Hypersurfaces of Spin$^c$ manifolds and Lawson type correspondence

TL;DR

This paper explores the use of Spin$^c$ structures and spinors to characterize hypersurfaces in certain 3D homogeneous manifolds and establishes a Lawson type correspondence for constant mean curvature surfaces.

Contribution

It introduces a Spin$^c$ geometric framework to characterize hypersurfaces and provides an elementary proof of the Lawson type correspondence for these surfaces.

Findings

Characterization of hypersurfaces via Spin$^c$ spinors

Elementary proof of Lawson type correspondence

Characterization of real hypersurfaces in complex spaces

Abstract

Simply connected 3-dimensional homogeneous manifolds $E(\kappa, \tau)$, with 4-dimensional isometry group, have a canonical Spin$^c$ structure carrying parallel or Killing spinors. The restriction to any hypersurface of these parallel or Killing spinors allows to characterize isometric immersions of surfaces into $E(\kappa, \tau)$. As application, we get an elementary proof of a Lawson type correspondence for constant mean curvature surfaces in $E(\kappa, \tau)$. Real hypersurfaces of the complex projective space and the complex hyperbolic space are also characterized via Spin$^c$ spinors.