We study how the spin structures on finite-volume hyperbolic nmanifolds restrict to cusps. When a cusp cross-section is a (n − 1)torus, there are essentially two possible behaviours: the spin structure is either bounding or Lie. We show that in every dimension n there are examples where at least one cusp is Lie, and in every dimension n ≤ 8 there are examples where all the cusps are bounding. By work of C. Bär, this implies that the spectrum of the Dirac operator is ℝ in the first case, and discrete in the second. We therefore deduce that there are cusped hyperbolic manifolds whose spectrum of the Dirac operator is ℝ in all dimensions, and whose spectrum is discrete in all dimensions n ≤ 8. HVBx7mXyfVswJeExosj/ASqxWOJhu.
The Dirac operator on cusped hyperbolic manifolds
Martelli, Bruno;
2025-01-01
Abstract
We study how the spin structures on finite-volume hyperbolic nmanifolds restrict to cusps. When a cusp cross-section is a (n − 1)torus, there are essentially two possible behaviours: the spin structure is either bounding or Lie. We show that in every dimension n there are examples where at least one cusp is Lie, and in every dimension n ≤ 8 there are examples where all the cusps are bounding. By work of C. Bär, this implies that the spectrum of the Dirac operator is ℝ in the first case, and discrete in the second. We therefore deduce that there are cusped hyperbolic manifolds whose spectrum of the Dirac operator is ℝ in all dimensions, and whose spectrum is discrete in all dimensions n ≤ 8. HVBx7mXyfVswJeExosj/ASqxWOJhu.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
