Ricci Flow of Regions with Curvature Bounded Below in Dimension Three

Abstract

We consider smooth complete solutions to Ricci flow with bounded curvature on manifolds without boundary in dimension three. Assuming an open ball at time zero of radius one has sectional curvature bounded from below by −1, then we prove estimates which show that compactly contained subregions of this ball will be smoothed out by the Ricci flow for a short but well-defined time interval. The estimates we obtain depend only on the initial volume of the ball and the distance from the compact region to the boundary of the initial ball. Versions of these estimates for balls of radius r follow using scaling arguments.

Appendix: Dimension of Gromov–Hausdorff Limits of Collapsing and Non-collapsing Spaces

We explain why some certain well-known properties of collapsing, respectively, non-collapsing manifolds, with curvature bounded from below hold. These properties follow from the results contained in [3] (see also [4]). Note that the definition of Alexandrov space with curvature bounded from below in [3] (Definition 2.3) and [4] (Proposition 10.1.1) agree.

Theorem 1.1

Let $(B_1(p_i),g_i)$, $(B_1(q_i),h_i)$, $i \in {\mathbb {N}}$ be balls whose closure is compactly contained in smooth Riemannian manifolds without boundary of dimension $n \in {\mathbb {N}}$ fixed. Assume that $\sec \ge -V$ on these balls and that $d_{GH}( (B_1(p_i),g_i), (B_1(q_i),h_i) ) \rightarrow 0$ as $i \rightarrow \infty$, and ${{\mathrm{vol}}}( (B_1(p_i),g_i)) \ge v_0>0$ for all $i \in {\mathbb {N}}$. Then it cannot be that ${{\mathrm{vol}}}(B_1(q_i),h_i) \rightarrow 0$ as $i \rightarrow \infty$.

Proof

Assume the theorem is false. We know that $(B_{1}(p_i),g_i,p_i)$ and $(B_1(q_i),h_i,q_i)$ Gromov–Hausdorff converge, after taking a subsequence, to the same space $(X=B_1(p),d,p)$ by the theorem of M. Gromov, and that (X, d, p) is an Alexandrov space (see Notes on Alexandrov Spaces below). Without loss of generality, we may assume that $\sec \ge -k^2$ on the balls we are considering, where $k^2>0$ is as small as we like. This can be seen as follows. Without loss of generality (renumber the indices i), we have ${{\mathrm{vol}}}(B_{1/i}(q_i),h_i) \le {{\mathrm{vol}}}(B_1(q_i),h_i) \le \frac{1}{i^{n+1}}.$ The Bishop–Gromov Comparison principle implies that ${{\mathrm{vol}}}(B_{1/i}(p_i),g_i) \ge c(v_0,n)\frac{1}{i^n}.$ Scaling both Riemannian metrics by $i^2$, we have (we also call the rescaled metrics $g_i$ and $h_i$) ${{\mathrm{vol}}}(B_{1}(p_i),g_i) \ge c(v_0,n)>0$ and ${{\mathrm{vol}}}(B_{1}(q_i),h_i) \le \frac{1}{i}$ and $\sec \ge - \frac{V}{i^2}.$ So we assume $\sec \ge -k^2$ with $k > 0$ arbitrarily small.

