# Spectral estimates on 2-tori

by

Bernd Ammann

**Spectral estimates on 2-tori**
(.dvi,
.ps)

*Preprint April 2000*

We prove upper and lower bounds for the eigenvalues of the Dirac
operator and the
Laplace operator on 2-dimensional tori. In particluar we give a lower bound
for the first eigenvalue of the Dirac operator for non-trivial spin structures.
It is the only explicit estimate for eigenvalues of the Dirac operator known so far
that uses information about the spin structure.
As a corollary we obtain lower bounds for the Willmore functional
of a torus embedded into S^{3}.
In the final section we compare Dirac spectra for two different spin structures
on an arbitrary Riemannian spin manifold.
53C27 (Primary), 58J50 53C80 (Secondary)
### Keywords

Dirac operator, Laplace operator, spectrum, conformal metrics,
two-dimensional torus, spin structures, Willmore functional

Bernd Ammann, 2.4.2000