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Space-filling surfaces: sharp Hölder continuous parameterizations from squares to cubes
Abstract
Following a hint of Semmes, we employ Stong's bijections between integer lattices to construct space-filling surfaces, which are higher-dimensional analogues of space-filling curves. For each m≥ 2 we build α-Hölder continuous parameterizations f:[0,1]m→[0,1]m+1 with sharp exponent α=m/(m+1). In particular, there exist (2/3)-Hölder continuous surjections from squares to cubes. This solves Arnold's problem 1988--5.
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BibTeX
@misc{badger2026spacefilling,
title = {{Space-filling surfaces: sharp Hölder continuous parameterizations from squares to cubes}},
author = {Matthew Badger and Kevin Palmer},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.21246},
eprint = {2608.21246},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.21246},
}
RIS
TY - UNPB
TI - Space-filling surfaces: sharp Hölder continuous parameterizations from squares to cubes
AU - Matthew Badger
AU - Kevin Palmer
PY - 2026
DA - 2026/09/18
PB - arXiv
DO - 10.48550/arxiv.2608.21246
UR - https://arxiv.org/abs/2608.21246
AB - Following a hint of Semmes, we employ Stong's bijections between integer lattices to construct space-filling surfaces, which are higher-dimensional analogues of space-filling curves. For each m≥ 2 we build α-Hölder continuous parameterizations f:[0,1]^m→[0,1]^m+1 with sharp exponent α=m/(m+1). In particular, there exist (2/3)-Hölder continuous surjections from squares to cubes. This solves Arnold's problem 1988--5.
ER -