Ricci flow smoothing for locally collapsing manifolds

Abstract

We show that for certain locally collapsing initial data with Ricci curvature bounded below, one could start the Ricci flow for a definite period of time. This provides a Ricci flow smoothing tool, with which we find topological conditions that detect the collapsing infranil fiber bundles over controlled Riemannian orbifolds among those locally collapsing regions with Ricci curvature bounded below. In the appendix, we also provide a local distance distortion estimate for certain Ricci flows with collapsing initial data.

Appendix A: Local distance distortion estimates for Ricci flows with collapsing initial data

The distance distortion estimates along Ricci flows is a crucial issue in view of its many natural applications (besides Lemma 4.1 in the proof of Theorem 1.4, see also e.g. [11] for a survey). In [29] a uniform distance distortion estimate for Ricci flows with collapsing initial data has been obtained. That estimate is for compact Ricci flow solutions and only compares the distance functions on nearby positive time slices. Here we present a new distance distortion estimate which can be seen both as an extension (to $t=0$) and a localization of the estimate in [29, Theorem 1.1] under some extra assumption on the Ricci curvature:

Theorem A.1

Given a positive integer m, positive constants ${\bar{C}}_0$, $C_R$, $T\le 1$ and $\alpha \in (0,\frac{1}{2(m-1)})$, there are constants $C_D({\bar{C}}_0,C_R,m)\ge 1$ and $T_D({\bar{C}}_0,C_R,m)\in (0,T]$ such that for an m-dimensional complete Ricci flow (M, g(t)) with bounded curvature for $t\in [0,T]$, if for some $x_0\in M$ and any $t\in [0,T]$ we have

$$\begin{aligned} {\mathbf {R}}_{g(0)}&\ge \ -C_R\ \text {in}\ B_{g(0)}(x_0,10), \end{aligned}$$

(A.1)

$$\begin{aligned} \left| {{\mathbf {R}}}{{\mathbf {c}}}_{g(t)}\right| _{g(t)}&\le \ 2(m-1)\alpha t^{-1}\ \text {in}\ B_{g(t)}\left( x_0,10+\sqrt{t}\right) , \end{aligned}$$

(A.2)

and the initial metric has a uniform bound ${\bar{C}}_0$ on the doubling and Poincaré constant for the geodesic ball $B_{g(0)}(x_0,10)$, then for any $x,y\in B_{g(0)}(x_0,\sqrt{T_D})$ and $t\in [0,T_D]$, we have

$$\begin{aligned} C_D^{-1}d_{g(0)}(x,y)^{1+4(m-1)\alpha }d_{g(0)}(x,y)\ \le \ d_{g(t)}(x,y)\ \le \ C_Dd_{g(0)}(x,y)^{1-4(m-1)\alpha }. \end{aligned}$$

(A.3)

The proof of the lemma is based on the theory of local entropy developed in [48, 49], as well as the local entropy lower bound by volume ratio obtained in [29]. It is inspired by the corresponding results for non-collapsing initial data in [28], and relies on a ball containment argument as surveyed in [11, §3].

Remark 9

Here we will rely on [41, Theorem 2.1], and we point out that the assumptions [41, (1) and (2)] are only needed locally, i.e. the same conclusion of [41, Theorem 2.1] holds even if these conditions on the doubling and Poincaré constants are assumed only within the geodesic ball $B_{g(0)}\left( x_0,10\right)$: notice that the only global result needed in the proof of this theorem is [41, Theorem 2.2], in which the constants involved only depend on the dimension of the manifold; whereas all other arguments leading to the application of [41, Theorem 2.2] only depend on the doubling and Poincaré constants within the geodesic ball in question.

