Enregistré dans:
| Auteurs principaux: | , , |
|---|---|
| Format: | Preprint |
| Publié: |
2025
|
| Sujets: | |
| Accès en ligne: | https://arxiv.org/abs/2501.10976 |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866910789876580352 |
|---|---|
| author | Feng, Yu Wang, Shuo Xu, Bin |
| author_facet | Feng, Yu Wang, Shuo Xu, Bin |
| contents | In 1987, Hitchin introduced the self-duality equations on rank-2 complex vector bundles over compact Riemann surfaces with genus greater than one as a reduction of the Yang-Mills equation and established the existence of solutions to these equations starting from a Higgs stable bundle. In this paper, we fill in some technical details in Hitchin's original proof by the following three steps. First, we reduce the existence of a solution of class $L_1^2$ to minimizing the energy functional within a Higgs stable orbit of the $L_2^2$ complex gauge group action. Second, using this transformation, we obtain a solution of class $L_1^2$ in this orbit. These two steps primarily follow Hitchin's original approach. Finally, using the Coulomb gauge, we construct a smooth solution by applying an $L_2^2$ unitary gauge transformation to the $L_1^2$ solution constructed previously. This last step provides additional technical details to Hitchin's original proof. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_10976 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A note On the existence of solutions to Hitchin's self-duality equations Feng, Yu Wang, Shuo Xu, Bin Differential Geometry Symplectic Geometry In 1987, Hitchin introduced the self-duality equations on rank-2 complex vector bundles over compact Riemann surfaces with genus greater than one as a reduction of the Yang-Mills equation and established the existence of solutions to these equations starting from a Higgs stable bundle. In this paper, we fill in some technical details in Hitchin's original proof by the following three steps. First, we reduce the existence of a solution of class $L_1^2$ to minimizing the energy functional within a Higgs stable orbit of the $L_2^2$ complex gauge group action. Second, using this transformation, we obtain a solution of class $L_1^2$ in this orbit. These two steps primarily follow Hitchin's original approach. Finally, using the Coulomb gauge, we construct a smooth solution by applying an $L_2^2$ unitary gauge transformation to the $L_1^2$ solution constructed previously. This last step provides additional technical details to Hitchin's original proof. |
| title | A note On the existence of solutions to Hitchin's self-duality equations |
| topic | Differential Geometry Symplectic Geometry |
| url | https://arxiv.org/abs/2501.10976 |