2. A priori estimates of Ricci flow

In this section, we will derive some a priori estimates of the Ricci flow, this being related to the curvature concentration.

Along the Ricci flow, it is more common to consider the log-Sobolev constant, which is measured by the $\nu$ -entropy. Due to the celebrated work of Perelman [Reference PerelmanPer02], it is well-known that the $\nu$ -entropy is monotone along a complete Ricci flow with bounded curvature (see also the complete non-compact case by Chau, Tam, and Yu [Reference Chau, Tam and YuCTY11]). We need a localized version by Wang [Reference WangWan18]. Following [Reference WangWan18], for a connected domain $\Omega$ with possibly empty boundary in M, we define

\begin{align*}\left\{\begin{array}{l}D_g(\Omega):=\{u\in W^{1,2}_0(\Omega): u\geq 0 \text{ and }\|u\|_{L^2(\Omega)}=1\},\\[9pt]W(\Omega, g, u, \tau):=\displaystyle\int_{\Omega}(\tau(\mathcal{R}\cdot u^2+4|\nabla u|^2)-2u^2\log u) \, d\mathrm{vol}_g -\displaystyle\frac{n}{2}\log(4\pi\tau)-n,\\[12pt]\nu(\Omega, g, \tau):= \inf\limits_{u\in D_g(\Omega), s \in (0, \tau]}W(\Omega, g, u, s),\end{array}\right.\end{align*}

where $\mathcal{R}$ denotes the scalar curvature.

The local entropy $\nu(\Omega,g,\tau)$ measures the optimal (weighed) log-Sobolev constant of $\Omega$ in scale of $\tau$ with respect to g. This also controls the Sobolev constant in some suitable scale by [Reference MartensMar25a, Lemma 3.3], which is built on [Reference YeYe15]. If g(t) is a complete Ricci flow on $\Omega$ with bounded curvature, it is well-known that $\nu(\Omega,g(t),\tau-t)$ is monotonically non-decreasing in t, again by the celebrated work of Perelman [Reference PerelmanPer02].

Although the (almost) monotonicity of the (localized) $\nu$ -entropy does not necessarily hold for incomplete local Ricci-flow solutions, we will construct local solutions in such a way that the entropy is still under control.

In this section, we will derive a priori estimates, assuming a bound on the localized entropy. We first need a local persistence of local curvature concentration, which is an essential key to the semi-preservation of small curvature concentration (see [Reference Chan, Chen and LeeCCL22, Reference Chan, Huang and LeeCHL24, Reference Chan and LeeCL24, Reference Chau and MartensCM26a, Reference Chau and MartensCM26b, Reference MartensMar25a, Reference MartensMar25b, Reference ChenChe22]). We need the following localized version of [Reference Chan and LeeCL24, Lemma 2.2].

Then, for all $t\in [0,T\wedge\tilde T]$ , we have

\begin{align*}\int_{B_{g(t)}(x_0,1)}|\mathrm{Rm}(g(t))|^{n/2}\, d\mathrm{vol}_{g(t)} \leq \int_{B_{g_0}(x_0,2)}|\mathrm{Rm}(g_0)|^{n/2}\, d\mathrm{vol}_{g_0}+ C_1\delta_0^{n/2} t.\end{align*}

Proof. Let $\varphi(s)$ be a smooth non-increasing function on $[0,+\infty)$ such that $\varphi\equiv 1$ on [0, 1], vanishes outside [0, 2], and satisfies $|\varphi'|^2\leq 10^3\varphi$ and $\varphi''\geq -10^3\varphi$ . Let

\begin{align*}\eta(x,t):=d_{g(t)}(x,x_0)+2({\alpha}+1)(n-1) \sqrt{t}\end{align*}

and define $\phi(x,t):=e^{-10^3t}\varphi(\eta(x,t))$ for $x\in B_{g(t)}(x_0,2)$ and $t\in [0,T]$ . It follows from [Reference PerelmanPer02, Lemma 8.3] that there exists $\tilde T=\tilde T(n,{\alpha})\gt0$ such that

\begin{align*}\bigg(\frac{\partial}{\partial t}-\Delta_{g(t)}\!\bigg) \phi \leq 0\end{align*}

for $t\in [0,T\wedge \tilde T]$ , in the barrier sense (for a detailed discussion, see also [Reference Simon and ToppingST22, Lemma 7.1]). The inequality also holds in the distribution sense by [Reference Mantegazza, Mascellani and UraltsevMMU14, Appendix A].

Furthermore, $|\nabla\phi|^2\leq 10^3 \phi,$ so that [Reference Chan, Chen and LeeCCL22, Lemma 2.1] implies

\begin{align*}\displaystyle\frac{d}{dt} \int_M \phi^2 |\mathrm{Rm}|^{n/2}\,d\mathrm{vol}_{g(t)}&\leq -C_0^{-1} \int_M |\nabla (\phi |\mathrm{Rm}|^{n/4})|^2 d\mathrm{vol}_{g(t)}\\&\quad +C_0\int_M \phi^2 |\mathrm{Rm}|^{n/2+1} \,d\mathrm{vol}_{g(t)}\\&\quad +C_0\int_M \phi |\mathrm{Rm}|^{n/2} \,d\mathrm{vol}_{g(t)}\\&:=-\mathbf{I}+\mathbf{II}+\mathbf{III}\end{align*}

for some dimensional constant $C_0(n)\gt0$ . We note that the evolution equation in [Reference Chan, Chen and LeeCCL22, Lemma 2.1] is purely local, and thus the completeness of the solution there is not necessary.

Since $\phi(\cdot,t)$ is compactly supported on $B_{g(t)}(x_0,2)$ for each $t\in [0,T]$ ,

\begin{align*}\mathbf{III}\leq C_0\delta_0^{n/2}.\end{align*}

Similarly, together with Hölder’s inequality, we have

\begin{align*}\mathbf{II}&\leq C_0\|\mathrm{Rm}\|_{L^{n/2}(B_{g(t)}(x_0,2))}\cdot \bigg( \int_M \phi^{{2n}/({n-2})}|\mathrm{Rm}|^{{n^2}/({2(n-2)})}\,d\mathrm{vol}_{g(t)}\!\bigg)^{\!({n-2})/{n}}\\&\leq C_0\delta_0 \cdot \bigg( \int_M \phi^{{2n}/({n-2})}|\mathrm{Rm}|^{{n^2}/({2(n-2)})}\,d\mathrm{vol}_{g(t)}\!\bigg)^{\!({n-2})/{n}}.\end{align*}

It remains to handle $\mathbf{I}$ . For any $t\in [0,T\wedge \tilde T]$ , we apply [Reference MartensMar25a, Lemma 3.3(2)] with $\Omega=B_{g(t)}(x_0,2)$ , $g=g(t)$ and $\tau=1$ to deduce that for some $\Lambda_0(n,A)\gt0$ , we have

(2.1) \begin{align}\bigg(\int_{B_{g(t)}(x_0,2)} |u|^{{2n}/({n-2})}\,d\mathrm{vol}_{g(t)} \!\bigg)^{\!({n-2})/{n}}\leq \Lambda_0 \int_{B_{g(t)}(x_0,2)} (|\nabla u|^2+ u^2) \,d\mathrm{vol}_{g(t)}\end{align}

for all $u\in W^{1,2}_0(B_{g(t)}(x_0,2))$ , provided that $\delta_0\leq \delta_1$ and that $\delta_1$ is small enough depending on A. Applying (2.1) to $u:=\phi |\mathrm{Rm}|^{n/4}$ yields

\begin{align*}\mathbf{I}&\geq (C_0\Lambda_0)^{-1} \bigg( \int_M \phi^{{2n}/({n-2})}|\mathrm{Rm}|^{{n^2}/({2(n-2)})}\,d\mathrm{vol}_{g(t)}\!\bigg)^{\!({n-2})/{n}}-C_0^{-1}\int_M \phi^2 |\mathrm{Rm}|^{n/2}\,d\mathrm{vol}_{g(t)}.\end{align*}

Combining the above estimates, we conclude that if $\delta_0\leq \delta_1\leq (C_0^2 \Lambda_0)^{-1}$ , then

\begin{align*}&\displaystyle\frac{d}{dt} \int_M \phi^2 |\mathrm{Rm}|^{n/2}\,d\mathrm{vol}_{g(t)}\\&\quad \leq (C_0+C_0^{-1}) \delta_0^{n/2}+( C_0\delta_0-(C_0\Lambda_0)^{-1} ) \cdot \bigg( \int_M \phi^{{2n}/({n-2})}|\mathrm{Rm}|^{{n^2}/({2(n-2)}})\,d\mathrm{vol}_{g(t)}\!\bigg)^{\!({n-2})/{n}}\\&\quad \leq C_1(n)\delta_0^{n/2}.\end{align*}

The assertion follows from integrating this from the initial time slice. This completes the proof.