Proof

By taking $r_0=\sqrt{t}$ in [39, Lemma 8.3] (see also [23, §17]), the Ricci curvature upper bound in (A.2) tells that if $x,y\in B_{g(t)}\left( x_0,10\right)$ whenever $t\le T$, then

$$\begin{aligned} d_{g(0)}(x,y)\ \le \ d_{g(t)}(x,y)+3(m-1)\sqrt{t}. \end{aligned}$$

(A.4)

To control the distance expansion, we first establish the following a priori estimate

Claim A.2

There exists a uniform constant ${\bar{C}}_1({\bar{C}}_0,C_R, m)>0$ such that if $x,y \in B_{g(t)}(x_0,10)$ for any $t\le t_0:=d_0^2$ with $d_0:=d_{g(0)}(x,y)\le \sqrt{T}$, then $d_{g(t_0)}(x,y)\le {\bar{C}}_1d_0$.

Proof of the claim

To see this, we let $\sigma :[0,1]\rightarrow B_{g(0)}\left( x,d_0\right)$ be a minimal g(0)-geodesic realizing $d_0$, and without loss of generality we may assume that $d_{g(t_0)}(x,y)\ge 10d_0$. Now let $\{\sigma (s_i)\}_{i=1}^N$ be a maximal collection of points on the image of $\sigma$ so that $d_{g(t_0)}(\sigma (s_i),\sigma (s_{j}))\ge 2d_0$ when $i\not =j$. By the maximality of $\left\{ \sigma (s_i)\right\}$, we see that $B_{g(t_0)}(\sigma (s_i), d_0)\cap B_{g(t_0)}(\sigma (s_j),d_0)=\emptyset$, and it is also clear that $\left\{ B_{g(t_0)}(\sigma (s_i),2d_0)\right\}$ cover $Image(\sigma )$. Especially, we have

$$\begin{aligned} d_{g(t_0)}(x,y)\le \sum _{i=1}^{N-1}d_{g(t_0)}(\sigma (s_i), \sigma (s_{i+1}))\ \le \ 4Nd_0. \end{aligned}$$

We therefore only need to bound N uniformly from above. Moreover, by (A.4) we see that for each $i=1,\ldots ,N$,

$$\begin{aligned} \forall y'\in B_{g(t_0)}(\sigma (s_i),d_0),\quad d_{g(0)}(\sigma (s_i),y')&\le \ d_{g(t_0)}(\sigma (s_i),y')+8(m-1)\sqrt{\alpha t_0}\\&\le \ (1+8(m-1)\sqrt{\alpha })d_0. \end{aligned}$$

Therefore, it is easily seen that each $B_{g(t_0)}(\sigma (s_i),d_0)\subset B_{g(0)}(\sigma (s_i),(1+8(m-1)\sqrt{\alpha })d_0)$, and consequently, as each $\sigma (s_i)\in B_{g(0)}(x,d_0)$, we have

$$\begin{aligned} \bigcup _{i=0}^kB_{g(t_0)}(\sigma (s_i),d_0)\ \subset \ B_{g(0)}(x, (2+8(m-1)\sqrt{\alpha })d_0). \end{aligned}$$

(A.5)

We now study the local entropy associated to the various metric balls. We first recall that the uniform bound ${\bar{C}}_0$ on the doubling and Poincaré constants gives a uniform bound on the Sobolev constant $C_S=C_S(m,{\bar{C}}_0)$, according to [41, Theorem 2.1] (see also Remark 9 and [29, Proposition 2.1]). Therefore, following the argument in [29, §3.1], we get to [29, (3.1)], and by (A.1) we have

$$\begin{aligned}&{\mathcal {W}}\left( B_{g(0)}(x,8md_0),g(0),v^2,\tau \right) \\&\quad \ge \ \log \left| B_{g(0)}(x,8md_0)\right| d_0^{-m} -\left( C_R+(8md_0)^{-2}\right) \tau -\frac{m}{2}\log \left( 512C_Sem^3\pi \right) . \end{aligned}$$

