  We prove lower Dirac eigenvalue bounds for closed surfaces with a spin
structure whose Arf invariant equals 1. Besides the area only one geometric
quantity enters in these estimates, the spin-cut-diameter which depends on the
choice of spin structure. It can be expressed in terms of various distances on
the surfaces or, alternatively, by stable norms of certain cohomology classes.
In case of the 2-torus we obtain a positive lower bound for all Riemannian
metrics and all nontrivial spin structures. The corresponding estimate also
holds for the $L^2$-spectrum of the Dirac operator on a noncompact complete
surface of finite area. As a corollary we get positive lower bounds on the
Willmore integral for all 2-tori embedded in $R^3$.
