Manifolds with small Dirac eigenvalues are nilmanifolds

The Dirac Operator on Nilmanifolds and Collapsing Circle Bundles by Bernd Ammann and Christian Bär November , 1997

We compute the spectrum of the Dirac operator on 3-dimensional Heisenberg manifolds. The behavior under collapse to the 2-torus is studied. Depending on the spin structure either all eigenvalues tend to ±∞ or there are eigenvalues converging to those of the torus. This is shown to be true in general for collapsing circle bundles with totally geodesic fibers. Using the Hopf fibration we use this...

Kähler manifolds with small eigenvalues of the Dirac operator and a conjecture of Lichnerowicz

Eigenvalues of the Dirac Operator on Manifolds with Boundary

Under standard local boundary conditions or certain global APS boundary conditions, we get lower bounds for the eigenvalues of the Dirac operator on compact spin manifolds with boundary. For the local boundary conditions, limiting cases are characterized by the existence of real Killing spinors and the minimality of the boundary.

Branson’s Q-curvature in Spin Geometry

Abstract. We first give an elementary proof of a result relating the eigenvalues of the Dirac operator to Branson’s Q-curvature on 4-dimensional spin compact manifolds. In the case of n-dimensional closed compact (spin) manifolds we then use the conformal covariance of the Dirac, Yamabe and Branson-Paneitz operators to compare appropriate powers of their first eigenvalues. Equality cases are al...

Dirac Eigenvalues for Generic Metrics on Three-manifolds

We show that for generic Riemannian metrics on a closed spin manifold of dimension three the Dirac operator has only simple eigenvalues.