Quantitative stratification and global regularity for 1/2-harmonic mappings
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| Main Authors: | , , , , |
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| Format: | Preprint |
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2026
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| _version_ | 1866917338138279936 |
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| author | Guo, Changyu Jiang, Guichun Wang, Changyou Xiang, Changlin Zheng, Gaofeng |
| author_facet | Guo, Changyu Jiang, Guichun Wang, Changyou Xiang, Changlin Zheng, Gaofeng |
| contents | In this paper, we extend the celebrated global regularity theory of Naber-Valtorta [Ann. Math. 2017] to 1/2-harmonic mappings into manifolds. Inspired by their work, we first adapt Lin's defect measure theory [Ann. Math. 1999] to such maps building on the partial regularity established by Millot-Pegon-Schikorra [Arch. Ration. Mech. Anal. 2021]. Then apply it to show that the set of singular points of such maps can be quantitatively stratified via a new notion of boundary symmetry with the aid of {the celebrated harmonic extension method by Caffarelli-Silverstre}. As in that of Naber-Valtorta, developing the necessary quantitative regularity estimates, and then combining it with the Reifenberg type theorems and a delicate covering argument allow us to get sharp growth estimates on the volume of tubular neighborhood around singular points and establish the rectifiability of each singular stratum. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_12709 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Quantitative stratification and global regularity for 1/2-harmonic mappings Guo, Changyu Jiang, Guichun Wang, Changyou Xiang, Changlin Zheng, Gaofeng Analysis of PDEs In this paper, we extend the celebrated global regularity theory of Naber-Valtorta [Ann. Math. 2017] to 1/2-harmonic mappings into manifolds. Inspired by their work, we first adapt Lin's defect measure theory [Ann. Math. 1999] to such maps building on the partial regularity established by Millot-Pegon-Schikorra [Arch. Ration. Mech. Anal. 2021]. Then apply it to show that the set of singular points of such maps can be quantitatively stratified via a new notion of boundary symmetry with the aid of {the celebrated harmonic extension method by Caffarelli-Silverstre}. As in that of Naber-Valtorta, developing the necessary quantitative regularity estimates, and then combining it with the Reifenberg type theorems and a delicate covering argument allow us to get sharp growth estimates on the volume of tubular neighborhood around singular points and establish the rectifiability of each singular stratum. |
| title | Quantitative stratification and global regularity for 1/2-harmonic mappings |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2603.12709 |