Lower bounds for the eigenvalues of the \(\mathrm{Spin}^c\) Dirac operator on submanifolds. (English) Zbl 1321.53056
The paper [N. Ginoux and B. Morel, Int. J. Math. 13, No. 5, 533–548 (2002; Zbl 1049.58028)] gives lower estimates for the eigenvalues of the submanifold Dirac operator \(D_H\) in any codimension in terms of the norm of the mean curvature, the energy momentum tensors of the related eigenspinor, and an adapted conformal change of the metric.
The current paper by the authors [Differ. Geom. Appl. 31, No. 1, 93–103 (2013; Zbl 1262.58008)] studies similar bounds for the submanifold Dirac operator in any codimension of \(\mathrm{Spin}^c\)-manifolds. In the limiting case the underlying geometric structures are special, possessing generalized twisted Killing spinors. However, the authors show that with natural assumptions on the codimension of the submanifold, these spinor fields are in fact twisted Killing spinors.
The current paper by the authors [Differ. Geom. Appl. 31, No. 1, 93–103 (2013; Zbl 1262.58008)] studies similar bounds for the submanifold Dirac operator in any codimension of \(\mathrm{Spin}^c\)-manifolds. In the limiting case the underlying geometric structures are special, possessing generalized twisted Killing spinors. However, the authors show that with natural assumptions on the codimension of the submanifold, these spinor fields are in fact twisted Killing spinors.
Reviewer: Felipe Leitner (Stuttgart)
MSC:
| 53C27 | Spin and Spin\({}^c\) geometry |
| 53C42 | Differential geometry of immersions (minimal, prescribed curvature, tight, etc.) |
| 58J50 | Spectral problems; spectral geometry; scattering theory on manifolds |
Keywords:
submanifolds; Dirac operator; eigenvalues, \(\mathrm{Spin}^c\) spinors; twisted Killing spinorsReferences:
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