The first conformal Dirac eigenvalue on 2-dimensional tori

Bernd Ammann (1) , Emmanuel Humbert (1)

1 IECN - Institut Élie Cartan de Nancy

Let M be a compact manifold with a spin structure \chi and a Riemannian metric g. Let \lambda_g^2 be the smallest eigenvalue of the square of the Dirac operator with respect to g and \chi. The \tau-invariant is defined as \tau(M,\chi):= sup inf \sqrt{\lambda_g^2} Vol(M,g)^{1/n} where the supremum runs over the set of all conformal classes on M, and where the infimum runs over all metrics in the given class. We show that \tau(T^2,\chi)=2\sqrt{\pi} if \chi is "the" non-trivial spin structure on T^2. In order to calculate this invariant, we study the infimum as a function on the spin-conformal moduli space and we show that the infimum converges to 2\sqrt{\pi} at one end of the spin-conformal moduli space.

Domaines: Géométrie différentielle [math.DG]

Bernd Ammann, Emmanuel Humbert. The first conformal Dirac eigenvalue on 2-dimensional tori. Journal of Geometry and Physics, 2006, 56, pp.623-642. ⟨hal-00101461⟩