Local bilipschitz conjecture for quasiconformal maps of Carnot groups

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Let GG be a Carnot group other than Rn\mathbb{R}^n or the Heisenberg group Hn\mathbb{H}_n for any nn, and let U,U′U,U' be open subsets of GG. Let f:U→U′f:U\to U' be a quasiconformal homeomorphism; when U=GU=G, regard ff as a self-map of GG. Xie's conjecture. The map ff is locally bilipschitz; moreover, if U=GU=G, then ff is bilipschitz. The conjecture is motivated by the fact that, apart from Euclidean spaces and Heisenberg groups, all known examples of quasiconformal homeomorphisms in this setting are locally bilipschitz, while the source gives no resolution.

References

Primary source

Bruce Kleiner, Stefan Muller and Xiangdong Xie, “Sobolev mappings between nonrigid Carnot groups”, arXiv:2112.01866 (2021).

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