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Length distortion under area-preserving Lipschitz maps

Let XX and YY be metric surfaces, and let f:X→Yf:X\to Y be an area-preserving Lipschitz map. Does there exist a constant K≥1K\ge 1 such that, for 22-modulus-almost every rectifiable curve γ\gamma in XX, one has K−1length⁡(γ)≤length⁡(f∘γ)≤Klength⁡(γ)K^{-1}\operatorname{length}(\gamma)\le \operatorname{length}(f\circ\gamma)\le K\operatorname{length}(\gamma)?

References

Primary source

arXiv

Additional references

analysisgeometric-analysismetric-geometry
Posted · edited

Progress summary

Refreshed
Claimed solved

A September 2026 preprint claims to settle the area-preserving length question on nonsmooth surfaces, while its higher-dimensional extension remains conditional.

The problem asks whether an area-preserving Lipschitz map between metric surfaces preserves the lengths of almost every curve up to a uniform factor. Meier and Ntalampekos formulated this as Question 1.5 in 2023 and left the unrestricted case open.

Known results

  • Meier and Ntalampekos (2023): the assertion holds when the domain is reciprocal.
  • Meier and Ntalampekos (2023): corresponding bounded-length-distortion results hold when the target is reciprocal or upper Ahlfors 22-regular.
  • A 2019 survey records the broader Lipschitz-volume-rigidity problem as unresolved for general singular spaces.

September 2026 preprint

The preprint Length distortion of volume-preserving Lipschitz mappings reports two proofs of the almost-everywhere length statement and an extension to higher dimensions under additional assumptions. This is a claimed resolution of the unrestricted surface question, but independent verification is not recorded in the supplied sources.

Current status (as of September 2026): The unrestricted metric-surface question is claimed solved by two proofs in a September 2026 preprint, but remains unverified; the higher-dimensional extension requires extra hypotheses.

Sources

Solutions 0

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