Loewner carpets and conformal dimension

Motivated by questions such as Cannon’s conjecture I am studying the possibilities for conformal dimension and its attainment. Indeed, even for simple fractals such as the Sierpiński carpet, the attainment question of conformal dimension remains unknown. This number measures how small a dimension can a given space be deformed into. If it is attained, then the minimizers would satisfy remarkable properties. Indeed, they would have many curves representing the masure. Until the work with Jeff Cheeger, there were no published examples of so called “Thin Loewner Carpets” which could serve as models for such minimizers. This shows that the minimization is topologically and geometrically possible — while it falls quite far from actually proving the specific case of the Sierpinski carpet. 

With Jasun Gong we studied another regime of “fat carpets”, and established natural conditions, related to uniformization, for their existence. Together with the work with Jeff Cheeger this gives a large class of examples of Loewner carpets in the full range of dimensions that they can exist. The work also gives some necessary conditions, general frameworks for their constructions and numerous tools of independent interest on rigidity, differentiation, and uniformization. Currently, I am further pursuing this direction by studying objects that can be defined using the hyperbolic filling — speficially interesting classes of curves.

Papers: 

• Thin Loewner carpets and their quasisymmetric embeddings in S2, Joint with Jeff Cheeger, submitted, 2020. 
• Almost uniform domains and Poincaré inequalities, joint with Jasun Gong, 2020.