for any $\tau >0$ and any $v\in W^{1,2}_0(B_{g(0)}(x,8md_0))$ with $\int _{M}v^2\text {d}V_g=1$. Here due to the selection of v, we have the Perelman’s ${\mathcal {W}}$-entropy equal to the local entropy ${\mathcal {W}}\left( B_{g(0)}(x,8md_0),g(0),v^2,\tau \right)$, as defined in [48, §2]. Taking the infimum of ${\mathcal {W}} \left( B_{g(0)}(x,8md_0),g(0),v^2,\tau \right)$ among all admissible v described above, we see that for any $\tau >0$,

$$\begin{aligned}&\varvec{\mu }\left( B_{g(0)}(x,8md_0),g(0),\tau \right) \\&\quad \ge \ \log \left| B_{g(0)}(x,8md_0)\right| d_0^{-m} -\left( C_R+(8md_0)^{-2}\right) \tau -\frac{m}{2}\log \left( 512C_Sem^3\pi \right) . \end{aligned}$$

Consequently, for $\varvec{\nu }\left( B_{g(0)}(x,8md_0),g(0),\tau \right) =\inf _{s\in (0,\tau ]}\varvec{\mu }\left( B_{g(0)}(x,8md_0),g(0),s\right)$ we have

$$\begin{aligned} \begin{aligned}&\varvec{\nu }\left( B_{g(0)}(x,8md_0),g(0),\tau \right) \\&\quad \ge \ \log \left| B_{g(0)}(x,8md_0)\right| d_0^{-m} -\left( C_R+(8md_0)^{-2}\right) \tau -\frac{m}{2}\log \left( 512C_Sem^3\pi \right) . \end{aligned} \end{aligned}$$

(A.6)

On the other hand, by (A.2) we have $\left| {\mathbf {R}}_{g(t)}\right| \le 2m(m-1)\alpha t^{-1}$ for $t>0$, and applying [48, Theorem 3.6], we can bound the local entropy from above by volume ratio:

$$\begin{aligned} \varvec{\nu }\left( B_{g(t)}(\sigma (s_i),d_0),g(t),d_0^2\right) \ \le \ \log \frac{\left| B_{g(t)}(\sigma (s_i),d_0)\right| }{\omega _md_0^m} +\left( 2^{m+7}+2m(m-1)\alpha t^{-1}d_0^2\right) ; \end{aligned}$$

(A.7)

while by (A.2) and the effective monotonicity of the local entropy ([48, Theorem 5.4]), we see that

$$\begin{aligned} \varvec{\nu }\left( B_{g(t)}(\sigma (s_i),d_0),g(t),d_0^2 \right) \ \ge \ \varvec{\nu }\left( B_{g(0)}(\sigma (s_i),3d_0),g(0),d_0^2+t \right) -1. \end{aligned}$$

(A.8)

Moreover, since $B_{g(0)}(\sigma (s_i),3d_0)\subset B_{g(0)}(x,8md_0)$ for each $i=1,\ldots ,N$, applying [48, Proposition 2.1] and (A.6) we see that

$$\begin{aligned} \begin{aligned}&\varvec{\nu }\left( B_{g(0)}(\sigma (s_i),3d_0),g(0),\tau \right) \\&\quad \ge \ \log \left| B_{g(0)}(x,8md_0)\right| d_0^{-m} -\left( C_R+(8md_0)^{-2}\right) \tau -\frac{m}{2}\log \left( 512C_Sem^3\pi \right) . \end{aligned} \end{aligned}$$

(A.9)

We now obtain a uniform bound on N. Denoting $\varvec{\nu }_{i,s}(r,\tau ) :=\varvec{\nu } (B_{g(s)}(\sigma (s_i),r),g(s),\tau )$ for each $i=1,\ldots ,N$, $r\le 8md_0$, $s\le t_0$ and $\tau \le d_0^2+t_0$, we consecutively apply (A.7), (A.8) and (A.9) with $t_0=d_0^2$ to see that

