1 Introduction

In this paper, we extend the estimates of [1], to prove the following result:

Theorem 1.1

For any \(0< A < \infty \) and \(n \in \mathbb {N}\) there is a constant \(C = C (A, n) < \infty \) such that the following holds:

Let \((\mathbf{M}^n, (g_t)_{t \in [0,1]} )\) be a Ricci flow (\(\partial _t g_t = -\,2 {{\,\mathrm{Ric}\,}}_{g_t}\)) on an n-dimensional compact manifold \(\mathbf{M}\) with the property that \(\nu [g_{0}, 1+A^{-1}] \ge - A\). Assume that the scalar curvature satisfies \(|R| \le R_0\) on \(\mathbf{M}\times [0, 1]\) for some constant \(0 \le R_0 \le A\).

Then for any \(0 \le t_1\le t_2 \le 1\) and \(x, y \in \mathbf{M}\) we have the distance bound

$$\begin{aligned} d_{t_1} (x,y) - C \sqrt{t_2 - t_1} \le d_{t_2} (x, y) \le \exp \big ({ C R_0^{1/2} \sqrt{t_2 - t_1} } \big ) d_{t_1} (x, y) + C \sqrt{t_2 - t_1}. \end{aligned}$$

In particular, if \(\min \{ d_{t_1} (x,y), d_{t_2} (x,y) \} \le D\) for some \(D < \infty \), then

$$\begin{aligned} \big | {d_{t_1} (x,y) - d_{t_2} (x,y) } \big | \le C' \sqrt{t_2 - t_1}, \end{aligned}$$

where \(C'\) may depend on A, D and n.

By parabolic rescaling, we obtain distance bounds on larger time-intervals. Note that Theorem 1.1 is a generalization of [1, Theorem 1.1], which only provides a bound on the distance distortion that does not improve for \(t_2\) close to \(t_1\). The constant \(\nu [g_0, 1+A^{-1}]\) is defined as the infimum of Perelman’s \(\mu \)-functional (cf [4]) \(\mu [g_0, \tau ]\) over all \(\tau \in (0,1+A^{-1})\). For more details see [1, Sect. 2]. The condition \(\nu [g_0, 1 + A^{-1}] \ge - A\), can be viewed as a non-collapsing condition. The exponential factor in the upper bound is necessary, as one can see for example in the case in which \((\mathbf{M}, (g_t)_{t \in [0,1]})\) is the Ricci flow on a hyperbolic manifold and the distance between x, y is very large. The proof of Theorem 1.1 will heavily use the results of [1], in particular the heat kernel bound, [1, Theorem 1.4].

As a consequence of Theorem 1.1, we obtain the following:

Corollary 1.2

Let \((\mathbf{M}, (g_t)_{t \in [0,T)} )\), \(T < \infty \) be a Ricci flow on a compact manifold and assume that the scalar curvature satisfies \(R< C < \infty \) on \(\mathbf{M}\times [0, T)\). Then the distance function

$$\begin{aligned} d : \mathbf{M}\times \mathbf{M}\times [0,T) \longrightarrow [0, \infty ), \qquad (x,y,t) \longmapsto d_t (x,y) \end{aligned}$$

can be extended continuously onto the domain \(\mathbf{M}\times \mathbf{M}\times [0,T]\).

Note that the corollary does not state that \(d_T : \mathbf{M}\times \mathbf{M}\rightarrow [0, \infty )\) is a metric on \(\mathbf{M}\). It only follows that \(d_T\) is a pseudometric, which means that we may have \(d_T (x, y) = 0\) for some \(x \ne y\). After taking the metric identification, however, \((\mathbf{M}/{ \sim }, d_T)\) is in fact the Gromov–Hausdorff limit of \((\mathbf{M}, g_t)\) as \(t \nearrow T\). Here \(x \sim y\) if and only if \(d_T (x,y) = 0\). Moreover, since the volume measure converges as well, the space \((\mathbf{M}/{ \sim }, d_T)\) becomes a metric measure space with doubling property and this space is the limit of \((\mathbf{M}, g_t)\) in the measured Gromov–Hausdorff sense.

More generally, we obtain the following consequence of Theorem 1.1.

Corollary 1.3

Let \((\mathbf{M}^i, (g^i_t)_{t \in [0,1]})\) be a sequence of Ricci flows on n-dimensional compact manifolds \(\mathbf{M}^i\) with the property that \(\nu [g^i_0, 1+A^{-1}] \ge - A\) and \(|R| < A\) on \(\mathbf{M}\times [0, 1]\) for some uniform \(A < \infty \). Let \(x_i \in \mathbf{M}^i\) be points. Then, after passing to a subsequence, we can find a pointed metric space \((\overline{\mathbf{M}}, \overline{d}, \overline{x})\), a continuous function

$$\begin{aligned} d^\infty : \overline{\mathbf{M}} \times \overline{\mathbf{M}} \times [0,1] \rightarrow [0, \infty ), \qquad (x,y,t) \mapsto d^\infty _t (x,y) \end{aligned}$$

and a continuous family of measures \((\mu _t)_{t \in [0,1]}\) such that for any \(x,y \in \overline{\mathbf{M}}\), the function \(t \mapsto d^\infty _t (x,y)\) is \(\frac{1}{2}\)-Hölder continuous and such that for any \(t \in [0,1]\), the metric identification \((\overline{\mathbf{M}} / {\sim _t}, d^\infty _t, \mu _t, \overline{x})\) is a metric measure space with doubling property for balls of radius less than \(\sqrt{t}\). Here \(x \sim _t y\) if and only if \(d^\infty _t (x,y) = 0\). Moreover, for any \(t \in [0,1]\) the sequence \((\mathbf{M}^i, g^i_t, dg^i_t, x_i)\) converges to \((\overline{\mathbf{M}} / {\sim _t}, d^\infty _t, \mu _t, \overline{x})\) in the pointed, measured Gromov–Hausdorff sense.

For the proof of Corollary 1.3 see Sect. 5.

Note that if we impose the extra assumption that \(|R| < R_i\) on \(\mathbf{M}\times [0,1]\) for some sequence \(R_i\) with \(\lim _{i \rightarrow \infty } R_i = 0\), then the limiting family of measures \((\mu _t)_{t \in [0,1]}\) is constant in time. Unfortunately, however, our results do not imply that \((d^\infty _t)_{t \in [0,1]}\) is constant in time as well.

Finally, we mention a direct consequence of Theorem 1.1, which can be interpreted as an analogue of the main result of [2] in the parabolic case.

Corollary 1.4

For any \(0< A < \infty \) and \(n \in \mathbb {N}\) there is a constant \(C = C (A, n) < \infty \) such that the following holds:

Let \((\mathbf{M}^n, (g_t)_{t \in [0,1]})\) be a Ricci flow on an n-dimensional compact manifold \(\mathbf{M}\) with the property that \(\nu [g_0, 1+ A^{-1}] \ge - A\). Assume that the scalar curvature satisfies \(|R| \le A\) on \(\mathbf{M}\times [0,1]\).

