On conical asymptotically flat manifolds

In this paper we study conical asymptotically flat ($\mathcal{AF}$) manifolds. By definition, a complete non-compact $m$ dimensional Riemannian manifold $(M^{m},g,p)$ is called

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  $\mathcal{AF}$ if the Riemannian curvature decays at the rate $|Rm_{g}|=o(r^{-2})$ as $r\rightarrow\infty$, where $r$ denotes the distance to $p$; in other words, the asymptotic curvature

  $$A(M,g)\equiv\limsup_{r\rightarrow\infty}r^{2}|Rm_{g}|$$

  is zero.

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  conical if it is asymptotic to a unique metric cone $\mathcal{C}_{\infty}$ at infinity, i.e., $(M,\lambda^{-2}g,p)$ converges in the pointed Gromov-Hausdorff topology to $\mathcal{C}_{\infty}$ as $\lambda\rightarrow\infty$. Here the asymptotic cone $\mathcal{C}_{\infty}$ may have a lower dimension.

We need to make a few remarks about the terminologies here. First the words “asymptotically flat” may refer to properties of varying generality in the literature. The notion of $\mathcal{AF}$ we use here is the same as the notion of asymptotically flat used in [35]. A related notion is what we shall call strongly $\mathcal{AF}$, which means that $|Rm_{g}|$ is bounded by a positive decreasing function $f(r)$ with $\int_{0}^{\infty}rf(r)dr<\infty$. An even stronger condition is that of faster than quadratic curvature decay, which means that $|Rm_{g}|\leq r^{-2-\epsilon}$ for some $\epsilon>0$ as $r\rightarrow\infty$. We also try to distinguish the notation $\mathcal{AF}$ from AF, which usually refers to being asymptotic to a specific flat model end (see Section 4).

There has been extensive work in Riemannian geometry studying topological and geometrical properties of $\mathcal{AF}$ manifolds. Recall that flat ends of Riemannian manifolds were completely classified by Eischenburg-Schroeder [18]. Ends of $\mathcal{AF}$ manifolds are much more flexible; for example, any 2 dimensional non-compact surface admits an $\mathcal{AF}$ metric [1]. It is known [23, 31] that strongly $\mathcal{AF}$ manifolds are automatically conical. However there are interesting examples of conical $\mathcal{AF}$ manifolds which are not strongly $\mathcal{AF}$. Examples include the family of ALG^∗ hyperkähler metrics [21].

Our interest in $\mathcal{AF}$ manifolds is motivated by the fact that they provide potential model ends for complete Ricci-flat metrics. In 4 dimensions a complete non-compact Ricci-flat manifold with $\int|Rm_{g}|^{2}<\infty$ is called a gravitational instanton. The readers should be warned that there are varying definitions of gravitational instantons in the literature. Some require the extra assumption of being hyperkähler, but we do not make this hypothesis in the current paper. All known examples of gravitational instantons are conical and they are all $\mathcal{AF}$ except the family of ALH^∗ hyperkähler metrics (see [21, 38]). The latter are known to have nilptoent asymptotic geometries.

For general conical $\mathcal{AF}$ manifolds, in 2001 Petrunin and Tuschmann [35] proved a structural result regarding the end structure. In particular, they proved that there are only finitely many ends and a simply connected end must have asymptotic cone the flat $\mathbb{R}^{m}$ if $m\neq 4$ and the flat $\mathbb{R}^{4},\mathbb{R}^{3}$ or the half plane $\mathbb{H}\equiv\mathbb{R}\times[0,\infty)$ when $m=4$. It follows that when $m\neq 4$ the metric has Euclidean volume growth and there is no collapsing phenomena along the convergence to the asymptotic cone. But when $m=4$ collapsing may indeed occur so the situation is more interesting. Clearly $\mathbb{R}^{4}$ can be realized. Also $\mathbb{R}^{3}$ can be realized as the asymptotic cone of the Taub-NUT gravitational instanton (see also [39]), whose end is diffeomorphic to $S^{3}\times[0,\infty)$. Along the convergence to the asymptotic cone $\mathcal{C}_{\infty}$, the $S^{3}$ collapses along the Hopf fibration to $S^{2}$. For the half plane $\mathbb{H}$, we know that it can be realized as the asymptotic cone of the flat space $\mathbb{R}^{4}/\mathbb{Z}$ (and of some members of the Kerr family of gravitational instantons), where the generator of the $\mathbb{Z}$-action is given by a translation in the first $\mathbb{R}^{2}$ factor and an irrational rotation in the second $\mathbb{R}^{2}$ factor. The end is diffeomorphic to $(S^{1}\times S^{2})\times[0,\infty)$; in particular it is not simply-connected. Petrunin-Tuschmann further made the following conjecture

