Parabolic gap theorems for the Yang-Mills energy
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866917412262117376 |
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| author | Dayaprema, Anuk Waldron, Alex |
| author_facet | Dayaprema, Anuk Waldron, Alex |
| contents | We prove parabolic versions of several known gap theorems in classical Yang-Mills theory. On an $\mathrm{SU}(r)$-bundle of charge $κ$ over the 4-sphere, we show that the space of all connections with Yang-Mills energy less than $4 π^2 \left( |κ| + 2 \right)$ deformation-retracts under Yang-Mills flow onto the space of instantons, allowing us to simplify the proof of Taubes's path-connectedness theorem. On a compact quaternion-Kähler manifold with positive scalar curvature, we prove that the space of pseudo-holomorphic connections whose $\mathfrak{sp}(1)$ curvature component has small Morrey norm deformation-retracts under Yang-Mills flow onto the space of instantons. On a nontrivial bundle over a compact manifold of general dimension, we prove that the infimum of the scale-invariant Morrey norm of curvature is positive. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_21050 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Parabolic gap theorems for the Yang-Mills energy Dayaprema, Anuk Waldron, Alex Differential Geometry Analysis of PDEs We prove parabolic versions of several known gap theorems in classical Yang-Mills theory. On an $\mathrm{SU}(r)$-bundle of charge $κ$ over the 4-sphere, we show that the space of all connections with Yang-Mills energy less than $4 π^2 \left( |κ| + 2 \right)$ deformation-retracts under Yang-Mills flow onto the space of instantons, allowing us to simplify the proof of Taubes's path-connectedness theorem. On a compact quaternion-Kähler manifold with positive scalar curvature, we prove that the space of pseudo-holomorphic connections whose $\mathfrak{sp}(1)$ curvature component has small Morrey norm deformation-retracts under Yang-Mills flow onto the space of instantons. On a nontrivial bundle over a compact manifold of general dimension, we prove that the infimum of the scale-invariant Morrey norm of curvature is positive. |
| title | Parabolic gap theorems for the Yang-Mills energy |
| topic | Differential Geometry Analysis of PDEs |
| url | https://arxiv.org/abs/2412.21050 |