Then for any \(r> 0\) and \(0 \le t_1 \le t_2 \le 1\) and \(x \in \mathbf{M}\) we have the following bound for Gromov–Hausdorff distance of r-balls

$$\begin{aligned} d_{\mathrm{GH}} (B(x, t_1, r), B(x,t_2, r)) \le C \sqrt{|t_1 - t_2|}. \end{aligned}$$

For the rest of the paper, we will fix the dimension \(n \ge 2\) of the manifold \(\mathbf{M}\). Most of our constants will depend on n. For convenience we will not mention this dependence anymore.

2 Upper volume bound

We first generalize the upper volume bound from [5] or [3].

Lemma 2.1

For any \(A < \infty \) there is a uniform constant \(C_0 = C_0 (A) < \infty \) such that the following holds:

Let \((\mathbf{M}^n, (g_t)_{t \in [-1,1]} )\) be a Ricci flow on a compact, n-dimensional manifold \(\mathbf{M}\) with \(|R| \le 1\) on \(\mathbf{M}\times [-1,1]\). Assume that \(\nu [g_{-1}, 4] \ge -A\). Then for any \((x, t) \in \mathbf{M}\times [0,1]\) and \(r > 0\) we have

$$\begin{aligned} |B(x, t, r)|_{t} < C_0 r^n e^{C_0 r}. \end{aligned}$$

Here \(|S|_t\) denotes the volume of a set \(S \subset \mathbf{M}\) with respect to the metric \(g_t\).

Proof

It follows from [3,4,5] (see also [1, Sect. 2]), that for any \(x \in \mathbf{M}\) and \(0 \le r \le 1\), we have

$$\begin{aligned} c r^n \le |B(x, t_0, r)|_{t_0} \le C r^n, \end{aligned}$$
(2.1)

for some constants c, C, which only depend on A.

Fix some \(x \in \mathbf{M}\) and let \(N < \infty \) be maximal with the property that we can find points \(x_1, \ldots , x_N \in B(x, t, \frac{1}{2})\) such that the balls \(B(x_1, t, \tfrac{1}{8} ), \ldots , B(x_N, t, \tfrac{1}{8} )\) are pairwise disjoint. Note that then

$$\begin{aligned} B\left( x_1, t, \tfrac{1}{8} \right) , \ldots , B\left( x_N, t, \tfrac{1}{8} \right) \subset B(x, t, 1). \end{aligned}$$

So, by (2.1), we have \(N \le C_* := (c (\frac{1}{8})^n)^{-1} C\). Moreover, by the maximality of N, we have

$$\begin{aligned} B\left( x_1, t, \tfrac{1}{4}\right) \cup \ldots \cup B\left( x_N, t, \tfrac{1}{4} \right) \supset B\left( x, t, \tfrac{1}{2} \right) . \end{aligned}$$
(2.2)

We now argue that for all \(r \ge \tfrac{1}{2}\)

$$\begin{aligned} B (x_1, t, r) \cup \ldots \cup B(x_N, t, r) \supset B\left( x, t, r + \tfrac{1}{4} \right) . \end{aligned}$$
(2.3)

Let \(y \in B(x, t, r+ \frac{1}{4})\) and consider a time-t minimizing geodesic \(\gamma : [0,l] \rightarrow \mathbf{M}\) between x and y that is parameterized by arclength. Then \(l < r + \frac{1}{4}\). By (2.2) we may pick \(i \in \{ 1, \ldots , N \}\) such that \(\gamma (\tfrac{1}{2}) \in \overline{B (x_i, t, \tfrac{1}{4} )}\). Then

$$\begin{aligned} {{\,\mathrm{dist}\,}}_{t} (x_i, y) \le \left( l- \tfrac{1}{2}\right) + {{\,\mathrm{dist}\,}}_{t} \left( \gamma \left( \tfrac{1}{2}\right) , x_i\right) \le l - \tfrac{1}{4} < r. \end{aligned}$$

So \(y \in B(x_i, t_0, r)\), which confirms (2.3).

Let us now prove by induction on \(k = 1, 2, \ldots \) that for any \(x \in \mathbf{M}\)

$$\begin{aligned} \left| B\left( x, t, \tfrac{1}{4} k \right) \right| _{t} < C_*^{k}. \end{aligned}$$
(2.4)

For \(k = 1\), the inequality follows from (2.1) (assuming \(c < 1\) and hence \(C_* > C\)). If the inequality is true for k, then we can use (2.3) to conclude

$$\begin{aligned} \left| B\left( x, t, \tfrac{1}{4} (k+1)\right) \right| _{t}\le & {} \left| B\left( x_1, t, \tfrac{1}{4} k \right) \right| _{t} + \ldots + \left| B\left( x_N, t, \tfrac{1}{4} k\right) \right| _{t} \\\le & {} N \cdot C_*^k \le C_* \cdot C_*^k = C_*^{k+1}. \end{aligned}$$

So (2.4) also holds for \(k+1\). This finishes the proof of (2.4).

The assertion of the lemma now follows from (2.1) for \(r < 1\). For \(r \ge 1\) choose \(k \in \mathbb {N}\) such that \(\frac{1}{4} (k-1) \le r < \frac{1}{4} k\). Then, by (2.4), we have

$$\begin{aligned} |B(x,t,r)|_t< \left| B\left( x, t, \tfrac{1}{4} k \right) \right| _t < C_*^k = C_* e^{( \log C_*) (k-1) } \le C_* e^{4(\log C_*) r}. \end{aligned}$$

This finishes the proof. \(\square \)

3 Generalized maximum principle

Consider a Ricci flow \((g_t)_{t \in I}\) on a closed manifold \(\mathbf{M}\). In the following we will consider the heat kernel K(x, t; y, s) on a Ricci flow background. That is, for any \((y,s) \in \mathbf{M}\times I\) the kernel \(K(\cdot ,\cdot ; y,s)\) is defined for \(t > s\) and \(x \in \mathbf{M}\) and satisfies

$$\begin{aligned} (\partial _t - \Delta _x ) K(x,t; y,s) = 0 \qquad \text {and} \qquad \lim _{t \searrow s} K(\cdot , t ; y,s) = \delta _y. \end{aligned}$$

Then, for fixed \((x,t) \in \mathbf{M}\times I\), the function \(K(x,t; \cdot , \cdot )\), which is defined for \(s < t\), is a kernel for the conjugate heat equation

$$\begin{aligned} (-\partial _s - \Delta _y + R(y,s)) K(x,t; y,s) = 0 \qquad \text {and} \qquad \lim _{s \nearrow t} K(x,t; \cdot , s) = \delta _x. \end{aligned}$$

Recall that for any \(s < t\) and \(x \in \mathbf{M}\) we have

$$\begin{aligned} \int \limits _\mathbf{M}K(x,t; y, s) dg_s (y) = 1. \end{aligned}$$
(3.1)

