Curvature growth of some 4-dimensional gradient Ricci soliton singularity models

TL;DR

This paper investigates curvature estimates for 4-dimensional gradient Ricci soliton singularity models using advanced geometric analysis techniques, contributing to understanding their structure and behavior near singularities.

Contribution

It introduces new curvature estimates for 4D gradient Ricci solitons by applying Perelman's point selection, Cheeger-Naber's results, and topological lemmas.

Findings

Derived curvature bounds for 4D gradient Ricci solitons.

Enhanced understanding of singularity models in Ricci flow.

Established links between topology and curvature estimates.

Abstract

In this note we discuss estimates for the curvature of 4-dimensional gradient Ricci soliton singularity models by applying Perelman's point selection, a fundamental result of Cheeger and Naber, and topological lemmas.

Equations

$$\left|\mathrm{Rm}\right|\le 4Q_0\text{in }{B}_{A_0{Q}_0^{-1/2}}(x_0).$$

$$O_1\doteqdot \left|\mathrm{Rm}\right|\left(y_1\right)>4O_0.$$

$$O_k\ge 4^k{O}_0=4^k{P}_0$$

$$d\left(y_k,y_0\right)$$

$$<\sum _{\ell =0}^{k-1}{A}_0{O}_{\ell }^{-1/2}$$

$$\le A_0\sum _{\ell =0}^{k-1}{O}_0^{-1/2}{2}^{-\ell }$$

$$<2A_0{O}_0^{-1/2}.$$

$$\left|\mathrm{Rm}\right|\le 4Q_i\text{in }{B}_{A_i{Q}_i^{-1/2}}(x_i),\text{where }{A}_i=\frac{1}{3}{P}_i^{1/2}\to \infty .$$

$$\frac{\partial }{\partial t}{\phi }_t\left(x\right)=\frac{1}{1-\lambda t}\left({\nabla }_{g(0)}f(0)\right)\left({\phi }_t\left(x\right)\right),$$

$$\frac{\partial }{\partial t}g(t)=-2\mathrm{Rc}=2{\nabla }_{g(t)}^{2}f(t)-\frac{\lambda }{1-\lambda t}g(t).$$

$${\left|\frac{\partial }{\partial t}{\phi }_t\left(x\right)\right|}_{g(0)}\le r^{-1}$$

$$d_{g(0)}({\phi }_t\left(x\right),x)\le {\int }_t^0{\left|\frac{\partial }{\partial t}{\phi }_{\bar{t}}\left(x\right)\right|}_{g(0)}d\bar{t}\le \frac{r}{2}.$$

$$\left|\mathrm{Rm}\right|\le 4Q_i\text{in }{B}_{A_i{Q}_i^{-1/2}}(x_i),\text{where }{A}_i=\frac{1}{3}{P}_i^{1/2}\to \infty .$$

$$\mathrm{Vol}{B}_{Q_i^{-1/2}}(x_i)\ge \kappa Q_i^{-2}\text{for all }i\ge 1.$$

$$d_{g(0)}({\phi }_t\left(x\right),x)\le \left|t\right|\le Q_i^{-1}\le Q_i^{-1/2},$$

$$\left|{\nabla }^m\mathrm{Rm}\right|\le C_m{Q}_i^{1+\frac{m}{2}}\text{in }{B}_{\frac{1}{2}{A}_i{Q}_i^{-1/2}}^{g(0)}(x_i)\times [-\frac{1}{2}{Q}_i^{-1},0].$$

$$\left|\mathrm{Rm}\right|(x)\le C(d(x,o)+1{)}^2\text{for all }x\in \mathcal{M}.$$

$$\left|\mathrm{Rm}\right|\le 4Q_i\text{in }{B}_{A_i{Q}_i^{-1/2}}(x_i),\text{where }{A}_i=\frac{1}{3}{P}_i^{1/2}\to \infty .$$

$${\left|\frac{\partial }{\partial t}{\phi }_t\left(x\right)\right|}_{g(0)}$$

$$\le \frac{1}{1-t}{f}^{1/2}\left({\phi }_t\left(x\right),0\right)$$

$$\le \frac{1}{1-t}\left(d_{g(0)}({\phi }_t\left(x\right),o)+C\right)$$

$$\le d_{g(0)}({\phi }_t\left(x\right),o)+C$$

$$d_{g(0)}({\phi }_t\left(x\right),x)$$

$$\le {\int }_t^0\left(d_{g(0)}({\phi }_t\left(x\right),o)+C\right)dt$$

$$\le C{Q}_i^{-1}\left(d_{g(0)}(x_i,o)+C\right)$$

$$\ll Q_i^{-1/2}$$

$$\left|\mathrm{Rm}\right|(y_i)\ge c\text{for all }i,$$

$${\mathrm{Rc}}_{\infty }+{\nabla }^2{f}_{\infty }$$

$$|\nabla f_{\infty }{|}^2+R_{\infty }$$

$$c\le |\mathrm{Rm}|(y_i)\to |{\mathrm{Rm}}_{\infty }|(y_{\infty })\ge c>0.$$

$${\nabla }^2{f}_{\infty }$$

$$|\nabla f_{\infty }{|}^2$$

$$\left|\mathrm{Rm}\right|(y_i)\ge c(d(y_i,o)+1{)}^2\text{for all }i,$$

$$d_{g(0)}({\phi }_t\left(x\right),x)\le C{Q}_i^{-1}\left(d_{g(0)}(x_i,o)+C\right)\le C{Q}_i^{-1/2}.$$

$${\mathrm{Rc}}_{g_i}+{\nabla }_{g_i}^2{f}_i=\frac{1}{2}{Q}_i^{-1}{g}_i.$$

$${\left|\nabla f_i\right|}_{g_i}^2+R_{g_i}=\frac{f}{Q_i}\ge \frac{\inf \left\{f(y):y\in B_{A{Q}_i^{-\frac{1}{2}}}^g(x_i)\right\}}{C(d_g(x_i,o)+1{)}^2}\ge c>0,$$