We now show that if we have a priori scaling-invariant curvature decay in time, then assumption (b) in Lemma 2.1 will be satisfied, provided that the initial metric is quantitatively Ahlfors n-regular.

1. (a) $B_{g(t)}(x_0,2)\Subset M$ ;

2. (b) $|\mathrm{Rm}(g(t))|\leq {\alpha}_0 t^{-1}$ on $B_{g(t)}(x_0,2)$ ;

3. (c) $ \nu(B_{g(t)}(x_0,2),g(t),1)\geq -A$ ;

4. (d) $\mathrm{Vol}_{g_0}( B_{g_0}(x,r))\leq v_0r^n$ for $x\in B_{g_0}(x_0,1)$ and $r\in (0,1]$ ; and

5. (e) $\|\mathrm{Rm}(g_0)\|_{L^{n/2}(B_{g_0}(x_0,2))}\lt\delta_0$ .

Then for all $t\in [0,T\wedge \tilde T]$ ,

\begin{align*}\bigg(\int_{B_{g(t)}(x_0,1)} |\mathrm{Rm}(g(t))|^{n/2} \,d\mathrm{vol}_{g(t)} \!\bigg)^{\!2/n}\leq \delta_1\end{align*}

where $\delta_1(n,A)$ is the constant from Lemma 2.1.

Let $\delta_0\in (0,\delta_1)$ be a constant to be specified. We claim that

(2.2) \begin{align}\bigg(\int_{B_{g(t)}(x,\rho(x,t))} |\mathrm{Rm}(g(t))|^{n/2}\,d\mathrm{vol}_{g(t)}\!\bigg)^{\!2/n}\lt\delta_1\end{align}

for all $ x\in B_{g(t)}(x_0,\tfrac32)$ and $t\in [0,T]$ if T is smaller than a dimensional constant.

By [Reference Simon and ToppingST22, Corollary 3.3], for some $\tilde T(n)\gt0,$ $ B_{g(t)}(x_0,\tfrac32)\Subset B_{g_0}(x_0,2)$ for $t\in [0,T\wedge \tilde T]$ and $\rho(x,t)\to 0$ as $ x\to \partial B_{g(t)}(x_0,\tfrac32)$ . Therefore, if the claim in (2.2) is not true, then there exists $t_1\in (0,T\wedge \tilde T]$ such that:

1. (i) $\|\mathrm{Rm}(g(t))\|_{L^{n/2}(B_{g(t)}(x,\rho(x,t)))}\lt\delta_1$ for all $t\in [0,t_1)$ and $ x\in B_{g(t)}(x_0,\tfrac32)$ ; and

2. (ii) $\|\mathrm{Rm}(g(t_1))\|_{L^{n/2}(B_{g(t_1)}(x_1,\rho(x_1,t_1)))}=\delta_1$ for some $ x_1\in B_{g(t_1)}(x_0,\tfrac32)$ .

We denote $ \rho_1:=\rho(x_1,t_1)\leq 1$ for convenience.

Proof of Claim 2.4. Given ${\beta}\gt1$ , we show that $\rho_1\geq {\beta}\sqrt{t_1}$ if ${\alpha}_0$ is sufficiently small. Suppose that this is not the case, i.e., $\rho_1\leq {\beta}\sqrt{t_1}$ . By (ii) and the assumption on curvature estimates,

\begin{align*}\delta_1^{n/2}&\leq ({\alpha} t_1^{-1})^{n/2} \cdot \mathrm{Vol}_{g(t_1)}(B_{g(t_1)}(x_1,\rho_1) )\\&\leq ({\alpha} t_1^{-1})^{n/2} \cdot \mathrm{Vol}_{g(t_1)}(B_{g(t_1)}(x_1,{\beta} \sqrt{t_1}) ).\end{align*}

By the volume comparison using $\operatorname{Ric}(g(t))\geq -(n-1){\alpha} t^{-1}$ and a simple scaling argument, we can estimate the volume from above as long as ${\alpha}_0{\beta}^2\leq 1$ , so that

\begin{align*}\delta_1^{n/2}\leq(\tfrac12C_0{\alpha}_0^{1/2}{\beta})^{n},\end{align*}

which is impossible if ${\alpha}_0\leq C_0^{-2}{\beta}^{-2}\delta_1$ .

On the other hand, for all $x\in B_{g(t_1)}(x_1,\rho_1)$ and $t\in [0,t_1]$ ,

\begin{align*}\tfrac32\rho(x,t)&=\tfrac32-d_{g(t)}(x_0,x)\\&\geq \tfrac32-d_{g(t)}(x_0,x_1)-d_{g(t)}(x_1,x)\\&\geq \tfrac32-d_{g(t_1)}(x_0,x_1)-d_{g(t_1)}(x_1,x) -C_1\sqrt{{\alpha}_0 t_1}\\&\ge \rho_1-C_1\sqrt{t_1}\end{align*}

for some $C_1\gt1$ , where we have used [Reference Simon and ToppingST22, Corollary 3.3]. Hence, by Claim 2.4, we might assume that $\rho(x,t)\geq \tfrac14 \rho_1$ for all $x\in B_{g(t_1)}(x_1,\rho_1)$ and $t\in [0,t_1]$ , if $ {\alpha}_0\leq (4C_0C_1)^{-2}\delta_1$ . Therefore, (i) implies that

(2.3) \begin{align}\int_{B_{g(t)}(x,(1/4)\rho_1)} |\mathrm{Rm}(g(t))|^{n/2}\,d\mathrm{vol}_{g(t)} \leq \delta_1^{n/2}\end{align}

for all $t\in [0,t_1]$ and $x\in B_{g(t_1)}(x_1,\rho_1)$ .

Fix $z\in B_{g(t_1)}(x_1,\rho_1)$ and consider the rescaled Ricci flow $\tilde g(t):=\lambda^2 g(\lambda^{-2}t),t\in [0,\lambda^2 t_1]$ , where $\lambda =8 \rho_1^{-1}$ . By (2.3) and the assumptions, for $t\in [0,\lambda^2 t_1],$ the rescaled flow satisfies:

• – $B_{\tilde g(t)}(z,2)=B_{g(\lambda^{-2}t)}(z,\tfrac14\rho_1)\Subset M$ ;

• – $\|\mathrm{Rm}(\tilde g(t))\|_{L^{n/2}(B_{\tilde g(t)}(z,2))}\lt\delta_1$ ;

• – $\nu(B_{\tilde g(t)}(z,2),\tilde g(t),1 )\geq -A;$ and

• – $|\mathrm{Rm}(\tilde g(t))|\leq {\alpha}_0 t^{-1}$ .

Here, we have used the monotonicity on the scale $\tau$ in $\nu$ -entropy from [Reference WangWan18, Proposition 3.2] and $\lambda^{-2}=8^{-2}\rho_1^2\leq 1$ .

We apply Lemma 2.1 to $\tilde g(t)$ to conclude that

\begin{align*}\int_{B_{\tilde g(t)}(z,1)}|\mathrm{Rm}(\tilde g(t))|^{n/2}\, d\mathrm{vol}_{\tilde g(t)}&\leq \int_{B_{\tilde g_0}(z,2)}|\mathrm{Rm}(\tilde g_0)|^{n/2}\, d\mathrm{vol}_{\tilde g_0}+ C_2\delta_1^{n/2} t\\&\leq \delta_0^{n/2}+ C_2\delta_1^{n/2} t\end{align*}

for all $t\in [0,\lambda^2t_1 \wedge \tilde T(n)]$ . Rescaling it back, we conclude that

(2.4) \begin{align}\int_{B_{g(s)}(z,(1/8)\rho_1)}|\mathrm{Rm}(g(s))|^{n/2}\,d\mathrm{vol}_{g(s)} &\leq \delta_0^{n/2}+64C_2\delta_1^{n/2} \rho_1^{-2} s\nonumber\\&=: \delta_0^{n/2}+ C_3\delta_1^{n/2} \rho_1^{-2} s\end{align}

for all $s\in [0,t_1\wedge 8^{-2}\tilde T \rho_1^2]$ and $z\in B_{g(t_1)}(x_1,\rho_1)$ . By Claim 2.4, if

\begin{align*} {\alpha}_0\leq \min\{(8C_0)^{-2}\tilde T\delta_1, (4C_1C_0)^{-2}\delta_1\},\end{align*}

then (2.4) holds at $s=t_1$ .

