0. \end{aligned}$$

(2.3)

This is motivated by an error term appearing in a result of Müller [22], Lemma 1.6]. For our case the error term reads; for any time-dependent vector field $X$ on $M$

$$\begin{aligned} \mathcal {D}(X) := 2(R_{ij} -h_{ij})(X, X) + 2 \langle 2 div \ h -\nabla \mathcal {H}, X\rangle + \partial _t \mathcal {H} - \Delta \mathcal {H} - 2|h_{ij}|^2, \end{aligned}$$

(2.4)

where $div$ is the divergence operator, i.e., $(div\ h)_k = g^{ij} \nabla _i h_{jk}$. Clearly the last three terms in (2.4) above is the same as the quantity $\beta - \Delta \mathcal {H}$. It does make sense to assume (2.3) whenever $\mathcal {D}(X)$ is nonnegative. The application of this is that we are on a steady or shrinking soliton (self-similar solution to the geometric flow) if the equality in (2.3) holds. Note that we can also express $|h_{ij}|^2 \ge \frac{1}{n} \mathcal {H}^2$ since $|g^{ij} h_{ij}|^2 = \mathcal {H}^2.$ Using the condition that $\beta - \Delta \mathcal {H} \ge 0$, we have a governing differential inequality for the evolution of $\mathcal {H}$ as follows

$$\begin{aligned} \frac{\partial }{\partial t} \mathcal {H} \ge \Delta \mathcal {H} + \frac{2}{n} \mathcal {H}^2. \end{aligned}$$

(2.5)

Suppose $\mathcal {H} \ge \mathcal {H}_{min},$ we can apply the maximum principle by comparing the solution of the differential inequality with that of the following ordinary differential equation

$$\begin{aligned} {\left\{ \begin{array}{ll} \displaystyle \frac{d \psi (t) }{d t} = \frac{2}{n} (\psi (t))^2 \\ \displaystyle \ \ \psi (0) = \mathcal {H}_{min}(0), \end{array}\right. } \end{aligned}$$

(2.6)

which is solved to

$$\begin{aligned} \psi (t) = \frac{ \mathcal {H}_{min}(0)}{ 1 - \frac{2}{n} \mathcal {H}_{min}(0)t }. \end{aligned}$$

Therefore

$$\begin{aligned} \mathcal {H}_{g(t)} \ge \psi (t) = \frac{\mathcal {H}_{min}(0)}{ 1 - \frac{2}{n} \mathcal {H}_{min}(0)t } \end{aligned}$$

(2.7)

for all $t \ge 0$ as long as the flow exists.

2.2 Examples of geometric flow

In the following, we give some examples of geometric flows where our results are valid. We remark that in these cases the error term $\mathcal {D}$ and the quantity $\beta - \Delta \mathcal {H}$ are nonnegative. More examples can be found in [22], Section 2].

2.2.1 Hamilton’s Ricci flow [18]

Let $(M, g(t))$ be a solution to the Ricci flow. This is the case where $h_{ij} =R_{ij}$ is the Ricci tensor and $\mathcal {H} =R$ is the scalar curvature on $M$. Here, the scalar curvature evolves by

$$\begin{aligned} \partial _t R = \Delta R + 2 |R_{ij}|^2. \end{aligned}$$

By twice contracted second Bianchi identity $g^{ij} \nabla _i R_{jk} = \frac{1}{2} \nabla _k R$, the quantity $\mathcal {D}(X)$ vanishes identically and $\beta -\Delta R \equiv 0$.

2.2.2 Ricci-harmonic map flow [23]

Let $(M, g)$ and $(N, \xi )$ be compact (without boundary) Riemannian manifolds of dimensions $m$ and $n$ respectively. Let a smooth map $\varphi : M \rightarrow N$ be a critical point of the Dirichlet energy integral $E( \varphi ) = \int _M | \nabla \varphi |^2 d \mu _g$, where $N$ is isometrically embedded in $\mathbb {R}^d, \ d \ge n,$ by the Nash embedding theorem. The configuration $(g(x ,t), \varphi (x, t)), t \in [0, T)$ of a one parameter family of Riemannian metrics $g(x, t)$ and a family of smooth maps $\varphi (x, t)$ is defined to be Ricci-harmonic map flow if it satisfies the coupled system of nonlinear parabolic equations

$$\begin{aligned} {\left\{ \begin{array}{ll} \displaystyle \frac{\partial }{\partial t} g(x ,t)= - 2 Rc(x ,t) + 2 \alpha \nabla \varphi (x ,t) \otimes \nabla \varphi (x ,t)\\ \displaystyle \frac{\partial }{\partial t} \varphi (x ,t) = \tau _g\varphi (x,t), \end{array}\right. } \end{aligned}$$

(2.8)

where $Rc(x ,t)$ is the Ricci curvature tensor for the metric $g$, $\alpha (t) \equiv \alpha > 0$ is a time-dependent coupling constant, $\tau _g \varphi$ is the intrinsic Laplacian of $\varphi$, which denotes the tension field of map $\varphi$ and $\nabla \varphi \otimes \nabla \varphi = \varphi ^* \xi$ is the pullback of the metric $\xi$ on $N$ via the map $\varphi$. See List [21] when the target manifold is one dimensional. Here $h_{ij} = R_{ij} - \alpha \nabla _i \varphi \otimes \nabla _j \varphi =: \mathcal {S}_{ij}$, $\mathcal {H}= R-\alpha |\nabla \varphi |^2 =: \mathbf{S}$ and

$$\begin{aligned} \partial _t \mathbf{S} = \Delta \mathbf{S} + 2 | \mathcal {S}_{ij}|^2 +2 \alpha |\tau _g \varphi |^2 - 2 \dot{\alpha } |\nabla \varphi |^2. \end{aligned}$$

(2.9)

Using the twice contracted second Bianchi identity, we have

$$\begin{aligned} \left( g^{ij} \nabla _i \mathcal {S}_{jk} - \frac{1}{2} \nabla _k \mathbf{S}\right) X_j = - \alpha \tau _g \varphi \nabla _j\varphi X^j. \end{aligned}$$

(2.10)

Then, $\mathcal {D}(\mathcal {S}_{ij}, X) = 2 | \tau _g \varphi - \langle \nabla \varphi , X \rangle |^2 - 2 \dot{\alpha } |\nabla \varphi |^2$ and $\beta - \Delta \mathbf{S} = 2 \alpha | \tau _g \varphi |^2 - \dot{\alpha } |\nabla \varphi |^2$ for all $X$ on $M$. Thus both $\mathcal {D}$ and $\beta - \Delta \mathbf{S}$ are nonnegative as long as $\alpha (t)$ is nonincreasing in time. (See [4] for more results in this direction.)