Full text

1. Introduction

1.1. Definition of a singularity model

For a finite-time singular solution to the Ricci flow on a closed oriented manifold $(\mathcal{M}^{n},g(t))$, $t\in[0,T)$, $T<\infty$, we have $\sup_{M\times[0,T)}|\operatorname{Rm}|=\infty$. An associated singularity model $(\mathcal{M}_{\infty}^{n},g_{\infty}(t))$, $t\in(-\infty,0]$, is a complete ancient solution which is a limit of pointed rescalings. More precisely, there exists a sequence of space-time points $(x_{i},t_{i})$ in $\mathcal{M}\times[0,T)$ with $K_{i}\doteqdot\left|\operatorname{Rm}\right|(x_{i},t_{i})\rightarrow\infty$ such that the sequence of pointed solutions $(\mathcal{M},g_{i}(t),(x_{i},0))$, where $g_{i}(t)=K_{i}g(K_{i}^{-1}t+t_{i})$ and $t\in[-K_{i}t_{i},0]$, converges in the $C^{\infty}$ pointed Cheeger–Gromov sense to the complete ancient solution $(\mathcal{M}_{\infty},g_{\infty}(t),(x_{\infty},0))$, $t\in(-\infty,0]$, for some $x_{\infty}\in\mathcal{M}_{\infty}$. Note that $g_{\infty}(t)$ is not assumed to have bounded curvature on each time slice.

A folklore conjecture is that any singularity model must have bounded curvature. In dimension $3$, this is true by the work of Perelman [47]. Observe that it is not obvious that singularity models are necessarily of finite (topological) type. Neither is it obvious that singularity models are embeddable in the compact manifold from which they arise. However, if a singularity model has finite type, then it is embeddable in the original compact manifold and thus has an orientation induced by the embedding.

1.2. Classification of $3$-dimensional singularity models

There is now a complete classification of 3-dimensional singularity models. We shall use GRS as an abbreviation for gradient Ricci soliton. Firstly, $3$-dimensional noncollapsed shrinking GRS with bounded curvature have been classified by Hamilton [26] and by Perelman [48], who proved nonexistence in the noncompact positive sectional curvature case. Naber [45] showed that shrinking GRS with bounded curvature must be noncollapsed. It was shown that $3$-dimensional shrinking GRS must have bounded curvature via the works of Cao, Chen, and Zhu [12] and Ni and Wallach [46], with related works by Chen [16] and Petersen and Wylie [49]. Secondly, Brendle [8] proved the assertion by Perelman that the only $3$-dimensional nonflat noncollapsed steady GRS is the rotationally symmetric Bryant soliton. Thirdly, the works of Hamilton [26], Perelman [47], and Brendle [10], prove that any $3$-dimensional singularity model must be either a shrinking GRS or a steady GRS. In particular, Brendle [10] proved Perelman’s conjecture that any $3$-dimensional noncollapsed ancient solution with positive bounded sectional curvature must be a steady GRS. See Bamler and Kleiner [5] for a later, alternative treatment related to their work on strong stability of $3$-dimensional Ricci flow and their proof of the generalized Smale conjecture [3, 4]. Finally, one obtains from Hamilton, Perelman, and Brendle’s results that the possible $3$-dimensional oriented singularity models are classified as: a shrinking spherical space form $S^{3}/\Gamma$, a round cylinder $S^{2}\times\mathbb{R}$ or its $\mathbb{Z}_{2}$-quotient, or the Bryant soliton.

There has been much progress on the understanding of higher-dimensional shrinking GRS, largely due to the works of Munteanu and Wang (see e.g. [43, 44] and the references therein), with the strongest results in dimension $4$. Wylie [51] proved that the fundamental group is finite. Kotschwar and Wang proved an important uniqueness result [31]; see also [30, 32]. Haslhofer and Müller [27, 28] have proved precompactness theorems. Y. Li and B. Wang [36] proved an improved no local collapsing theorem for shrinking GRS.

For recent progress on steady GRS, see Appleton [1], Brendle [9], Cao and Cui [13], Deng and Zhu [21, 22, 23], Deruelle [24], Munteanu and Sesum [38], and Munteanu, Sung, and Wang [39], Munteanu and Wang [40, 41, 42], and the references therein. For the classification of $4$-dimensional shrinking GRS with positive isotropic curvature, see Li, Ni, and Wang [35]. Very recently, Chen, Deruelle, and Sun [20] proved that any $4$-real-dimensional Kähler shrinking GRS with scalar curvature tending to zero at infinity must be the Feldman–Ilmanen–Knopf shrinker [25].

In the case where the potential function is constant, a steady GRS is a Ricci-flat manifold. Assuming also that this manifold is a singularity model, by Perelman’s no local collapsing theorem it is necessarily $\kappa$-noncollapsed on all scales. In particular, it has Euclidean volume growth. In dimension 4, by a fundamental result of Cheeger and Naber [15], it must be a Ricci-flat asymptotically locally Euclidean (ALE) manifold. We remark that singularity models of 4-dimensional singular solutions with bounded scalar curvature have been shown to be such ALE manifolds by Bamler and Zhang [6] and Simon [50].

1.3. Main results

In this paper we study the curvature growth of 4-dimensional GRS singularity models.

Theorem 1**.**

Any $4$-dimensional steady GRS singularity model $(\mathcal{M},g,f)$ must have bounded curvature, that is, there exists a constant $C$ depending on the GRS such that $|\operatorname{Rm}|\leq C$ on $\mathcal{M}$.

For similar reasons, we have:

Theorem 2**.**

Any $4$-dimensional shrinking GRS singularity model must have curvature which grows at most quadratically. That is, for any $o\in\mathcal{M}$ there exists a constant $C$ depending only on the GRS and $o$ such that $|\operatorname{Rm}|(x)\leq C(d(x,o)+1)^{2}$ for all $x\in\mathcal{M}$.

The paper is organized as follows. In Section 2 we first recall Perelman’s point selection method on Riemannian manifolds, which is generally used to obtain limits. We then discuss under what conditions the local derivative of curvature estimates of Shi yield instantaneous estimates for GRS. Next, we prove the main Theorems 1 and 2 modulo the topological lemmas proved in Section 3.

For a survey of $4$-dimensional Ricci flow, see [18] by Gompf, Hillman, and two of the authors.

2. Curvature estimates for GRS singularity models

2.1. Point selection

The following is Perelman’s point selection method; see [47]. Since the method is crucial to our results, we include its proof for the sake of completeness.