Lemma 3.1

Let \((\mathbf{M}, (g_t)_{t \in [0, 1]})\) be a Ricci flow on a compact manifold \(\mathbf{M}\) with \(|R| \le R_0\) on \(\mathbf{M}\times [0, 1]\) for some constant \(R_0 \ge 0\). Then for any \((x,t) \in \mathbf{M}\times (0,1]\) we have

$$\begin{aligned} \int \limits _0\limits ^t \int \limits _\mathbf{M}K(x,t; y,s) |{{{\,\mathrm{Ric}\,}}}|^2 (y,s) dg_s (y) ds \le R_0. \end{aligned}$$

Proof

This follows from the identities

$$\begin{aligned} R(x,t) = \int \limits _\mathbf{M}K(x,t; y,0) R (y,0) dg_0 (y) + 2 \int \limits _0\limits ^t \int \limits _\mathbf{M}K(x,t; y,s) |{{{\,\mathrm{Ric}\,}}}|^2 (y,s) dg_s (y) ds \end{aligned}$$

and (3.1) as well as \(R(x,t) \le R_0\) and \(R(\cdot , 0) \ge - R_0\) on \(\mathbf{M}\). \(\square \)

We will now use the Gaussian bounds from [1] to bound the forward heat kernel in terms of the backwards conjugate heat kernel based at a certain point and time. Note that in the following Lemma we only obtain estimates on the time-interval [0, 1], but we need to assume that the flow exists on \([-1,1]\). This is due to an extra condition in [1, Theorem 1.4].

Lemma 3.2

For any \(A < \infty \) there are uniform constants \(C_1 = C_1 (A), Y = Y (A) < \infty \) such that the following holds:

Let \((\mathbf{M}^n, (g_t)_{t \in [-1,1]})\) be a Ricci flow on a compact, n-dimensional manifold \(\mathbf{M}\) with the property that \(\nu [g_{-1}, 4 ] \ge - A\). Assume that \(|R| \le 1\) on \(\mathbf{M}\times [-1,1]\). Let \(0 \le t_1< t_2 < t_3 \le 1\) such that

$$\begin{aligned} Y (t_2 - t_1) \le t_3 - t_2 \le 10 Y (t_2 - t_1). \end{aligned}$$

Then for all \(x, y \in \mathbf{M}\)

$$\begin{aligned} K(x,t_2; y, t_1) < C_1 K(y, t_3; x, t_2). \end{aligned}$$

Proof

Recall that, by [1, Theorem 1.4] and the remark afterwards, there are constants \(C^*_1 = C^*_1 (A), C^*_2 = C^*_2 (A) < \infty \) such that for any \(0 \le s < t \le 1\)

$$\begin{aligned} \frac{1}{C^*_1 (t-s)^{n/2}} \exp \Big ({ - \frac{C^*_2 d_s^2 (x,y)}{t-s} }\Big )< K (x,t; y,s) < \frac{C^*_1}{ (t-s)^{n/2}} \exp \Big ({ - \frac{d_t^2 (x,y)}{C^*_2 (t-s)} }\Big ). \end{aligned}$$
(3.2)

Set now

$$\begin{aligned} Y := (C^*_2)^2 \qquad \text {and} \qquad C_1 := (C^*_1)^2 (10 Y)^{n/2} . \end{aligned}$$

Then

$$\begin{aligned} K(x,t_2 ; y,t_1)&< \frac{C^*_1}{ (t_2- t_1)^{n/2}} \exp \Big ({ - \frac{ d_{t_2}^2 (x,y)}{C^*_2 (t_2-t_1)} }\Big ) \\&\le \frac{C^*_1}{(10 Y)^{-n/2} (t_3-t_2)^{n/2}} \exp \Big ({ - \frac{ d_{t_2}^2 (x,y)}{C^*_2 (t_2- t_1)} }\Big ) \\&\le C_1 \frac{1}{C^*_1 (t_3-t_2)^{n/2}} \exp \Big ({ - \frac{ d_{t_2}^2 (x,y)}{C^*_2 Y^{-1} (t_3-t_2)} }\Big ) \\&= C_1 \frac{1}{C^*_1 (t_3-t_2)^{n/2}} \exp \Big ({ - \frac{ C^*_2 d_{t_2}^2 (x,y)}{ (t_3-t_2)} }\Big ) < C_1 K(y, t_3, x, t_2). \end{aligned}$$

This finishes the proof. \(\square \)

Next, we combine Lemmas 3.1 and 3.2 to obtain the following bound.

Lemma 3.3

For any \(A < \infty \) there are uniform constants \(C_2 = C_2 (A) < \infty \), \(\theta _2 = \theta _2 (A) > 0\) such that the following holds:

Let \((\mathbf{M}^n, (g_t)_{t \in [-1,1]})\) be a Ricci flow on a compact, n-dimensional manifold \(\mathbf{M}\) with the property that \(\nu [g_{-1}, 4 ] \ge - A\). Assume that \(|R| \le R_0\) on \(\mathbf{M}\times [-1,1]\) for some constant \(0 \le R_0 \le 1\). Then for any \(0 \le t < 1\) and \(0 < a \le \theta _2 (1-t)\) and \(x \in \mathbf{M}\) we have

$$\begin{aligned} \int \limits _{t+a}\limits ^{t+2a} \int \limits _\mathbf{M}K(y,s ; x,t) |{{{\,\mathrm{Ric}\,}}}| (y,s) dg_s (y) ds < C_2 R_0^{1/2} \sqrt{a}. \end{aligned}$$

Proof

Choose \(\theta _2 := \frac{1}{2} Y^{-1}\) and set

$$\begin{aligned} t_3 := t + 2Y a \le 1. \end{aligned}$$

So for any \(s \in [t+a, t+2a]\) we have

$$\begin{aligned} Y (s - t) \le Y \cdot 2a = t_3 - t \le 10 Y a \le 10 Y (s-t). \end{aligned}$$

So by Lemma 3.2, we have for any \((y,s) \in \mathbf{M}\times [t+a, t+2a]\)

$$\begin{aligned} K(y,s ; x,t) < C_1 K(x,t_3 ; y,s). \end{aligned}$$

We can then conclude, using Cauchy-Schwarz, (3.1) and Lemma 3.1, that

$$\begin{aligned} \int \limits _{t+a}\limits ^{t+2a} \int \limits _\mathbf{M}K(y,s&; x,t) |{{{\,\mathrm{Ric}\,}}}| (y,s) dg_s (y) ds \\&\le C_1 \int \limits _{t+a}\limits ^{t+2a} \int \limits _\mathbf{M}K(x,t_3 ; y,s) |{{{\,\mathrm{Ric}\,}}}| (y,s) dg_s (y) ds \\&\le C_1 \bigg ( \int \limits _{t+a}\limits ^{t+2a} \int \limits _\mathbf{M}K(x,t_3 ; y,s) dg_s (y) ds \bigg )^{1/2} \\&\qquad \qquad \cdot \bigg ( \int \limits _{t+a}\limits ^{t+2a} \int \limits _\mathbf{M}K(x,t_3 ; y,s) |{{{\,\mathrm{Ric}\,}}}|^2 (y,s) dg_s (y) ds \bigg )^{1/2} \\&= C_1 \sqrt{a} \bigg ( \int \limits _{t+a}\limits ^{t+2a} \int \limits _\mathbf{M}K(x,t_3 ; y,s) |{{{\,\mathrm{Ric}\,}}}|^2 (y,s) dg_s (y) ds \bigg )^{1/2} \\&\le C_1 R_0^{1/2} \sqrt{a}. \end{aligned}$$

