QUASISYMMETRIC SPHERES CONSTRUCTED OVER
QUASIDISKS
VYRON VELLIS AND JANG-MEI WU
Abstract. In this paper we provide some concrete examples of quasispheres and quasisymmetric spheres. We present two different constructions of surfaces in R3 over a quasidisk Ω ⊂ R2 . In the first construction,
the surface is the graph of a function of dist(·, ∂Ω). In the second, the
level sets of the surface are images of {rS1 }r<1 under a quasiconformal
function that maps the unit disk onto Ω. We examine the properties
of the quasidisks and that of the height functions under which these
surfaces are quasispheres or quasisymmetric equivalent to S2 .
An n-dimensional quasisphere is the image of the unit sphere Sn under
a quasiconformal self map of Rn+1 . Unlike the case n = 1, where various
characterizations of quasicircles have been found, little is known of quasispheres in higher dimensions. F. Gehring [4] and later J. Väisälä [10] proved
that if Σ ⊂ Rn+1 is a topological sphere and both components of Rn+1 \ Σ
are quasiconformally equivalent to Bn+1 , then Σ is a quasisphere. Examples of quasispheres constructed by snowflake procedures are given by C.
Bishop, [1], G. David and T. Toro [3], J. Lewis and A. Vogel [7] and D.
Meyer [8]. Examples from classical geometric topology are due to P. Pankka
and J.-M. Wu [9, Theorem 8.2]. The search for intrinsic characterizations of
quasispheres remains a longstanding problem in the study of quasiconformal
geometry.
An n-dimensional quasisymmetric sphere is defined to be a metric nsphere which is quasisymmetric to Sn . When n = 1, this notion coincides
with that of quasicircles while for n ≥ 2 it is weaker than that of quasispheres. M. Bonk and B. Kleiner [2] established necessary and sufficient
conditions for 2-dimensional topological spheres which admit quasisymmetric parametrization. In the same paper the authors proved that if Σ is
a 2-topological sphere which is Ahlfors 2-regular and linearly locally contractible, then it is a quasisymmetric sphere [2, Theorem 1.1]. For n ≥ 3,
there is no known characterization of quasisymmetric spheres.
In this paper, we present two different constructions of surfaces over Jordan domains Ω. For the height of these surfaces we are using homeomorphisms of [0, ∞) to itself. Define
F = {h : [0, +∞) → [0, +∞) , h is a homeomorphism}.
Such homeomorphisms are called height functions. We investigate the conditions on Ω and h, for which the resulting surface is quasisymmetric to
S2 .
Date: November 8, 2013.
Research supported in part by the NSF grants DMS-0653088 and DMS-1001669.
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VYRON VELLIS AND JANG-MEI WU
In the first construction, to which we refer as the Geometric construction,
the height of the surface is a function of the distance of x ∈ Ω from the
boundary ∂Ω. Let Γ = ∂Ω and h ∈ F. Define the 2-dimensional surface
Σ(Γ, h) = {(x, z) : x ∈ Ω, |z| = h(dist(x, Γ))}.
Väisälä [11] proved that if Γ is a Jordan arc and I is an interval, then Γ×I
is quasisymmetric embeddable in R2 if and only if Γ satisfies the chord-arc
condition. For the height functions of the form h(t) = tα with α ∈ (0, 1),
the surface Σ(Γ, h), close to Γ × {0}, resembles Γ × I for some small interval
I. In view of Väisälä’s result, one expects that Σ(Γ, tα ) is quasisymmetric
to S2 if and only if Γ is a chord-arc curve. It turns out that the chord-arc
property is necessary but not sufficient.
The geometry of the level sets γ of Γ plays a key role in the properties
of Σ(Γ, h). The -level set of Γ is defined to be
γ = {x ∈ Ω : dist(x, Γ) = }.
A Jordan curve Γ is said to satisfy the level chord-arc property (or LCA
property), if there exist 0 > 0 and C ≥ 1 such that γ is a C-chord-arc
curve for every 0 ≤ ≤ 0 .
The behavior of h close to 0 is another important factor of the behavior of
Σ(Γ, h). More specifically, if lim inf t→0 h(t)/t = 0 then, for any Jordan curve
Γ, the surface Σ(Γ, h) is not quasisyymetrically equivalent to S2 . Thus, it
is enough to consider the following sub-collection of F in the study of these
surfaces:
F1 = {h ∈ F : lim inf h(t)/t > 0 and h is lipschitz in [r, +∞) for all r > 0}.
t→0
The main result for this construction is the following theorem.
Theorem 1. Let Γ be a Jordan curve.
(1) If Γ has the level chord-arc property then Σ(Γ, h) is quasisymmetric
to S2 for all height functions h ∈ F1 .
(2) If Γ does not have the level chord-arc property, then there exists
h ∈ F1 such that Σ(Γ, h) is not quasisymmetric to S2 . In fact, h can
be chosen so that limt→0 h(t)/t = +∞.
The assumption that h is lipschitz away from zero is necessary for the
first claim.