$$\begin{aligned} \begin{aligned} \left| B_{g(t_0)}(\sigma (s_i),d_0)\right| d_0^{-m}&\ge \ \omega _me^{\varvec{\nu }_{i,t_0}\left( d_0,d_0^2\right) -2^{m+7}-2m(m-1)\alpha }\\&\ge \ \omega _m e^{\varvec{\nu }_{i,0}\left( 3d_0,d_0^2+t_0\right) -2^{m+7}-2m(m-1)\alpha -1}\\&\ge \ \frac{\omega _m e^{-2^{m+7}-2C_Rd_0^2-2m(m-1)\alpha -2}}{\left( 512C_S em^3\pi \right) ^{\frac{m}{2}}} \left| B_{g(0)}(x,8md_0)\right| d_0^{-m}. \end{aligned} \end{aligned}$$

(A.10)

Therefore, by the mutual disjointness of $\left\{ B_{g(t_0)}(\sigma (s_i),d_0)\right\}$ and (A.5) we have

$$\begin{aligned} \begin{aligned} \left| B_{g(0)}(x,8md_0)\right| _{g(0)}&\ge \ \sum _{i=1}^N\left| B_{g(t_0)}(\sigma (s_i),d_0)\right| _{g(0)}\\&\ge \ \sum _{i=1}^N\left| B_{g(t_0)}(\sigma (s_i),d_0)\right| _{g(t)}\\&\ge \ \sum _{i=1}^N C\left| B_{g(0)}(x,8md_0)\right| _{g(0)}, \end{aligned} \end{aligned}$$

(A.11)

where $C=\omega _m e^{-2^{m+7}-2C_Rd_0^2-2m(m-1)\alpha -2}(512C_S em^3\pi )^{-\frac{m}{2}}$. Consequently, we easily see that

$$\begin{aligned} N\ \le \ \omega _m^{-1}e^{2^{m+7}+C_R+2m(m-1) +2} (288C_S em\pi )^{\frac{m}{2}}+10\ =:\ \frac{1}{4}{\bar{C}}_1({\bar{C}}_0,C_R,m), \end{aligned}$$

(A.12)

since $\alpha <1$, and we emphasize that $C_S$ is solely determined by ${\bar{C}}_0$ and m. $\square$

Now let $T_0\le T$ be the first time when some point $y\in B_{g(0)}(x_0,\sqrt{T_0})$ sees $d_{g(T_0)}(x_0,y)\ge 10$, then applying the estimate established by the claim with $x_0,y$, we have $T_0\ \ge \ 100{\bar{C}}_1^{-2}=:T_D({\bar{C}}_0,C_R,m)$. Especially, for any $x,y\in B_{g(0)}(x_0,\sqrt{T_D})$, we have (A.4) holds without any extra assumption, and for any such points,

$$\begin{aligned} \forall 0<s<t\le T_D,\quad \left( \frac{s}{t}\right) ^{4(m-1)\alpha }\ \le \ \frac{d_{g(t)}(x,y)}{d_{g(s)}(x,y)}\ \le \ \left( \frac{t}{s}\right) ^{4(m-1)\alpha }. \end{aligned}$$

(A.13)

Therefore, reasoning in the same way as in [28, Appendix], we have

$$\begin{aligned} \forall x,y\in B_{g(0)}(x_0,\sqrt{T_D}),\quad d_{g(0)}(x,y) \le \ 3m d_{g(t)}(x,y)^{\frac{1}{4(m-1)\alpha +1}}. \end{aligned}$$

(A.14)