Notice that there is no curvature equation imposed. It is easy to see that if a conical $\mathcal{AF}$ 4-manifold has an end with asymptotic cone $\mathbb{H}$, then the end must be diffeomorphic to $N\times[0,\infty)$ where $N$ is diffeomorphic to either $S^{1}\times S^{2}$ or $S^{3}/\Gamma$. So the confirmation of Conjecture 1.1 would give a topological classification of such ends. On the other end, if Conjecture 1.1 is false then potentially there could be new interesting gravitational instantons with this asymptotics.

The results in [35] were proved by investigating the geometry of the rescaled annuli $(M,\lambda^{-2}g,p)$ as $\lambda\rightarrow\infty$ when they collapse to lower dimensions. Foundational results on collapsing were obtained by Cheeger, Fukaya, Gromov, and others (see for example [7, 36]). Specifically, [35] makes use of the continuously collapsing techniques developed by Petrunin-Rong-Tuschmann [34]. In particular, the argument is of local nature, i.e., it focuses on the local collapsing geometry of the family of almost flat annuli of fixed size (with respect to the rescaled metrics), and it does not use the fact that the annuli come from the end of a fixed $\mathcal{AF}$ manifold. In fact as mentioned in [35] (see also Section 3), one may construct a sequence of Riemannian metrics $g_{i}$ on $\mathfrak{A}=S^{3}\times[0,1]$ with $\sup_{A}|Rm_{g_{i}}|\rightarrow 0$ which collapse to the annulus $\mathbb{A}\equiv\{2^{-1}\leq r\leq 2\}$ in $\mathbb{H}$, where $r$ is the distance function to the origin in $\mathbb{H}$. Therefore Conjecture 1.1 is of global nature. We remark that even under the stronger assumption of faster than quadratic curvature decay Conjecture 1.1 was unknown.

In this paper we confirm the conjecture of Petrunin-Tuschmann.

We will see that the standard $\mathbb{R}^{4}$ admits a complete conical Riemannian metric $g_{j}$ with asymptotic curvature $A(\mathbb{R}^{4},g_{j})\leq j^{-1}$ and asymptotic cone $\mathbb{H}$, but by Theorem 1.2 there is no complete conical $\mathcal{AF}$ metric on $\mathbb{R}^{4}$ with asymptotic cone $\mathbb{H}$. This also gives a negative answer to a version of the gap question in [35] (Question 2).

Now we briefly sketch the idea behind the proof of Theorem 1.1. Suppose $(M,g)$ is a conical $\mathcal{AF}$ 4-manifold with a simply-connected end whose asymptotic cone is $\mathbb{H}$, we will derive a contradiction. General theory allows us to reduce to the case when the end admits a $T^{2}$ action and the metric $g$ is $T^{2}$ invariant. Such a metric $g$ then yields a family of flat metrics $G$ on the 2-torus parametrized by $(\rho,z)\in\mathbb{H}=\mathbb{R}_{\geq 0}\times\mathbb{R}$ for $\rho^{2}+z^{2}\geq R^{2}$. For fixed $z$ as $\rho\rightarrow 0$ the metric $G$ degenerates as in a specific model situation that we can understand. Fix an interval $I\subset(0,\pi)$ and consider the sector region $z=\rho\cot\theta$ for $\theta\in I$. As $\rho\rightarrow\infty$, the 2-torus endowed with the rescaled metric $\rho^{-1}G$ converges to a point in the Gromov-Hausdorff sense. We introduce two quantities $\sigma$ and $\tau$ to each metric $G$. Roughly speaking, $\sigma$ measures the area of the 2-torus under the metric $\rho^{-1}G$, and $\tau$ measures the deviation of the metric $\rho^{-1}G$ from a model flat metric. The fact that the asymptotic cone is $\mathbb{H}$ gives control on the asymptotics of quantities involving $\tau$ and $\sigma$ as $\rho\rightarrow\infty$. Then an elementary calculus argument yields a contradiction.

Acknowledgements: Both authors were partially supported by the Simons Collaboration Grant in Special Holonomy. The first author is grateful to the IASM at Zhejiang University for hospitality during his visit in Spring 2024, and would like to thank Prof. Guofang Wei for sharing Abresch’s paper [2]. We thank John Lott for helpful comments that improved the exposition of the paper.

