A metric sphere not a quasisphere but for which every weak tangent is Euclidean
We show that for all , there exists a doubling linearly locally contractible metric space that is topologically a -sphere such that every weak tangent is isometric to but is not quasisymmetrically equivalent to the standard -sphere. The same example shows that -Ahlfors regularity in Theorem 1.1 of [BK02] on quasisymmetric uniformization of metric -spheres is optimal.
Let and be two metric spaces. A homeomorphism is a quasisymmetry if there exists a homeomorphism such that for all , with , we have
In this case we say that and are quasisymmetrically equivalent.
In [Kin17] Kinneberg characterized metric circles that are quasisymmetric to the standard circle in terms of weak tangents (defined in Section 4 below):
Theorem 1.1 (Kinneberg, [Kin17]).
A doubling metric circle is quasisymmetrically equivalent to the standard circle if and only if every weak tangent of quasisymmetrically equivalent to the real line based at .
Here a metric space is said to be doubling if for every , every ball of radius can be covered by balls of radius , where is a positive integer that does not depend on the choice of the ball . In this paper we prove that Kinneberg’s result cannot be extended to higher dimensions:
For every , there exists a doubling, linearly locally contractible metric space that is topologically an -sphere such that every weak tangent is isometric to but is not quasisymmetrically equivalent to the standard -sphere.
When , one can compare our result with the following Theorem:
Theorem 1.3 (Bonk and Kleiner, [Bk02]).
Let be an 2-Ahlfors regular metric space homeomorphic to . Then is quasisymmetric to if and only if is linearly locally contractible.
Our study is also related to the following theorem, proven in [Lin17]:
Let be a doubling metric space homeomorphic to . The following are equivalent:
is quasisymmetrically equivalent to the standard -sphere.
For every , there exists an open neighborhood of in such that is quasisymmetrically equivalent to .
An alternative proof of Theorem 1.4 can be given using ideas in [GW18]. Roughly speaking, Theorem 1.4 says local geometry properties promote to global property. Since weak tangents are local, one could ask the following question:
Suppose is doubling and linearly locally connected. Are the following two statements equivalent?
is quasisymmetrically equivalent to the standard -sphere
Every weak tangent of is quasisymmetrically equivalent to .
When is a doubling and linearly locally connected metric sphere, statement (i) implies statement (ii). However, our construction shows that statement (ii) does not imply statement (i).
The author would like to thank Mario Bonk and John Garnett for their support and many conversations. The author thanks Jeff Lindquist for helpful discussions and Wenbo Li for his comments and corrections.
2. The Modification of the Metric on
We first construct a metric on so that every weak tangent of is isometric to , where is the Euclidean metric on , and so that there are segments of so that . Here, and in the remaining of the paper, denotes the length of the curve with respect to the metric . Once the metric is constructed, we show that for all , the weak tangents of the product space are isometric to the Euclidean space , but because , the product space cannot be quasisymmetric to the Euclidean space .
To construct , we glue together segments that look like snowflake curves of Hausdorff dimension i.e. intervals equipped with the metric . Such snowflake curves will have infinite length, so we modify the metric on the segment to get a metric so that looks like the Euclidean metric when two points are close together, but it looks like when the two points are far apart.
Note that our construction of only modifies the geometry of a bounded subset of , and therefore we can embed our -dimensional construction into a topological -sphere.
To begin the construction let and and define
The function is the only function that has the following properties:
is linear on ,
There exists , with , such that when ,
is continous and differentiable at .
In addition to the above properties that uniquely define , we observe that has the following extra properties:
is a homeomorphism,
is concave on .
For any , , and , we have
The function is decreasing in to , therefore if , then
The function meets condition (1)-(5) listed above for some , therefore . This implies that for all . Taking , we have
For any fixed , for any , and whenever , we have
When , for all , we have . When is fixed, is maximized when
which is possible only if i.e. . Then we have
When , we have
By the concavity of , we have
In any case, we have
3. The One Dimensional Construction
For any , we have
Let be an increasing sequence in such that . By (1), we can choose such that , and
Let . Choose a sequence so that for all , , such that is decreasing and .
For all , let . Let , and equip with the metric , where is the usual Euclidean metric on . Note that
The distance between the two endpoints of is .
is rectifiable and the length of is .
We construct a metric on so that if ,
when restricted to ,
when and ,
when and , and
when , .
4. The Weak Tangents of
A pointed metric space is a a triplet , where is a metric space and is a point in . Let . A map between two pointed metric spaces and is a -rough isometry if
for all ,
Note that a -rough isometry may not be continuous. The pointed Gromov-Hausdorff distance between 2 pointed metric spaces, denoted , is defined as the infimum of for which we can find a -rough isometry .
