PreprintarXiv licencearXiv
An elementary proof of a tangent-cone identity for rank-one planar Lipschitz maps
Abstract
We give an elementary proof of a tangent-cone identity which might be known to specialists in geometric measure theory. It concerns Lipschitz mappings g from the closed unit ball B⊂ℝ2 to ℝ2. Let x be an interior point of differentiability of g at which Dg(x) has rank one. If the upper one-dimensional Hausdorff density of g(B) at g(x) is at most one, then the tangent cone to g(B) at g(x) coincides with the image of Dg(x). This property is used in the proof of rigidity for the four-gradient problem. We provide a detailed, self-contained proof based on differentiability, connectedness and elementary estimates for one-dimensional Hausdorff measure, without using advanced methods from geometric measure theory.
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BibTeX
@misc{buchowiec2026elementary,
title = {{An elementary proof of a tangent-cone identity for rank-one planar Lipschitz maps}},
author = {Igor Buchowiec},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.05161},
eprint = {2611.05161},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.05161},
}
RIS
TY - UNPB
TI - An elementary proof of a tangent-cone identity for rank-one planar Lipschitz maps
AU - Igor Buchowiec
PY - 2026
DA - 2026/11/03
PB - arXiv
DO - 10.48550/arxiv.2611.05161
UR - https://arxiv.org/abs/2611.05161
AB - We give an elementary proof of a tangent-cone identity which might be known to specialists in geometric measure theory. It concerns Lipschitz mappings g from the closed unit ball B⊂ℝ^2 to ℝ^2. Let x be an interior point of differentiability of g at which Dg(x) has rank one. If the upper one-dimensional Hausdorff density of g(B) at g(x) is at most one, then the tangent cone to g(B) at g(x) coincides with the image of Dg(x). This property is used in the proof of rigidity for the four-gradient problem. We provide a detailed, self-contained proof based on differentiability, connectedness and elementary estimates for one-dimensional Hausdorff measure, without using advanced methods from geometric measure theory.
ER -