Suppose $(M,g,p)$ is a conical $\mathcal{AF}$ manifold with a simply-connected end and with asymptotic cone $\mathcal{C}_{\infty}=\mathbb{H}$. We will derive a contradiction.

First we introduce some notations. Let $(\rho,z)$ be the standard coordinates on $\mathbb{H}\equiv\mathbb{R}_{\geq 0}\times\mathbb{R}$. Denote by $g_{0}=d\rho^{2}+dz^{2}$ the standard metric on $\mathbb{H}$ and by $r_{0}$ the distance function to the origin. For $j>0$ write $A_{j}\equiv\overline{B}_{g}(p,2^{j+1})\setminus B_{g}(p,2^{j-1})$ and denote by $\widetilde{A}_{j}$ the manifold $A_{j}$ endowed with the rescaled metric $g_{j}\equiv 2^{-2j}g$. By Theorem A of [35] we know $A_{j}$ is simply connected for $j$ large.

As initial steps we make a few reductions using general theory. We first prove a smoothing result.

We will assume from now on that $g^{\prime}=g$. Next we apply the Cheeger-Fukaya-Gromov theory to reduce the case when the end of $g$ is $T^{2}$ invariant.

Now we fix $k_{0}=10$. Without loss of generality we will assume $\widecheck{g}=g$ in the following. The above discussion only makes use of the topology of the end and of the asymptotic cone. Now we investigate the asymptotical flat geometry in more detail. Introduce the following notations.

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  We write the decomposition $G=G_{1}\times G_{2}$ and denote by $\partial_{\alpha}$ the infinitesimal generator of $G_{\alpha}$, normalized to have period $2\pi$.

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  Fix $\epsilon>0$ small, we may cover $\mathbb{H}$ by 2 open subsets $\mathbb{U}_{1}=\{0\leq\theta<\frac{\pi}{2}+\epsilon\}$ and $\mathbb{U}_{2}=\{\frac{\pi}{2}-\epsilon<\theta\leq\pi\}$. Then we get an open cover of $M^{\circ}$ by $U_{\alpha}\equiv F^{-1}(\mathbb{U}_{\alpha}),\alpha=1,2$.

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  Denote $U_{\alpha,j}\equiv U_{\alpha}\cap\widetilde{A}_{j}$, which is endowed with the rescaled metric $g_{j}$, so $\widetilde{A}_{j}=U_{1,j}\cup U_{2,j}$. Note that $U_{\alpha,j}$ is converging to $\mathbb{U}_{\alpha}\cap\mathbb{A}$.

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  Denote $H_{\alpha}=\pi_{1}(U_{\alpha,j})$. Then $H_{\alpha}$ is isomorphic to $\mathbb{Z}$ and is generated by the orbit of the $G_{3-\alpha}$ action. Notice that since the $G_{\alpha}$ action on $U_{\alpha,j}$ has fixed points the $G_{\alpha}$ orbits are homotopically trivial in $U_{\alpha,j}$.

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  Denote $\mathbb{V}\equiv\mathbb{U}_{1}\cap\mathbb{U}_{2}$ and $V\equiv F^{-1}(\mathbb{V})$. Denote the overlap by $V_{j}\equiv U_{1,j}\cap U_{2,j}$, then $\pi_{1}(V_{j})=H_{1}\times H_{2}$. Note that $V_{j}$ is converging to $\mathbb{V}\cap\mathbb{A}$.

See Figure 1 and Figure 2 for pictures of the singular fibration $F$.

Now consider the convergence of $\widetilde{A}_{j}$ to $\mathbb{A}=\{1/4\leq\rho^{2}+z^{2}\leq 4\}\subset\mathbb{H}$. Denote the universal cover of $U_{\alpha,j}$ by $\widehat{U}_{\alpha,j}$. Then we see that for each $\alpha$, $\widehat{U}_{\alpha,j}$ (with the induced metric from $g_{j}$) is volume non-collapsing. Furthermore, $\partial_{1}$ and $\partial_{2}$ both lift to Killing fields on $\widehat{U}_{\alpha,j}$, which we denote by $\widehat{\partial}_{1;\alpha,j}$ and $\widehat{\partial}_{2;\alpha,j}$ respectively. Notice that on $\widehat{U}_{\alpha,j}$ the vector field $\widehat{\partial}_{\alpha;\alpha,j}$ still generates an $S^{1}$ action which descends to the action of $G_{\alpha}$ on $U_{\alpha,j}$. Passing to subsequences we may assume that $(\widehat{U}_{\alpha,j},H_{\alpha})$ converges equivariantly in the $C^{k_{0}}$ Cheeger-Gromov topology to a 4-dimensional flat manifold $(\widehat{U}_{\alpha,\infty},H_{\alpha,\infty})$ so that $\mathbb{U}_{\alpha}\cap\mathbb{A}$ is isometric to $\widehat{U}_{\alpha,\infty}/H_{\alpha,\infty}$.

