Hypersurfaces immersed in special Spin${}^c$ manifolds by first eigenspinors

Abstract

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 ${\lambda }_1(/D)$ (with the least absolute value) of the induced Dirac operator $/D$ 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.

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