$L^2$-harmonic forms and spinors on stable minimal hypersurfaces
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866908801181941760 |
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| author | Bei, Francesco Pipoli, Giuseppe |
| author_facet | Bei, Francesco Pipoli, Giuseppe |
| contents | Let $f:N\rightarrow (M,g)$ be an oriented (or spin), complete, stable, minimal, immersed hypersurface. In this paper we establish various vanishing theorems for the space of $L^2$-harmonic forms and spinors (in the spin case) under suitable positive curvature assumptions on the ambient manifold. Our results in the setting of forms extend to higher dimensions and more general ambient Riemannian manifolds previous vanishing theorems due to Tanno \cite{Tanno} and Zhu \cite{Zhu}. In the setting of spin manifolds our results allow to conclude, for instance, that any oriented, complete, stable, minimal, immersed hypersurface of $\mathbb{R}^m$ or $\mathbb{S}^m$ carries no non-trivial $L^2$-harmonic spinors. Finally, analogous results are proved for strongly stable constant mean curvature hypersurfaces. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2407_18836 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | $L^2$-harmonic forms and spinors on stable minimal hypersurfaces Bei, Francesco Pipoli, Giuseppe Differential Geometry 53C42, 53C27, 58J05 Let $f:N\rightarrow (M,g)$ be an oriented (or spin), complete, stable, minimal, immersed hypersurface. In this paper we establish various vanishing theorems for the space of $L^2$-harmonic forms and spinors (in the spin case) under suitable positive curvature assumptions on the ambient manifold. Our results in the setting of forms extend to higher dimensions and more general ambient Riemannian manifolds previous vanishing theorems due to Tanno \cite{Tanno} and Zhu \cite{Zhu}. In the setting of spin manifolds our results allow to conclude, for instance, that any oriented, complete, stable, minimal, immersed hypersurface of $\mathbb{R}^m$ or $\mathbb{S}^m$ carries no non-trivial $L^2$-harmonic spinors. Finally, analogous results are proved for strongly stable constant mean curvature hypersurfaces. |
| title | $L^2$-harmonic forms and spinors on stable minimal hypersurfaces |
| topic | Differential Geometry 53C42, 53C27, 58J05 |
| url | https://arxiv.org/abs/2407.18836 |