2.3 Lorentzian mean curvature flow

Let $M^n(t) \subset L^{n+1}$ be a family of space-like hypersurfaces in ambient Lorentzian manifold evolving by Lorentzian mean curvature flow. Then, the induced metric evolves by

$$\begin{aligned} \partial _tg_{ij} = 2 H \Pi _{ij}, \end{aligned}$$

where $\Pi _{ij}$ denotes the components of the second fundamental form $\Pi$ on $M$ and $H = g^{ij} \Pi _{ij}$ denotes the mean curvature of $M$. In this case $h_{ij} = - H \Pi _{ij}$ and $\mathcal {H} = - H^2$. Also $\partial _tH = \Delta H - H (|\Pi |^2 + \widetilde{Rc}(\nu , \nu )$, $\beta - \Delta H = 2 H^2 |\Pi |^2 + |\nabla H|^2 + 2 H \widetilde{Rc}(\nu , \nu )$ and

$$\begin{aligned} \mathcal {D}{X} = 2|\nabla H - \Pi (X, \cdot )|^2 + 2 \widetilde{(}H \nu - X, H \nu - X) + 2 \langle \widetilde{Rm}(X, \nu )\nu , X \rangle , \end{aligned}$$

(2.11)

where $\widetilde{Rc}$ and $\widetilde{Rm}$ denote the Ricci and Riemman curvature tensor of $L^{n+1}$ respectively. $\nu$ denotes future-oriented timelike unit normal vector on $M$. Obviously both $\mathcal {D}(X)$ and $\beta -\Delta H$ are nonnegative when assuming nonnegativity on sectional curvature of $L^{n+1}$.

3 Sobolev-type inequalities along the flow

In this section, we give a brief discussion on the version of Sobolev embedding that will be used in the proof of the main theorems. The main ingredients used here are logarithmic Sobolev inequalities and ultracontractivity property of the heat semigroup. It is well known that Gross logarithmic Sobolev inequality [15] is equivalent to Nelson’s hypercontractive inequality [24], both of which may imply ultracontractivity of the heat semigroup (see also [12, 13]).

3.1 The Sobolev embedding

Let $(M,g)$ be an $n$-dimensional ($n \ge 3$) Riemannian manifold without boundary, it is well known that when $M$ is compact the Sobolev space $H_1^q(M)$ is continuously embedded in $L^{q^*}(M)$ for any $1 \le q < n$ and $\frac{1}{q^*} = \frac{1}{q} - \frac{1}{n}$. Here $H_1^q(M)$ is the completion of $C^\infty (M)$ with respect to the standard norm

$$\begin{aligned} \Vert u \Vert _q = \left( \int _M | \nabla u |^q d \mu (g) \right) ^{\frac{1}{q}}+ \left( \int _M | u |^q d \mu (g) \right) ^{\frac{1}{q}} \end{aligned}$$

(3.1)

and the embedding of $H_1^q(M)$ in $L^{q^*}(M)$ is critical. Similarly, the following Sobolev embedding inequality holds true; there exists a positive constant $B_q$ depending on $q$ such that for any $u \in H_1^q(M)$

$$\begin{aligned} \left( \int _M |u|^{q^*} d\mu (g) \right) ^{\frac{1}{q^*}} \le K(n, q)\left( \int _M | \nabla u |^q d \mu (g) \right) ^{\frac{1}{q}} + B_q)\left( \int _M | u |^q d \mu (g) \right) ^{\frac{1}{q}}, \end{aligned}$$

(3.2)

where $K(n, q)$, an explicit constant depending on $n$ and $q$ is the smallest constant having this property, $K(n, q)$ is the best constant in the Sobolev embedding for $\mathbb {R}^n$). See Aubin [5] and Hebey [19] and Talenti [26]. In other words, there exist positive constants $A$ and $B$ such that for all $u \in W^{1, 2} (M, g)$, we have

$$\begin{aligned} \left( \int _M u^{\frac{2n}{n-2}} d\mu (g) \right) ^{\frac{n-2}{n}} \le A\int _M | \nabla u |^2 d \mu (g) + B \int _M u^2 d \mu (g). \end{aligned}$$

(3.3)

On the compact manifold whose metric evolves along the Ricci flow, Zhang, [29], Hsu [20] and Ye [27] have adopted Perelman $\mathcal {W}$-entropy monotonicity formula to derive various Sobolev embedding that holds for the case $n \ge 3$. In this section we shall make use of Zhang’s version to prove the following:

Theorem 1

Let $(M, g)$ be a compact Riemannian manifold with dimension $n \ge 3$ whose metric evolves by the geometric flow in the interval $t \in [0, T].$ Let there exist positive constants $A$ and $B$ for the initial metric $g_0$ such that the following Sobolev inequality holds for any $u \in W^{1, 2} (M, g_0)$

$$\begin{aligned} \left( \int _M u^{\frac{2n}{n-2}} d\mu (g_0) \right) ^{\frac{n-2}{n}} \le A \int _M | \nabla u |^2 d \mu (g_0) + B \int _M u^2 d \mu (g_0). \end{aligned}$$

(3.4)

Then, there exist positive functions of time $A(t)$ and $B(t)$ depending only on the initial metric $g_0$ and $t$ such that for $u \in W^{1, 2} (M, g(t)), t>0$, it holds that

$$\begin{aligned} \left( \int _M u^{\frac{2n}{n-2}} d\mu (g(t)) \right) ^{\frac{n-2}{n}}\le A(t) \int _M \left( | \nabla u |^2 + \frac{1}{4} \mathcal {H} u^2\right) d \mu (g(t)) + B(t) \int _M u^2 d \mu (g(t)), \end{aligned}$$

(3.5)

Moreover, if $\mathcal {H}(x, 0) > 0$, then $A(t)$ and $B(t)$ are in independent of $t$.

Here we take $u =u(x,t)$ as an $L^2(M)$-solution of the heat type equation and then prove the above theorem using ultracontractive estimates for heat kernel semigroup. Note that we have from the Sobolev embedding for $1 \le q < n$ that $W^{1, q}(M)$ can be continuously embedded in $L^{q^*}(M)$, i.e, there exists a constant $C= C(n, q),$ such that

$$\begin{aligned} \Vert u\Vert _{L^{q^*}(M)} \le C(n,q) \Vert u \Vert _{W^{1, q}(M)} \end{aligned}$$

for all $u \in W^{1, q}(M)$. So by Holder’s inequality we have ($p \ge q$)

$$\begin{aligned} \int _M |u|^p d \mu = \int _M |u|^q | u |^{p-q} \le \left( \int _M |u|^{\frac{qn}{n-q}} d \mu \right) ^{\frac{n-q}{n}} \left( \int _M|u|^{\frac{n}{q}(p-q)} d\mu \right) ^{\frac{q}{n}}. \end{aligned}$$

(3.6)

