Skip to content
DraconiticResearch
Search Sources API
203,234 studies31 sourcesUpdated 29 min ago
PreprintarXiv licencearXiv

Length distortion of volume-preserving Lipschitz mappings

  • Damaris Meier
  • Kai Rajala
Open at sourcePDFDOICite

Abstract

We show that every area-preserving Lipschitz map between metric surfaces distorts the length of almost every curve by at most a multiplicative factor, answering a question posed by the first author and Ntalampekos. We give two different proofs. The first is entirely intrinsic and extends to higher dimensions under additional assumptions. The second relies on non-smooth uniformization theory and yields the conclusion that the family of curves intersecting the purely 2-unrectifiable part of a metric surface in a set of positive length is exceptional.

Related

  1. Additive mappings of unweighted directed graphs preserving individual values of the cycle length
    Alexander Vlasov, Alexander Guterman, Svetlana Zhilina
    arXivPreprintarXiv licenceFull textCombinatorics & discrete mathematicsarXiv
  2. Differentiability of Lipschitz mappings into metric spaces: area and co-area formulas
    Behnam Esmayli, Paweł Goldstein, Piotr Hajłasz
    arXivPreprintCC BY 4.0Full textAnalysis & PDEsGeometry & topologyarXiv
  3. Weakly Lipschitz mappings in higher dimensions
    Anatoly Golberg
    arXivPreprintCC BY-NC-ND 4.0Full textAnalysis & PDEsarXiv
  4. Affine balancing of complex dilatations: affine invariants and optimal distortion of harmonic quasiconformal mappings
    Zhi-Gang Wang, Deguang Zhong
    arXivPreprintCC BY 4.0Full textAnalysis & PDEsGeometry & topologyarXiv
  5. Fixed points for uniformly asymptotically Hölder-Lipschitz mappings
    Luciano G. Josino, Carlos Sérgio R. da Silva
    arXivPreprintarXiv licenceFull textAnalysis & PDEsarXiv

Cite

BibTeXRIS
BibTeX
@misc{meier2026length,
  title = {{Length distortion of volume-preserving Lipschitz mappings}},
  author = {Damaris Meier and Kai Rajala},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.09842},
  eprint = {2609.09842},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.09842},
}
RIS
TY  - UNPB
TI  - Length distortion of volume-preserving Lipschitz mappings
AU  - Damaris Meier
AU  - Kai Rajala
PY  - 2026
DA  - 2026/09/09
PB  - arXiv
DO  - 10.48550/arxiv.2609.09842
UR  - https://arxiv.org/abs/2609.09842
AB  - We show that every area-preserving Lipschitz map between metric surfaces distorts the length of almost every curve by at most a multiplicative factor, answering a question posed by the first author and Ntalampekos. We give two different proofs. The first is entirely intrinsic and extends to higher dimensions under additional assumptions. The second relies on non-smooth uniformization theory and yields the conclusion that the family of curves intersecting the purely 2-unrectifiable part of a metric surface in a set of positive length is exceptional.
ER  -