We now use (2.4) to contradict (ii), the equality at $(x_1,t_1)$ . By choosing a maximal collection of $\{z_i\}_{i=1}^N$ in $B_{g(t_1)}(x_1,\tfrac{39}{40}\rho_1)$ such that $B_{g(t_1)}(z_i,\tfrac{1}{40}\rho_1)$ are pair-wise disjoint, we have a collection of points $\{z_i\}_{i=1}^N$ such that

(2.5) \begin{align}\coprod_{i=1}^N B_{g(t_1)}\bigg(z_i,\frac1{40}\rho_1\!\bigg)\subseteq B_{g(t_1)}(x_1,\rho_1) \subseteq \bigcup_{i=1}^NB_{g(t_1)}\bigg(z_i,\frac18\rho_1\!\bigg).\end{align}

Then (ii) and (2.4) imply that

(2.6) \begin{align}\delta_1^{n/2}&\leq N ( \delta_0^{n/2}+ C_3\delta_1^{n/2} \rho_1^{-2}t_1 ).\end{align}

Proof of Claim 2.5. This follows from [Reference MartensMar25b, Lemma 2.1] (see also the proof of [Reference Chan, Chen and LeeCCL22, Theorem 1.2]). For readers’ convenience, we include its proof.

By [Reference Simon and ToppingST22, Corollary 3.3] and Claim 2.4, we might assume that $B_{g(t_1)}(x_1,\rho_1)\subseteq B_{g_0}(x_1,2\rho_1)$ and $B_{g(t_1)}(z_i,\tfrac{1}{40}\rho_1)\subseteq B_{g(s)}(z_i, \rho_1/4)$ for $s\in[0,t_1]$ . From (2.5) and the assumption of Ahlfors n-regularity, we have

(2.7) \begin{align}\sum_{i=1}^N\mathrm{Vol}_{g_0}\bigg(\!B_{g(t_1)}\bigg(\!z_i,\frac1{40}\rho_1\!\bigg)\! \bigg)&\leq \mathrm{Vol}_{g_0}(B_{g(t_1)}(x_1,\rho_1) )\nonumber\\&\leq \mathrm{Vol}_{g_0}(B_{g_0}(x_1,2\rho_1) )\leq v_0(2\rho_1)^n.\end{align}

It remains to control each $\mathrm{Vol}_{g_0}(B_{g(t_1)}(z_i,\tfrac{1}{40}\rho_1) )$ from below. We compare it with volume at $t=t_1$ as follows. For $s\in [0,t_1]$ and any i,

\begin{align*}&\displaystyle\frac{d}{ds}\mathrm{Vol}_{g(s)}\bigg(\!B_{g(t_1)}\bigg(\!z_i,\frac1{40}\rho_1\!\bigg)\!\bigg)\\[10pt] &\quad=-\int_{B_{g(t_1)}(z_i,(1/{40})\rho_1)}\mathcal{R}(g(s))\,d\mathrm{vol}_{g(s)}\\[10pt]&\quad\leq\bigg(\int_{B_{g(t_1)}(z_i,(1/{40})\rho_1)}|\mathrm{Rm}(g(s))|^{{n}/{2}}\,d\mathrm{vol}_{g(s)}\!\bigg)^{\!2/n}\bigg[\mathrm{Vol}_{g(s)}\bigg(\!B_{g(t_1)}\bigg(\!z_i,\frac1{40}\rho_1\!\bigg)\!\bigg) \bigg]^{1-(2/n)}\\[10pt]&\quad\leq \delta_1\bigg[\mathrm{Vol}_{g(s)}\bigg(\!B_{g(t_1)}\bigg(\!z_i,\frac1{40}\rho_1\!\bigg)\!\bigg) \bigg]^{1-(2/n)} ,\end{align*}

where we have used (2.3) and $z_i\in B_{g(t_1)}(x_1,\rho_1)$ .

By integrating it from $s=0$ to $s=t_1$ , we conclude that

(2.8) \begin{align}\bigg[\mathrm{Vol}_{g(t_1)}\bigg(\!B_{g(t_1)}\bigg(\!z_i,\frac1{40}\rho_1\!\bigg)\! \bigg)\bigg]^{2/n}&\leq\bigg[\mathrm{Vol}_{g_0}\bigg(\!B_{g(t_1)}\bigg(\!z_i,\frac1{40}\rho_1\!\bigg)\! \bigg)\bigg]^{2/n} +\frac{2\delta_1}n t_1.\end{align}

On the other hand, by assumption (c) and the monotonicity of the $\nu$ -entropy (in terms of domains) [Reference WangWan18, Proposition 2.1], $\nu(B_{g(t_1)}(z_i,\tfrac{1}{40} \rho_1, g(t_1),1)\geq -A.$ Thus, (2.3), [Reference MartensMar25a, Lemmas 3.1 and 3.3], and Remark 2.2 imply that

(2.9) \begin{align}\mathrm{Vol}_{g(t_1)}(B_{g(t_1)}(z_i,\tfrac1{40}\rho_1))\geq \gamma_0\cdot \rho_1^n\end{align}

for some $\gamma_0(n,A)\gt0$ .

By Claim 2.4 with ${\beta}=2\gamma_0^{-1/n}$ , we then have

\begin{align*}\gamma_0^{2/n} \rho_1^2\geq 4 t_1\gt \frac{4}{n} \delta_1 t_1\end{align*}

provided that ${\alpha}_0\leq C_0^{-2}\delta_1\cdot \min\{2\gamma_0^{-1/n}, 8^{-2}\tilde T\}^{ -2}=:\gamma_1(n,A) \cdot \delta_1$ . Using this with (2.9) and (2.8), we estimate the left-hand side of (2.7) by

\begin{align*}\sum_{i=1}^N\mathrm{Vol}_{g_0}\bigg(\!B_{g(t_1)}\bigg(\!z_i,\frac1{40}\rho_1\!\bigg)\! \bigg)&\geq \sum_{i=1}^N\bigg(\gamma_0^{2/n} \rho_1^2 -\frac{2\delta_1}{n}t_1\!\bigg)^{\!n/2}\\&\geq N2^{-n/2} \gamma_0 \rho_1^n ,\end{align*}

so that $N\leq 4^n\gamma_0^{-1} v_0=: N_0(n,A,v_0)$ from (2.7).

We can finish the proof of Proposition 2.3. We now choose $\delta_0(n,v_0,A):= (4N_0)^{-2/n}\delta_1$ so that from Claim 2.5, we can deduce from (2.6) that

\begin{align*}\rho_1^2\lt 2C_3N_0 t_1.\end{align*}

By Claim 2.4, this is impossible if we choose

\begin{align*}{\alpha}_0:=\delta_1\cdot\min\{(8C_0)^{-2}\tilde T, \gamma_1,(4C_0C_1)^{-2}, {C_0^{-2}} (2C_3N_0)^{-1}\}.\end{align*}

This proves the claim (2.2). The proposition follows by the claim using $\rho(x_1,t)=1$ .

The upshot of Proposition 2.3 is that it allows us to re-apply Lemma 2.1 to obtain an improved estimate of curvature.

Proposition 2.6. For $n,A\gt0$ , there exists $\Lambda_0(n,A)\gt0$ such that the following holds. Under the assumptions in Proposition 2.3, if, in addition, we have $\|\mathrm{Rm}(g_0)\|_{L^{n/2}(B_{g_0}(x_0,1))}\leq \varepsilon$ , then we have

\begin{align*}|\mathrm{Rm}(x_0,t)|\leq \Lambda_0 t^{-1}(\varepsilon +t^{2/n})\end{align*}

for $t\in (0,T\wedge \tilde T]$ .