This is reduced to (from the interpolation inequality)

$$\begin{aligned} \int _M u^2 d \mu \le \left( \int _M u^{\frac{2n}{n-2}} d \mu \right) ^{\frac{n-2}{n+2}} \left( \int _M u d\mu \right) ^{\frac{4}{n+2}} \end{aligned}$$

(3.7)

for the case $q =2$ and $n \ge 3$. Then by Sobolev inequality for manifold evolving by the Ricci flow we have

$$\begin{aligned} \displaystyle \int _M u^2 d \mu&\le \bigg ( A(t) \int _M \bigg ( | \nabla u |^2 + \frac{1}{4} \mathcal {H} u^2 \bigg ) d \mu (g(t)) + B(t) \int _M u^2 d \mu (g(t)) \bigg )^{\frac{n}{n+2}} \\ \displaystyle&\quad \times \left( \int _M u d\mu (g(t))\right) ^{\frac{4}{n+2}}. \end{aligned}$$

Let $h(t) := \Big ( \int _M u d\mu (g(t)) \Big )^{\frac{4}{n+2}},$ the last inequality becomes

$$\begin{aligned} \displaystyle \int _M | \nabla u|^2 d\mu (g(t))&\ge \frac{1}{A(t)} \Bigg ( h^{-1}(t) \int _M u^2 d \mu (g(t)) \Bigg )^{\frac{n+2}{n}} \nonumber \\&\quad \displaystyle - \frac{B(t)}{A(t)} \int _M u^2 \mu (g(t)) -\frac{1}{4} \int _M \mathcal {H} u^2 \mu (g(t)). \end{aligned}$$

(3.8)

Thus, we have proved the following by using the bound on the scalar curvature (2.7) as discussed in the Sect. 1.

Lemma 2

With the hypothesis of Theorem 1 the following inequality holds

$$\begin{aligned} \left. \begin{array}{ll} \displaystyle \int _M | \nabla u|^2 d\mu (g(t)) &{}\ge \frac{1}{A(t)} \Bigg ( h^{-1}(t) \int _M u^2 d \mu (g(t)) \Bigg )^{\frac{n+2}{n}}\\ &{}\quad \displaystyle - \Bigg ( \frac{B(t)}{A(t)} + \frac{\psi (t)}{4}\Bigg ) \int _M u^2 d \mu (g(t)) \end{array} \right\} , \end{aligned}$$

(3.9)

where $\psi (t)$ is as defined in (2.7).

3.2 Log-Sobolev inequalities and ultracontractive estimates

By the results cited above. Let there exist positive constants $A_0, B_0 < \infty$ such that for all $u \in W^{1, 2}(M, g_0),$

$$\begin{aligned} \Vert u \Vert _{\frac{2n}{n-2}} \le A_0 \Vert \nabla u \Vert _2 + B_0 \Vert u \Vert _2, \end{aligned}$$

(3.10)

where $A_0$ and $B_0$ depends only on $n, g_0$, lower bound for the Ricci curvature and injectivity radius. We can then write (3.10) as

$$\begin{aligned} \Bigg ( \int _M u^{\frac{2n}{n-2}} d\mu _{g_0} \Bigg )^{\frac{n-2}{n}}\le A \int _M ( 4 | \nabla u |^2 + \mathcal {H}_g u^2 ) d\mu _{g_0} + B \int _M u^2 d \mu _{g_0}, \end{aligned}$$

(3.11)

where

$$\begin{aligned} A = \frac{1}{4} A_0, \quad and \quad B = \frac{1}{4} A_0 \sup \mathcal {H}_g^-(\cdot , 0) + B_0 \end{aligned}$$

since $\mathcal {H}_g(x, 0) + \sup \mathcal {H}_g^-(\cdot , 0) = \mathcal {H}_g^+(x, 0) - \mathcal {H}_g^-(x, 0).$ We will assume that (3.11) holds uniformly for $g(t), t >0$ and different $A$ and $B$ in order to prove the logarithmic Sobolev inequalities.

The usual way of deriving logarithmic Sobolev inequality follows from a careful application of Hölder’s and Jensen’s inequalities, since $\log v$ is a concave function in which case

$$\begin{aligned} \int u^2 \ln u^{q-2} d \mu \le \ln \int u^q d \mu \end{aligned}$$

with the assumption that $\int u^2 d \mu = 1$, then

$$\begin{aligned} \int u^2 \ln u \ d \mu \le \frac{q}{q-2} \ln \Bigg ( \int u^q d \mu \Bigg )^{\frac{1}{q}}. \end{aligned}$$

Taking $q = \frac{2n}{n-2},$ we have

$$\begin{aligned} \int u^2 \ln u \ d \mu \le \frac{n}{2} \ln \Bigg ( \int u^{\frac{2n}{n-2}} d \mu \Bigg )^{\frac{n-2}{2n}}, \end{aligned}$$

multiplying both sides by $2$ we obtain the following

Lemma 3

For any $u \in W^{1,2}(M, g_0)$ with $\Vert u \Vert _2 =1$

$$\begin{aligned} \int _M u^2 \ln u^2 d\mu _{g_0} \le \frac{n}{2} \ln \Bigg ( A \int _M (4 | \nabla u |^2 + \mathcal {H}_g u^2 ) d \mu _{g_0} + B \Bigg ). \end{aligned}$$

(3.12)

See [20, 27, 29] for similar proofs. Inequalities in (3.12) are usually estimated further by the application of an elementary inequality of the form $\ln y \le \theta y - \ln \theta - 1, \ \ \theta , y, \ge 0$. Precisely, taking $y = A \int _M ( 4 | \nabla u |^2 + \mathcal {H}_g u^2 ) d \mu _{g_0} + B$ in (3.12) gives us

$$\begin{aligned} \int _M u^2 \ln u^2 d\mu _{g_0} \le \frac{ n \theta }{2} \Bigg \{ A\int _M ( 4 | \nabla u |^2 + \mathcal {H}_g u^2 ) d \mu _{g_0} + B \Bigg \} - \frac{n}{2} ( 1 + \ln \theta ) \end{aligned}$$

(3.13)

It is well known that log Sobolev inequalities and ultracontractivity are equivalent and both may imply sharp upper bound for the heat kernel, see Gross [15, 16] and Davies [12]. We now use the ultracontractive estimates on heat semigroup to prove Theorem 1.

3.3 Proof of Theorem 1

This section discusses how one obtains a uniform Sobolev-type inequality form global bounds on the heat kernel along the geometric flow. The proof of this type is standard as contained in [12], Chapter 2], the same procedures have been adapted in [29] for Kähler–Ricci flow, See also [27] and [20]. For completeness we give the summary of the approach.

