Abstract for Sfb Preprint No. 330

Upper bounds for the first eigenvalue of the Dirac operator on surfaces I. Agricola and T. Friedrich In this paper we will prove new extrinsic upper bounds for the eigenvalues of the Dirac operator on a surface isometrically immersed into R^3 as well as intrinsic bounds for 2-dimensional compact manifolds of genus zero and genus one. Moreover, we compare the different estimates of the eigenvalue of the Dirac operator for special families of metrics.