Let $\overline{B_{a}(y)} \subseteq B_{1-10a}(p)$ and let $\{B_{R/3}(s_j) \}_{j\in \{1, \ldots ,N \} }$ be any maximally pairwise disjoint collection of balls with $R<<a<1/(10)$ and centres $s_j$ in $B_{a}(y)$. By maximally pairwise disjoint we mean that if we try and add a ball $B_{R/3}(z)$ to the collection, where $z \in B_{a}(y)$, then the new collection is not pairwise disjoint. Then clearly $\{ B_{R}(s_j) \}_{j\in \{1, \ldots N\} }$ must cover $B_{a}(y)$. Let $\tilde{s}_j$, respectively, $\tilde{p} = p_i$, $\tilde{y}$ be the corresponding points in $(B_1(p_i),g_i,p_i)$ which one obtains by mapping $s_j$, respectively, p,y back to $(B_1(p_i),g_i,p_i)$ using the Gromov–Hausdorff approximation $f_i:(B_1(p),d,p) \rightarrow (B_1(p_i),g_i ,p_i)$: we write $p_i = \tilde{p}$, and so on, suppressing the dependence of the points on i sometimes, in order to make this explanation more readable. For i large enough, $\{ B_{2R}(\tilde{s}_j) \}_{j\in \{1, \ldots N\} }$ must cover $B_{a}(\tilde{y})$ and $\{B_{R/4}(\tilde{s}_j)\}_{j\in \{ 1,\ldots ,N\}}$ must be pairwise disjoint and contained in $B_{2a}(\tilde{y}) \subseteq B_1(\tilde{p})$. The Bishop–Gromov volume comparison principle implies that $\frac{c_2(v_0,a,n)}{R^n} \le N \le \frac{c_1(v_0,a,n)}{R^n},$ for some fixed $0<c_0(v_0,V,a,n),c_1(v_0,a,V,n) < \infty$ and hence the rough dimension of $B_{a}(y)$ (see Definition 6.2 in [3]) must be n.

This means that the Hausdorff dimension and burst index of $B_s(p)$ is also n for all $s<1$ (see Lemma 6.4 and Definition 6.1 in [3]). Assume $\varepsilon \le \frac{1}{1000n}$ in all that follows. Now let $z \in B_{1/4}(p)$ be a point for which there is an $(n,\varepsilon )$ explosion (Definition 5.2 in [3]: an $(n,\varepsilon )$ explosion is called an $(n,\varepsilon )$ strainer in [4], see Definition 10.8.9 there). Note that for any $\frac{1}{1000n} \ge \varepsilon >0$ such a point exists (see Corollary 6.7 in [3]). Let $(a_k,b_k)_{k \in \{1,\ldots n\}}$ be such an $(n,\varepsilon )$ explosion at z and assume that $a_k,b_k \in B_{s}(z)$ for all $k =1, \ldots ,n$ with $s<<1$: as pointed out in [3] (just after Definition 5.2), we can always make this assumption, see also Proposition 10.8.12 in [4]. Then there exists a small ball $B_r(z)$ such that $(a_k,b_k)_{k \in \{1,\ldots n\}}$ is an $(n,\varepsilon )$ explosion at x for all $x \in B_{r}(z)$ and $(a_k,b_k)_{k \in \{1,\ldots n\}}$ is in $B_{s}(p) \backslash B_{2 {\hat{r}}}(z)$ where $s>>{\hat{r}}>> r>0$: distance is continuous in X and comparison angles (which are measured in $M^2(-V):=$ hyperbolic space with curvature equal $-V$) change continuously as distances change continuously and stay away from zero (see [11], equation (44)). With $s>>{\hat{r}}>> r$, we mean $\frac{{\hat{r}}}{s}<< 1$ and $\frac{ r}{{\hat{r}}}<< 1.$ Going back to $(B_1(q_i),h_i,q_i)$ with our Gromov–Hausdorff approximation, we see (once again dropping dependence on i for readability) that there exists a ball $B_r(\tilde{z}) \subseteq B_{1/2}(q_i)$ and an explosion $(\tilde{a}_k,\tilde{b}_k)_{k \in \{1,\ldots n\}}$ in $B_{2s}(\tilde{z}) \backslash B_{\hat{r}}(\tilde{z})$ (if i is large enough) such that $(\tilde{a}_k,\tilde{b}_k)_{k \in \{1,\ldots n\}}$ is an $(n,4\varepsilon )$ explosion at x for all $x \in B_r(\tilde{z})$: once again, this follows from the fact that angle comparisons change continuously as distances change continuously and stay away from zero, and distance changes at most by $\delta (i)$, with $\delta (i) \rightarrow 0$ as $i \rightarrow \infty$, under our Gromov–Hausdorff approximation. There are no $((n+1),\varepsilon )$ explosions in $(B_{1}(q_i),h_i,q_i)$, as the Hausdorff dimension of the manifold (and hence the burst index) is n (see Theorem 5.4 in [3] or Proposition 10.8.15 in [4]). Fix $0<\varepsilon (n)<< \frac{1}{2000n}$. But then, using Theorem 5.4 in [3], see also Theorem 10.8.18 in [4], (more explicitly, using the proofs thereof) we see that there is a $\tilde{r} =\tilde{r}(n,r)>0$ and a bi-Lipschitz homeomorphism from $f: B_{\tilde{r}}( \tilde{z} ) \rightarrow f( B_{\tilde{r}}(\tilde{z})) \subseteq {\mathbb {R}}^n,$ where the bi-Lipschitz constant may be estimated by $\frac{1}{c(n)}d_i(x,y) \le |f(x) - f(y)| \le c(n) d_i(x,y)$ for some $c(n) >0$, and hence ${{\mathrm{vol}}}( B_1(q_i),h_i,q_i) \ge \varepsilon (n,r) >0$ for i large enough, as r, n do not depend on i. This shows, that after taking a subsequence, we must have ${{\mathrm{vol}}}(B_1(q_i),h_i,q_i) \ge \varepsilon (n,r)>0$. $\square$

