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

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 $\lambda_1(\not\hspace{-0.1cm}D)$ (with the least absolute value) of the induced Dirac operator $\not\hspace{-0.1cm}D$ of $M$ satisfies the Spin$^c$ B\"{a}r inequality \begin{eqnarray*} \lambda_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$^c$ B\"{a}r 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.