Proof. By Proposition 2.3, we have $\|\mathrm{Rm}(g(t))\|_{L^{n/2}(B_{g(t)}(x_0,1))}\leq \delta_1$ for $t\in [0,T\wedge \tilde T]$ , where $\delta_1(n,A)$ is the constant from Lemma 2.1. Applying Lemma 2.1 to the rescaled Ricci flow $\tilde g(t):= 4g(4^{-1}t),t\in [0,4(T\wedge \tilde T)]$ and then rescaling it back to g(t), we see that

\begin{align*}\int_{B_{g(t)}(x_0,1/2 )}|\mathrm{Rm}(g(t))|^{n/2}\, d\mathrm{vol}_{g(t)}& \le \int_{B_{g_0}(x_0, 1)}|\mathrm{Rm}(g_0)|^{n/2}\, d\mathrm{vol}_{g(0)} + 4C_1\delta_1^{n/2} t\\& \leq \varepsilon^{n/2}+ 4C_1\delta_1^{n/2} t\end{align*}

for all $t\in [0,T\wedge\tilde T]$ . Due to the curvature assumption and the volume non-collapsing from [Reference WangWan18, Theorem 5.4], standard Moser iteration (see, e.g., [Reference LiLi12, Chapter 19]) yields

\begin{align*}t|\mathrm{Rm}(x_0,t)|\leq \Lambda_0 (\varepsilon +t^{2/n})\end{align*}

for some $\Lambda_0(n,A)\gt0$ . This completes the proof.

In particular, Proposition 2.6 asserts that the Ricci flow has an improved estimate from the smallness of the initial local curvature concentration.

The next lemma states that for a complete bounded-curvature Ricci flow, the local $\nu$ -functional is (almost) preserved. This has first been considered by Wang [Reference WangWan18], localizing the work of Perelman [Reference PerelmanPer02]. The following refined version is by Cheng [Reference ChengChe25].

Lemma 2.7 (Lemma 2.5 in [Reference ChengChe25]). Fix $0\leq s_1\leq s_2\leq 1$ . Suppose that g(t) is a smooth solution to the Ricci flow on $M\times [s_1,s_2]$ with bounded curvature such that for all $t\in [s_1,s_2]$ and $x\in B_{g(t)}(x_0,\sqrt{t}),$

\begin{align*}\operatorname{Ric}(g(t))\leq (n-1)\hat {\alpha} t^{-1}.\end{align*}

Then, for any $\tau\gt0$ , ${\beta}\in (0,\tfrac{1}{10}),$ and $\gamma\in [0,8-20{\beta})$ , we have

\begin{align*}&\nu(\Omega_2,g(s_2),\tau+1-s_2)-\nu(\Omega_1,g(s_1),\tau+1-s_1)\\&\quad \geq -\bigg[ \frac{1+\tau}{10 \hat {\alpha}^2{\beta}^2}+\frac1e\bigg]\cdot \bigg[ \exp\bigg(\frac{s_2-s_1}{10\hat {\alpha} ^2{\beta}^2}\!\bigg)-1\bigg] ,\end{align*}

where

\begin{align*}\left\{\begin{array}{l}\Omega_1:=B_{g(s_1)}(x_0,10\hat {\alpha} -2\hat {\alpha}\sqrt{s_1}-\hat {\alpha} \gamma );\\[3pt]\Omega_2:=B_{g(s_2)}(x_0,10\hat {\alpha} (1-2{\beta})-2\hat {\alpha} \sqrt{s_2}-\hat {\alpha} \gamma).\end{array}\right.\end{align*}

3. Local smoothing using Ricci flow

In this section, we will construct local solutions to the Ricci flow that locally regularize metrics with small local curvature concentration, i.e., Theorem 1.1. We adapt the idea of Cheng [Reference ChengChe25] with the estimates in § 2 to construct local Ricci flow, building on the idea of Hochard [Reference HochardHoc19]. To prove Theorem 1.1, we need to specify some constants precisely. We need two more ingredients. The first is a result of Hochard that allows us to construct a local Ricci flow on regions with bounded curvature.

1. (i) $h(0)=h_0$ on $N_\rho=\{x\in N: B_{h_0}(x,\rho)\Subset N\}$ ;

2. (ii) $|\mathrm{Rm}(h(t))|\leq \Lambda \rho^{-2}$ throughout $N\times [0,\gamma \rho^2]$ .

The Ricci flow obtained in [Reference Lee and ToppingLT25, Proposition 4.2] is constructed by modifying the incomplete metric at its extremities in order to make it complete, and followed by running Shi’s classical solution. Thus, the solution h(t) is by construction a complete solution on the whole N.

We also recall the shrinking-balls lemma, which is one of the local ball inclusion results based on the distance-distortion estimates of Hamilton and Perelman from [Reference PerelmanPer02, Lemma 8.3].

Lemma 3.2 [Reference Simon and ToppingST22, Corollary 3.3]. For $n\geq 2$ , there exists a constant $\beta(n)\geq 1$ such that the following is true. Suppose that $(N^n,g(t))$ is a Ricci flow for $t\in [0,S]$ and $x_0\in N$ with $B_{g(0)}(x_0,r)\Subset N$ for some $r\gt0$ , and $\operatorname{Ric}_{g(t)}\leq a(n-1)/t$ on $B_{g_0}(x_0,r)$ for each $t\in (0,S]$ . Then, for each $t\in (0,S],$

\begin{align*}B_{g(t)}(x_0,r-\beta\sqrt{a t})\subseteq B_{g_0}(x_0,r).\end{align*}

We are now ready to prove Theorem 1.1.

Proof of Theorem 1.1. We first prove the existence when $\|\mathrm{Rm}(g_0)\|_{L^{n/2}(B_{g_0}(x_0,1),g_0)}\leq \varepsilon_1$ for $\varepsilon_1\gt0$ , which is fixed below. By [Reference MartensMar25a, Lemma 3.2], we might assume that $\nu(B_{g_0}(x,1),g_0,2 )\geq -A_0$ for some $A_0(n,C_s)$ , by working on a slightly smaller ball if necessary. We need to specify how the positive constants are chosen, since the precise forms of the constants will play a crucial role in the construction:

• – $\gamma(n)$ and $\Lambda(n)$ from Proposition 3.1;

• – ${\beta}(n)$ from Lemma 3.2;

• – $A:=A_0+1$ ;

• – $\tilde T(n),\delta_0(n,v_0,A)$ and ${\alpha}_0(n,v_0,A)$ from Proposition 2.3;

• – $\Lambda_0(n,A)$ from Proposition 2.6;

• – $\varepsilon_1(n,v_0,A):=\min\{ \delta_0, {\alpha}_0 (2\Lambda_0)^{-1}, {\alpha}_0 (4\Lambda\Lambda_0)^{-1}\}$ ;

• – $\mu(n,v_0,A):=\min\{ 10^{-2},\gamma (2\varepsilon_1 \Lambda_0)^{-1}\}$ ;

• – $\sigma(n,v_0,A):=10^{-3} (1-(1+\mu)^{-1/5} )$ ;

• – $L(n,v_0,A):=10^4{\alpha}_0^{-1/2}\cdot \max\{ {\beta},\tilde T ^{-1/2},\varepsilon_1^{-n} ,n, \mu\sigma^{-2} , \sigma^{-2}[1-(1+\mu)^{-1/5} ]^{-1}\}$ ; and

• – $\hat T(n,v_0,A):=\min\{ 10^{-1}{\alpha}_0^{-1}L^{-2},(\tfrac14 L^2{\alpha}_0+\mu+1)^{-1}\}.$

Let $\rho_0\in (0,\tfrac12)$ be small such that $|\mathrm{Rm}(g_0)|\leq \rho_0^{-2}$ on $N_0:=B_{g_0}(x_0,2+\bar r)$ . Then, from Proposition 3.1, there exists a smooth complete Ricci flow h(t) on $N_0\times [0,\gamma\rho_0^2]$ such that:

• – $h(0)=g_0$ on $(N_0)_{\rho_0} := B_{g_0}(x_0,2+\bar r-\rho_0)$ ; and

• – $|\mathrm{Rm}(h(t))|\leq \Lambda \rho_0^{-2}$ on $N_0\times [0,\gamma \rho_0^2]$ .