Since we know the collapsing to $\mathbb{A}$ is along the $G$ orbits, we see that $H_{\alpha,\infty}$ is generated by two commuting Killing fields both of which are limits of linear combinations of $\widehat{\partial}_{1;\alpha,j}$ and $\widehat{\partial}_{2;\alpha,j}$. Now since the $G_{\alpha}$ action lifts to $\widehat{U}_{\alpha,j}$ it also acts on the limit $\widehat{U}_{\alpha,\infty}$, hence $\widehat{\partial}_{\alpha;\alpha,j}$ naturally converges to an element $\widehat{\partial}_{\alpha,\infty}$ in the Lie algebra $\mathfrak{h}$ of ${H}_{\alpha,\infty}$ with period $2\pi$. Locally it is given by a rotation field in a 2-dimensional plane $W\subset\mathbb{R}^{4}$ after immersion into $\mathbb{R}^{4}$. Given any other Killing field $\xi\in\mathfrak{h}$, since it commutes with $\widehat{\partial}_{\alpha,\infty}$, locally it must be given by $\lambda\widehat{\partial}_{\alpha,\infty}+\xi^{\prime}$, where $\xi^{\prime}$ is a Euclidean motion in the plane orthogonal to $W$. We claim that $\xi^{\prime}$ can not be a rotation. One way to see this is to apply the Key Lemma in [35] (again, in the statement of the Key Lemma in [35] we know $l<m$). Therefore we may identify a unique element (up to $\pm 1$) $\widehat{\xi}_{\alpha}\in\mathfrak{h}$ with $\|\widehat{\xi}_{\alpha}\|=1$, which is locally a translation vector field, such that $\widehat{\partial}_{\alpha,\infty}$ and $\widehat{\xi}_{\alpha}$ generate $\mathfrak{h}$.

For $x\in\widehat{U}_{\alpha,\infty}$ where $\widehat{\partial}_{\alpha,\infty}$ does not vanish, we denote by $\kappa(x)$ the geodesic curvature of the orbit of the flow of $\widehat{\partial}_{\alpha,\infty}$ at $x$.

Let $\widecheck{V}_{j}$ be the universal cover of $V_{j}$. Passing to subsequences again we may assume $(\widecheck{V}_{j},H_{1}\times H_{2})$ converges equivariantly in the $C^{k_{0}}$ Cheeger-Gromov topology to a 4-dimensional flat manifold $(\widecheck{V}_{\infty},\widecheck{H})$, where $\widecheck{H}$ is abstractly isomorphic to $\mathbb{R}^{2}$. Denote by $\widehat{V}_{\alpha,j}$ the intermediate covering of $V_{j}$ associated to the subgroup $H_{\alpha}\subset H_{1}\times H_{2}$. Then $\widehat{V}_{\alpha,j}$ can be naturally identified as an open subset of $\widehat{U}_{\alpha,j}$. We may assume $(\widehat{V}_{\alpha,j},H_{\alpha})$ converges to $(\widehat{V}_{\alpha,\infty},H_{\alpha,\infty})\subset(\widehat{U}_{\alpha,\infty},H_{\alpha,\infty})$ and there is a $\mathbb{Z}$-covering map from $\widecheck{V}_{\infty}$ to $\widehat{V}_{\alpha,\infty}$. It follows that $\widecheck{H}$ is also generated by a rotation vector field and a translation vector field. See Figure 3 for a topological picture of the intermediate covers and the universal cover for $V_{j}$.

