The first conformal Dirac eigenvalue on 2-dimensional tori

Details The first conformal Dirac eigenvalue on 2-dimensional tori The first conformal Dirac eigenvalue on 2-dimensional tori

2004-12-20

arxiv.org/abs/math/0412409v2

J. Geom. Phys. 56, 623-642 (2006)

Mathematics

Differential Geometry

published version (typos removed, bibliography updated)

Scientific paper

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.

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