Lemma 3**.**

Let $(\mathcal{M}^{n},g)$ be a complete Riemannian manifold. For any $y_{0}\in\mathcal{M}$ (let $P_{0}\doteqdot\left|\mathop{\mathrm{R}m}\right|(y_{0})$) and $A_{0}\in\mathbb{R}^{+}$ there exists $x_{0}\in B_{2A_{0}P_{0}^{-1/2}}(y_{0})$ such that $Q_{0}\doteqdot\left|\mathop{\mathrm{R}m}\right|(x_{0})\geq P_{0}$ and

[TABLE]

For example, we may choose $A_{0}=\frac{1}{3}P_{0}^{1/2}$, in which case we also have $A_{0}Q_{0}^{-1/2}\leq\frac{1}{3}$, so that $x_{0}\in B_{2/3}(y_{0})$ and $B_{A_{0}Q_{0}^{-1/2}}(x_{0})\subset B_{1}(y_{0})$.

Proof.

If (1) holds for $x_{0}=y_{0}$, then we are done. So suppose (1) does not hold for $x_{0}=y_{0}$. Let $O_{0}\doteqdot\left|\mathop{\mathrm{R}m}\right|\left(y_{0}\right)=P_{0}$. Since (1) does not hold for $x_{0}=y_{0}$, there exists $y_{1}\in B_{A_{0}O_{0}^{-1/2}}(y_{0})$ such that

[TABLE]

By induction, suppose that $y_{0},\ldots,y_{j}$ have been chosen such that (1) does not hold for $x_{0}=y_{k}$ for all $0\leq k\leq j-1$ and with $O_{k}\doteqdot\left|\mathop{\mathrm{R}m}\right|\left(y_{k}\right)$ for $0\leq k\leq j$ we have $y_{k+1}\in B_{A_{0}O_{k}^{-1/2}}(y_{k})$ satisfies $O_{k+1}>4O_{k}$ for $0\leq k\leq j-1$. Then, for $0\leq k\leq j$ we have

[TABLE]

and

[TABLE]

Hence $y_{k}\subset B_{2A_{0}O_{k}^{-1/2}}(y_{0})$ and $B_{A_{0}O_{k}^{-1/2}}(y_{k})\subset B_{3A_{0}O_{k}^{-1/2}}(y_{0})$ for $0\leq k\leq j$.

For the sequence $y_{0},y_{1},\ldots$ there exists a first $j_{0}$ for which (1) holds for $x_{0}=y_{j_{0}}$, for otherwise we would have an infinite sequence of points $\{y_{k}\}_{k=0}^{\infty}$ for which $\left|\mathop{\mathrm{R}m}\right|\left(y_{k}\right)\geq 4^{k}P_{0}$ and $y_{k}\in B_{2A_{0}O_{0}^{-1/2}}(y_{0})$ for all $k\geq 0$, a contradiction. The lemma follows from taking $x_{0}=y_{j_{0}}$ since $\left|\mathop{\mathrm{R}m}\right|\left(y_{j_{0}}\right)\geq 4^{j_{0}}P_{0}\geq P_{0}$ and since (3) implies $d\left(y_{j_{0}},y_{0}\right)<2A_{0}P_{0}^{-1/2}$.

To summarize, given any $A_{0}>0$ and point $y_{0}$, Perelman’s point selection method finds a nearby point $x_{0}$ such that $\left|\mathop{\mathrm{R}m}\right|$ in the ball centered at $x_{0}$ of scaled radius $A_{0}$ is bounded by $4$ times its value at $x_{0}$. This is effective since $A_{0}$ is arbitrary. So, given a sequence $\{y_{i}\}$ with $\left|\mathop{\mathrm{R}m}\right|(y_{i})\rightarrow\infty$, we may choose $\left\{A_{i}\right\}$ so that $A_{i}\rightarrow\infty$. In particular, as an immediate consequence of the lemma, we have:

Proposition 4**.**

Let $(\mathcal{M}^{n},g)$ be a complete Riemannian manifold. For any sequence $\{y_{i}\}_{i=1}^{\infty}$ in $\mathcal{M}$ with $P_{i}\doteqdot\left|\mathop{\mathrm{R}m}\right|(y_{i})\rightarrow\infty$ there exists a sequence $\{x_{i}\}_{i=1}^{\infty}$ in $\mathcal{M}$ such that for each $i\geq 1$ we have $x_{i}\in B_{2/3}(y_{i})$, $Q_{i}\doteqdot\left|\mathop{\mathrm{R}m}\right|(x_{i})\geq P_{i}$, and

[TABLE]

2.2. Instantaneous local derivative estimates

We have the following (instantaneous in time) local derivative estimates. This is useful for rescalings about points where $\left|\operatorname{Rm}\right|$ is bounded below by a positive constant.

Lemma 5**.**

Let $(\mathcal{M}^{n},g,f)$ be a complete steady or shrinking GRS. Suppose that $p\in\mathcal{M}$, $r>0$, and $C$ are such that in $B_{2r}(p)$ we have $\left|\mathop{\mathrm{R}m}\right|\leq Cr^{-2}$ and $|\nabla f|\leq r^{-1}$. Then $\left|\nabla^{m}\mathop{\mathrm{R}m}\right|\leq C_{m}r^{-2-m}$ in $B_{r}(p)$, where $C_{m}$ depends only on $m$, $C$, and $n$.

Proof.

Let $(\mathcal{M},g(t),f(t))$, $\lambda t<1$, be the canonical form associated to the steady GRS; see e.g. Theorem 4.1 in [19]. Here $\lambda=0$ in the steady case and $\lambda=1$ in the shrinking case, so that $\lambda\geq 0$. By hypothesis, $\left|\mathop{\mathrm{R}m}\right|(x,0)\leq Cr^{-2}$ for $x\in B_{2r}^{g(0)}(p)$. By the definition of the canonical form,

[TABLE]

$g(t)=(1-\lambda t)g(0)$, and $f(t)=f(0)\circ\varphi_{t}$ satisfy

[TABLE]

That is, $(\mathcal{M},g(t))$ is a solution to the Ricci flow evolving by scaling and by the pullback by a $1$-parameter family of diffeomorphisms. Hence, by the assumption that $|\nabla f(0)|_{g(0)}\leq r^{-1}$ in $B^{g(0)}_{2r}(p)$, we have the inequality

[TABLE]

whenever $\varphi_{t}(x)\in B^{g(0)}_{2r}(p)$ and $t\leq 0$. Hence, if $x\in B_{3r/2}^{g(0)}(p)$ and $t\in[-\frac{r^{2}}{2},0]$, then

[TABLE]

Thus $\varphi_{t}\left(x\right)\in B_{2r}^{g(0)}(p)$, so that $\left|\mathop{\mathrm{R}m}\right|(x,t)=\left|\mathop{\mathrm{R}m}\right|(\varphi_{t}(x),0)\leq Cr^{-2}$. By Shi’s local derivative of curvature estimates, $\left|\nabla^{m}\mathop{\mathrm{R}m}\right|(x,0)\leq C_{m}r^{-2-m}$ for $x\in B_{r}^{g(0)}(p)$.