This proves the desired result. \(\square \)

Lemma 3.4

For any \(A < \infty \) there are constants \(C_3 = C_3 (A) < \infty \), \(\theta _3 = \theta _3(A) > 0\) such that the following holds:

Let \((\mathbf{M}^n, (g_t)_{t \in [0,1]})\) be a Ricci flow on a compact, n-dimensional manifold \(\mathbf{M}\) with the property that \(\nu [ g_{-1}, 4 ] \ge - A\). Assume that \(|R| \le R_0\) on \(\mathbf{M}\times [-1,1]\) for some constant \(0 \le R_0 \le 1\). Then for any \(0 \le s < t \le 1\) with \(t-s \le \theta _3 (1-s)\) and any \(x \in \mathbf{M}\), we have

$$\begin{aligned} \int \limits _s\limits ^t \int \limits _\mathbf{M}K(y,s; x, t) |{{{\,\mathrm{Ric}\,}}}| (y,s) dg_s (y) ds < C_3 R_0^{1/2} \sqrt{t-s}. \end{aligned}$$

Proof

Choose \(\theta _3 (A) = \theta _2(A)\). Then, using Lemma 3.3,

$$\begin{aligned}&\int \limits _s\limits ^t \int \limits _\mathbf{M}K(y,s; x,t) |{{{\,\mathrm{Ric}\,}}}| (y,s) dg_s (y) \\&\quad = \sum _{k=1}^\infty \int \limits _{s + (t-s) 2^{-k}}\limits ^{s + 2 (t-s) 2^{-k}} \int \limits _\mathbf{M}K(y,s; x,t) |{{{\,\mathrm{Ric}\,}}}| (y,s) dg_s (y) ds \\&\quad \le \sum _{k=1}^\infty C_2 R_0^{1/2} \sqrt{ (t-s) 2^{-k}} \\&\quad = C_2 R_0^{1/2} \sqrt{t-s} \sum _{k=1}^\infty 2^{-k/2} \\&\quad \le C C_2 R_0^{1/2} \sqrt{t-s}. \end{aligned}$$

This proves the desired estimate. \(\square \)

Proposition 3.5

For every \(A < \infty \) there are constants \(\theta _4 = \theta _4 (A) > 0\) and \(C_4 = C_4 (A) < \infty \) such that the following holds:

Let \((\mathbf{M}^n, (g_t)_{t \in [-1, 1]})\) be a Ricci flow on a compact, n-dimensional manifold \(\mathbf{M}\) with the property that \(\nu [g_{-1}, 4] \ge - A\). Assume that \(|R| \le R_0\) on \(\mathbf{M}\times [-1, 1]\) for some constant \(0 \le R_0 \le 1\). Let \(H > 1\) and \([t_1, t_2] \subset [0,1)\) be a sub-interval with \(t_2 - t_1 \le \theta _4 \min \{ (1- t_1), H^{-1} \}\) and consider a non-negative function \(f \in C^\infty (\mathbf{M}\times [t_1,t_2])\) that satisfies the following evolution inequality in the barrier sense:

$$\begin{aligned} - \partial _t f \le \Delta f + H |{{{\,\mathrm{Ric}\,}}}| f - R f. \end{aligned}$$

Then

$$\begin{aligned} \max _\mathbf{M}f (\cdot , t_1) \le \big ( 1 + C_4 H R_0^{1/2} \sqrt{t_2 - t_1} \big ) \max _\mathbf{M}f (\cdot , t_2). \end{aligned}$$

Note that with similar techniques, we can analyze the evolution inequality \(- \partial _t f \le \Delta f + H |{{{\,\mathrm{Ric}\,}}}|^p f\) for any \(p \in (0,2)\).

Proof

We first find that that for any \((x,t) \in \mathbf{M}\times [-1,1)\) and \(t < s \le 1\)

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}s} \int \limits _{\mathbf{M}} K(y,s; x,t) dg_s (y)= & {} \int \limits _{\mathbf{M}} \big ( \Delta _y K(y,s; x,t) - K(y,s; x,t) R(y,s) \big ) dg_s (y) \\\le & {} R_0 \int \limits _{\mathbf{M}} K(y,s; x,t) dg_s (y), \end{aligned}$$

which implies

$$\begin{aligned} \int \limits _{\mathbf{M}} K(y,s; x,t) dg_s (y) \le e^{R_0 (s-t)}. \end{aligned}$$

So for any \((x,t) \in \mathbf{M}\times [t_1, t_2]\) we have by Lemma 3.4, assuming \(\theta _4 \le \theta _3\) and \(C_3 > 1\),

$$\begin{aligned} f(x,t)&\le \int \limits _\mathbf{M}K(y, t_2; x, s) f (y, t_2) dg_{t_2} (y) \\&\qquad \qquad + \int \limits _{t}\limits ^{t_2} \int \limits _\mathbf{M}K(y,s; x, t) \cdot H |{{{\,\mathrm{Ric}\,}}}| (y,s) \cdot f(y,s) dg_s (y) ds \\&\le e^{R_0 (t_2 - t)} \max _\mathbf{M}f (\cdot , t_2) + H \big ( \max _{\mathbf{M}\times [t, t_2]} f \big ) \int \limits _t\limits ^{t_2} \int \limits _\mathbf{M}K(y,s; x,t) |{{{\,\mathrm{Ric}\,}}}| (y,s) dg_s (y) ds \\&\le e^{R_0 (t_2 - t)} \max _\mathbf{M}f (\cdot , t_2) + H \big ( \max _{\mathbf{M}\times [t, t_2]} f \big ) \cdot C_3 R_0^{1/2} \sqrt{t_2 - t} . \end{aligned}$$

It follows that

$$\begin{aligned} \max _{\mathbf{M}\times [t, t_2]} f \le e^{R_0 (t_2 - t)} \max _\mathbf{M}f (\cdot , t_2) + \big ( \max _{\mathbf{M}\times [t, t_2]} f \big ) \cdot C_3 H R_0^{1/2} \sqrt{t_2 - t}. \end{aligned}$$

So if \(t_2 - t < (2C_3 H)^{-2}\), then

$$\begin{aligned} \max _{\mathbf{M}\times [t, t_2]} f \le \frac{e^{R_0 (t_2 - t)} \max _\mathbf{M}f(\cdot , t_2) }{1 - C_3 H R_0^{1/2} \sqrt{t_2 - t}} \le \big (1+ 10 C_3 H R_0^{1/2} \sqrt{t_2 - t} \big ) \max _\mathbf{M}f(\cdot , t_2). \end{aligned}$$

