Upper bounds for the first eigenvalue of the Dirac operator on surfaces

In this paper we will prove new extrinsic upper bounds for the eigenvalues of the Dirac operator on an isometrically immersed surface M-2 hooked right arrow R-3 as well as intrinsic bounds for two-dimensional compact manifold s of genus zero and genus one. Moreover, we compare the different estimates of the eigenvalue of the Dirac operator for special families of metrics. ( C) 1999 Elsevier Science B.V. All rights reserved.