The nonuniqueness of the tangent cones at infinity of Ricci-flat manifolds
3-14-1 Hiyoshi, Kohoku, Yokohama 223-8522, Japan
For a complete Riemannian manifold with nonnegative Ricci curvature, it is shown by Gromov’s Compactness Theorem that if one take a sequence
such that , then there is a subsequence such that converges to a pointed metric space as in the sense of the pointed Gromov-Hausdorff topology . The limit is called the tangent cone at infinity of . In general, the pointed Gromov-Hausdorff limit might depend on the choice of or its subsequences.
The tangent cone at infinity is said to be unique if the isometry classes of the limits are independent of the choice of and its subsequences, and Colding and Minicozzi showed the next uniqueness theorem under the certain assumptions.
Theorem 1.1 ().
Let be a Ricci-flat manifold with Euclidean volume growth, and suppose that one of the tangent cone at infinity has a smooth cross section. Then the tangent cone at infinity of is unique.
Among the assumptions in Theorem 1.1, the Ricci-flat condition is essential since there are several examples of complete Riemannian manifolds with nonnegative Ricci curvature and Euclidean volume growth, of whom one of the tangent cones at infinity has smooth cross section, but the tangent cones at infinity is not unique .
Here, let be the set of all of the isometry classes of the tangent cones at infinity of . In this paper, the isometry between pointed metric spaces means the bijective map preserving the metrics and the base points. It is known that is closed with respect to the pointed Gromov-Hausdorff topology, and has the natural continuous -action defined by the rescaling of metrics. The uniqueness of the tangent cones at infinity means that consists of only one point.
In this paper, we show that the assumption for the volume growth in Theorem 1.1 is essential. More precisely, we obtain the next main result.
There is a complete Ricci-flat manifold of dimension such that is homeomorphic to . Moreover, -action on fixes , , , where is the Euclidean metric, , and is the completion of the Riemannian metric
and acts freely on
Here, is the Cartesian coordinate on .
Here, we mention more about the metric spaces appearing in Theorem 1.2. For , denote by the metric on induced by the Riemannian metric
For in Theorem 1.2, we show that contains , and . Here, we can check easily that and are homothetic to and , respectively. We can show that
Both of and can be regarded as the Riemannian cones with respect to the dilation on . Although the dilation also pulls back to , does not become the metric cone with respect to this dilation since is not a ray. In fact, any open intervals contained in have infinite length with respect to .
In general, tangent cones at infinity of complete Riemannian manifolds with nonnegative Ricci curvature and Euclidean volume growth are metric cones . In our case, it is shown in Section 9 that never become the metric cone of any metric space.
The Ricci-flat manifold appeared in Theorem 1.2 is one of the hyper-Kähler manifolds of type , constructed by Anderson, Kronheimer and LeBrun in  applying Gibbons-Hawking ansatz, and by Goto in  as hyper-Kähler quotients. Combining Theorems 1.1 and 1.2, we can see that the volume growth of should not be Euclidean. In fact, the author has computed the volume growth of the hyper-Kähler manifolds of type in , and showed that they are always greater than cubic growth and less than Euclidean growth. To construct , we “mix” the hyper-Kähler manifold of type whose volume growth is for some , and equipped with the standard hyper-Kähler structure. Unfortunately, the author could not compute the volume growth of in Theorem 1.2 explicitly.
In this paper, we can show that a lot of metric spaces may arise as the Gromov-Hausdorff limit of hyper-Kähler manifolds of type . Let
and denote by the metric on induced by the Riemannian metric . Then we have the following result.
There is a complete Ricci-flat manifold of dimension such that contains
Since and are contained in in the above theorem, then their limits and are also contained in . The author does not know whether any other metric spaces are contained in .
Theorems 1.2 and 1.3 are shown along the following process. The above-mentioned hyper-Kähler manifolds are constructed from infinitely countable subsets in such that . We denote it by and fix the base point . From the construction, has a natural -action preserving and the hyper-Kähler structure, then we obtain a hyper-Kähler moment map such that , which is a surjective map whose generic fibers are . There is a unique distance function on such that is a submetry. Here, submetries are the generalization of Riemannian submersions to the category of metric spaces. For we can see , hence by taking such that , we obtain a sequence of submetries . Now, assume that converges to a metric space for some in the pointed Gromov-Hausdorff topology, and the diameters of fibers of converges to in some sense. Then we can show is the Gromov-Hausdorff limit of . We raise a concrete example of and sequences , then obtain several limit spaces. Among them, it is shown in Section 9 that is not a polar space in the sense of Cheeger and Colding .
This paper is organized as follows. We review the construction of hyper-Kähler manifolds of type and hyper-Kähler moment map in Section 2. Then we review the notion of submetry in Section 3, and the notion of Gromov-Hausdorff topology in Section 4. In Section 5, we construct a submetry from to by using and dilation, where is the metric induced by the Riemannian metric . Here, is a positive valued harmonic function determined by and some constants. Then we see that the convergence of can be reduced to the convergence of . In Sections 6 and 7, we raise concrete examples of and fix , then estimate the difference of and another positive valued harmonic function , which induces the metric on . In Section 8, we observe some examples by applying the results in Sections 6 and 7, then show Theorems 1.2 and 1.3. In Section 9, we prove that is not a polar space.
The author would like to thank Professor Shouhei Honda who invited the author to this attractive topic, and also thank him for giving the several advice on this paper. The author also would like to thank the referee for careful reading and several useful comments. Thanks to his pointing out, the author could make the main results much stronger. The author was supported by Grant-in-Aid for Young Scientists (B) Grant Number 16K17598.