For any $t \in [0, T)$ we define the operator

$$\begin{aligned} \mathcal {A} := - \Delta _g + \frac{\mathcal {H}_g + \sup _M \mathcal {H}^-_g}{4}. \end{aligned}$$

(3.14)

Since $\mathcal {H}_g(\cdot , \tau ) \ge - \sup _M \mathcal {H}_g(\cdot , \tau ),$ we know that $\Phi = \frac{1}{4}(\mathcal {H}_g + \sup _M \mathcal {H}^-_g) \ge 0$, $\Phi \in L^\infty (M)$, then $\mathcal {A} \ge 0$ and essentially a self-adjoint operator on $L^2(M)$ with the associated quadratic form

$$\begin{aligned} \mathbf {Q}(u) = \int _M ( | \nabla u|^2 + \Phi u^2) d \mu _g, \quad \forall u \in W^{1, 2}(M). \end{aligned}$$

(3.15)

By the heat kernel convolution property we have

$$\begin{aligned} e^{- t\mathcal {A}} w_0 = \int _M F(x, t; y) w_0(y) d \mu _g(y), \end{aligned}$$

(3.16)

where $e^{- t\mathcal {A}}$ is a self-adjoint positivity preserving semigroup for all $t \ge 0$. It is also a contraction on $L^\infty (M)$ and $L^1(M)$ for all $t \ge 0$, then

$$\begin{aligned} \Vert e^{- t\mathcal {A}} w_0 \Vert _\infty \le C_0 t^{-\frac{n}{2}} \Vert w_0\Vert _1. \end{aligned}$$

(3.17)

The next is to apply a theorem in [12] which we state below as a lemma.

Lemma 4

If $n \ge 2$, then a bound of the form

$$\begin{aligned} \Vert e^{- t\mathcal {A}} w_0 \Vert _\infty \le C_1 t^{-\frac{n}{4}} \Vert w_0\Vert _2. \end{aligned}$$

(3.18)

for all $t > 0$ and all $w_0 \in L^2(M)$ is equivalent to a bound of the form

$$\begin{aligned} \Vert w_0 \Vert ^2_{\frac{2n}{n-2}} \le C_2 \mathbf {Q}(w_0) \quad \forall \ \ w_0 \in W^{1, 2}(M). \end{aligned}$$

(3.19)

By Lemma 4 we can prove that

$$\begin{aligned} \Bigg ( \int _M u^{\frac{2n}{n-2}} d \mu _g \Bigg )^{\frac{n-2}{2}} \le A_0 \int _M \Bigg ( | \nabla u |^2 + \frac{1}{4} \Bigg (\mathcal {H}_g + \sup _M \mathcal {H}^-_g\Bigg ) u^2 \Bigg ) d \mu _g \end{aligned}$$

(3.20)

using the estimate of the form (3.22) below. The only thing remaining for us to show is that estimate (3.17) and (3.18) are equivalent. We do this via the following lemma and Hölder inequality.

Lemma 5

Suppose $n \ge 2$ and $T < \infty$. Let $C_1 > 0$ be the same as $C_1$ in (3.18), then we have

$$\begin{aligned} \Vert e^{- t\mathcal {A}} w_0 \Vert _2 \le C_1 t^{-\frac{n}{4}} \Vert w_0 \Vert _1\quad \forall \ \ w_0 \in L^1(M). \end{aligned}$$

(3.21)

Now write $e^{- t\mathcal {A}} w_0 = e^{- \frac{1}{2} t\mathcal {A}} e^{- \frac{1}{2} t\mathcal {A}} w_0$ and by assuming (3.18) we have

$$\begin{aligned} \Vert e^{- t\mathcal {A}} w_0 \Vert _\infty \le C_1 t^{-\frac{n}{4}} \Vert e^{- \frac{1}{2} t\mathcal {A}} w_0 \Vert _2 \le C_1^2 t^{-\frac{n}{2}} \Vert w_0 \Vert _1. \end{aligned}$$

Similarly, combining the fact that $e^{- t\mathcal {A}}$ is a contraction on $L^\infty (M)$ with bound (3.17) gives us (3.18). Indeed,

$$\begin{aligned} \displaystyle \Vert e^{- t\mathcal {A}} w_0 \Vert _\infty&= \Bigg | \int _M F(x, t; y) w_0(y) d \mu _g(y) \Bigg | \\ \displaystyle&\le \Bigg ( \int _M F^{q'} (x, t; y) d\mu _g(y) \Bigg )^{\frac{1}{q'}} \Bigg ( \int _M w_0^q d\mu _g(y) \Bigg )^{\frac{1}{q}}\\ \displaystyle&= \Bigg ( \int _M F^{q' -1} F d\mu _g(y) \Bigg )^{\frac{1}{q'}} \Bigg ( \int _M w_0^q d\mu _g(y) \Bigg )^{\frac{1}{q}}\\ \displaystyle&\le \Bigg (C_0 t^{-\frac{n}{2}(q' -1)} \int _M F d\mu _g(y) \Bigg )^{\frac{1}{q'}} \Bigg ( \int _M w_0^q d\mu _g(y) \Bigg )^{\frac{1}{q}}\\&\le C t^{-\frac{n}{2 q}} \Vert w_0 \Vert _q, \end{aligned}$$

$\forall \ w_0 \in L^q(M)$ with $1/q = 1 - 1/q'$ and $\int _M F(x, t: y) d \mu _g \le 1$. Here, we take $q$ to satisfy $1 \le q < n$ for obvious reason (though, by Riez–Thorin interpolation theorem, the above holds for any $1 \le q < \infty$ since $e^{- t\mathcal {A}}$ is a contraction on $L^1(M)$ and $L^\infty (M)$).

The main point here is the following

Theorem 2

With the condition of Theorem 1 we claim that estimate of the form

$$\begin{aligned} F(x, T; y ) \le CT^{- \frac{n}{2}} \end{aligned}$$

(3.22)

where $C$ depends on $n, t, T, A_0, B_0$ and $\sup \mathcal {H}_g(\cdot , 0)$, implies the uniform Sobolev inequality (3.11) which is essential the same as (3.5) which we wanted to proof.

Proof

Based on the previous argument and modification of the calculation in [29] we define the operator $\widetilde{\mathcal {A}} = \mathcal {A}+ 1$, which also has all the properties of $\mathcal {A}$ ($\widetilde{\mathcal {A}} \ge 0$ and generates a symmetric Markov semigroup). Then for any positive constant $c$ depending on $n$, $T$, a lower bound for $\mathcal {H}_{g_0}$ and upper bound for $A_0$, such that for all $t \in [0, T)$ and $v \in Dom(\tilde{\mathcal {A}}) \subseteq W^{1, q}(M)$, there holds for $n \ge 3$

$$\begin{aligned} \Vert \widetilde{\mathcal {A}}^{-\frac{1}{2}} w \Vert _{\frac{ nq }{n-q}}\le c \Vert w \Vert _q \quad \forall \ w \in W^{1, 2}_0 (M). \end{aligned}$$

(3.23)

Since $\widetilde{\mathcal {A}}^{-\frac{1}{2}}$ is of weak type $(p, q), \ p= \frac{ nq }{n-q}$ for any $1 < q < n$, a simple analysis and the Marcinkiewicz interpolation theorem tell us that $\widetilde{\mathcal {A}}^{- \frac{1}{2}}$ is a bounded operator from $L^q$ to $L^p$ and that (3.23) holds true.

