New estimates for eigenvalues of the basic Dirac operator. (English) Zbl 1197.53036
For Riemannian foliations with transverse spin structure, there exists the notion of a basic Dirac operator \(D_b\) acting on transverse spinor fields. There are lower bounds known for the square of the first eigenvalue of \(D_b\) in terms of the transversal scalar curvature, the mean curvature and the first eigenvalue of the basic Yamabe operator [cf. S. D. Jung, J. Geom. Phys. 39, No. 3, 253–264 (2001; Zbl 1024.53019) and S. D. Jung, B. H. Kim and J. S. Pak, J. Geom. Phys. 51, No. 2, 166–182 (2004; Zbl 1076.58022)]. These bounds generalise known results due to Th. Friedrich and O. Hijazi for the first eigenvalue of the Dirac operator on a Riemannian spin manifold.
In the current work, the authors improve (the mean curvature part of) these lower bounds by using a modified transverse spinor connection. In the limiting case, the underlying foliation is transversally Einstein with minimal leaves.
In the current work, the authors improve (the mean curvature part of) these lower bounds by using a modified transverse spinor connection. In the limiting case, the underlying foliation is transversally Einstein with minimal leaves.
Reviewer: Felipe Leitner (Stuttgart)
MSC:
| 53C12 | Foliations (differential geometric aspects) |
| 53C27 | Spin and Spin\({}^c\) geometry |
| 53C43 | Differential geometric aspects of harmonic maps |
| 58J50 | Spectral problems; spectral geometry; scattering theory on manifolds |
Keywords:
transverse spin foliations; basic Dirac operator; eigenvalue estimates; transverse Einsteinian; minimal leavesReferences:
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