2 Hyper-Kähler manifolds of type
Here we review shortly the construction of hyper-Kähler manifolds of type , along .
Let be a countably infinite subset satisfying the convergence condition
and take a positive valued harmonic function over defined by
Then is a closed -form where is the Hodge’s star operator of the Euclidean metric, and we have an integrable cohomology class , which is equal to the st Chern class of a principal -bundle . For every , we can take a sufficiently small open ball centered at which does not contain any other elements in . Then is isomorphic to Hopf fibration as principal -bundles, hence there exists a -manifold and an open embedding , and can be extended to an -fibration
Moreover we may write and . Next we take an -connection on , whose curvature form is given by . Then is uniquely determined up to exact -form on . Now, we obtain a Riemannian metric
on , which can be extended to a smooth Riemannian metric over by taking appropriately.
Theorem 2.1 ().
Let be as above. Then it is a complete hyper-Kähler (hence Ricci-flat) metric of dimension .
Since acts on isometrically, it is easy to check that
is a Riemannian submersion, where is the Euclidean metric on .
Next we consider the rescaling of . For , put . Then it is easy to see
and , hence holds. Thus we have
Throughout of this paper, the distance between and in a metric space is denoted by . If it is clear which metric is used, we often write
The map appeared in the previous section is not a Riemannian submersion, since degenerates on and does not defined on the whole of . However we can regard as a submetry, which is a notion introduced in .
Definition 3.1 ().
Let be metric spaces, and be a map, which is not necessarily to be continuous. Then is said to be a submetry if holds for every and , where is the closed ball of radius centered at .
Any proper Riemannian submersions between smooth Riemannian manifolds are known to be submetries. Conversely, a submetry between smooth complete Riemannian manifolds becomes a Riemannian submersion .
Now we go back to the setting in Section 2. Denote by the metric on defined as the completion of the Riemannian distance induced from . Since is a Riemannian submersion, we have the following proposition.
Let be a hyper-Kähler manifolds of type . The map is a submetry, where is the Riemannian distance induced from . Moreover, we have
4 The Gromov-Hausdorff convergence
In this section, we discuss with the pointed Gromov-Hausdorff convergence of a sequence of pointed metric spaces equipped with submetries. First of all, we review the definition of the pointed Gromov-Hausdorff convergence of pointed metric spaces. Denote by the open ball of radius centered at in a metric space .
Let and be pointed metric spaces, and be positive real numbers. is said to be an -isometry from to if , holds for any , contains .
Let be a sequence of pointed metric spaces. Then is said to converge to a metric space in the pointed Gromov-Hausdorff topology, or , if for any there exists an positive integer such that -isometry from to exists for every .
For metric spaces , and a map , define by
Let and be pointed metric spaces equipped with submetries satisfying , and be another pointed metric space. Assume that and we have an -isometry from to . Then there exists an -isometry from to .
Now, there is an -isometry from to . Then it is easy to check that the composition is an -isometry from to . ∎
5 Tangent cones at infinity
Let be a metric space and be a decreasing sequence of positive numbers converging to . If is the pointed Gromov-Hausdorff limit of , then it is called an tangent cone at infinity of . It is clear that the limit does not depend on , but may depend on the choice of a sequence .
In this paper we are considering the tangent cones at infinity of . In Section 2 we have seen for , hence is a submetry. By taking and the dilation defined by , we have another submetry
Here, is the completion of the Riemannian distance of
therefore we obtain which is the completion of the Riemannian metric , where
In other words, is given by
where is the set of smooth paths in joining , and
By the definition of , one can see that the diameter of the fiber is given by . Accordingly, the diameter of is given by .
For a metric on and constants , we introduce the next assumptions.
The identity map
is an -isometry.
Then we obtain the next proposition by Proposition 4.3.
Let and be as above, satisfy and be a metric on . If (A1-2) are satisfied for given constants , then is an -isometry from to .
Fix , and let
Take an increasing sequence of integers .
In this paper many constants will appear, and they may depend on or . However, we do not mind the dependence on these parameters.
Since , we can see that , accordingly we obtain a hyper-Kähler manifold .
From now on we fix , and , then put and
Let and put
Here, is given by
for . For , define a positive valued function by
Throughout this section, we put
for any .
Then we obtain
The above inequality holds since the function has at most one critical point and
for all . ∎
Next we obtain the lower estimate of as follows.
First of all one can see
if , and
Next we have
and the similar argument to the proof of Proposition 6.1 gives
Combining these inequalities one can the second assertion if . If , then we have
and by the similar argument we obtain the assertion.
By Proposition 6.2, we have the following.
Let be as above. Then
holds for every .
In the previous section we have estimated from the above on .
In this section we introduce more general positive functions and , and induced metric on respectively. What we hope to show in this section is that if we fix a very large and assume that holds for a very small and every , then the identity map of becomes the -isometry from to , for a large and a small . Here, we explain the difficulty to show it.
We hope to show that is small for every and . By the estimate of , it is easy to see that is small, where (resp. ) is the Riemannian distance of the Riemannian metric (resp. ). However, may not equal to in general since the geodesic of joining two points in might leave from . To see that is sufficiently small, we have to observe that a path joining two points in which leaves can be replaced by a shorter path included in .
In this section we consider positive valued functions satisfying the following conditions for given constants , and