The condition $|\nabla f|\leq r^{-1}$ puts some restriction on the scales $r$ for which we may apply this lemma. For instance, on a steady GRS it is only known that $|\nabla f|^{2}=1-R\leq 1$; hence this results works naturally for $0<r\leq 1$ (or less than any fixed constant). On the other hand, on a shrinking GRS, since we only have $|\nabla f|\leq\sqrt{f-R}\leq\frac{1}{2}d(o,x)+C$, this lemma works naturally for $0<r\lesssim d^{-1}(o,x)$. In particular, for the steady case, we have:

Corollary 6**.**

Let $(\mathcal{M}^{n},g,f)$ be a complete steady GRS with $\left|\mathop{\mathrm{R}m}\right|\leq Cr^{-2}$ in $B_{2r}(p)$, where $r\in(0,1]$. Then $\left|\nabla^{m}\mathop{\mathrm{R}m}\right|\leq C_{m}r^{-2-m}$ in $B_{r}(p)$, where $C_{m}$ depends only on $m$, $C$, and $n$.

2.3. Four-dimensional Ricci-flat ALE manifolds

We say that a complete noncompact oriented Riemannian $4$-manifold $(\mathcal{M}^{4},g)$ is asymptotically locally Euclidean (ALE) if there exists $\tau>0$, a compact subset $K$, a finite subgroup $\Gamma$ of $SO(4)$ acting freely on $S^{3}$, and an orientation-preserving diffeomorphism $\Phi:(\mathbb{R}^{4}-B_{C}(0))/\Gamma\rightarrow\mathcal{M}-K$ for some $C$ such that the pullback $\tilde{h}$ to the cover $\mathbb{R}^{4}-B_{C}(0)$ of $h=\Phi^{*}g$ satisfies $\partial_{I}(\tilde{h}_{jk}-\delta_{jk})=O(|x|^{-\tau-|I|})$ for each multi-index $I=(i_{1},i_{2},i_{3},i_{4})$. By Bando, Kasue, and Nakajima [7, Theorem 1.5], if an ALE manifold is Ricci flat, then there exists $\Phi$ with $\tau=4$.

It is conjectured that any simply-connected Ricci-flat ALE $4$-manifold must be hyperkähler, the latter of which has been classified by Kronheimer [33, 34]. For progress in this direction, see Lock and Viaclovsky [37]. Appleton [2] has shown that the Eguchi–Hanson space, which is a hyperkähler ALE, can occur as a singularity model of $4$-dimensional Ricci flow.

2.4. Steady GRS singularity models

Our first main result is the following.

Proposition 7**.**

If $(\mathcal{M}^{4},g,f)$ is a steady GRS which is also a singularity model, then $\left|\mathop{\mathrm{R}m}\right|$ is bounded.

Proof.

Suppose $\left|\mathop{\mathrm{R}m}\right|$ is not bounded. Then there exists $\{y_{i}\}_{i=1}^{\infty}$ in $\mathcal{M}$ with $P_{i}\doteqdot\left|\mathop{\mathrm{R}m}\right|(y_{i})\rightarrow\infty$. By Proposition 4, there exists a sequence $\{x_{i}\}_{i=1}^{\infty}$ in $\mathcal{M}$ such that for each $i\geq 1$ we have $x_{i}\in B_{2/3}(y_{i})$, $Q_{i}\doteqdot\left|\mathop{\mathrm{R}m}\right|(x_{i})\geq P_{i}\geq 100$, and

[TABLE]

Since $(\mathcal{M}^{4},g,f)$ is a singularity model, it is $\kappa$-noncollapsed on all scales for some $\kappa>0$. Hence (4) implies

[TABLE]

Let $(\mathcal{M}^{4},g(t),f(t))$, $t\in(-\infty,\infty)$, be the associated canonical form, where $g(t)=\varphi_{t}^{\ast}g$ and $f(t)=f\circ\varphi_{t}$. Since $\frac{\partial}{\partial t}\varphi_{t}\left(x\right)=\left(\nabla_{g(0)}f(0)\right)\left(\varphi_{t}\left(x\right)\right)$, we have $\left|\frac{\partial}{\partial t}\varphi_{t}\left(x\right)\right|_{g(0)}\leq 1$. Hence, if $x\in B_{(A_{i}-1)Q_{i}^{-1/2}}^{g(0)}(x_{i})$ and $t\in[-Q_{i}^{-1},0]$, then

[TABLE]

so that $\varphi_{t}\left(x\right)\in B_{A_{i}Q_{i}^{-1/2}}^{g(0)}(x_{i})$ and hence $\left|\mathop{\mathrm{R}m}\right|(x,t)=\left|\mathop{\mathrm{R}m}\right|(\varphi_{t}(x),0)\leq 4Q_{i}$. Hence, by Shi’s local derivative estimates, we have

[TABLE]

By the above and by Hamilton’s Cheeger–Gromov compactness theorem, there exists a subsequence such that $(\mathcal{M}^{4},Q_{i}g(Q_{i}^{-1}t),x_{i})$ converges in the $C^{\infty}$ pointed Cheeger–Gromov sense to a complete solution to the Ricci flow $(\mathcal{M}_{\infty}^{4},g_{\infty}(t),x_{\infty})$, $t\in[-\frac{1}{2},0]$, with $\left|\mathop{\mathrm{R}m}_{g_{\infty}}\right|\leq 4$ and $\left|\mathop{\mathrm{R}m}_{g_{\infty}}\right|(x_{\infty},0)=1$. In particular, $g_{\infty}(t)$ is nonflat. Since $(\mathcal{M}^{4},g,f)$ is a steady GRS, we have $0\leq R\leq 1$. This and $Q_{i}\rightarrow\infty$ imply that $R_{g_{\infty}}\equiv 0$, which in turn implies that $\mathop{\mathrm{R}c}_{g_{\infty}}\equiv 0$ by the Ricci flow equation $\frac{\partial R_{g_{\infty}}}{\partial t}=\Delta_{g_{\infty}}R_{g_{\infty}}+2\left|\mathop{\mathrm{R}c}_{g_{\infty}}\right|^{2}$.