A pointed metric space is called a weak tangent of another metric space if there exist points and positive integers such that converges to in pointed Gromov-Hausdorff sense, i.e. for all , and for all there exists such that for all ,
In particular, we get if for all , and for all there exists such that for all ,
Let M be a set of separable, uniformly doubling, and uniformly linearly locally contractible pointed metric spaces. The pointed Gromov Hausdorff convergence on the set induces a topology on that is metrizable[Don11]. In particular, if a sequence of pointed metric space converges to a Gromov-Hausdorff limit, then the limit is unique.
The goal of this section is to prove the following proposition that describe all the weak tangents of .
For all , and for all positive integers , converges in pointed Gromov-Hausdorff sense to .
Note that Proposition 4.1 guarentees the existence of weak tangents of . To prove the above proposition we will use the following three lemmas:
Suppose are three points so that . Suppose . Then
If , then . Otherwise, let
We have . By definition of , we have
Bu our choice of and , either for some , or . In the latter case, we have
In the former case, Lemma 2.2 gives
Since , we have our desired conclusion. ∎
Let and be arbitrary. Suppose . Let and . The map
is a -rough isometry, where .
Since is surjective and fixes , it remains to check that for all , we have
Suppose . If , then
The second to last inequality is a consequence of Lemma 4.2. If , then
If , then following a similar arguement as when , we get
This verifies that is a -rough isometry. ∎
For , we have
Let and be arbitrary. Let . If , then on . We have
If , we consider 2 cases:
Case 1: . As , , therefore . In this case Lemma 4.3 implies
Case 2: . In this case, is a length metric on , and is an isometry between two length spaces. We have
5. Linear Local Contractibility and Assouad Dimension of
In this section we establish two properties of the space . These properties often appear in the study of quasisymmetry classes of metric spheres. Both properties are discussed in detail in [Hei01].
Let be a constant. A metric space is -linearly locally contractible if every small ball is contractible inside a ball whose radius is times larger. A metric space is linearly locally contractible if it is -linearly locally contractible for some .
Let . A metric space is -doubling if for all , every open ball of can be covered by balls of radius . A metric space is doubling if it is -doubling for some .
Both doubling and linear local contractibility are preserved under quasisymmetry. The Euclidean spaces like or are doubling and linearly locally contractible. The doubling property also ensures the existence of weak tangents.
The space is -linearly locally contractible.
Any open ball in is an open interval . Denote the center of (in .) Note that the map is increasing on , and decreasing on . Therefore the map is a homotopy of to in . This proves that is -linearly locally contractible. ∎
The space is doubling.
If a metric space is doubling, then there exists and such that for all and , any set of diameter in can be covered by at most subsets of diameter at most . The function is called the covering function of . The Assouad dimension of is defined to be the infimum of all so that a covering function of the form of exists. Conversely, any metric space of finite Assouad dimension is doubling. Proposition 5.4 will follow from the stronger proposition below.
For each , the function is a covering function of .
Every subinterval of of -diameter has -diameter . Thus our goal is to show that for every , and every , every subset of of -diameter can be covered by no more than -many subintervals of of -diameter at most . The number of subintervals we need can be bounded from above by
We claim that the last supremum is attained when . This is equivalent to
Suppose . When , we have , and
When , we have and . Since , we have
In any case, we can take the covering function of to be
The Assouad dimension of is .
Let be arbitrary. There exists such that when , . Let . Then the function , where , is a covering function of for all . Thus is a covering function of .
Since has Assouad dimension , the proposition follows. ∎
6. Higher Dimension Construction
Let . We will denote by the Euclidean metric on . Let
be the product of and . Write . Here are some facts about .
Every weak tangent of is isometric to .
is doubling and linearly locally contractible.
Every weak tangent of is of the form , where is a weak tangent of .
Every finite segment in is rectifiable. Let be a the measure on given by length. For , let be the product measure on , where is the -dimensional Lebesgue measure on .
In the remaining of this section we show that is not quasisymmetrically equivalent to . To do that we consider a geometric quantity that is roughly preserved under quasisymmetry called modulus. Given a family of curves in a measured metric space , we say that a function is admissible if for all ,
Let . We define the -modulus of as
Let be two disjoint nondegerate continua in . Let to be the collection of all rectifiable curves joining and . We write .
Moduli behave nicely under quasisymmetry, as illustrated by the following theorem.
Theorem 6.2 (Tyson, [Tys98]).
Let be locally compact, connected, -Ahlfors regular metric spaces, where , and be a quasisymmetric homeomorphism. Then there exists such that for all curve family , we have
See the next section for the definition of -Ahlfors regularity.
We can now prove that is not quasisymmetrically equivalent to . The idea is that if the two spaces are quasisymmetrically equivalent, then for any curve family , and should have comparable moduli. We know that has the property that any disjoint nondegerate continua in satisfy
for some non-increasing function . However, Proposition 6.3 shows that some sequence disjoint nondegerate continua in do not behaviour as in (3). The only problem is that is not -Ahlfors regular, so we cannot apply Theorem 6.2 directly. In Proposition 6.4, however, we will show that the inequality
holds for some independent of .
From now on, we will denote .
There exists such that and .
Take . Then , and , therefore
For each , and for and , let be the path