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A brief introduction to the regularity theory of minimising harmonic maps
Abstract
We introduce various classical, foundational results on the regularity theory of harmonic maps, with focuses on the energy minimising harmonic maps. Topics include inner and outer variations, the monotonicity identity, the ε-regularity theorem, blow-ups and tangent maps, and the dimension estimate and stratification of singular sets. We also discuss the analogous theory for stationary harmonic maps and mention some significant recent developments.
Related
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- Minimising the harmonic sum of cycle lengths
- Partial Regularity of Stable Stationary Harmonic Maps into Compact Lie Groups
- Boundary regularity of harmonic map heat flows from RCD spaces into CAT(0) spaces
- Transversal Hölder Criteria and Dini--Zygmund Endpoint Regularity for Hyperbolic Harmonic Mappings
BibTeX
@misc{li2026brief,
title = {{A brief introduction to the regularity theory of minimising harmonic maps}},
author = {Siran Li},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.23497},
eprint = {2609.23497},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.23497},
}
RIS
TY - UNPB
TI - A brief introduction to the regularity theory of minimising harmonic maps
AU - Siran Li
PY - 2026
DA - 2026/09/20
PB - arXiv
DO - 10.48550/arxiv.2609.23497
UR - https://arxiv.org/abs/2609.23497
AB - We introduce various classical, foundational results on the regularity theory of harmonic maps, with focuses on the energy minimising harmonic maps. Topics include inner and outer variations, the monotonicity identity, the ε-regularity theorem, blow-ups and tangent maps, and the dimension estimate and stratification of singular sets. We also discuss the analogous theory for stationary harmonic maps and mention some significant recent developments.
ER -