By Perelman’s no local collapsing theorem and since $R_{g_{\infty}(0)}=0$ and we are on a singularity model (a limit of a limit is a limit), there exists $\kappa>0$ such that $\operatorname{Vol}_{g_{\infty}(0)}B^{g_{\infty}(0)}_{r}(x_{\infty})\geq\kappa r^{4}$ for all $r>0$ (see e.g. Theorems 28.6 and 28.9 in [17]);111This follows from Perelman’s no local collapsing using the scalar curvature and the fact that the rescaling factors tend to infinity. hence by definition $g_{\infty}(0)$ has Euclidean volume growth. By Cheeger and Naber’s theorem (Corollary 8.86 of [15]), $(\mathcal{M}_{\infty},g_{\infty}(0))$ is asymptotically locally Euclidean (ALE).

Since $(\mathcal{M}_{\infty}^{4},g_{\infty}(0))$ is a nonflat Ricci-flat ALE $4$-manifold, there exists a finite subgroup $\Gamma$ of $\mathop{\mathrm{S}O}(4)$ such that the end of $\mathcal{M}_{\infty}$ is diffeomorphic to $(\mathcal{S}^{3}/\Gamma)\times(0,1)$.222A nonflat Ricci-flat ALE $4$-manifold has only one end since otherwise it will split as the product of a line and a Ricci flat three-manifold, which in turn implies that it is flat. Let $\mathcal{S}^{3}/\Gamma\subset\mathcal{M}_{\infty}$ be embedded so that the noncompact region in $\mathcal{M}_{\infty}$ bounded by $\mathcal{S}^{3}/\Gamma$ is diffeomorphic to $(\mathcal{S}^{3}/\Gamma)\times[0,1)$. Let $\Omega$ denote the compact region in $\mathcal{M}_{\infty}$ bounded by $\mathcal{S}^{3}/\Gamma$. Then $\Omega^{\circ}$ is diffeomorphic to $\mathcal{M}_{\infty}$. By the definition of Cheeger–Gromov convergence, there exist an exhaustion $U_{i}$ of $\mathcal{M}_{\infty}$ and embeddings $\psi_{i}:U_{i}\rightarrow B_{\frac{1}{2}A_{i}Q_{i}^{-1/2}}^{g(0)}(x_{i})$. In particular, $\psi_{i}(\Omega)$ are embeddings of $\Omega$ with $\psi_{i}(\Omega)\subset B_{1}^{g(0)}(x_{i})$ for each $i$ sufficiently large. So we may assume that the $\psi_{i}(\Omega)$ are disjoint from each other in the steady GRS $\mathcal{M}$.

By hypothesis, there exists a solution to the Ricci flow on a closed $4$-manifold $(\mathcal{N}^{4},h(t))$, $t\in[0,T)$, $T<\infty$, for which the steady GRS $(\mathcal{M},g,f)$ is a singularity model. Hence there exist an exhaustion $\{V_{j}\}$ of $\mathcal{M}$ and embeddings $\phi_{j}:V_{j}\rightarrow\mathcal{N}$. In particular, for any $I\in\mathbb{N}$ there exists $j(I)$ such that $V_{j(I)}$ contains $\psi_{i}(\Omega)$ for all $1\leq i\leq I$. Then the $\Omega_{i}\doteqdot(\phi_{j(I)}\circ\psi_{i})(\Omega)$ are disjoint embeddings into $\mathcal{N}$ for $1\leq i\leq I$. Since the $\Omega_{i}^{\circ}$ are each diffeomorphic to $\mathcal{M}_{\infty}$ and are pairwise disjoint and since $I$ is arbitrary, we obtain a contradiction to Theorem 12 below.

2.5. Shrinking GRS singularity models

By a similar argument we can prove the following.

Proposition 8**.**

If $(\mathcal{M}^{4},g,f)$ is a shrinking GRS which is also a singularity model and $o\in\mathcal{M}$, then there exists a constant $C$ such that

[TABLE]

Proof.

Suppose there exists a sequence of points in the shrinking GRS $\{y_{i}\}_{i=1}^{\infty}$ in $\mathcal{M}$ with $P_{i}(d(y_{i},o)+1)^{-2}\rightarrow\infty$, where $P_{i}\doteqdot\left|\mathop{\mathrm{R}m}\right|(y_{i})$. By a previous proposition, there exists a sequence $\{x_{i}\}_{i=1}^{\infty}$ in $\mathcal{M}$ such that for each $i\geq 1$ we have $x_{i}\in B_{2/3}(y_{i})$, $Q_{i}\doteqdot\left|\mathop{\mathrm{R}m}\right|(x_{i})\geq P_{i}$, and

[TABLE]

Let $(\mathcal{M}^{4},g(t),f(t))$, $t\in(-\infty,1)$, be the associated canonical form, where $g(t)=\left(1-t\right)\varphi_{t}^{\ast}g$ and $f(t)=f\circ\varphi_{t}$. Since $\frac{\partial}{\partial t}\varphi_{t}\left(x\right)=\frac{1}{1-t}\left(\nabla_{g(0)}f(0)\right)\left(\varphi_{t}\left(x\right)\right)$, we have

[TABLE]

for $t\leq 0$ since $|\nabla(f^{1/2}(0))|_{g(0)}\leq 1$ by $R\geq 0$. Hence, if $x\in B_{\frac{3}{4}A_{i}Q_{i}^{-1/2}}^{g(0)}(x_{i})$ and $t\in[-Q_{i}^{-1},0]$, then

[TABLE]

since $Q_{i}(d_{g(0)}(x_{i},o)+1)^{-2}\gtrsim P_{i}(d_{g(0)}(y_{i},o)+1)^{-2}\rightarrow\infty$. Thus $\varphi_{t}\left(x\right)\in B_{A_{i}Q_{i}^{-1/2}}^{g(0)}(x_{i})$ and hence $\left|\mathop{\mathrm{R}m}\right|(x,t)=\left|\mathop{\mathrm{R}m}\right|(\varphi_{t}(x),0)\leq 4Q_{i}$. Hence we may apply Shi’s local derivative estimates and Hamilton’s Cheeger–Gromov compactness theorem to obtain that there exists a subsequence such that $(\mathcal{M}^{4},Q_{i}g(Q_{i}^{-1}t),x_{i})$ converges in the $C^{\infty}$ pointed Cheeger–Gromov sense to a complete solution to the Ricci flow $(\mathcal{M}_{\infty}^{4},g_{\infty}(t),x_{\infty})$, $t\in[-\frac{1}{2},0]$, with $\left|\mathop{\mathrm{R}m}_{g_{\infty}}\right|\leq 4$ and $\left|\mathop{\mathrm{R}m}_{g_{\infty}}\right|(x_{\infty},0)=1$. Since $(\mathcal{M}^{4},g,f)$ is a shrinking GRS, we have $0<R\leq f$. This and $Q_{i}f^{-1}(x_{i})\rightarrow\infty$ imply that $R_{g_{\infty}}\equiv 0$, which in turn implies that $\mathop{\mathrm{R}c}_{g_{\infty}}\equiv 0$ and so $g_{\infty}(0)$ is ALE. The remainder of the proof will be the same as in the steady GRS case.