This finishes the proof. \(\square \)

4 Proof of theorem 1.1

We will first establish a lower bound on the distortion of the distance:

Lemma 4.1

For every \(A < \infty \) there is a constant \(C_5 = C_5 (A) < \infty \) such that the following holds:

Let \((\mathbf{M}^n, (g_t)_{t \in [-1, 1]})\) be a Ricci flow on a compact, n-dimensional manifold \(\mathbf{M}\) with the property that \(\nu [g_{-1}, 4] \ge - A\). Assume that \(|R| \le 1\) on \(\mathbf{M}\times [-1, 1]\). Let \([t_1, t_2] \subset [0,1]\) be a sub-interval and consider two points \(x_1, x_2 \in \mathbf{M}\). Then

$$\begin{aligned} d_{t_2} (x_1,x_2) \ge d_{t_1} (x_1,x_2) - C_5 \sqrt{t_2 - t_1}. \end{aligned}$$

Proof

Set \(d := d_{t_1} (x_1,x_2)\) and let \(u \in C^0 (\mathbf{M}\times [t_1, t_2]) \cap C^\infty (\mathbf{M}\times (t_1, t_2])\) be a solution to the heat equation

$$\begin{aligned} \partial _t u = \Delta u, \qquad u (\cdot , t_1 ) = d_{t_1} (x_1, \cdot ). \end{aligned}$$

Then for any \((x,t) \in \mathbf{M}\times [t_1, t_2]\)

$$\begin{aligned} u(x,t) = \int \limits _\mathbf{M}K(x,t; y,t_1) u(t_1) dg_{t_1}(y) = \int \limits _\mathbf{M}K(x,t; y,t_1) d_{t_1} (x_1, y) dg_{t_1} (y). \end{aligned}$$

Using [1, Theorem 1.4] (compare also with (3.2)), we find that by Lemma 2.1

$$\begin{aligned} u(x_1, t_2)&\le \int \limits _\mathbf{M}\frac{C_1^*}{(t_2 - t_1)^{n/2}} \exp \bigg ({ -\frac{d^2_{t_1}(x_1, y)}{C^*_2 (t_2 - t_1)}} \bigg ) d_{t_1} (x_1, y) dg_{t_1} (y) \\&= \sum _{k = -\infty }^\infty \int \limits _{B(x_1, t_1, 2^{k}) \setminus B(x_1, t_1, 2^{k-1})} \frac{C_1^*}{(t_2 - t_1)^{n/2}} \exp \bigg ({ - \frac{d_{t_1}^2 (x_1, y)}{C^*_2 (t_2 - t_1)}} \bigg ) \\&\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \cdot d_{t_1} (x_1, y) dg_{t_1} (y) \\&\le \sum _{k = - \infty }^\infty |B(x_1, t_1, 2^{k})|_{t_1} \frac{C_1^*}{(t_2 - t_1)^{n/2}} \exp \bigg ({ - \frac{2^{2k-2}}{C^*_2 (t_2 - t_1)}} \bigg ) \cdot 2^k \\&\le \sum _{k = -\infty }^\infty C_0 (2^k)^n e^{C_0 2^k} \frac{C_1^*}{(t_2 - t_1)^{n/2}} \exp \bigg ({ - \frac{2^{2k}}{4C^*_2 (t_2 - t_1)}} \bigg ) \cdot 2^k \\&\le \int \limits _{\mathbb {R}^n} \frac{C C_0 C^*_1}{(t_2 - t_1)^{n/2}} \exp \bigg ({2C_0 |x| - \frac{|x|^2}{4 C^*_2 (t_2 - t_1)} } \bigg ) |x| dx \\&= \sqrt{t_2 - t_1} \int \limits _{\mathbb {R}^n} C C_0 C^*_1 \exp \bigg ({ 2C_0 |x| \sqrt{t_2 - t} - \frac{|x|^2}{4 C^*_2} } \bigg ) |x| dx \le C \sqrt{t_2 - t_1} \end{aligned}$$

On the other hand, using (3.1),

$$\begin{aligned} |d - u(x_2, t_2)|= & {} \bigg | \int \limits _\mathbf{M}K(x_2,t; y,t_1) (d - d_{t_1} (x_1, y) ) dg_{t_1} (y) \bigg | \\\le & {} \int \limits _\mathbf{M}K(x_2,t; y,t_1) | d_{t_1} (x_1, x_2) - d_{t_1} (x_1, y) | dg_{t_1} (y) \\\le & {} \int \limits _\mathbf{M}K(x_2,t; y,t_1)d_{t_1} (x_2, y) dg_{t_1} (y). \end{aligned}$$

So similarly,

$$\begin{aligned} | d - u(x_2, t_2) | \le C \sqrt{t_2 - t_1}. \end{aligned}$$

It follows that

$$\begin{aligned} | u(x_1, t_2) - u(x_2, t_2 ) | \ge d - 2 C \sqrt{t_2 - t_1}. \end{aligned}$$
(4.1)

Next, consider the quantity \(|\nabla u |\) on \(\mathbf{M}\times [t_1, t_2]\). It is not hard to check that, in the barrier sense,

$$\begin{aligned} \partial _t |\nabla u| \le \Delta |\nabla u|. \end{aligned}$$
(4.2)

Since \(|\nabla u | (\cdot , t_1) \le 1\), we have by the maximum principle that \(|\nabla u| \le 1\) on \(\mathbf{M}\times [t_1, t_2]\). So

$$\begin{aligned} | u (x_1, t_2) - u(x_2, t_2 ) | \le d_{t_2} (x_1, x_2). \end{aligned}$$

Together with (4.1) this gives us

$$\begin{aligned} d_{t_2} (x_1, x_2) \ge d - 2 C \sqrt{t_2 - t_1} = d_{t_1} (x_1, x_2) - 2 C \sqrt{t_2 - t_1}. \end{aligned}$$

This finishes the proof. \(\square \)

For the upper bound on the distance distortion, we will argue similarly, by reversing time. The derivation of the bound on \(|\nabla u|\) will now be more complicated, since the equation (4.2) will have an extra \(4 |{{{\,\mathrm{Ric}\,}}}| |\nabla u|\) term. We will overcome this difficulty by applying the generalized maximum principle from Proposition 3.5.

