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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2021
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2112.03640 |
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| _version_ | 1866910537265184768 |
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| author | Sire, Yannick Xu, Tian |
| author_facet | Sire, Yannick Xu, Tian |
| contents | Let $M$ be a closed spin manifold of dimension $m\geq6$ equipped with a Riemannian metric $\ig$ and a spin structure $\sa$. Let $\lm_1^+(\tilde\ig)$ be the smallest positive eigenvalue of the Dirac operator $D_{\tilde\ig}$ on $M$ with respect to a metric $\tilde\ig$ conformal to $\ig$. The Bär-Hijazi-Lott invariant is defined by $\lm_{min}^+(M,\ig,\sa)=\inf_{\tilde\ig\in[\ig]}\lm_1^+(\tilde\ig)\Vol(M,\tilde\ig)^\frac{1}{m}$. In this paper, we show that \[ \lm_{min}^+(M,\ig,\sa)<\lm_{min}^+(S^m,\ig_{S^m},\sa_{S^m})=\frac m2\Vol(S^m,\ig_{S^m})^{\frac1m} \] provided that $\ig$ is not locally conformally flat. This estimate is a spinorial analogue to an estimate by T. Aubin, solving the Yamabe problem in this setting. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2112_03640 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | On the Bär-Hijazi-Lott invariant for the Dirac operator and a spinorial proof of the Yamabe problem Sire, Yannick Xu, Tian Differential Geometry Let $M$ be a closed spin manifold of dimension $m\geq6$ equipped with a Riemannian metric $\ig$ and a spin structure $\sa$. Let $\lm_1^+(\tilde\ig)$ be the smallest positive eigenvalue of the Dirac operator $D_{\tilde\ig}$ on $M$ with respect to a metric $\tilde\ig$ conformal to $\ig$. The Bär-Hijazi-Lott invariant is defined by $\lm_{min}^+(M,\ig,\sa)=\inf_{\tilde\ig\in[\ig]}\lm_1^+(\tilde\ig)\Vol(M,\tilde\ig)^\frac{1}{m}$. In this paper, we show that \[ \lm_{min}^+(M,\ig,\sa)<\lm_{min}^+(S^m,\ig_{S^m},\sa_{S^m})=\frac m2\Vol(S^m,\ig_{S^m})^{\frac1m} \] provided that $\ig$ is not locally conformally flat. This estimate is a spinorial analogue to an estimate by T. Aubin, solving the Yamabe problem in this setting. |
| title | On the Bär-Hijazi-Lott invariant for the Dirac operator and a spinorial proof of the Yamabe problem |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2112.03640 |