The first conformal Dirac eigenvalue on 2-dimensional tori

Let M be a compact manifold with a spin structure χ and a Riemannian metric g. Let λ2g be the smallest eigenvalue of the square of the Dirac operator with respect to g and χ. The τ -invariant is defined as τ(M,χ) := sup inf q λ2gVol(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 τ(T , χ) = 2 √ π if χ is “the” non-trivial spin structure on T . 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 √ π at one end of the spin-conformal moduli space. 1 MSC 2000: 53 A 30, 53C27 (Primary) 58 J 50 (Secondary)