2.6. Steady GRS models with curvature not decaying

Proposition 9**.**

Let $(\mathcal{M}^{4},g,f)$ be a steady GRS which is also a singularity model. If there exist a constant $c>0$ and a sequence $y_{i}\rightarrow\infty$ such that

[TABLE]

then $(\mathcal{M},g,y_{i})$ subconverges to a non-Ricci-flat steady GRS with bounded curvature.

Proof.

Observe that we may assume that $(\mathcal{M}^{4},g)$ is not Ricci flat, since otherwise there does not exist a sequence $\{y_{i}\}$ as in the hypothesis by Cheeger and Naber’s aforementioned theorem. Since the singularity model has bounded curvature by Proposition 7, it follows from Shi’s local derivative estimates (or Lemma 5) that the covariant derivative of curvature of each order is uniformly bounded. Consider the sequence $\{(\mathcal{M},g,f_{i},y_{i})\}_{i=1}^{\infty}$, where $f_{i}(x)=f(x)-f(y_{i})$. We have $|\nabla f_{i}|=|\nabla f|=\sqrt{1-R}$ and $|\nabla^{2}f|=|\mathop{\mathrm{R}c}|$ are both uniformly bounded. By applying Shi’s Bernstein-type estimates and by the covariant derivatives of curvature bounds, it is not hard to show that the covariant derivatives of $f_{i}$ of each order are bounded independent of $i$. Taking into account the noncollapsing condition, we may extract a subsequence from $\{(\mathcal{M},g,f_{i},y_{i})\}_{i=1}^{\infty}$ which converges in the pointed smooth Cheeger–Gromov sense to $(\mathcal{M}_{\infty}^{4},g_{\infty},f_{\infty},y_{\infty})$. It then follows from the smooth convergence that

[TABLE]

The limit is evidently a steady GRS. It is also nonflat since

[TABLE]

To see that it is not Ricci flat, we assume for a contradiction that $g_{\infty}$ is Ricci flat. Then immediately we have

[TABLE]

Hence $(\mathcal{M}_{\infty},g_{\infty})$ splits as the product of a line and a three-dimensional Ricci-flat manifold, which is flat; a contradiction.

2.7. Shrinking GRS models with quadratic curvature growth

The following result is proved using similar methods.

Proposition 10**.**

Let $(\mathcal{M}^{4},g,f)$ be a shrinking GRS which is also a singularity model and let $o\in\mathcal{M}$. If there exist a constant $c>0$ and a sequence $y_{i}\rightarrow\infty$ such that

[TABLE]

then there exists $x_{i}\rightarrow\infty$ with associated rescalings limiting to a non-Ricci-flat steady GRS with bounded curvature.

Proof.

Let $\left\{x_{i}\right\}$, $Q_{i}$, and $A_{i}$ be as in Proposition 4. By (5), we have that if $x\in B_{\frac{3}{4}A_{i}Q_{i}^{-1/2}}^{g(0)}(x_{i})$ and $t\in[-Q_{i}^{-1},0]$, then

[TABLE]

Since $A_{i}\rightarrow\infty$, we have $\varphi_{t}\left(x\right)\in B_{A_{i}Q_{i}^{-1/2}}^{g(0)}(x_{i})$ for $i$ sufficiently large. As in the proof of Proposition 8, there exists a subsequence such that $(\mathcal{M},Q_{i}g(Q_{i}^{-1}t),x_{i})$ converges in the $C^{\infty}$ pointed Cheeger–Gromov sense to a complete solution to the Ricci flow $(\mathcal{M}_{\infty}^{4},g_{\infty}(t),x_{\infty})$, $t\in[-\frac{1}{2},0]$, with $\left|\mathop{\mathrm{R}m}_{g_{\infty}}\right|\leq 4$ and $\left|\mathop{\mathrm{R}m}_{g_{\infty}}\right|(x_{\infty},0)=1$. Let $g_{i}=Q_{i}g$ and let $f_{i}(x)=f\left(x\right)-f\left(x_{i}\right)$. We have

[TABLE]

For any $A>0$ and for any $x\in B_{A}^{g_{i}}(x_{i})$, we have

[TABLE]

for all $i$ large enough, where $c>0$ is independent of $A$ and where we have used the lower bound for the potential function of Cao and Zhou [14]. Furthermore, we have

[TABLE]

[TABLE]

and

[TABLE]

Hence, by taking $i\rightarrow\infty$ we obtain $f_{i}\rightarrow f_{\infty}$ in $C^{\infty}$ with respect to the Cheeger–Gromov convergence of $g_{i}$ to $g_{\infty}=g_{\infty}(0)$, where

[TABLE]

Notice that the left-hand-side of the first equation is constant because the steady GRS equation holds by the second equation; it follows from (6) that this constant is positive. According to our point selection procedure, the limit steady GRS $(\mathcal{M}^{4}_{\infty},g_{\infty},f_{\infty})$ is nonflat and has bounded curvature. To see that it is non-Ricci-flat, we assume the contrary and obtain

[TABLE]

It follows immediately that $(\mathcal{M}_{\infty},g_{\infty})$ splits as the product of a line and a three-dimensional Ricci-flat manifold and hence $g_{\infty}$ must be flat; this is a contradiction.

One would like to show that steady GRS limits as in the proposition above are not possible, in which case it would follow for a 4-dimensional shrinking GRS singularity model that $|\mathop{\mathrm{R}m}|=\mathop{\mathrm{o}}(d^{2})$ (this would imply finite topological type).

