ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

5. Differential Geometry and Global Analysis

1991 MS Classification: 58G25 (prim.), 53C42, 53A50, 53A30 (sec.) Ammann, Bernd, Universität Freiburg New estimates for the Laplace and Dirac spectrum of 2-Tori and applications to the Willmore functional We use conformal methods to estimate geometric data on the 2-torus T^{2} in terms of the L^{p}-norm (p > 1) of the Gaussian curvature, the 1-systole and the area. This method yields estimates for the eigenvalues of the Laplace and Dirac operators. If T^{2} carries a non-trivial spin structure, then we give a lower bound for the smallest eigenvalue of the square of the Dirac operator in terms of the 1-systole, the spin systole, the area and the L^{p}- norm of the curvature. It is the first such estimate depending on the spin structure. We apply this estimate to the Willmore functional for embeddings T^{2} → S^{3}. For many spin-conformal equivalence classes of T^{2} we prove the Willmore conjecture under the assumption that the L^{p}-norm of the Gaussian curvature is sufficiently small.