Quantitative stratification and optimal regularity for stationary J-holomorphic maps
Abstract
In this paper, we give an alternate proof of the ε-regularity theorem of Wang [Comm. Cont. Math. 2003] for weakly J-holomorphic maps by adapting the conservation law approach of Rivière [Invent. Math. 2007]. Then we adapt the powerful quantitative stratification method of Naber--Valtorta [Ann. Math. 2017] to obtain optimal volume estimates and rectifiability for the quantitative strata of stationary J-holomorphic maps. In contrast with general harmonic maps, the almost complex equation gives rise to an additional symmetry improvement: Every odd-dimensional translation symmetry extends to an even-dimensional one. Consequently, consecutive even and odd strata coincide, and the k-th quantitative stratum satisfies a tubular-neighborhood estimate with exponent 2m−2⌊ k/2⌋, where 2m is the real dimension of the domain. The corresponding strata are 2⌊ k/2⌋-rectifiable, with almost-everywhere unique complex planes of symmetry. Finally, we prove that every stationary J-holomorphic map is smooth if the target { manifold} admits no non-constant J-holomorphic sphere, improving the corresponding theorem of Wang.
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BibTeX
@misc{guo2026quantitative,
title = {{Quantitative stratification and optimal regularity for stationary $J$-holomorphic maps}},
author = {Chang-Yu Guo and Ming-Lun Liu and Chang-You Wang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.25813},
eprint = {2610.25813},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.25813},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - Quantitative stratification and optimal regularity for stationary J-holomorphic maps
AU - Chang-Yu Guo
AU - Ming-Lun Liu
AU - Chang-You Wang
PY - 2026
DA - 2026/10/19
PB - arXiv
DO - 10.48550/arxiv.2610.25813
UR - https://arxiv.org/abs/2610.25813
AB - In this paper, we give an alternate proof of the ε-regularity theorem of Wang [Comm. Cont. Math. 2003] for weakly J-holomorphic maps by adapting the conservation law approach of Rivière [Invent. Math. 2007]. Then we adapt the powerful quantitative stratification method of Naber--Valtorta [Ann. Math. 2017] to obtain optimal volume estimates and rectifiability for the quantitative strata of stationary J-holomorphic maps. In contrast with general harmonic maps, the almost complex equation gives rise to an additional symmetry improvement: Every odd-dimensional translation symmetry extends to an even-dimensional one. Consequently, consecutive even and odd strata coincide, and the k-th quantitative stratum satisfies a tubular-neighborhood estimate with exponent 2m−2⌊ k/2⌋, where 2m is the real dimension of the domain. The corresponding strata are 2⌊ k/2⌋-rectifiable, with almost-everywhere unique complex planes of symmetry. Finally, we prove that every stationary J-holomorphic map is smooth if the target { manifold} admits no non-constant J-holomorphic sphere, improving the corresponding theorem of Wang.
ER -