A paper by Dimitrios Ntalampekos has recently been accepted for publication in Inventiones Mathematicae (Ntalampekos 2026). The paper gives a new quasiconformal uniformization theorem for subsets of the 2-sphere. In particular, it gives an exact criterion for when a Sierpiński carpet can be transformed by a quasiconformal homeomorphism into a round Sierpiński carpet.
We first introduce the objects appearing in the theorem. It is convenient to identify the 2-sphere with the Riemann sphere We equip it with the usual spherical metric, that is, the great-circle distance after identifying with the unit sphere in by stereographic projection. For and , the open ball is called a geometric disk. Its boundary is a circle on the sphere.
Recall that a Jordan curve is a subset homeomorphic to a circle. By the Jordan curve theorem, the complement of a Jordan curve in the sphere has exactly two connected components. Each of them is called a Jordan region.
A Sierpiński carpet is a compact set with empty interior such that the connected components of are Jordan regions whose closures are pairwise disjoint, and whose diameters tend to zero. More precisely, the last condition means that for every , only finitely many satisfy The regions are called the peripheral regions of .
A Sierpiński carpet is called round if all its peripheral regions are geometric disks.
The familiar standard Sierpiński carpet, obtained by repeatedly removing the middle square from a subdivision of every remaining square, is therefore not round: its complementary regions are squares rather than disks. Topologically, however, the particular shapes of the holes make no difference. In fact, a classical theorem of Whyburn says that all Sierpiński carpets are homeomorphic. The question here is much more geometric: can one pass from a given carpet to a round one by a homeomorphism whose distortion is uniformly controlled?
This leads to quasiconformal maps. Let be a homeomorphism. For and , put and Thus a small circle of radius around is mapped to a curve whose points have distances between and from .
For , we call -quasiconformal if for every . We call quasiconformal if it is -quasiconformal for some . Thus a quasiconformal map may distort shapes, but its infinitesimal distortion is bounded uniformly over the sphere.
Two subsets are called quasiconformally equivalent if there is a quasiconformal homeomorphism of the sphere such that
We can now formulate the condition discovered by Ntalampekos. Let and be two disjoint Jordan regions. We say that the pair is -quasiconformally circularizable if there exists a -quasiconformal homeomorphism such that both and are geometric disks.
A collection is called uniformly quasiconformally pairwise circularizable if there exists one constant such that every pair , , is -quasiconformally circularizable.
There is an important subtlety in this definition. We do not require one map to circularize all the . For each pair , we may use a completely different quasiconformal map. Only the bound on the distortion has to be independent of the chosen pair.
Ntalampekos proved that this apparently pairwise condition is already enough to guarantee one global map which circularizes every peripheral region simultaneously.
Theorem. Let be a Sierpiński carpet with peripheral regions . Then is quasiconformally equivalent to a round Sierpiński carpet if and only if the family is uniformly quasiconformally pairwise circularizable.
One direction of the theorem is immediate. Suppose that a -quasiconformal homeomorphism transforms into a round carpet. Then every is a geometric disk. Consequently, the same map circularizes every pair , and the family is uniformly pairwise circularizable.
The converse is the remarkable part. We assume only that every pair of holes can be made round, with a uniform bound on the distortion, using maps which may depend on the pair. The theorem concludes that there is a single quasiconformal homeomorphism which makes all countably many holes round at once.
The result gives a sharp answer to a natural uniformization question for Sierpiński carpets, substantially extends the earlier theory, and at the same time forms part of a more general theorem covering Schottky sets and gaskets.