Define $u(x, t) = \widetilde{\mathcal {A}}^{- \frac{1}{2}} w(x, t)$ which implies $w(x, t) =\widetilde{\mathcal {A}}^{\frac{1}{2}} u(x, t)$. Taking $q =2$ we have

$$\begin{aligned} \Vert w \Vert ^2_2 = \int _M \widetilde{\mathcal {A}}^{ \frac{1}{2}} u \ \widetilde{\mathcal {A}}^{\frac{1}{2}} u \ d \mu _g = \int _M (\widetilde{\mathcal {A}} u) u \ d \mu _g = \int _M ((\mathcal {A} +1) u) u \ d \mu _g. \end{aligned}$$

Combining with (3.23) and (3.19) we obtain the Sobolev inequality

$$\begin{aligned} \Vert u \Vert ^2_{\frac{2n}{n-2}} \le c \cdot C_2\ \Bigg (\mathbf {Q}(u) +\int _M u^2 \mu _g \Bigg ), \end{aligned}$$

(3.24)

whereby (3.5) follows with $A = c \cdot C_2$ and $B = \frac{1}{4} c \cdot C_2 (\sup _M \mathcal {H}_g + 1).$

Remark 1

Fixing $t_0$ during geometric flow, it is clear that $\widetilde{H} = e^{-1} H$ is the heat kernel generated by $\widetilde{\mathcal {A}}$ and that

$$\begin{aligned} \int _M \widetilde{F} (x, t; y) d \mu _g(y) \le \int _M F(x, t; y) d \mu _g(y) \le 1. \end{aligned}$$

By the upper bound for $F$, we are sure that $\widetilde{F}$ obeys global upper bound

$$\begin{aligned} \widetilde{F} (x, t; y) d \mu _g(y) \le \tilde{C} t^{-\frac{n}{2}}, \quad t > 0, \end{aligned}$$

where $\tilde{C}$ depends on $n, A_0, B_0, t_0$ and $T$. Similarly

$$\begin{aligned} \Vert e^{- t \widetilde{\mathcal {A}}} w \Vert _\infty = \Vert e^{- t } e^{- t \mathcal {A}} w\Vert _\infty \le e^{- t } C t^{-\frac{n}{2}} \Vert w \Vert _1 = \widetilde{C} t^{-\frac{n}{2}} \Vert w \Vert _1. \end{aligned}$$

4 Pointwise upper bound with Sobolev inequality

In this section, we prove an upper estimate on the heat kernel of the manifold evolving by the geometric flow, it turns out that the estimate depends on the best constants in Sobolev-type inequalities (3.5) for the geometric flow and the bound on the metric trace of $h_{ij}$. The main result of this section is the following

Theorem 3

Let $(M, g(x,t)), t \in [0, T]$ be a solution to the geometric flow with $n \ge 3$ and $F(x, t; y,s)$ be the fundamental solution to the heat-type equation. Then for a constant $C_n$ depending on $n$ only, the following estimate holds

$$\begin{aligned} F(x,t;y,s) \le \frac{C_n}{\Bigg ( \int _s^{\frac{t+s}{2}} \frac{e^{\frac{2}{n} P(\tau )}}{\alpha (\tau ) A(\tau )} d\tau \cdot \int ^s_{\frac{t+s}{2}} \frac{e^{- \frac{2}{n} P(\tau )}}{ A(\tau )} d\tau \Bigg )^{\frac{n}{4}}} \end{aligned}$$

(4.1)

for $0 \le s < t \le T$, where $\alpha ( \tau )= \frac{\rho ^{-1}- \frac{2}{n} \tau }{\rho ^{-1}}$, $\mathcal {H}(g_0) \ge \rho$ being the infimum of the metric trace of $h_{ij}$ at the initial time, $P(\tau ) = \int _s^t (B(\tau ) A^{-1}(\tau ) - \frac{1}{2} \phi (\tau )) d \tau$, with $A(t)$ and $B(t)$ being positive constants in the Zhang–Ricci–Sobolev inequality and $\phi (t)$ is the lower bound for the scalar curvature.

4.1 Proof of Theorem 3

Proof

We suppose here and thereafter that $s=0$ without loss of generality. Since $F(x,t; y,s)$ is the fundamental solution, it then follows from its semigroup property and Cauchy–Schwarz inequality that

$$\begin{aligned}&F(x,t;y,0) = \int _M F\left( x, t; z , \frac{t}{2}\right) F \left( z , \frac{t}{2};y, 0\right) d \mu (g(z, t)) \\&\quad \displaystyle \le \Bigg ( \int _M F^2\left( x, t; z ,\frac{t}{2} \right) d \mu \left( g\left( z, \frac{t}{2}\right) \right) \Bigg )^{ \frac{1}{2}} \Bigg (\int _M F^2\left( z , \frac{t}{2}; y, 0\right) d \mu \left( g\left( z, \frac{1}{2}\right) \right) \Bigg )^{\frac{1}{2}}. \end{aligned}$$

Traditionally, deriving an upper bound for each of the terms in the right hand side of the last inequality suffices to settle the proof, the nature of the bound to obtain depends largely upon the ingredient. In the present, we rely on estimates from Sobolev embedding theorems on the manifold evolving by the geometric flow. Now denote, say

$$\begin{aligned} V(t)&= \int _M F^2 (x, t; y,s ) d \mu (g(x, t)) \\ W(t)&= \int _M F^2 (x, t; y,s ) d \mu (g(y, s)). \end{aligned}$$

Thus, the pointwise estimate on the quantities $V(t)$ and $W(t)$ will determine an upper bound for the fundamental solution $F (x, t; y,s )$. Approaches to obtaining bound for each of the quantities $V(t)$ and $W(t)$ differ slightly due to the interpolation of the heat kernel between the heat-type equation in the variables $(x, t)$ and the heat equation in the variables $(y, s)$, i.e.,

$$\begin{aligned} ( - \partial _t - \Delta _x + \mathcal {H}(x,t) ) F(x, t; \cdot , \cdot ) = 0 \\ ( \partial _s - \Delta ) F( \cdot , \cdot ; y, s) = 0. \end{aligned}$$

We first treat the case when $F(x, t; y, s)$ solves the heat-type equation, that is, we want to estimate $V(t)$. The idea is to find an inequality involving $V(t)$. Hence

$$\begin{aligned} V'(t)&= \int _M ( 2 F \partial _t F - \mathcal {H}F^2 ) d \mu (x, t) \\ \displaystyle&= \int _M 2F ( - \Delta F + \mathcal {H} F ) d \mu (x, t) - \int _M \mathcal {H}F^2 d \mu (x, t) \\ \displaystyle&= 2 \int _M | \nabla F|^2 d \mu (x, t) + \int _M \mathcal {H}F^2 d \mu (x, t). \end{aligned}$$

