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

TL;DR

This paper classifies hypersurfaces in special Spin$^c$ manifolds that achieve equality in a geometric eigenvalue inequality, focusing on cones over manifolds with Killing spinors, extending known results to Spin$^c$ settings.

Contribution

It extends the classification of hypersurfaces satisfying the equality case of the Spin$^c$ Bär inequality to cones over Spin$^c$ manifolds with Killing spinors, under specific curvature conditions.

Findings

Hypersurfaces satisfying the equality are slices of the cone.

The classification applies to cones over manifolds with real Killing spinors.

Results generalize previous classifications in the Riemannian spin case.

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 $\lambda_1(\not\hspace{-0.1cm}D)$ (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.