Lemma 4.2

For every \(A < \infty \) there are constants \(\theta _6 = \theta _6 (A) > 0\) and \(C_6 = C_6 (A) < \infty \) such that the following holds:

Let \((\mathbf{M}^n, (g_t)_{t \in [-1, 1]})\) be a Ricci flow on a compact, n-dimensional manifold \(\mathbf{M}\) with the property that \(\nu [g_{-1}, 4] \ge - A\). Assume that \(|R| \le R_0\) on \(\mathbf{M}\times [-1, 1]\) for some constant \(0 \le R_0 \le 1\). Let \([t_1, t_2] \subset [0,1)\) be a sub-interval with \(t_2 - t_1 \le \theta _6 (1- t_1)\) and consider two points \(x_1, x_2 \in \mathbf{M}\). Then

$$\begin{aligned} d_{t_2} (x_1, x_2) \le \exp \big ({ C_6 R_0^{1/2} \sqrt{t_2 - t_1} } \big ) d_{t_1} (x_1, x_2) + C_6 \sqrt{t_2 - t_1}. \end{aligned}$$

Proof

Set \(d := d_{t_2} (x_1, x_2)\). For \(i = 1, 2\) let \(u_i \in C^0 (\mathbf{M}\times [t_1, t_2]) \cap C^\infty (\mathbf{M}\times [t_1, t_2))\) be a solution to the backwards (not the conjugate!) heat equation

$$\begin{aligned} - \partial _t u_i = \Delta u_i, \qquad u_i (\cdot , t_2) = d_{t_2} (x_i, \cdot ) \end{aligned}$$
(4.3)

and let \(v_i \in C^0 (\mathbf{M}\times [t_1, t_2]) \cap C^\infty (\mathbf{M}\times [t_1, t_2))\) be a solution to the conjugate heat equation

$$\begin{aligned} - \partial _t v_i = \Delta v_i - R v_i, \qquad v_i(\cdot , t_2) = d_{t_2} (x_i, \cdot ). \end{aligned}$$

Note that by the maximum principle, we have on \(\mathbf{M}\times [t_1, t_2]\)

$$\begin{aligned} u_1 + u_2 \ge \min _{\mathbf{M}} \big ( u_1 (\cdot , t_2) + u_2 (\cdot , t_2) \big ) \ge \min _{\mathbf{M}} \big ( d_{t_2} (x_1, \cdot ) + d_{t_2} (x_2, \cdot ) \big ) \ge d. \end{aligned}$$
(4.4)

We also claim that we have for all \(t \in [t_1, t_2]\)

$$\begin{aligned} u_i (\cdot , t) \le e^{R_0 (t_2 - t)} v_i ( \cdot , t). \end{aligned}$$
(4.5)

This inequality follows by the maximum principle and by the fact that whenever \(v_i \ge 0\), we have

$$\begin{aligned} (- \partial _t - \Delta ) \big ( e^{R_0 (t_2 - t)} v_i (\cdot , t) \big ) = e^{R_0 (t_2 - t)} R_0 v_i (\cdot , t) - e^{R_0 (t_2 - t)} R(\cdot , t) v_i (\cdot , t) \ge 0. \end{aligned}$$

We now make use of the fact that for any \(x \in \mathbf{M}\),

$$\begin{aligned} v_i (x, t_1) = \int \limits _\mathbf{M}K(y,t_2; x,t_1) v_i (y, t_2) dg_{t_2} (y) = \int \limits _\mathbf{M}K(y,t_2; x,t_1) d_{t_2} (x_i, y ) dg_{t_2} (y) \end{aligned}$$

and

$$\begin{aligned} K(y,t_2; x, t_1) < \frac{C^*_1}{(t_2 - t_1)^{n/2}} \exp \bigg ({ - \frac{d^2_{t_2} (x, y)}{C^*_2 (t_2 - t_1)} }\bigg ), \end{aligned}$$

for some constants \(C^*_1, C^*_2\), which depend only on A. Note that the latter inequality is similar to (3.2) except that the distance between x, y is taken at time \(t_2\). This inequality follows from [1, Theorem 1.4] and the subsequent comment in that paper. We can hence estimate, similarly as in the proof of Lemma 4.1,

$$\begin{aligned} v_i (x_i, t_1) \le \int \limits _\mathbf{M}\frac{C^*_1}{(t_2 - t_1)^{n/2}} \exp \bigg ({ - \frac{ d^2_{t_2} (x_i, y)}{C^*_2 (t_2 - t_1)} }\bigg ) d_{t_2}( x_i, y) dg_{t_2} (y) \le C \sqrt{t_2 - t_1}. \end{aligned}$$

So, using (4.5), we have

$$\begin{aligned} u_i (x_i, t_1) \le C e^{R_0 (t_2 -t_1)} \sqrt{t_2 - t_1} \le 10 C \sqrt{t_2 - t_1} . \end{aligned}$$

So by (4.4) we have

$$\begin{aligned} u_1 (x_2, t_1) \ge d - u_2 (x_2, t_1) \ge d - 10 C \sqrt{t_2 - t_1}. \end{aligned}$$

This implies

$$\begin{aligned} | u_1 (x_1, t_1) - u_1 (x_2, t_2)| \ge d - 20 C \sqrt{t_2 - t_1}. \end{aligned}$$
(4.6)

Taking derivatives of (4.3), we obtain the evolution inequality

$$\begin{aligned} - \partial _t |\nabla u_1 | \le \Delta |\nabla u_1 | + 4 |{{{\,\mathrm{Ric}\,}}}| \cdot |\nabla u_1 | \le \Delta |\nabla u_1 | + (4 + \sqrt{n}) |{{{\,\mathrm{Ric}\,}}}| \cdot |\nabla u_1 | - R |\nabla u_1 |, \end{aligned}$$

which holds in the barrier sense. Note that by definition \(|\nabla u_1 (\cdot , t_2)| \le 1\). So, by Proposition 3.5, we have for sufficiently small \(\theta _6\)

$$\begin{aligned} | \nabla u_1 (\cdot , t_1 ) | \le 1 + C R_0^{1/2} \sqrt{t_2 - t_1} . \end{aligned}$$

So, using (4.6), we obtain

$$\begin{aligned}&d_{t_2} (x_1, x_2) - 10 C \sqrt{t_2 - t_1} \le | u(x_1, t_1) - u(x_2, t_2) | \\&\quad \le \big ( 1 + CR_0^{1/2} \sqrt{t_2 - t_1} \big ) d_{t_1} (x_1, x_2) \le \exp \big ( CR_0^{1/2} \sqrt{t_2 - t_1} \big ) d_{t_1} (x_1, x_2). \end{aligned}$$

This finishes the proof. \(\square \)

Next, we remove the assumption \(t_2 - t_1 \le \theta _6 (1- t_1)\) from Lemma 4.2.

Lemma 4.3

For every \(A < \infty \) there is a constant \(C_7 = C_7 (A) < \infty \) such that the following holds:

Let \((\mathbf{M}^n, (g_t)_{t \in [-1, 1]})\) be a Ricci flow on a compact, n-dimensional manifold \(\mathbf{M}\) with the property that \(\nu [g_{-1}, 4] \ge - A\). Assume that \(|R| \le R_0\) on \(\mathbf{M}\times [-1, 1]\) for some constant \(0 \le R_0 \le 1\). Let \(0 \le t_1 \le t_2 \le 1\) and consider two points \(x, y \in \mathbf{M}\). Then

$$\begin{aligned} d_{t_2} (x, y) \le \exp \big ({ C_7 R_0^{1/2} \sqrt{t_2 - t_1} } \big ) d_{t_1} (x, y) + C_7 \sqrt{t_2 - t_1}. \end{aligned}$$

Proof

In the case in which \(t_2 - t_1 \le \theta _6 (1-t_1)\), the bound follows immediately from Lemma 4.2. Let us now assume that \(t_2 - t_1 > \theta _6 (1- t_1 )\). By continuity we may also assume without loss of generality that \(t_2 < 1\).