Using Lemma 2, we arrive at

$$\begin{aligned} V'(t)&\ge 2 A^{-1}(t) \Bigg ( h^{-1}(t) \int _M F^2 d\mu (x, t) \Bigg )^{\frac{n+2}{n}} - 2 \Bigg ( B(t) A^{-1}(t) + \frac{1}{4} \psi (t) \Bigg ) \int _M F^2 d \mu (x, t) \\ \displaystyle&\quad + \psi (t) \int _M F^2 d \mu (x, t)\\ \displaystyle&= 2 A^{-1}(t) \Bigg (h^{-1}(t) \int _M F^2 d\mu (x, t)\Bigg )^{\frac{n+2}{n}} {-}\Bigg ( 2 B(t) A^{-1}(t) {-} \frac{1}{2} \psi (t)\Bigg ) \int _M F^2 d \mu (x, t). \end{aligned}$$

The problem is reduced to solving the following ODE

$$\begin{aligned} V'(t) + q(t) V(t) \ge 2 A^{-1}(t) V(t) ^{\frac{n+2}{n}}, \end{aligned}$$

(4.2)

where $q(t) = 2 B(t) A^{-1}(t) - \frac{1}{2} \psi (t) .$ Equation (4.2) is due to the fact that under variables $(x, t)$, the fundamental solution $F$ satisfies

$$\begin{aligned} \int _M F(x,t;y,s) d \mu (x, t) = 1 \end{aligned}$$

and consequently then

$$\begin{aligned} h(t) = \Bigg ( \int _M F d \mu \Bigg )^{\frac{4}{n+2}} = 1. \end{aligned}$$

Notice that the resulting ODE (4.2) is true for any $\tau \in [s, t]$, we then solve it by using integrating factor method. Denote $Q(\tau ) = \int q(\tau ) d \tau$, the integrating factor is $e^{Q(\tau ) }$, therefore we have

$$\begin{aligned} ( e^{Q(\tau ) } V(\tau ) )' \ge 2 A^{-1}( \tau )( e^{Q(\tau ) } V(\tau ) )^{\frac{n+2}{n}} e^{-\frac{2}{n} Q(\tau )} \end{aligned}$$

integrating from $s$ to $t$ since it is true for all $\tau \in [s, t]$, with the facts that

$$\begin{aligned} \displaystyle \int _s^t \frac{( e^{Q(\tau ) } V(\tau ) )' }{( e^{Q(\tau ) } V(\tau ) )^{\frac{n+2}{n}} } d \tau = - \frac{n}{2}( e^{Q(\tau ) } V(\tau ) )^{-\frac{2}{n}} \Bigg |_s^t \end{aligned}$$

and

$$\begin{aligned} \lim _{\tau \searrow s} V(t) =\int _M \lim _{\tau \searrow s} F^2 (x,t; y,s ) d \mu (x, t) = \int _M \delta ^2_y(x) d \mu (x, t) = 0 \end{aligned}$$

we obtain the bound as follows

$$\begin{aligned} V(t)&\le \frac{ \Bigg (\frac{2}{n}\Bigg )^{\frac{n}{2}}e^{-Q(t)}}{\Bigg ( 2 \int _s^t \frac{e^{-\frac{2}{n} Q(\tau )}}{A(\tau )}d \tau \Bigg )^{\frac{n}{2}}} = \frac{\Bigg (\frac{1}{n}\Bigg )^{\frac{n}{2}} e^{-Q(t)}}{\Bigg ( \int _s^t\frac{e^{-\frac{2}{n} Q(\tau )}}{A(\tau )} d \tau \Bigg )^{\frac{n}{2}}}. \end{aligned}$$

Taking $C_n := \Big (\frac{1}{n}\Big )^{\frac{n}{2}}$, we arrive at

$$\begin{aligned} \int _M F^2 (x, t; y,s ) d \mu (x, t) = V(t) \le \frac{ C_n e^{-Q(t)}}{\Bigg ( \int _s^t A^{-1}(\tau ) e^{-\frac{2}{n} Q(\tau )} d\tau \Bigg )^{\frac{n}{2}}}. \end{aligned}$$

(4.3)

The next is to estimate

$$\begin{aligned} W(s) = \int _M F^2 (x, t; y,s ) d \mu (y,s). \end{aligned}$$

Due to the asymmetry of the equation, the computation is slightly different. We recall that $F (x, t; y,s )$ satisfies the heat equation in the variables $(y, s)$, then we similarly have

$$\begin{aligned} W'(s)&= \int _M ( 2 F \partial _s F - \mathcal {H}F^2 ) d \mu (y,s) \\ \displaystyle&= \int _M 2F ( \Delta F ) - \mathcal {H} F^2 d \mu (y,s) \\ \displaystyle&= - 2 \int _M | \nabla F|^2 d \mu (y,s) - \int _M \mathcal {H}F^2 d \mu (y,s). \end{aligned}$$

Using Lemma 2 again we arrive at

$$\begin{aligned} \displaystyle W'(s)&\le - 2 A^{-1}(s) \Bigg ( h^{-1}(s) \int _M F^2 d \mu (y,s) \Bigg )^{\frac{n+2}{n}} + 2 \Bigg ( B(s) A^{-1}(s)+ \frac{1}{4} \psi (s) \Bigg ) \nonumber \\ \displaystyle&\quad \int _M F^2 d \mu (y,s)- \psi (t) \int _M F^2 d\mu (y,s) \nonumber \\ \displaystyle&= - 2 A^{-1}(s) \Bigg ( h^{-1}(s) \int _M F^2 d \mu (y,s) \Bigg )^{\frac{n+2}{n}}\nonumber \\ \displaystyle&\quad + \Bigg ( 2 B(s) A^{-1}(t) - \frac{1}{2} \psi (s)\Bigg ) \int _M F^2 d \mu (y,s). \end{aligned}$$

(4.4)

We can further estimate the quantity $h(s) = ( \int _M F d \mu )^{\frac{4}{n+2}}.$ Notice that contrary to what was obtainable in the variable $(x, t)$, $\int _M F( x, t; y, s) d \mu (y,s) \ne 1$, since the coordinate $(x, t)$ are kept fixed here and we only integrate in $(y,s)$. Therefore

$$\begin{aligned} \lambda '(s)&= \frac{d}{ds} \Bigg ( \int _M F( x, t; y, s) d\mu (y,s) \Bigg ) \\ \displaystyle&= \int _M \partial _s F( x, t; y, s) d\mu (y,s) - \int _M \mathcal {H}(y,s) F d\mu (y,s) \\ \displaystyle&= \int _M \Delta _{y,s} F( x, t; y, s) d\mu (y,s) - \int _M \mathcal {H} (y,s) F( x, t; y, s) d\mu (y,s) \\ \displaystyle&\le - \psi (s) \int _M F( x, t; y, s) d\mu (y,s). \end{aligned}$$

