Conformal invariants for the zero mode equation
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915716345626624 |
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| author | Wang, Guofang Zhang, Mingwei |
| author_facet | Wang, Guofang Zhang, Mingwei |
| contents | For non-trivial solutions to the zero mode equation on a closed spin manifold \[D φ=iA\cdot φ,\] we first provide a simple proof for the sharp inequality \eq{ \norm{A}_{L^n}^2 \ge \frac {n}{4(n-1)} Y(M,[g]), } where $Y(M,[g])$ is the Yamabe constant of $(M,g)$, which was obtained by Frank-Loss and Reuss. Then we classify completely the equality case by proving that equality holds if and only if $φ$ is a Killing spinor, and if and only if $(M,g)$ is a Sasaki-Einstein manifold with $A$ (up to scaling) as its Reeb field and $φ$ a vacuum up to a conformal transformation. More generalizations have been also studied. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_17854 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Conformal invariants for the zero mode equation Wang, Guofang Zhang, Mingwei Differential Geometry Mathematical Physics Analysis of PDEs For non-trivial solutions to the zero mode equation on a closed spin manifold \[D φ=iA\cdot φ,\] we first provide a simple proof for the sharp inequality \eq{ \norm{A}_{L^n}^2 \ge \frac {n}{4(n-1)} Y(M,[g]), } where $Y(M,[g])$ is the Yamabe constant of $(M,g)$, which was obtained by Frank-Loss and Reuss. Then we classify completely the equality case by proving that equality holds if and only if $φ$ is a Killing spinor, and if and only if $(M,g)$ is a Sasaki-Einstein manifold with $A$ (up to scaling) as its Reeb field and $φ$ a vacuum up to a conformal transformation. More generalizations have been also studied. |
| title | Conformal invariants for the zero mode equation |
| topic | Differential Geometry Mathematical Physics Analysis of PDEs |
| url | https://arxiv.org/abs/2512.17854 |