Choose times

$$\begin{aligned} t'_k := 1 - (1- \theta _6)^k (1- t_1) \end{aligned}$$

and observe that \(t'_0 = t_1\) and

$$\begin{aligned} t'_{k+1} - t'_k = \theta _6 ( 1- \theta _6)^k (1-t_1) = \theta _6 (1- t'_k). \end{aligned}$$

So by Lemma 4.2

$$\begin{aligned} d_{t'_k} (x,y)\le & {} \exp \Big ({ C_6 R_0^{1/2} \sum _{l=1}^k \sqrt{t'_l - t'_{l-1}} }\Big ) d_{t_1} (x,y) \\&+ C_6 \sum _{l = 1}^k \exp \Big ({ C_6 R_0^{1/2} \sum _{j=l+1}^k \sqrt{t'_j - t'_{j-1}} }\Big ) \sqrt{t'_l - t'_{l-1}}. \end{aligned}$$

Since

$$\begin{aligned} \sum _{l=1}^k \sqrt{ t'_l - t'_{l-1} } = \sum _{l=1}^k \sqrt{\theta _6} (1-\theta _6)^{l/2} \sqrt{1-t_1} \le C' \sqrt{1-t_1} \end{aligned}$$

and

$$\begin{aligned}&\sum _{l = 1}^k \exp \Big ({ C_6 R_0^{1/2} \sum _{j=l+1}^k \sqrt{t'_j - t'_{j-1}} }\Big ) \sqrt{t'_l - t'_{l-1}} \\&\quad \le \sum _{l = 1}^k \exp \Big ({ C_6 C' R_0^{1/2} \sqrt{1-t_1} }\Big ) \sqrt{t'_l - t'_{l-1}} \le C'' \sqrt{1-t_1}, \end{aligned}$$

we find that for a generic constant \(C < \infty \)

$$\begin{aligned} d_{t'_k} (x,y) \le \exp \big ({ C R_0^{1/2} \sqrt{1 - t_1} } \big ) d_{t_1} (x,y) + C \sqrt{1 - t_1}. \end{aligned}$$

Choose now k such that \(t'_k \le t_2 < t'_{k+1}\). Then \(t_2 - t'_k \le t'_{k+1} - t'_k \le \theta _6 (1- t'_1)\), so again by Lemma 4.2, we have

$$\begin{aligned} d_{t_2} (x, y)&\le \exp \big ({ C_6 R_0^{1/2} \sqrt{t_2 - t'_k} }\big ) d_{t'_k} (x,y) + C_6 \sqrt{t_2 - t'_k} \\&\le \exp \big ({ (C+C_6)R_0^{1/2} \sqrt{1- t_1} }\big ) d_{t_1} (x,y) + C \exp ( 1 + C_6 ) \sqrt{1- t_1} + C_6 \sqrt{1-t_1}. \end{aligned}$$

The claim now follows using \(\sqrt{1-t_1} < \theta _6^{-1/2} \sqrt{t_2 - t_1}\). \(\square \)

We can finally prove Theorem 1.1.

Proof of Theorem 1.1

Consider the Ricci flow \((\mathbf{M}^n, (g_t)_{t \in [0,1]})\) with \(\nu [ g_0, 1+ A^{-1}] \ge - A\) and \(|R| \le R_0\) for \(0 \le R_0 \le A\). After replacing A by \(4A +2\), we may assume without loss of generality that \(A > 2\) and that we even have \(\nu [g_0, 1+ 4A^{-1}] \ge - A\).

We will first prove the distance bounds for the case in which \(t_1 > 0\) and \(t_2 \le (1+A^{-1} ) t_1\). By monotonicity of \(\nu \) (compare with [1, Sect. 2]), we find that for any \(t \in [0,1]\) we have

$$\begin{aligned} \nu [g_t , 4A^{-1}] \ge \nu [g_0, 1+4A^{-1}] \ge - A. \end{aligned}$$

Restrict the flow to the time-interval \([(1-A^{-1})t_1, (1+A^{-1})t_1]\) and parabolically rescale by \(A^{1/2} t_1^{-1/2}\) to obtain a flow \((\widetilde{g}_t)_{t \in [A-1,A+1]}\). Then \(\nu [\widetilde{g}_{A-1} , 4] \ \ge - A\) and \(| \widetilde{R} | \le \widetilde{R}_0 := A^{-1} t_1 R_0 \le 1\). Then \(t_1, t_2\) correspond to times \(\widetilde{t}_1 := A , \widetilde{t}_2 := A t_1^{-1} t_2\) and we have

$$\begin{aligned} \widetilde{R}_0^{1/2} \sqrt{\widetilde{t}_2 - \widetilde{t}_1} = R_0^{1/2} \sqrt{t_2 - t_1}. \end{aligned}$$

So the distance bounds follow from Lemmas 4.1 and 4.3.

Consider now the case in which \(t_2 > (1+ A^{-1} ) t_1\). So \(t_1 < \lambda t_2\), where \(\lambda := (1 + A^{-1})^{-1} < 1\). By continuity we may assume without loss of generality that \(t_1 > 0\). Then we can find \(1 \le k_2 < k_1\) such that \(t_1 \in [\lambda ^{k_1}, \lambda ^{k_1 -1}]\) and \(t_2 \in [\lambda ^{k_2}, \lambda ^{k_2-1}]\). Using our previous conclusions, we find

$$\begin{aligned} d_{t_2} (x, y) \ge d_{\lambda ^{k_2}} (x,y) - C \sqrt{\lambda ^{k_2}} \ge \ldots \ge d_{t_1} (x,y) - C \sum _{l=k_1}^{k_2} \sqrt{\lambda ^{l}} \ge d_{t_1} (x,y) - C' C \lambda ^{k_2/2}. \end{aligned}$$

Since \(t_1 < \lambda t_2\), we have \(\sqrt{t_2 - t_1}> \sqrt{(1- \lambda ) t_2} > \sqrt{1-\lambda } \sqrt{\lambda ^{k_2 }}\). So

$$\begin{aligned} d_{t_2} (x,y) \ge d_{t_1} (x,y) - C' C (1-\lambda )^{-1/2} \sqrt{t_2 - t_1}. \end{aligned}$$

This establishes the lower bound.