We choose $t_0:= \rho_0^2\cdot \min\{\gamma, \Lambda^{-1}{\alpha}_0\}$ so that

\begin{align*}|\mathrm{Rm}(h(t))|\leq \Lambda \rho_0^{-2}\leq {\alpha}_0 t_0^{-1} \leq {\alpha}_0t^{-1}\end{align*}

on $N_0\times (0,t_0]$ . We also denote $t_{-1}=0$ for notional consistency. We define a sequence of times $t_k$ and radii $r_k$ inductively as follows:

1. (i) $r_0:=2+\bar r-\rho_0$ ;

2. (ii) $t_{k+1}:=(1+\mu) t_k$ for $k\geq 0$ ; and

3. (iii) $r_{k+1}=r_k-8L\sqrt{{\alpha}_0 t_k}$ for $k\geq 0$ .

We consider the statement $\mathcal{P}(k)$ :

There exists a smooth Ricci flow g(t) defined on $B_{g_0}(x_0,r_k)\times [0,t_k]$ such that

(I) $g(0)=g_0$ ,

(II) $|\mathrm{Rm}(g(t))|\leq {\alpha}_0 t^{-1}$ for $t\in (0,t_k]$ , and

(III) $ \nu(B_{ g(t)}(x,L\sqrt{{\alpha}_0 t_k}), g(t),\tfrac14 L^2 {\alpha}_0 t_k)\geq -A$ for all $(x,t)\in B_{g_0}(x_0,r_k)\times (0,t_k]$ .

From the above discussion, since h(t) is smooth and $A\gt A_0$ , we might assume that $\mathcal{P}(0)$ is true by shrinking $\rho_0$ if necessary, for instances using the continuity of entropy [Reference WangWan18, Corollary 2.5]. We want to show that $\mathcal{P}(k)$ is true if $r_k\gt0$ and $t_k\lt\hat T$ , both of which hold when $k=0$ .

We prove this by induction on k. $\mathcal{P}(0)$ is true by the above construction. Suppose that for some $k\ge 0$ with $r_{k+1}\gt0$ and $t_{k+1}\lt \hat T,$ $\mathcal{P}(i)$ is true for $0\le i\le k.$ We let g(t) be the Ricci flow on $B_{g_0}(x_0,r_k)\times [0,t_k]$ from $\mathcal{P}(k)$ . We first improve the curvature estimate on a smaller domain.

Claim 3.3. We have

\begin{align*}|\mathrm{Rm}(g(t))|\leq 2\varepsilon_1\Lambda_0t^{-1}\end{align*}

on $B_{g_0}(x_0,r_k-2L\sqrt{{\alpha}_0 t_k})\times [0,t_k]$ .

Proof of Claim 3.3. Let $z\in B_{g_0}(x_0,r_k-2L\sqrt{{\alpha}_0 t_k})$ so that Lemma 3.2 implies

\begin{align*}B_{g(t)}(z,L\sqrt{{\alpha}_0 t_k})&\subseteq B_{g_0}(z,2L\sqrt{{\alpha}_0 t_k})\subseteq B_{g_0}(x_0,r_k)\end{align*}

for all $t\in [0,t_k]$ . We consider the rescaled Ricci flow $\tilde g(t):=\lambda^{-2}g(\lambda^2 t)$ for $t\in [0,\lambda^{-2}t_k]$ , where $\lambda=\tfrac12 L\sqrt{{\alpha}_0 t_k}$ , and denote $\tilde g_0:=\tilde g(0)$ . Then, for all $t\in [0,4L^{-2}{\alpha}_0^{-1}]$ , we have:

1. (A) $B_{\tilde g(t)}(z,2)=B_{g(\lambda^2 t)}(z,2\lambda)\Subset M$ ;

2. (B) $|\mathrm{Rm}(\tilde g(t))|\leq {\alpha}_0 t^{-1}$ on $B_{\tilde g(t)}(z,2)$ ;

3. (C) $ \nu(B_{\tilde g(t)}(z,2),\tilde g(t),1)\geq -A$ ;

4. (D) $\mathrm{Vol}_{\tilde g_0}( B_{\tilde g_0}(x,r))\leq v_0r^n$ for $x\in B_{\tilde g_0}(z,1)$ and $r\in (0,1]$ ; and

5. (E) $\|\mathrm{Rm}(\tilde g_0)\|_{L^{n/2}(B_{\tilde g_0}(z,2) , \tilde g_0)}\lt\varepsilon_1$ .

Here, we have used $2\lambda=L\sqrt{{\alpha}_0 t_k}\lt1$ so that $B_{\tilde g_0}(x,2)\subseteq B_{g_0}(x,1)$ where the initial estimates hold. The entropy lower bound (C) follows from (III):

\begin{align*}\nu(B_{\tilde g(t)}(z,2),\tilde g(t),1)&= \nu(B_{ g(s)}(z,L\sqrt{{\alpha}_0 t_k}), g(s),\tfrac14 L^2{\alpha}_0 t_k)\geq -A\end{align*}

for $s=\lambda^2t\in [0,t_k]$ . Thus, the rescaled flow $\tilde g(t)$ satisfies all the conditions of Proposition 2.3. It follows from Proposition 2.6 and our choice of L and $\varepsilon_1$ that for $t\in (0,t_k]$ ,

\begin{align*}|\mathrm{Rm}(z,t)|\leq 2\varepsilon_1\Lambda_0 t^{-1}.\end{align*}

This proves the claim.

We now use the improved estimate from Claim 3.3 to extend the solution.

Proof of Claim 3.4. Denote ${\alpha}_1:= 2\varepsilon_1\Lambda_0\le \alpha_0$ for convenience. We apply Proposition 3.1 with a translation in time, on $N=N_k:=B_{g_0}(x_0,r_k-2L\sqrt{{\alpha}_0t_k})$ and $h_{0,k}=g(t_k)$ with $\rho=\rho_k:=\sqrt{{\alpha}_1^{-1}t_k}$ to obtain a complete Ricci flow $h_k(t)$ on $N_k\times [t_k,t_k+\gamma {\alpha}_1^{-1} t_k]$ such that:

• – $h_k(t_k)=h_{0,k}$ on $(N_k)_{\rho_k}=\{ x \in N_k : B_{g(t_k)}(x,\rho_k)\Subset N_k \}$ ; and

• – $|\mathrm{Rm}(h_k(t))|\leq \Lambda \rho_k^{-2}= \Lambda {\alpha}_1 t_k^{-1}$ on $N_k\times [t_k,(1+\gamma {\alpha}_1^{-1} )t_k]$ .

It remains to show that $(N_k)_{\rho_k}\supseteq B_{g_0}(x_0,r_{k}-4L\sqrt{{\alpha}_0 t_k})$ . Let $x\in B_{g_0}(x_0,r_{k}-4L\sqrt{{\alpha}_0 t_k})$ ; hence we have $B_{g_0}(x,2L\sqrt{{\alpha}_1 t_k})\subseteq N_k$ . Due to (II) in the induction hypothesis $\mathcal{P}(k)$ , Lemma 3.2 implies that

\begin{align*}B_{g(t_k)}(x,L\sqrt{{\alpha}_0 t_k})\Subset B_{g_0}(x,2L\sqrt{{\alpha}_0 t_k})\subseteq N_k ,\end{align*}

so that $x\in (N_k)_{\rho_k}$ . Thus, $h_k({ t_k})=h_{0,k}=g(t_k)$ on $B_{g_0}(x_0,r_{k}-4L\sqrt{{\alpha}_0 t_k})$ .

By our choice of $\mu$ , we know that $t_{k+1}-t_k=\mu t_k\leq \gamma {\alpha}_1^{-1} t_k$ . Therefore, we might patch the Ricci flow together on $B_{g_0}(x_0,r_k-4L\sqrt{{\alpha}_0t_k})\times [0,t_{k+1}]$ by taking

\begin{align*}g(t):=\begin{cases} g(t)&\text{if } t\in[0, t_k], \\ h_k(t)&\text{if } t\in [t_k,t_{k+1}] .\end{cases}\end{align*}

By our choice of $\varepsilon_1$ , we see that for $t\in [t_k,t_{k+1}]$ ,

\begin{align*}|\mathrm{Rm}(g_k(t))|\leq \Lambda {\alpha}_1 t^{-1}_k& = 2\varepsilon_1 \Lambda_0\Lambda(1+\mu) t^{-1}_{k+1}\\&\leq 4\varepsilon_1 \Lambda_0\Lambda t^{-1}\leq {\alpha}_0 t^{-1}.\end{align*}

This finishes the proof of the claim.

