The paper under review applies to Kähler-Einstein manifolds the program of studying rigidity problems using the spectral properties of the Dirac operator on a Riemannian spin manifold. Such results were given in [O. Hijazi et al., Ann. Global Anal. Geom. 23, No. 3, 247–264 (2003; Zbl 1032.53040); Math. Res. Lett. 8, No. 1–2, 195–208 (2001; Zbl 0988.53019); O. Hijazi and S. Montiel, Math. Z. 244, No. 2, 337–347 (2003; Zbl 1035.53066)], using S. Raulot’s correspondence [Lett. Math. Phys. 86, No. 2–3, 177–192 (2008; Zbl 1204.53037)] of parallel spinors on an \((n+2)\)-dimensional manifold \(N\) with solutions \(\phi\) of the Dirac equation \(D_M \phi = \frac{n+1}{n}H_0 \phi\) on its boundary \(M\). In [J. Geom. 107, No. 3, 533–555 (2016; Zbl 1362.53058)] the authors extended this correspondence to the case of a spin manifold \(N\) such that its boundary \(M\) is endowed with a Riemannian flow given by a unit vector field. They obtained similar rigidity results in this case.
In this paper the authors extend this correspondence to the case of a \(\mathrm{spin}^c\) manifold \(N\). They formulate a \(\mathrm{spin}^c\) and basic version of the Dirac equation and obtain a certain estimate for an arbitrary solution, relating it with the mean curvature of \(M\) and the O’Neill tensor field associated with the flow. It follows from the classification of \(\mathrm{spin}^c\) manifolds [A. Moroianu, Commun. Math. Phys. 187, No. 2, 417–427 (1997; Zbl 0888.53035)] stating that, in the limiting case, the following cases can occur: When \(n\) is even, \(N\) is a domain in some Kähler-Einstein manifold with nonnegative scalar curvature. When \(n\) is odd, \(N\) is the product of a Kähler-Einstein manifold with \(\mathbb R\) or \(S^1\). Then they establish the appropriate version of the correspondence with the parallel spinors in each case.