Quasisymmetric parametrizations of two-dimensional metric spheres

By M. Bonk, Bruce Kleiner. 2001

202 citations recorded by the source corpus.

We study metric spaces homeomorphic to the 2-sphere, and find conditions under which they are quasisymmetrically homeomorphic to the standard 2-sphere. As an application of our main theorem we show that an Ahlfors 2-regular, linearly locally contractible metric 2-sphere is quasisymmetrically homeomorphic to the standard 2-sphere, answering a question of Heinonen and Semmes.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/13c6c87577ba18c2be9b4aca8d094512e832b3da

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

1. S. Wenger
2. A. Lytchak
3. Bruce Kleiner
4. Kai Rajala
5. Enrico Le Donne
6. S. Keith
7. Renjin Jiang
8. U. Lang

In Geometry and Topology

1. A. Lytchak
2. Bruce Kleiner
3. S. Wenger
4. Kai Rajala
5. S. Keith
6. Enrico Le Donne
7. U. Lang
8. M. Bonk

Citations in the corpus

• Modulus and the Poincaré inequality on metric measure spaces (2003, background): “Bonk and Kleiner [ BK02 ] have shown that an Ahlfors regular space which is homeomorphic to the 2-sphere, and which contains sufficiently many curves, is actually quasisymmetric to the 2-sphere.”
• Uniformization of two-dimensional metric surfaces (2014, methodology): “iprocal spaces can be constructed that do not satisfy (4) even locally. Theorems 1.4, 1.5 and 1.6 hold also when Xis homeomorphic to the Riemann sphere S2, with obvious modifications. Bonk and Kleiner [11] gave an excellent characterization for quasispheres among the topological spheres satisfying (4”
• The asymptotic geometry of negatively curved spaces: uniformization, geometrization and rigidity (2006, background): “Using arguments from [38], [5] is not hard to see that if Z is aQ-Loewner space, then Z satisfies the combinatorial Q-Loewner property.”
• A Primer on Carnot Groups: Homogenous Groups, Carnot-Carathéodory Spaces, and Regularity of Their Isometries (2018, background): “For this reason, nonAbelian Carnot groups, together with the classical fractals and boundaries of hyperbolic groups, are the main examples in an emerging eld called ‘non-smooth analysis’, in addition see [14, 15, 17, 18, 24, 28, 42, 43, 47, 48, 53, 54, 59, 66, 73, 92].”
• Intrinsic structure of minimal discs in metric spaces (2016, methodology): “3 and to Bonk and Kleiner [4] and Wildrick [40], the space Z is locally quasisymmetric to the unit disc if areas of balls in Z have a quadratic upper bound in terms of the radius.”