We will assume that g(t) for $t\leq t_k$ , is constructed by inductively using Claim 3.4, so that on each sub-interval $[t_i,t_{i+1}]$ , the Ricci flow g(t) is restriction of some complete bounded-curvature Ricci flow.

It remains to show that (III) in $\mathcal{P}(k+1)$ holds. For each i, from Claim 3.4 and the induction hypothesis, there is a complete bounded-curvature Ricci flow $h_i(t)$ on $B_{g_0}(x_0,r_i-2L\sqrt{{\alpha}_0 t_i})\times [t_i,t_{i+1}]$ such that:

• – $h_i(t)|_{B_{g_0}(x_0,r_i-4L\sqrt{{\alpha}_0 t_i})}=g(t)$ ; and

• – $|\mathrm{Rm}(h_i(t))|\leq {\alpha}_0 t^{-1}$ , where ${\alpha}_0\lt 1\lt10^3n$ .

We now consider the rescaled Ricci flow $\tilde g(t):=t_k^{-1}g(t_k t)$ for $t\in [0,1+\mu]$ . For each i, we also define the rescaled Ricci flow $\tilde h_i(t):=t_k^{-1}h_i(t_k t)$ for $t\in [\tilde t_i,\tilde t_{i+1}]$ , where $\tilde t_i:=t_k^{-1}t_i$ . For $z\in B_{g_0}(x_0,r_{k+1})$ , we apply Lemma 2.7 to deduce that

(3.1) \begin{align}& \nu(B_{\tilde h_i(\tilde t_{i+1})}(z, (10(1-2\hat{\beta})- \gamma-2\sqrt{\tilde t_{i+1}}) \hat{\alpha}),\tilde h_i(\tilde t_{i+1}),\tau+1-\tilde t_{i+1} )\nonumber\\&\quad \geq \nu(B_{\tilde h_i(\tilde t_{i})}(z, (10- \gamma-2\sqrt{\tilde t_{i}}) \hat{\alpha}),\tilde h_i(\tilde t_{i}),\tau+1-\tilde t_{i})\nonumber\\&\qquad -\bigg( \frac{1+\tau}{10 \hat {\alpha} ^2\hat{\beta}^2}+\frac1e\bigg)\cdot \bigg[ \exp\bigg(\frac{\mu \tilde t_i}{10 \hat {\alpha} ^2\hat{\beta}^2}\bigg)-1\bigg]\end{align}

for any $\hat{\alpha}\geq 10^3n$ , $\tau\gt0$ , $\hat{\beta}\in(0,\tfrac{1}{10})$ , and $\gamma\in [0,8-20\hat{\beta})$ . The analogous estimate also holds if $\tilde t_{i+1}$ is replaced by any $t\in [\tilde t_i,\tilde t_{i+1}]$ .

We start to prove (III) in $\mathcal{P}(k+1).$ Suppose that $\tilde t\in [\tilde t_\ell,\tilde t_{\ell+1}]$ for some $-1\le \ell\le k.$ We fix

\begin{align*}\hat{\alpha}:=5^{-1}L\sqrt{{\alpha}_0}\geq10^3n\quad\mathrm{and}\quad \tau:=\mu+\tfrac14 L^2{\alpha}_0.\end{align*}

For $-1\le i\le \ell,$ we define

\begin{align*}\hat{\beta}_i&:= \sigma\cdot \tilde t_i^{1/5},\\\gamma_{i}&:=\gamma_{i-1}+20\hat{\beta}_{i-1},\end{align*}

and

\begin{align*} \nu_i&:=\nu(B_{\tilde h_i(\tilde t_{i})}(z, (10- \gamma_i-2\sqrt{\tilde t_{i}})\hat{\alpha}),\tilde h_i(\tilde t_{i}),\tau+1-\tilde t_{i}),\end{align*}

where $\gamma_i$ is defined inductively starting from the final data below. Here, $\sigma$ is the constant defined at the beginning of the proof. Recall that we also set $t_{-1}:=0.$ We also define the ‘initial data’:

\begin{align*}\hat {\beta}_{\tilde t}&:= \sigma\, \tilde t^{1/5},\\\gamma_{\tilde t}&:= 10(1-2\hat {\beta}_{\tilde t})-2\sqrt{\tilde t}-L\hat {\alpha}^{-1}\sqrt{{\alpha}_0},\,\\ \gamma_{\ell}&:= \gamma_{\tilde t}+20\hat{\beta}_{\tilde t},\end{align*}

and

\begin{align*} \nu_{\tilde t}&:= { \nu(B_{\tilde h_{\ell}(\tilde t)}(z, (10- \gamma_{\tilde t}-2\sqrt{\tilde t})\hat{\alpha}),\tilde h_{\ell}(\tilde t),\tau+1-\tilde t).}\end{align*}

The definition of $\gamma_{\tilde t}$ implies that

(3.2) \begin{align}10(1-2\hat {\beta}_{\tilde t})-\gamma_{\tilde t}-2\sqrt{\tilde t}= L\hat {\alpha}^{-1}\sqrt{{\alpha}_0}=5.\end{align}

Since $B_{g_0}(z,4L\sqrt{{\alpha}_0t_k})\Subset B_{g_0}(x_0,r_k-4L\sqrt{{\alpha}_0t_k})$ , where ${ g(t)= h_i(t)}$ for $t\in [t_i,t_{i+1}]$ from Claim 3.4 and the remark after it, Lemma 3.2 implies that for $t\in [t_i,t_{i+1}]$ ,

\begin{align*}B_{ g(t)}( z,3L\sqrt{{\alpha}_0t_k} )\Subset B_{g_0}(x_0,r_i-4L\sqrt{{\alpha}_0t_i})\end{align*}

and thus,

\begin{align*}\nu_i=\nu(B_{\tilde g(\tilde t_{i})}(z, (10- \gamma_i-2\sqrt{\tilde t_{i}})\hat{\alpha}),\tilde g(\tilde t_{i}),\tau+1-\tilde t_{i}).\end{align*}

By the monotonicity of the $\nu$ -entropy [Reference WangWan18, Propositions 2.1], due to $t_k\leq \hat T$ and our choice of $\hat T$ , we have

(3.3) \begin{align}\nu_{-1}&\geq \nu(B_{\tilde g_0}(z, 10\hat{\alpha}),\tilde g_0,\tau+1)\nonumber\\&=\nu(\!B_{ g_0}(z, 2L\sqrt{{\alpha}_0 t_k}), g_0,(\tfrac14L^2{\alpha}_0+\mu+1)t_k)\nonumber\\&\geq \nu(B_{g_0}(x_0,1),g_0,2) )\geq -A_0.\end{align}

We show that $\sigma$ is chosen so that the assumptions on $\hat {\beta}_i$ and $\gamma_i$ in (3.1) are justified. Clearly, $\hat {\beta}_i\leq \sigma (1+\mu)^{1/5}\leq 10^{-2}$ if $\sigma\leq 10^{-3}$ . We now estimate $\gamma_i$ . The upper bound is trivial from decreasing property so that $\gamma_i\leq 5\lt8-20\hat {\beta}_i$ . To estimate the lower bound, we claim that $\gamma_{-1}\gt0$ .

(3.4) \begin{align} \gamma_{\tilde t}-\gamma_{-1}&=\gamma_{\tilde t}-\gamma_{\ell} +\sum^\ell_{i=0} (\gamma_{i}-\gamma_{i-1}) \nonumber\\ &\leq 20 \sigma\cdot \sum_{i=0}^{k} (\tilde t_{k-i})^{1/5}\nonumber\\ &\leq 20\sigma \sum_{i=0}^\infty (1+\mu)^{-i/5}= \frac{20\sigma}{1-(1+\mu)^{-1/5}}.\end{align}

From (3.2), the definition of $\hat\beta_{\tilde t},$ and $\tilde t\le 1+\mu,$ we have

\begin{align} 5\leq 2(1+\mu)^{1/2}+20\sigma (1+\mu)^{1/5}+\gamma_{\tilde t} ,\end{align}

so that (3.4) implies that

\begin{align} \gamma_{i}\geq \gamma_{-1}\geq 5-2(1+\mu)^{1/2}-20\sigma (1+\mu)^{1/5}-\frac{20\sigma}{1-(1+\mu)^{-1/5}}\gt1\end{align}

for all $-1\leq i\leq \ell$ by our choice of $\mu$ .

