PreprintarXiv licencearXiv
Lipschitz regularity of Sobolev maps from RCD(K,N) spaces to CAT(0) spaces
Abstract
We establish a quantitative criterion for local Lipschitz regularity of Sobolev maps in the sense of Korevaar and Schoen from finite-dimensional RCD(K,N) spaces into complete CAT(0) spaces, without assuming harmonicity. We quantify the failure of local energy minimality by the harmonic replacement potential, a dyadic sum of the square roots of normalized energy gaps between a map and its harmonic replacements. We prove that this potential, together with an averaged energy term, controls the minimal weak upper gradient pointwise, with constants depending only on the curvature-dimension bounds and the scale. In particular, local boundedness of the replacement potential implies that the map admits a locally Lipschitz representative.
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BibTeX
@misc{han2026lipschitz,
title = {{Lipschitz regularity of Sobolev maps from $\mathrmRCD(K,N)$ spaces to $\mathrmCAT(0)$ spaces}},
author = {Bang-Xian Han and Zeyu Wang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.39644},
eprint = {2610.39644},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.39644},
}
RIS
TY - UNPB
TI - Lipschitz regularity of Sobolev maps from RCD(K,N) spaces to CAT(0) spaces
AU - Bang-Xian Han
AU - Zeyu Wang
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2610.39644
UR - https://arxiv.org/abs/2610.39644
AB - We establish a quantitative criterion for local Lipschitz regularity of Sobolev maps in the sense of Korevaar and Schoen from finite-dimensional RCD(K,N) spaces into complete CAT(0) spaces, without assuming harmonicity. We quantify the failure of local energy minimality by the harmonic replacement potential, a dyadic sum of the square roots of normalized energy gaps between a map and its harmonic replacements. We prove that this potential, together with an averaged energy term, controls the minimal weak upper gradient pointwise, with constants depending only on the curvature-dimension bounds and the scale. In particular, local boundedness of the replacement potential implies that the map admits a locally Lipschitz representative.
ER -