The last inequality is due to the fact that we are on compact manifold, where $\int _M \Delta F d \mu = 0$ and by the lower bound on quantity $\mathcal {H}$ due to the maximum principle. Now for any $\tau \in [s, t]$ and by lower bound (2.7)

$$\begin{aligned} \lambda '(\tau )&\le - \psi (\tau ) \lambda (\tau ) \\ \displaystyle \frac{ \lambda '(\tau )}{ \lambda (\tau )}&\le - \psi (\tau ) = -\frac{1}{\rho ^{-1} - \frac{2}{n} \tau }, \end{aligned}$$

integrating this from $s$ to $t$ we get

$$\begin{aligned} \ln \lambda (t) - \ln \lambda (s) \le \frac{n}{2} \ln ( \rho ^{-1} - \frac{2}{n} \tau )\Bigg |_s^t \end{aligned}$$

$$\begin{aligned} \frac{\lambda (t)}{\lambda (s)} \le \Bigg ( \frac{\rho ^{-1} - \frac{2}{n} t }{\rho ^{-1} - \frac{2}{n}s}\Bigg )^{\frac{n}{2}} \implies \lambda (t) \le \Bigg ( \frac{\rho ^{-1} - \frac{2}{n} t }{\rho ^{-1} - \frac{2}{n}s}\Bigg )^{\frac{n}{2}}\lambda (s) , \end{aligned}$$

we can show that $\lambda (s) \equiv 1$ as follows

$$\begin{aligned} \lambda (s) = \lim _{t \rightarrow s} \int _M F(x, t; y, s) d \mu ( y,s)&= \int _M \lim _{t \rightarrow s} F(x, t; y, s) d \mu ( y,s) \\ \displaystyle&= \int _M \delta _x(y) = 1, \end{aligned}$$

combining these we have

$$\begin{aligned} h(t) = \Bigg ( \frac{\rho ^{-1} - \frac{2}{n} t }{\rho ^{-1} - \frac{2}{n}s}\Bigg )^{\frac{n}{2} \cdot \frac{4}{n +2} } = \Bigg ( \frac{\rho ^{-1} - \frac{2}{n} t }{\rho ^{-1} - \frac{2}{n}s}\Bigg )^{ \frac{2n}{n +2} } =: \alpha ^{ \frac{2n}{n +2}}. \end{aligned}$$

By this (4.4) is now reduced to the following

$$\begin{aligned} \displaystyle W(s)&\le - 2 A^{-1}(s) \alpha ^{-2}(s) \Bigg ( \int _M F^2 d \mu (y,s) \Bigg )^{ \frac{n+2}{n}} \\&\quad \displaystyle + \left( 2 B(s) A^{-1}(s) - \frac{1}{2} \psi (s) \right) \int _M F^2 d \mu (y,s),\nonumber \end{aligned}$$

(4.5)

we are then to solve the following ODE

$$\begin{aligned} W'(s) \le - 2 A^{-1} \alpha ^{-2} W(s)^{ \frac{n+2}{n}} + r(s) W(s), \end{aligned}$$

(4.6)

where $r(s) = 2 B(s) A^{-1}(s) - \frac{1}{2} \psi (s)$. In the similar vein to the previous estimate, we also solve (4.6) using integrating factor method. Denote $R(\tau ) = \int r(\tau ) d\tau$, the integrating factor is $e^{- R(\tau )}$. Therefore we have

$$\begin{aligned} ( e^{- R(\tau )} W(\tau ))' \le - 2 A^{-1} \alpha ^{-2} ( e^{- R(\tau )} W(\tau ))^{ \frac{n+2}{n}}e^{\frac{2}{n} R(\tau )}, \end{aligned}$$

integrating from $s$ to $t$ since it is true for any $\tau \in [s, t]$ we have immediately

$$\begin{aligned} W'(s)&\le \frac{ \Bigg (\frac{2}{n}\Bigg )^{\frac{n}{2}}e^{R(s)}}{\Bigg ( 2 \int _s^t \frac{e^{\frac{2}{n} R(\tau )}}{\alpha ^2(\tau )A(\tau )} d \tau \Bigg )^{\frac{n}{2}}} = \frac{\Bigg (\frac{1}{n}\Bigg )^{\frac{n}{2}} e^{R(s)}}{\Bigg ( \int _s^t\alpha ^{-2}(\tau ) A^{-1}(\tau ) e^{\frac{2}{n} R(\tau )} d \tau \Bigg )^{\frac{n}{2}}}, \end{aligned}$$

hence

$$\begin{aligned} \int _M F^2 (x, t; y,s ) d \mu (y, s) = W(s) \le \frac{ C_n e^{R(s)}}{\Bigg ( \int _s^t \alpha ^{-2}(\tau ) A^{-1}(\tau )e^{\frac{2}{n} R(\tau )} d \tau \Bigg )^{\frac{n}{2}}}. \end{aligned}$$

(4.7)

We can then see from the computation above that

$$\begin{aligned} V\left( \frac{t}{2}\right) = \int _M F^2 \Bigg (x, t; z, \frac{t}{2}\Bigg ) d \mu \Bigg (z, \frac{t}{2}\Bigg ) = \frac{ C_n e^{-Q\left( \frac{t}{2}\right) }}{\Bigg (\int _s^t A^{-1}(\tau ) e^{-\frac{2}{n} Q(\tau )} d \tau \Bigg )^{\frac{n}{2}}} \end{aligned}$$

and

$$\begin{aligned} W\left( \frac{t}{2}\right) = \int _M F^2 \Bigg (z, \frac{t}{2}; y, 0 \Bigg ) d \mu \Bigg (z, \frac{t}{2}\Bigg ) = \frac{ C_n e^{R\left( \frac{t}{2}\right) }}{\Bigg (\int _s^t \Bigg (\frac{\rho ^{-1} - \frac{2}{n} \tau }{\rho ^{-1} }\Bigg )^{-2} A^{-1}(\tau ) e^{\frac{2}{n} R(\tau )} d \tau \Bigg )^{\frac{n}{2}}}. \end{aligned}$$

Here we choose

$$\begin{aligned}&P\left( \frac{t}{2}\right) = \int _0^{\frac{t}{2}} \Bigg [ B(\tau ) A^{-1}(\tau ) - \frac{1}{2} \phi (\tau ) \Bigg ] d \tau = Q\left( \frac{t}{2}\right) = R\left( \frac{t}{2}\right) \\&\quad with \quad \phi (t) := \frac{1}{\rho ^{-1} - \frac{2}{n} t}. \end{aligned}$$

Finally we obtain the bound

$$\begin{aligned} F(x,t;y,s) \le \frac{C_n}{\Bigg ( \int _s^{\frac{t+s}{2}} \Bigg (\frac{\rho ^{-1} - \frac{2}{n} \tau }{\rho ^{-1} } \Bigg )^{-2} A^{-1}(\tau ) e^{\frac{2}{n} P(\tau )} \cdot \int ^s_{\frac{t+s}{2}} A^{-1}(\tau ) e^{- \frac{2}{n} P(\tau )} d\tau \Bigg )^{\frac{n}{4}}}. \end{aligned}$$

(4.8)

The required estimate follows immediately.