Without loss of generality, we might assume that $\tilde t=\tilde t_{\ell+1}$ . By our choices of $\hat {\beta}_i$ and $\gamma_i$ , (3.1) implies that for all $ -1\leq i\leq \ell$ ,

\begin{align*}\nu_{i+1}-\nu_i &\geq -\bigg( \frac{(1+\tau)}{ 10\sigma^{2}\hat {\alpha} ^2(\tilde t_i)^{2/5}}+\frac1e\bigg)\cdot \bigg[ \exp\bigg(\frac{\mu (\tilde t_i)^{3/5}}{ 10\sigma^2\hat {\alpha} ^2}\bigg)-1\bigg].\end{align*}

Since $ 10^{-1}\mu\hat {\alpha}^{-2}\sigma^{-2}\tilde t_i^{3/5} \leq \tfrac{{25}}{10}\mu L^{-2}{\alpha}_0^{-1}\sigma^{-2} \leq 1$ , using $e^z-1\le 2z$ for $z\in (0,1),$ we can simplify it as

\begin{align*}\nu_{i+1}-\nu_i &\geq -2\bigg( \frac{10(1+\mu+\frac14 L^2{\alpha}_0)}{ \sigma^2 L^2 {\alpha}_0(\tilde t_i)^{2/5}}+\frac1e\bigg)\cdot \bigg( \frac{10\mu (\tilde t_i)^{3/5}}{ \sigma^2L^2{\alpha}_0}\bigg) \\&\geq -10^{4}\sigma^{-2}L^{-2}{\alpha}_0^{-1} (1+\mu)^{(i-\ell)/5}.\end{align*}

Combining it with (3.3), we conclude that

\begin{align*}\nu_{\tilde t}&\geq \nu_{-1}-10^{4}\sigma^{-2}L^{-2}{\alpha}_0^{-1}\sum_{j=0}^{\ell+1} (1+\mu)^{-j/5}\\&\geq -A_0- \frac{10^{4}\sigma^{-2}}{1-(1+\mu)^{-1/5}}\cdot L^{-2}{\alpha}_0^{-1}\\&\geq -A_0-1=-A.\end{align*}

By (3.2) and the monotonicity of the $\nu$ -entropy [Reference WangWan18, Propositions 2.1] again, this proves that (III) in $\mathcal{P}(k+1)$ holds. By induction, this shows that $\mathcal{P}(k)$ is true whenever $r_k\gt0$ and $t_k\leq \hat T$ .

We let $k_0$ be the maximal k such that $r_k\geq 1+\bar r$ , i.e., $r_{k_0+1}\lt1+\bar r\leq r_{k_0}$ . We claim that $t_{k_0}$ is uniformly bounded from below. To see this, by the construction, we have

\begin{align*} 1+\bar r&\gt r_{k_0+1}=r_0+\sum^{k_0}_{i=0} (r_{i+1}-r_i)\\ &=2+\bar r -\rho_0-8L\sqrt{{\alpha}_0}\cdot \sum^{k_0}_{i=0} \sqrt{t_{k_0-i}}\\ &\geq \frac32 +\bar r-8L\sqrt{{\alpha}_0 t_{k_0}} \cdot \sum^{\infty}_{i=0} (1+\mu)^{-i/2}.\end{align*}

Thus, $t_{k_0}\geq \tilde T(n,v_0,A)\gt0$ . Replacing $\hat T$ with $\hat T\wedge \tilde T$ , we prove the existence with (1), (3), and (4) on $B_{g_0}(x_0,1+\bar r)$ using $\mathcal{P}(k_0)$ and the monotonicity of entropy (on scales and domains) after relabeling the constants. The injectivity radius lower bound (2) follows from (4) and the result of Cheeger, Gromov, and Taylor [Reference Cheeger, Gromov and TaylorCGT82]. By enlarging $\Lambda_1$ if necessary, the Hölder equivalence of distances follows from the proof of [Reference MartensMar25b, Claim 3.2] (see also the proof of [Reference Chan, Huang and LeeCHL24, Lemma 5.1]).

With the existence of Ricci flow, the general case in which $\|\mathrm{Rm}(g_0)\|_{L^{n/2}(B_{g_0}(x_0,1),g_0)}\leq \varepsilon \lt\varepsilon_1$ follows from Lemma 2.1 and Proposition 2.6 after we shrink $\hat T$ if necessary (see the proof of Proposition 2.6).

By letting $\bar r\to+\infty$ , we obtain a short-time solution to the Ricci flow on $M\times [0,\hat T]$ with the desired estimates.

1. (a) $ \mathrm{Vol}_{g_0}(B_{g_0}(x,r))\leq v_0 r^n$ for all $r\in (0,1]$ ;

2. (b) for all $u\in W^{1,2}_0( B_{g_0}(x,1), g_0)$ ,

   \begin{align*}\bigg(\int_{ B_{g_0}(x,1)} u^{{2n}/({n-2})}\,d\mathrm{vol}_{g_0}\!\bigg)^{\!({n-2})/{n}}\leq C_s \int_{ B_{g_0}(x,1)} (|\nabla u|^2+u^2\bigg)\, d\mathrm{vol}_{g_0};\end{align*}
   and

3. (c) $\|\mathrm{Rm}(g_0)\|_{L^{n/2}(B_{g_0}(x,1),g_0)}\leq \varepsilon$ for some $\varepsilon\in (0,\delta_1]$ .

Then there exists a smooth complete Ricci-flow solution g(t) on $M\times [0,\hat T]$ such that $g(0)=g_0$ and:

1. (1) $|\mathrm{Rm}(g(t))|\leq \Lambda_1 \varepsilon t^{-1}$ ;

2. (2) $\mathrm{inj}(g(t))\geq \sqrt{t}$ ;

3. (3) $\|\mathrm{Rm}(g(t))\|_{L^{n/2}(B_{g(t)}(x,1/2))}\leq \Lambda_1\varepsilon $ ; and

4. (4) $\nu (B_{g(t)}(x,2L\sqrt{\hat T},g(t),L^2\hat T )\geq -\Lambda_1$

for all $(x,t)\in M\times (0,\hat T]$ .

4. Compactness of manifolds with bounded curvature concentration

In this section, we use the local-smoothing result from Theorem 1.1 to discuss the compactness of the space of compact manifolds with bounded curvature concentration, i.e., Theorem 1.3. We start with a simple observation that the collection of manifolds satisfying the assumptions in Theorem 1.3 is pre-compact with respect to the Gromov–Hausdorff topology.

1. (1) there exists $C_s\gt0$ so that for all $u\in W^{1,2}(M)$ ,

   \begin{align*}\bigg(\int_M u^{{2n}/({n-2})}\,d\mathrm{vol}_{g} \!\bigg)^{\!({n-2})/{n}}\leq C_s\int_M (|\nabla u|^2+ u^2) \,d\mathrm{vol}_{g} ;\end{align*}
   and

2. (2) $\mathrm{diam}(M,g)\leq D_0$ .

Then there exists $v_1(n,C_s,D_0)\gt0$ such that for all $0\lt r\leq \mathrm{diam}(M,g)$ and $x\in M$ ,

\begin{align*}\mathrm{Vol}_g(B_g(x,r) )\geq v_1 r^n.\end{align*}

The basic idea in proving the compactness without Ricci geometry is based on finding a local homeomorphism from an open ball around a regular point to an open ball in a smooth manifold. This is based on mollifying the initial metrics locally using Theorem 1.1 to a metric with regularity.