Remark 11**.**

For the exclusion of Ricci-flat ALE limits in the proofs of Propositions 9 and 10, we may alternatively use Theorem 12 below.

3. Excluding an unbounded number of copies of a

Ricci-flat ALE 4-manifold

3.1. Statement of the result

In this section we prove the result which we applied in the previous section to exclude there existing an unbounded number of disjoint copies of a Ricci-flat ALE 4-manifold in a closed 4-manifold.

Theorem 12**.**

If $\mathcal{N}$ is a closed $4$-manifold and if $\mathcal{A}$ is a Ricci-flat ALE $4$-manifold, then there can exist at most a bounded number of disjoint copies of $\mathcal{A}$ embedded in $\mathcal{N}$.

3.2. Spherical space form ends of $4$-manifolds having an unbounded number of disjoint embeddings

We first prove the following.333One of the many references for the standard results from algebraic topology used in this section is Hatcher’s book [29].

Proposition 13**.**

Let $\mathcal{M}_{0}$ be a noncompact $4$-manifold without boundary and with a single end diffeomorphic to $S^{3}/\Gamma\times[0,\infty)$, where $S^{3}/\Gamma$ is a spherical space form. If there exists a compact $4$-manifold $\mathcal{N}$ containing an unbounded number of disjoint copies of $\mathcal{M}_{0}$, then $H_{1}(S^{3}/\Gamma,\mathbb{Z})$ is a direct double, i.e., isomorphic to $A\oplus A$ for some abelian group $A$.

Let $\mathcal{M}_{0}$ be as in the hypotheses of the proposition. Add $S^{3}/\Gamma$ to $\mathcal{M}_{0}$ as its boundary to obtain a compact manifold $\mathcal{M}$ with boundary $\partial\mathcal{M}=S^{3}/\Gamma\doteqdot\partial$; so, $\mathcal{M}_{0}$ is the interior of $\mathcal{M}$.

Lemma 14**.**

Under the hypotheses of Proposition 13, the homomorphism $H_{m}(\partial;R)\rightarrow H_{m}(\mathcal{M};R)$ induced by $\partial\hookrightarrow\mathcal{M}$ is onto for $m=0,1,2,3$ and any coefficient ring $R$. The cases $R\cong\mathbb{Z}$ and $R\cong\mathbb{Z}_{p}$ will be useful for us. We suppress writing the coefficients, $R$, in the proof.

Proof.

This is clear for $m=0$. Applying Mayer–Vietoris to one copy of $\mathcal{M}$ and to the complement of its interior $\overline{\mathcal{N}-\mathcal{M}}$, which have intersection $\partial$, we obtain the following exact sequence:

[TABLE]

where $i:\partial\hookrightarrow\mathcal{M}$, $j:\partial\hookrightarrow\overline{\mathcal{N}-\mathcal{M}}$, $k:\mathcal{M}\hookrightarrow\mathcal{N}$, and $l:\overline{\mathcal{N}-\mathcal{M}}\hookrightarrow\mathcal{N}$ are the inclusion maps. Suppose that $i_{\ast}:H_{m}(\partial)\rightarrow H_{m}(\mathcal{M})$ is not onto. Then there exists a class $\alpha\in H_{m}(\mathcal{M})$ which is not in the image of $i_{\ast}$. Thus, for any such $\alpha$ and any $\beta\in H_{m}(\overline{\mathcal{N}-\mathcal{M}})$, we have $k_{\ast}(\alpha)-l_{\ast}(\beta)\neq 0$.

Thus the cokernel of $i_{*}$, $\operatorname{coker}(i_{*})=H_{m}(\mathcal{M})/i_{*}(H_{m}(\partial))$, injects into $H_{m}(\mathcal{N})/l_{*}(H_{m}(\overline{\mathcal{N}-\mathcal{M}})$ by the map induced by $k_{*}$. Indeed, firstly observe that this map is well defined since $k_{*}(i_{*}(H_{m}(\partial)))=l_{*}(j_{*}(H_{m}(\partial))\subset l_{*}(H_{m}(\overline{\mathcal{N}-\mathcal{M}})$. Secondly, suppose that there is a nonzero coset $\alpha\,i_{*}(H_{m}(\partial))$ which maps to zero. Then we have that $\alpha\in H_{m}(\mathcal{M})$ is not in the image of $i_{\ast}$ and that $k_{*}(\alpha)\in l_{*}(H_{m}(\overline{\mathcal{N}-\mathcal{M}}))$, which is a contradiction.

Now suppose that $\mathcal{N}$ contains a finite sequence of disjoint embedded copies of $\mathcal{M}$, denoted by $\{\mathcal{M}_{p}\}_{p=1}^{I}$, for $I$ arbitrarily large. Let $\mathcal{M}^{I}=\bigcup_{p=1}^{I}\mathcal{M}_{p}$ and $\partial^{I}=\partial\mathcal{M}^{I}=\cup_{p=1}^{I}\partial\mathcal{M}_{p}$. Let $i^{I}:\partial^{I}\hookrightarrow\mathcal{M}^{I}$ and $l^{I}:\overline{\mathcal{N}-\mathcal{M}^{I}}\hookrightarrow\mathcal{N}$ denote the inclusion maps. Since, in the argument above we may replace $\mathcal{M}$ by $\mathcal{M}^{I}$ (we did not assume connectedness), we have that

[TABLE]

injects into $H_{m}(\mathcal{N})/l_{*}^{I}(H_{m}(\overline{\mathcal{N}-\mathcal{M}^{I}})$, where $i_{p}:\partial\mathcal{M}_{p}\hookrightarrow\mathcal{M}_{p}$ is the inclusion map. In this way, we obtain an injection from modules of increasing rank into $H_{m}(\mathcal{N})/l_{*}^{I}(H_{m}(\overline{\mathcal{N}-\mathcal{M}^{I}})$. This contradicts the finite dimensionality of $H_{m}(\mathcal{N})$.

Remark 15**.**

One may easily picture the workings of the preceding formal argument: If there is any class $\alpha$ (of any dimension $m$) in a copy of $\mathcal{M}$ in $\mathcal{N}$ which does not come from the boundary $\partial$, it is effectively ‘locked into’ that copy of $\mathcal{M}$; such classes can neither die in $\mathcal{N}$ nor participate in relations among themselves, because intersecting such null-homologies or homologies (respectively) with $\partial$ would yield a cycle in $\partial$ mapping onto $\alpha$, contrary to assumption. In other words, the intuition is that the cokernel in each copy of $\mathcal{M}$ is isolated like an island and cannot have any relation with other classes.