We now show the other side of the estimate following the same argument as in [28, Appendix]. Given $x,y\in B_{g(0)}(x_0,\sqrt{T_D})$ satisfying $d_0=d_{g(0)}(x,y)\le \sqrt{T_D}$ and given $t\le T_D$, we begin with setting $N:= \left\lceil d_0t^{-\frac{1}{2}}\right\rceil +1$ so that $d_0<N\sqrt{t} \le 2\sqrt{T}\le 2$. Dividing a minimal g(0)-geodesic $\sigma$ that realizes $d_0$ into N pieces of equal length, i.e. $\left| \sigma |_{[s_i,s_{i+1}]}\right| =N^{-1}d_0$ for $0=s_0<\cdots <s_N=1$, by the claim above we see that

$$\begin{aligned} \forall i=0,1,\ldots ,N-1,\quad d_{g(N^{-2}d_0^2)}(\sigma (s_i),\sigma (s_{i+1}))\ \le \ {\bar{C}}_1N^{-1}d_0; \end{aligned}$$

moreover, since $t>N^{-2}d_0^2$, by (A.13) we have

$$\begin{aligned} d_{g(N^{-2}d_0^2)}(x,y)\ \ge \ \left( N^{-2}d_0^{2}t^{-1}\right) ^{2(m-1)\alpha } d_{g(t)}(x,y). \end{aligned}$$

Now adding through $i=0,1,\ldots ,N-1$, by these inequalities we have

$$\begin{aligned} d_0&= \sum _{i=0}^{N-1}\left| \sigma |_{[s_i,s_{i+1}]}\right| \\&\ge \ {\bar{C}}_1^{-1}\sum _{i=0}^{N-1}d_{g(N^{-2}d_0^2)} (\sigma (s_i),\sigma (s_{i+1}))\\&\ge \ {\bar{C}}_1^{-1}d_{g(N^{-2}d_0)}(\sigma (0),\sigma (1))\\&\ge \ {\bar{C}}_1^{-1}\left( N^{-2}d_0^2t^{-1} \right) ^{2(m-1)\alpha } d_{g(t)}(x,y)\\&\ge \ {\bar{C}}_1^{-1}16^{-(m-1)\alpha }d_0^{4(m-1)\alpha } d_{g(t)}(x,y), \end{aligned}$$

and consequently, we have $d_{g(t)}(x,y)\le 16^m{\bar{C}}_1d_0^{1-4(m-1)\alpha }$.

Combining this with (A.14) we have the desired estimate

$$\begin{aligned} \forall x,y\in B_{g(0)}(x_0,\sqrt{T_D}),\ \forall t&\le T_D,\quad C_D^{-1}d_{g(0)}(x,y)^{1+4(m-1)\alpha }\\&\le \ d_{g(t)}(x,y)\ \le \ C_Dd_{g(0)}(x,y)^{1-4(m-1)\alpha }, \end{aligned}$$

where $C_D:=\max \left\{ (8m)^{4m},16^m{\bar{C}}_1\right\}$, only depending on ${\bar{C}}_0$ and $C_R$. $\square$

Arguing in the same way, if (A.2) can be assumed globally on M, then we have the following

Corollary A.3

With the same assumptions as in Theorem A.1, but with (A.2) replaced by

$$\begin{aligned} \forall t\le T,\quad \sup _M\left| {{\mathbf {R}}}{{\mathbf {c}}}_{g(t)}\right| _{g(t)}\ \le \ 2(m-1)\alpha t^{-1}, \end{aligned}$$

(A.15)

then we have for any $t\le T$ and $x,y\in B_{g(0)}(x_0,5)$ satisfying $d_{g(0)}(x,y)\le 1$,

$$\begin{aligned} C_D^{-1}d_{g(0)}(x,y)^{1+4(m-1)\alpha }\ \le \ d_{g(t)}(x,y)\ \le \ C_Dd_{g(0)}(x,y)^{1-4(m-1)\alpha }. \end{aligned}$$

(A.16)

Proof

By (A.15), we see that (A.4) holds for any $x,y\in B_{g(0)}(x_0,5)$, without any extra assumption, and thus Claim A.2 holds for all such points. Therefore, the rest of the arguments follow without needing to confine ourselves in a smaller geodesic ball. $\square$

Remark 10

The referee kindly pointed out to us that a similar distance distortion should hold under curvature bound and scaling invariant injectivity radius lower bond $r_{inj}\ge ct^{\frac{1}{2}}$ using a similar argument in the proof.