Proof of Theorem 1.3. We will use $g_{i,0}$ to denote $g_i$ , to emphasize that it is the initial metric whenever smoothing is used. We closely follow the argument in [Reference NakajimaNak88] with modifications using ideas from [Reference Simon and ToppingST21] and the new smoothing result, i.e., Theorem 1.1. It is more convenient to first obtain a convergent sequence in metric geometry. By Lemma 4.2, we may assume that $(M_i,g_{i,0})$ converges to a metric space $(X,d_\infty)$ in the Gromov–Hausdorff topology after passing to subsequence. We also let $\varphi_i: (M_i,d_{g_{i,0}})\to (X,d_\infty)$ be a sequence of $\varepsilon_i$ —Gromov–Hausdorff approximations for some $\varepsilon_i\to 0$ as $i\to+\infty$ .

We let $\delta_1(n,v_0,C_s)$ be the constant from Theorem 1.1. We define the singular set $\mathcal{S}_r$ of scale $r\gt0$ by

\begin{align*} \mathcal{S}_r := \{ x\in X: \liminf_{i\to\infty} \|\mathrm{Rm}(g_{i,0})\|_{L^{n/2}(B_{g_{i,0}}(x_i,r))}\ge \delta_1 \text{ for any }x_i\in M_i \text{ such that } d_\infty(\varphi_i(x_i), x)\to 0 \}\end{align*}

and we consider

\begin{align*} \mathcal{S}:=\bigcap_{r\gt0}\mathcal{S}_r.\end{align*}

Proof of Claim 4.3. For any fixed $r\gt0,$ we take a collection $\{x^j\}_{j\in J}\subseteq\mathcal{S}$ such that $\{B_{d_\infty}(x^j, 2r):j\in J\}$ is a disjoint collection of balls satisfying

\begin{align*} \mathcal{S}\subseteq \bigcup_{j\in J} B_{d_\infty}(x^j, 3r).\end{align*}

For each $j\in J$ , $x^j\in \mathcal{S}_r$ for $r\gt0$ . We can find an approximating sequence $x^j_i\in M_i$ of $x^j$ such that $d_\infty(\varphi_i(x_i),x)\to 0$ and for large enough i, we have

\begin{align*} \bigg(\int_{B_{g_i}(x^j_i,r)} |\mathrm{Rm}(g_{i,0})|^{n/2} d\mathrm{vol}_{g_{i,0}} \!\bigg)^{\!2/n} \gt \frac12\delta_1 .\end{align*}

By the disjoint property in X and the approximating property of $x^j_i$ ’s, we know that for large i, $\{B_{g_i}(x_i^j, r):j\in J\}$ is a disjoint collection in $M_i.$ Hence, in particular, we derive

\begin{align*} |J| \le (2\Lambda_0 \delta_1^{-1})^{n/2} ,\end{align*}

where $\Lambda_0$ is from the assumption of Theorem 1.3. This bound is independent of r and it particularly implies that

\begin{align*} |\mathcal{S}|\le (2\Lambda_0 \delta_1^{-1})^{n/2}.\end{align*}

This proves the claim.

To show that the topological space M induced by $(X\setminus \mathcal{S},d_\infty)$ is in fact a manifold with bi-Hölder charts, we must show that given an arbitrary $x\in X\setminus\mathcal{S}$ , there is a neighborhood of x that is bi-Hölder homeomorphic to an open subset of a complete Riemannian manifold. We follow the treatment in [Reference Simon and ToppingST21].

Let $x\in X\setminus\mathcal{S}$ . Then there exist $r\gt0$ and $x_i\in M_i$ such that $d_\infty(\varphi_i(x_i),x)\to 0$ and

\begin{align*}\|\mathrm{Rm}(g_{i,0})\|_{L^{n/2}(B_{g_{i,0}}(x_i,r))}\lt \delta_1\end{align*}

for large enough i. By considering $\tilde g_{i,0}:=(3r)^{-2}g_{i,0}$ , we might assume that $r=3$ . By Theorem 1.1 with $\bar r=0$ , there exists a Ricci flow $g_i(t)$ on $B_{g_{i,0}}(x_i,1)\times [0,\hat T]$ such that $g_i(0)=g_{i,0}$ and:

1. (1) $|\mathrm{Rm}(g_i(t))|\leq \Lambda_1 \delta_1 t^{-1}$ ;

2. (2) $\mathrm{inj}(g_i(t))\geq \sqrt{t}$ ;

3. (3) $\|\mathrm{Rm}(g_i(t))\|_{L^{n/2}(B_{g_i(t)}(x_i,1/2))}\leq \Lambda_1 \varepsilon $ ; and

4. (4) $\nu (B_{g_i(t)}(x_i,2L\sqrt{\hat T},g_i(t),L^2\hat T )\geq -\Lambda_1$

for all $(x,t)\in B_{g_{i,0}}(x_i,1)\times (0,\hat T]$ . Moreover, we might assume that for some $C_1(n,v_0,C_s)\gt0,\gamma(n,v_0,C_s)\in (0,1),$ we have

(4.1) \begin{align} C_1^{-1} (d_{g_i(t)}(x,y))^{1/\gamma}\leq d_{g_{i,0}}(x,y)\leq C_1 (d_{g_i(t)}(x,y))^{\gamma}\end{align}

for all $x,y\in B_{g_{i,0}}(x_i,1)$ and $t\in [0,\hat T]$ .

We now have all the necessary ingredients to carry out the proof in [Reference Simon and ToppingST21, Theorem 1.4 and Corollary 1.5]. We only give a sketch. We use Shi’s estimates and apply smooth compactness on $g_i(\hat T)$ to obtain a family of local diffeomorphisms $F_i$ such that $F_i^*g_i(t)$ sub-converges to $g_\infty(t)$ on $B_{g_\infty(\hat T)}(x_\infty,\sigma)$ for $\sigma\leq C_1^{-1}\sigma^{-\gamma}$ . Using $F_i$ , we might use the argument in the proof of [Reference Chan, Chen and LeeCCL22, Theorem 1.4] to show that $F_i^*d_{g_{i,0}}$ sub-converges to a distance function $\tilde d_\infty$ on $B_{g_\infty(\hat T)}(x_\infty,\sigma)$ that is locally bi-Hölder to the distance $d_{g_\infty(\hat T)}$ due to (4.1). The topological manifold structure in proximity to x follows from the uniqueness of the pointed Gromov–Hausdorff limits, which identifies $\tilde d_\infty$ in proximity to $x_\infty$ with $d_\infty$ in proximity to x.

5. Application to topological gap theorems

In this section, we will use Theorem 1.1 to prove the gap theorem in a differentiable structure of manifolds with small curvature concentration, i.e., Theorem 1.4. We start with the long-time existence. If the initial metric is assumed to have bounded curvature, this has been obtained earlier by [Reference Chau and MartensCM26a] (see also [Reference ChenChe22]).

1. (i) $|\mathrm{Rm}(g(t))|\leq \Lambda\varepsilon t^{-1}$ ;

2. (ii) $\mathrm{inj}(g(t))\geq \sqrt{ t}$ ;

3. (iii) $\|\mathrm{Rm}(g(t))\|_{L^{n/2}(M,g(t))}\leq \varepsilon$ ; and

4. (iv) $\nu(M,g(t))\geq -\Lambda$ .

Furthermore, we have $\sup_M|\mathrm{Rm}(g(t))|=o(1)/t$ as $t\to +\infty$ .

Using Theorem 5.1, the diffeomorphic gap theorem is immediate from the existence of Ricci flow and its estimates.

Proof of Theorem 1.4. This follows from Theorem 5.1 and [Reference Huang and PengHP26, Theorem 1.1] (see also [Reference He and LeeHL21, Theorem 1.1] and [Reference WangWan20, Theorem 5.7]).

Basic examples of manifolds satisfying the assumptions in Theorem 1.4 are compact perturbations of Euclidean metrics. By the rigidity case in positive-mass theorems, we observe that the additional condition on scalar curvature $\mathcal{R}\geq 0$ is strong enough to rule them out. Motivated by this, we conjecture the following.

1. (a) $C_{sob}(M)\geq C_s^{-1}$ ;

2. (b) $\mathrm{Vol}_{g_0}(B_{g_0}(x,r))\leq v_0r^n$ for all $(x,r)\in M\times \mathbb{R}_+$ ;

3. (c) $\|\mathrm{Rm}(g_0)\|_{L^{n/2}(M)}\leq \delta_0(n,C_s,v_0)$ ; and

4. (d) $\mathcal{R}(g_0)\geq 0$ .

Then $(M,g_0)$ is isometric to $\mathbb{R}^n$ .