Notes on Alexandrov Spaces

The fact that $(B_{1}(p_i),g_i,p_i)$ and $(B_1(q_i),h_i,q_i)$ Gromov–Hausdorff converge to some metric space $(X = B_1(p),d)$ after taking a subsequence follows from Gromov’s Convergence Theorem (we apply the theorem to the closed balls $\overline{B_{1-\frac{1}{i}}(p)} \subseteq B_1(p)$ with $i \in {\mathbb {N}}$, and then take a diagonal subsequence). See 10.7.2 in [4]. The limit space has the property that $\overline{B_s(p)}$ is complete for all $0<s<1$ (by construction), and $\overline{B_s(p)}$ is compact for all $0<s<1$, since it is also totally bounded (due to the Bishop–Gromov comparison principle: see the argument on the rough dimension of $\overline{B_a(y)}$ at the beginning of the proof above).

In order to guarantee that $(X=B_1(p),d,p)$ is an Alexandrov space, a local version of the Globalisation Theorem of Alexandrov–Toponogov–Burago–Gromov–Perelman (Theorem 3.2 in [3]) is necessary, as the spaces we are considering are not complete. Such a local version of the theorem exists, as pointed out in Remark 3.5 in [3]. Proofs of the Globalisation Theorem can be found in the book [1] and a similar proof, obtained independently, is given in the paper [10]. Examining the proofs of the Globalisation Theorem (in the case $\sec \ge -1$) in any of the proofs mentioned above, we see that the proofs are local. Examining any of the proofs mentioned above, we see that the following is true: if $(B_1(x_0),g)$ is compactly contained in a smooth manifold, and $\sec \ge -1$ on $(B_1(x_0),g)$ and $z \in B_1(x_0)$ has $d(x_0,z) = 1-r$, then the quadruple condition (or the hinge condition , or any of the other equivalent conditions, see sect. 2 in [3] or 8.2.1 in [1], or the discussion on page 3 of [10] to see why these conditions are equivalent) hold on the ball $B_{rc}(z) \subseteq B_1(x_0)$ for some fixed constant $0< c<< 1$ independent of z or r. Note that the space $(X=B_1(p),d)$ we obtain this way is locally intrinsic: for all $x \in X$, for all $z,q \in B_{\varepsilon }(x)$ for all $B_{5\varepsilon }(x) \subseteq B_{1-\alpha }(p)$ for all $1>\alpha ,\varepsilon >0$ there exists a length minimising geodesic between z and q which is contained in $B_{5\varepsilon }(x)$: see the proof of Theorem 2.4.16 in [4].