In [13], sufficient conditions for a curve to satisfy the LCA property are
given in terms of local flatness. These conditions are sharp. For x, y ∈ Γ
denote with Γ(x, y) the subarc of Γ connecting x and y that has a smaller
diameter and denote with lx,y the infinite straight line passing from x, y.
Define
1
sup dist(z, lx,y )
ζΓ (x, y) =
|x − y| z∈Γ(x,y)
and
ζΓ = lim
sup
r→0 x,y∈Γ,|x−y|≤r
ζΓ (x, y).
Theorem 2.
(1) Suppose that Γ is a chord-arc curve with ζΓ < 1/2.
Then, Σ(Γ, tα ) is quasisymmetric to S2 for all α ∈ (0, 1].
QUASISYMMETRIC SPHERES CONSTRUCTED OVER QUASIDISKS
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(2) There exists a chord-arc curve with ζΓ = 1/2 such that for any α ∈
(0, 1), the surface Σ(Γ, tα ) is not quasisymmetric to S2 .
If h ∈ F1 is bilipschitz in a neighborhood of 0 (e.g. h(t) = t) and Γ is a
quasicircle, the surface Σ(Γ, h) is a quasisphere. A partial converse is also
true.
Theorem 3. Suppose that Γ is a Jordan curve.
(1) If Γ is a quasicircle then the surface Σ(Γ, h) is a quasisphere for
every h ∈ F1 which is bilipschitz in a neighborhood of 0.
(2) If the surface Σ(Γ, h) is quasisymmetric to S2 for some h ∈ F1 which
is bilipschitz in a neighborhood of 0 then Γ is a quasicircle.
The proof of Theorem 3 involves methods used by Väisälä in the study
of cylindrical domains [12] and methods employed by Gehring in the study
of slit domains [5]. One should expect the use of these ideas, since the
cylindrical domain Ω × R is essentially a component of R3 \ Σ(∂Ω, h) with
h ≡ ∞ and the slit domain R3 \ Ω is essentially R3 \ Σ(∂Ω, h) with h ≡ 0.
An iteration of this construction with height functions h1 , h2 , · · · ∈ F1 ,
which are bilipschitz in a neighborhood of 0, yields quasispheres of any
dimension.
It follows from Theorem 1 that if Γ is not a chord-arc curve, then it does
not satisfy the LCA property and Σ(Γ, h) is not quasisymmetric to S2 for
some h ∈ F1 with limt→∞ h(t)/t = ∞. In some sense, having Assouad
dimension bigger than 1 (see [6, p. 81]), is the least degree of complexity
for a curve which is not a chord-arc curve.
Theorem 4. Let Γ be a quasicircle with Assouad dimension bigger than 1.
Then, for any height function of the form h(t) = tα , the surface Σ(Γ, h) is
not quasisymmetric to S2 .
Consequently, if Γ has Hausdorff dimension greater than 1 then Σ(Γ, tα )
fails to be a quasisymmetric sphere when α ∈ (0, 1).
In the second construction, to which we refer as the Analytic construction,
the level sets of the surface are images of {rS1 }r<1 under a quasiconformal
function that maps the unit disk onto Ω. More precisely, suppose that f is a
quasiconformal mapping that maps B2 onto Ω. For a function h ∈ F, define
the surface
Σ̃(f, h) = {(f (rη), z) ∈ R2 × R : |z| = h(1 − r), r ∈ [0, 1], η ∈ S1 }.
The dependence of the surface on Ω is suppressed.
As with the Geometric construction, the behavior of h close to zero is a
crucial factor of the behavior of Σ̃(f, h). Define the following sub-collection
of F
h(1) − h(t)
< ∞}.
F2 = {h ∈ F : lim inf h(t)/t > 0 and lim sup
t→0
1−t
t→1
The main result for this construction is the following theorem.
Theorem 5. Suppose that Ω is a Jordan domain and Γ = ∂Ω.
(1) If Γ is a chord-arc curve then for any bilipschitz mapping f from B2
onto Ω and for any h ∈ F2 , the surface Σ̃(f, h) is bilipschitz to S2 .
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VYRON VELLIS AND JANG-MEI WU
(2) If Γ is not a chord-arc curve then for any quasiconformal mapping f from B2 onto Ω there exists a function h ∈ F2 such that
Σ̃(f, h) is not quasisymmetric to S2 . In fact, h can be chosen so that
limt→0 h(t)/t = ∞.
Both limit conditions of functions h ∈ F2 are necessary for the first claim
of Theorem 5.
A special case of the Analytic construction is given in the following theorem.
Theorem 6. Suppose that f is a bilipschitz map of B2 onto Ω. If h ∈ F is
lipschitz in a neighborhood of 1 and h−1 is lipschitz in a neighborhood of 0
then Σ̃(f, h) is the image of S2 under a bilipschitz self map of R3 .
References
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Department of Mathematics, University of Illinois, 1409 West Green Street,
Urbana, IL 61820, USA
E-mail address: [email protected]
E-mail address: [email protected]