Hypersurfaces immersed in special Spin$^c$ manifolds by first eigenspinors
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915463084113920 |
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| author | Nakad, Roger |
| author_facet | Nakad, Roger |
| contents | Let $M$ be a closed orientable hypersurface of dimension $n$, with nonwhere vanishing mean curvature $H$, immersed into a Riemannian Spin$^c$ manifold $\mathcal Z$ carrying a parallel spinor field. The first eigenvalue $λ_1(\not\hspace{-0.1cm}D)$ (with the least absolute value) of the induced Dirac operator $\not\hspace{-0.1cm}D$ of $M$ satisfies the Spin$^c$ Bär inequality \begin{eqnarray*} λ_1^2 (\not\hspace{-0.1cm}D) \leq \frac{n^2}{4 \ \mathrm{vol}(M)}\int_M H^2 dV, \end{eqnarray*} where $\mathrm{vol}(M)$ is the volume of $M$ and $dV$ is the volume form of the manifold $M$. In this paper, we classify hypersurfaces $M$ that satisfy the equality case in the Spin$^c$ Bär inequality when $\mathcal Z = (0,+\infty) \times P$ is the cone over a Riemannian Spin$^c$ manifold $P$ carrying a real Killing spinor, under two conditions: one being a Ricci condition on $\mathcal Z$, and the second one the curvature of the auxiliary line bundle associated with the Spin$^c$ structure on $\mathcal Z$. More precisely, we prove that $M$ are the slices $\{s\} \times P$, where $s \in (0,+\infty)$. In the special case, when $\mathcal Z=\mathbb R^{n+1}$, i.e., the cone over the sphere, which is a Spin manifold with a parallel spinor, the classification result was previously obtained by Hijazi and Montiel. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2508_18472 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Hypersurfaces immersed in special Spin$^c$ manifolds by first eigenspinors Nakad, Roger Differential Geometry Spectral Theory 53C27, 53C40, 53C80, 58G25 Let $M$ be a closed orientable hypersurface of dimension $n$, with nonwhere vanishing mean curvature $H$, immersed into a Riemannian Spin$^c$ manifold $\mathcal Z$ carrying a parallel spinor field. The first eigenvalue $λ_1(\not\hspace{-0.1cm}D)$ (with the least absolute value) of the induced Dirac operator $\not\hspace{-0.1cm}D$ of $M$ satisfies the Spin$^c$ Bär inequality \begin{eqnarray*} λ_1^2 (\not\hspace{-0.1cm}D) \leq \frac{n^2}{4 \ \mathrm{vol}(M)}\int_M H^2 dV, \end{eqnarray*} where $\mathrm{vol}(M)$ is the volume of $M$ and $dV$ is the volume form of the manifold $M$. In this paper, we classify hypersurfaces $M$ that satisfy the equality case in the Spin$^c$ Bär inequality when $\mathcal Z = (0,+\infty) \times P$ is the cone over a Riemannian Spin$^c$ manifold $P$ carrying a real Killing spinor, under two conditions: one being a Ricci condition on $\mathcal Z$, and the second one the curvature of the auxiliary line bundle associated with the Spin$^c$ structure on $\mathcal Z$. More precisely, we prove that $M$ are the slices $\{s\} \times P$, where $s \in (0,+\infty)$. In the special case, when $\mathcal Z=\mathbb R^{n+1}$, i.e., the cone over the sphere, which is a Spin manifold with a parallel spinor, the classification result was previously obtained by Hijazi and Montiel. |
| title | Hypersurfaces immersed in special Spin$^c$ manifolds by first eigenspinors |
| topic | Differential Geometry Spectral Theory 53C27, 53C40, 53C80, 58G25 |
| url | https://arxiv.org/abs/2508.18472 |