4.2 The special case of nonnegative $\mathcal {H}(x,t)$

Note that if $\mathcal {H}(x, 0) \ge 0$, the maximum principle shows that it remains so as long as the geometric flow exists. For this case we obtain a Sobolev type embedding from Lemma 2

$$\begin{aligned} \int _M | \nabla u|^2 d\mu (g(t)) \ge \frac{1}{A} \Bigg ( \int _M u^2 d\mu (g(t)) \Bigg )^{\frac{n+2}{2}} - \frac{B}{A} \int _M u^2 d\mu (g(t)), \end{aligned}$$

(4.9)

where $A$ and $B$ are absolute constant independent of time, in fact $A = K(n, 2)^2$ is the best constant in Euclidean Sobolev embedding and $B$ can be taken to be equivalent to zero when $\mathcal {H}(x, 0)= 0.$

In the case $\mathcal {H}(x, 0) >0$, we have $\lambda '(s) \le 0$ showing that $\lambda (s)$ is decreasing, that is $\lambda (s) \le \lambda (t)$. This implies that $h(s) \le h(t) = 1$, then (4.6) becomes

$$\begin{aligned} W'(s) \le - 2 A^{-1} W(s)^{ \frac{n+2}{n}} + r(s) W(s) \end{aligned}$$

with $\tilde{r} = \frac{2B}{A}$ and we obtain the estimate

$$\begin{aligned} W(s) \le \frac{ C_n e^{\tilde{R}(s)}}{A^{-1} \Bigg ( \int _s^t (\tau ) e^{\frac{2}{n} R(\tau )} d \tau \Bigg )^{\frac{n}{2}}}, \end{aligned}$$

similarly

$$\begin{aligned} V(t) \le \frac{ C_n e^{- \tilde{R}(t)}}{A^{-1} \Bigg ( \int _s^t (\tau ) e^{- \frac{2}{n} R(\tau )} d \tau \Bigg )^{\frac{n}{2}}}. \end{aligned}$$

Putting these together we have a counterpart estimate to (4.8) as follows

$$\begin{aligned} F(x,t;y,s) \le \frac{C_n}{ \Bigg [A^{-2} \Bigg (\int _s^{\frac{t+s}{2}} e^{\frac{2}{n} \tilde{R}(\tau )} d\tau \cdot \int ^s_{\frac{t+s}{2}}e^{- \frac{2}{n} \tilde{R}(\tau )} d\tau \Bigg )\Bigg ]^{\frac{n}{4}}}. \end{aligned}$$

(4.10)

Here, the denominator in the right hand side of the inequality (4.10) is simplified to

$$\begin{aligned}&\Bigg [A^{-2} \Bigg ( \int _s^{\frac{t+s}{2}} e^{\frac{2}{n} \tilde{R}(\tau )} d\tau \cdot \int ^s_{\frac{t+s}{2}}e^{- \frac{2}{n} \tilde{R}(\tau )} d\tau \Bigg ) \Bigg ]^{\frac{n}{4}} \\&\quad = \Bigg [ \frac{n^2}{16 B^2}\Bigg ( e^{\frac{4B}{nA} \cdot \frac{t+s}{2}} - e^{\frac{4B}{nA} \cdot s} \Bigg )\Bigg ( e^{- \frac{4B}{nA} \cdot \frac{t+s}{2}} - e^{-\frac{4B}{nA} \cdot t} \Bigg ) \Bigg ]^{\frac{n}{4}}\\&\quad = \Bigg [ \frac{n^2}{16 B^2}\Bigg ( 1 - e^{-\frac{4B}{nA} \cdot \frac{t-s}{2}} \Bigg )^2 \Bigg ]^{\frac{n}{4}}. \end{aligned}$$

Therefore

$$\begin{aligned} F(x,t;y,s) \le \frac{C_n}{ \Bigg [ \frac{n}{4 B}\Bigg ( 1 -e^{-\frac{4B}{nA} \cdot \frac{t-s}{2}} \Bigg ) \Bigg ]^{\frac{n}{2}}}\le \frac{\tilde{C}_n}{ ( t-s )^{\frac{n}{2}}} \end{aligned}$$

by Taylor series expansion (i.e., $1- e^{-z} \lesssim z$), where $\tilde{C}_n = C_n \cdot (2 A)^{\frac{n}{2}}.$

In the case $\mathcal {H}(x, 0) = 0$, $B(t) \equiv 0$, $\tilde{R}(t) = \frac{B}{A} t \equiv 0$ and

$$\begin{aligned} F(x,t;y,s) \le \frac{C_n}{ \Bigg [A_0^{-2} \Bigg (\int _s^{\frac{t+s}{2}} d\tau \cdot \int ^s_{\frac{t+s}{2}} d\tau \Bigg ) \Bigg ]^{\frac{n}{4}}} = \frac{C_n}{ \Bigg [A_0^{-1} \Bigg (\frac{t-s}{2}\Bigg ) \Bigg ]^{\frac{n}{2}}} = \frac{\tilde{C}_n}{ ( t-s)^{\frac{n}{2}}}, \end{aligned}$$

(4.11)

where $\tilde{C}_n = C_n \cdot (2 A_0)^{\frac{n}{2}} = ( \frac{2}{n} k(n, 2) )^{\frac{n}{2}}.$

5 Conclusion

We have obtained uniform Sobolev-type inequalities which are valid on a compact Riemannian manifold whose metric solves the abstract geometric flow. The input of these inequalities are taken to be $W^{1,2}(M)$-solutions of the heat-type equation and as a consequence we derive upper bound for the minimal positive solution (heat kernel) without any explicit restriction on the curvature of the underlying manifold. The semigroup property of the heat kernel plays a crucial role in this result. However, the estimates hold for any positive solution with some normalization condition. In application, the heat-type equation could be an adjoint heat equation which needs to be solved backward in time as the geometric flow is being solved forward in time. This will allow us gain some control on singularities as we reach maximum time T, since the fundamental solution will tend to $\delta$-function as $t$ tends to $T$. The examples of geometric flows mentioned in Sect. 2 show that our results are more general and valid in various cases. As a by-product of our approach in this paper, we have established the equivalence of Sobolev inequalities, log-Sobolev inequalities, contractive estimates and heat kernel upper bounds.