By Lemma 14, the long exact sequence of the pair $(\mathcal{M},\partial)$ yields the three short exact sequences. Setting $R\cong\mathbb{Z}$, we have:

[TABLE]

Since $\partial$ is a spherical space form, $H_{1}(\partial;\mathbb{Z})$ is torsion, and by (7) so is $H_{1}(\mathcal{M};\mathbb{Z})$. By Poincaré duality and the cohomology universal coefficient theorem (see [29, Theorem 3.2 and Corollary 3.3]), we have that $H_{2}(\partial;\mathbb{Z})\cong H^{1}(\partial;\mathbb{Z})\cong\mathbb{Z}^{b_{1}(\partial)}\oplus\operatorname{Tor}H_{0}(\partial,\mathbb{Z})\cong 0$, where $b_{1}(\partial)$ is the first Betti number of $\partial$. Thus (8) implies

[TABLE]

We also have that $H_{3}(\partial;\mathbb{Z})\cong\mathbb{Z}$, but maps to zero in $H_{3}(\mathcal{M};\mathbb{Z})$ by (9) since $H_{4}(\mathcal{M},\partial;\mathbb{Z})\rightarrow H_{3}(\partial;\mathbb{Z})$ is an isomorphism. Thus $H_{3}(\mathcal{M};\mathbb{Z})\cong 0$.

By Lemma 14 with $R=\mathbb{Z}_{p}$, that is, by (7) with coefficients $\mathbb{Z}_{p}$ instead of $\mathbb{Z}$, we have

[TABLE]

By Lefschetz duality (see [29, Theorem 3.43]), $H_{2}(\mathcal{M},\partial;\mathbb{Z}_{p})\cong H^{2}(\mathcal{M};\mathbb{Z}_{p})$. By the cohomology universal coefficient theorem,

[TABLE]

As observed below (9), $H_{1}(\mathcal{M};\mathbb{Z})$ is torsion; thus

[TABLE]

But this agrees with the usual444See for example the computations on the Wikipedia page Ext functor or Hatcher’s book [29]. According to these formulae, $\operatorname{Ext}^{1}_{\mathbb{Z}}(\mathbb{Z},\mathbb{Z}_{p})\cong 0$ and $\operatorname{Ext}^{1}_{\mathbb{Z}}(\mathrm{torsion},\mathbb{Z}_{p})\cong\operatorname{Hom}_{\mathbb{Z}}(\mathrm{torsion},\mathbb{Z}_{p})$. We use these formulae and the fact that $\operatorname{Ext}$ commutes with direct sums. formula which calculates $\operatorname{Ext}^{1}_{\mathbb{Z}}H_{1}(\mathcal{M};\mathbb{Z}),\mathbb{Z}_{p})$. Putting this together with (12) yields:

[TABLE]

These observations applied to (11) yield:

[TABLE]

To bring this information back towards integer coefficients, apply the homology universal coefficient theorem, for $X=\mathcal{M}$ and $\partial$:

[TABLE]

to get

[TABLE]

This still does not quite prove Proposition 13. We need to exclude possibilities like $H_{1}(\partial;\mathbb{Z})\cong\mathbb{Z}_{4}\oplus\mathbb{Z}_{2}$ and $H_{1}(\mathcal{M};\mathbb{Z})\cong\mathbb{Z}_{2}$. This can be done by returning to line (7): $\operatorname{order}H_{1}(\partial;\mathbb{Z})=(\operatorname{order}H_{2}(\mathcal{M},\partial;\mathbb{Z}))\times(\operatorname{order}H_{1}(\mathcal{M};\mathbb{Z}))$, but $H_{2}(\mathcal{M},\partial;\mathbb{Z})\cong H^{2}(\mathcal{M};\mathbb{Z})\cong\operatorname{Ext}^{1}_{\mathbb{Z}}(H_{1}(\mathcal{M};\mathbb{Z}),\mathbb{Z})\cong\operatorname{torsion}H_{1}(\mathcal{M};\mathbb{Z})\cong H_{1}(\mathcal{M};\mathbb{Z})$, so $\operatorname{order}H_{1}(\partial;\mathbb{Z})=(\operatorname{order}H_{1}(\mathcal{M};\mathbb{Z}))^{2}$. Proposition 13 follows from this, (14), and the fundamental theorem of finite abelian groups. $\Box$

3.3. Proof of Theorem 12

We may now prove Theorem 12 using Bamler’s argument in §6 of [52]. For a contradiction, suppose that there exists a nonflat Ricci-flat ALE $4$-manifold $\mathcal{A}$ that admits unbounded collections of disjoint embeddings in a compact $4$-manifold $\mathcal{N}$. By Proposition 13, the spherical space form $S^{3}/\Gamma$, $\Gamma$ nontrivial, of the end of $\mathcal{A}$ satisfies $H_{1}(S^{3}/\Gamma,\mathbb{Z})\cong A\oplus A$ for some abelian group $A$. This implies (see e.g. Lemma 6.3 in [52]) that either $\Gamma$ is the binary dihedral group $D_{n}^{*}$ with $n$ even or $\Gamma$ is the binary icosahedral group of order $120$. Now, by Lemma 6.5 in [52], $b_{2}(\mathcal{A})\geq 1$, which contradicts (10). $\Box$

Remark 16**.**

We may also rule out the case where $\Gamma$ is the binary icosahedral group $I^{*}$ of order $120$, that is, where $\partial=S^{3}/I^{*}$ is the Poincaré homology sphere, as follows. By Rochlin’s theorem, if $\partial\mathcal{M}=S^{3}/I^{*}$, then $H_{*}(\mathcal{M};\mathbb{Z})\not\cong H_{*}(\operatorname{pt};\mathbb{Z})$. As in Lemma 14, the facts that $H_{*}(S^{3}/I^{*};\mathbb{Z})\cong H_{*}(S^{3};\mathbb{Z})$ and $H_{*}(\mathcal{M};\mathbb{Z})\not\cong H_{*}(\operatorname{pt};\mathbb{Z})$ imply, via Mayer–Vietoris, that $H_{*}(\mathcal{N};\mathbb{Z})$ is infinitely generated, a contradiction.

Acknowledgment. We would like to thank the referee for a number of helpful suggestions.