Denote by $\widecheck{\partial}_{\alpha,j}$ the lift to $\widecheck{V}_{j}$ of $\widehat{\partial}_{\alpha;\alpha,j}$. Since $\widehat{\partial}_{\alpha;\alpha,j}$ converges to $\widehat{\partial}_{\alpha,\infty}$ on $\widehat{V}_{\alpha,\infty}$, we see that $\widecheck{\partial}_{\alpha,j}$ on $\widecheck{V}_{j}$ also converges to the lift $\widecheck{\partial}_{\alpha,\infty}$ of $\widehat{\partial}_{\alpha,\infty}$ on $\widecheck{V}_{\infty}$. Obviously both $\widecheck{\partial}_{1,\infty}$ and $\widecheck{\partial}_{2,\infty}$ belong to the Lie algebra of $\widecheck{H}$. It follows that $\widecheck{\partial}_{2,\infty}=\lambda\widecheck{\partial}_{1,\infty}$ for some $\lambda\neq 0$. A crucial observation is that we must have $\lambda=\pm 1$. Indeed, this follows from (2.1), noticing that $|\kappa(x)|$ depends on the local orbit through $x$.

We remark that in the above we have passed to subsequences at various stages, but the derived properties hold on all the possible limits. By the continuity of angle between $\partial_{1},\partial_{2}$, we may without loss of generality assume that $\lambda=1$ on all possible limits. On any limit we also denote $\widecheck{\partial}\equiv\widecheck{\partial}_{1,\infty}=\widecheck{\partial}_{2,\infty}$. Then $\widecheck{H}$ is generated by $\widecheck{\partial}$ and a translation vector field $\widecheck{\xi}$ with $\|\widecheck{\xi}\|=1$. Furthermore, the quotient $\widecheck{V}_{\infty}/\widecheck{H}$ is the subset $\mathbb{V}\cap\mathbb{A}$ of $\mathbb{H}$, and $\|\widecheck{\partial}\|=\rho$.

In the rest of the proof we focus on the set $V=F^{-1}(\mathbb{V})$. This is a (trivial) $T^{2}$ bundle over $\mathbb{V}$. The function $\rho$ on $\mathbb{H}^{\circ}$ can be naturally viewed as a function on $V$. The Riemannian metric $g$ gives rise to a family of flat metrics on $G=T^{2}$ parametrized by $\mathbb{V}$. We introduce the Gram matrix $\mathbb{G}$ with $\mathbb{G}_{\alpha\beta}\equiv g(\partial_{\alpha},\partial_{\beta})$ for $1\leq\alpha,\beta\leq 2$. Let $\tau$ be the function on $V$ such that ${\xi}\equiv\partial_{1}-(1-\tau)\partial_{2}$ is pointwise orthogonal to $\partial_{2}$, and let $\sigma\equiv\|\xi\|_{g}$. Notice that $\tau$ and $\sigma$ are both $T^{2}$ invariant so can be viewed as functions on $\mathbb{V}$.

As $j\rightarrow\infty$, passing to subsequences we know that under the convergence of $\widecheck{V}_{j}$ to $\widecheck{V}_{\infty}$, the Killing fields $\widecheck{\partial}_{1,j}$ and $\widecheck{\partial}_{2,j}$ both converge naturally to the same vector field $\widecheck{\partial}$ on $\widecheck{V}_{\infty}$. On $\widecheck{V}_{j}$, the function $\rho$ is uniformly comparable to $2^{j}$. Denote $\xi_{j}\equiv\partial_{1,j}-(1-\tau)\partial_{2,j}$ in the rescaled annulus $\widetilde{A}_{j}$.

By construction since $\widecheck{\xi}$ is a translation vector field we must have

Denote $f\equiv\|\partial_{2}\|_{g}$. Then from the convergence of $\widecheck{\partial}_{2,j}$ to $\widecheck{\partial}$ one can see that

Observe that $2^{-2j}f\sigma$ is the area of the $T^{2}$ orbit in $V_{j}$. By assumption we have

Next we derive a few key limiting properties.

Now using (2.3), (2.5), (2.6), (2.7), (2.8) and the L’Hospital’s rule we get

This is a contradiction hence completes the proof of Theorem 1.2.

In this section we construct an explicit sequence of Riemannian metrics $g_{i}$ on $\mathfrak{A}=S^{3}\times[2^{-1},2]$ with $\sup_{\mathfrak{A}}|Rm_{g_{i}}|\rightarrow 0$ which collapse to the annulus $\mathbb{A}\equiv\{2^{-1}\leq r\leq 2\}$ in $\mathbb{H}$. This shows that Theorem 1.2 is of a global nature. We will also explain the relevance to the asymptotic curvature gap conjecture of Petrunin-Tuschmann [35].