For the upper bound, set \(t'_0 := t_1\), \(t'_1 := \lambda ^{k_1 -1}\), ..., \(t'_{k_1 - k_2} := \lambda ^{k_2}\), \(t'_{k_1 - k_2 + 1} := t_2\). Then we have by our previous conclusions

$$\begin{aligned} d_{t_2} (x,y)\le & {} \exp \Big ({ C R_0^{1/2} \sum _{l=1}^{k_1 - k_2 + 1} \sqrt{t'_{l} - t'_{l-1}} }\Big ) d_{t_1} (x,y) \\&+ C \sum _{l=1}^{k_2 - k_1 + 1} \exp \Big ({ C R_0^{1/2} \sum _{j=l+1}^{k_1 - k_2 + 1} \sqrt{t'_j - t'_{j-1}} }\Big ) \sqrt{t'_l - t'_{l-1}} \end{aligned}$$

Similarly as in the proof of Lemma 4.3, we conclude

$$\begin{aligned} d_{t_2} (x,y) \le \exp \Big ({ C R_0^{1/2} \sqrt{\lambda ^{k_2}} }\Big ) d_{t_1} (x,y) + C \sqrt{\lambda ^{k_2}}. \end{aligned}$$

Again, using \(\sqrt{t_2 - t_1} > \sqrt{1-\lambda } \sqrt{\lambda ^{k_2}}\), we get the desired bound. \(\square \)

5 Proof of corollary 1.3

Proof of Corollary 1.3

For each i consider the metric \(\overline{d}^i\) on \(\mathbf{M}^i\) with

$$\begin{aligned} \overline{d}^i (x,y) := \int _0^1 d^i_t (x,y) dt. \end{aligned}$$

Note that by the Hölder bound in Theorem 1.1 there is a uniform constant \(c' > 0\) such that for all \(t, t' \in [0,1]\) we have \(d^i_{t'} (x,y) > \frac{1}{2} d^i_t (x,y)\) whenever \(|t-t'| \le c' (d^i_t (x,y))^2\). So there is a uniform constant \(c > 0\) such that for all \(t \in [0,1]\)

$$\begin{aligned} \overline{d}^i (x,y) \ge c \big ( \min \{ d^i_t (x,y), 1 \} \big )^3. \end{aligned}$$
(5.1)

So by the triangle inequality and Theorem 1.1, for any \(A < \infty \) there is a constant \(C < \infty \) such that for any \(x,y, x', y' \in \mathbf{M}\) and \(t, t' \in [0,1]\) with \(\overline{d}^i (x,y) + \overline{d}^i (x, x') + \overline{d}^i (y, y') < A\) we have

$$\begin{aligned} \big |{ d^i_t (x,y) - d^i_{t'} (x', y') }\big | \le C \big ( \overline{d}^i(x,x') \big )^{1/3} + C \big ( \overline{d}^i (y, y') \big )^{1/3} + C |t- t'|^{1/2}. \end{aligned}$$
(5.2)

We first argue that the sequence \((\mathbf{M}^i, \overline{d}^i)\) is uniformly totally bounded in the following sense: For any \(0< a < b\) there is a number \(N = N(a,b) < \infty \) such that for any i and any \(x \in \mathbf{M}^i\), the ball \(\overline{B}^i (x, b) := \{ x \in \mathbf{M}^i \;\; : \;\; \overline{d}^i (x,z) < b \}\) contains at most N pairwise disjoint balls \(\overline{B}^i (y_j, a)\), \(j = 1, \ldots , m\). Fix \(0< a < b\) and assume without loss of generality that \(a < 1\). By (5.1) there is a constant \(b' = b'(b) < \infty \) such that \(\overline{B}^i (x,b) \subset B^i(x, t, b')\) for all \(t \in [0,1]\).

Assume that \(y_1, \ldots , y_m \in \overline{B}^i(x,b)\) such that the balls \(\overline{B}^i (y_j, a)\) are pairwise disjoint. This implies \(\overline{d}^i (y_{j_1}, y_{j_2}) \ge 2a\) for all \(j_1 \ne j_2\). By the Hölder bound in Theorem 1.1, we may find a large integer \(L = L(a) < \infty \) such that whenever \(\overline{d}^i (y, y') \ge 2a\) for some points \(y, y' \in \mathbf{M}^i\), then \(d^i_{\frac{l}{L}} (y, y') > a\) for some \(l \in \{ 1, \ldots , L \}\). So for any \(j_1 \ne j_2\), there is an \(l_{j_1, j_2} \in \{ 1, \ldots , L \}\) such that

$$\begin{aligned} d^i_{\frac{l_{j_1, j_2}}{L}} (y_{j_1}, y_{j_2}) > a. \end{aligned}$$

This implies the following statement: If we form the L-fold Cartesian product \(\mathbf{M}^{i, L} := (\mathbf{M}^i)^L = \mathbf{M}\times \ldots \times \mathbf{M}\) equipped with the metric \(g^i_{\frac{1}{L}} \oplus \ldots \oplus g^i_{\frac{L-1}{L}}\) and if we define \(y^L_j := (y_j, \ldots , y_j) \in \mathbf{M}^{i,L}\), then \(d^{\mathbf{M}^{i,L}} (y^L_{j_1}, y^L_{j_2} ) > a\) for any \(j_1 \ne j_2\). So the \(\frac{1}{2} a\)-balls around \(y^L_{j_1}\) are pairwise disjoint and contained in \(B^i (x, \frac{1}{L}, b' + a) \times \ldots \times B^i (x, \frac{L-1}{L}, b' +a)\). Using (2.1) and Lemma 2.1, we conclude that

$$\begin{aligned} \Big (c \Big ( \frac{a}{\sqrt{L}} \Big )^n \Big )^{L} \cdot m \le \Big ( C_0 (b')^n e^{C_0 b'} \Big )^L, \end{aligned}$$

which yields an upper bound on m. So the sequence \((\mathbf{M}^i, \overline{d}^i)\) is in fact uniformly totally bounded.

We may now pass to a subsequence and assume that \((\mathbf{M}^i, \overline{d}^i, x_i)\) converges to some metric space \((\overline{\mathbf{M}}, \overline{d}, \overline{x})\) in the pointed Gromov–Hausdorff sense. By (5.2) and Arzelá–Ascoli and after passing to another subsequence, the sequence of time-dependent metrics \((d^i)_{t \in [0,1]}\) converges locally uniformly to a time-dependent, continuous family of pseudometrics \((d^\infty _t)_{t \in [0,1]}\) on \(\overline{\mathbf{M}}\). So for any \(t \in [0,1]\), the pointed metric spaces \((\mathbf{M}^i, d^i_t, x_i)\) converge to \((\overline{\mathbf{M}} /{\sim _t}, d^\infty _t, \overline{x})\) in the pointed Gromov–Hausdorff sense. Passing to another subsequence once again, and using (2.1), we can ensure that also the volume forms \(dg^i_t\) converge uniformly for every rational \(t \in [0,1]\). Since \(e^{-A |t_2 - t_1|} dg^i_{t_1} \le dg^i_{t_2} \le e^{A |t_2 - t_1|} dg^i_{t_1}\), the convergence holds for any \(t \in [0,1]\). The doubling property for balls of radius less than \(\sqrt{t}\) follows from (2.1) after parabolic rescaling by \((\frac{1}{2} t)^{-1/2}\). \(\square \)