Speaker: Bernd Ammann (MSRI)

Title: A nonlinear Dirac equation of Yamabe type and 
       constant mean curvature surfaces
       (joint work with E. Humbert and B. Morel, Nancy)


Abstract:

The subject of the talk is a conformally invariant functional
on a compact Riemannian spin manifold $(M^n,g,\sigma)$.
We show that the functional attains its infimum if the first positive
eigenvalue $\lambda_1^+(M,g,\sigma)$ of the Dirac operator on
$(M,g,\sigma)$ satisfies
$$
\lambda_1^+(M,g,\sigma)\vol(M,g)^{1/n}
    < \lambda_1^+(S^n)\vol(S^n)^{1/n}.
$$
We provide two applications of this result. As a first application,
using the spinorial Weierstrass representation, the minimizer provides
periodic constant mean curvature surfaces in the three-dimensional
spaces of constant curvature $R^3$, $S^3$ and $H^3$.  As a second
application, we show that the first Dirac eigenvalue is bounded from
below in a spin-conformal class, and the infimum is attained by a
metric with certain singularities.

In the last part of the talk, we present some theorems showing that the above
inequality holds on a large class of manifolds, including all compact
Riemann surfaces of genus $\geq 1$ with a suitable spin structure, all
non-conformally-flat spin manifolds of dimension $\geq 7$, and all
conformally flat spin manifolds whose Dirac operator has a non-trivial
mass endomorphism or a non-trivial kernel.

The subject of the talk has many analogies and relations
to the solution of Yamabe's problem, which amounts to finding
a constant scalar curvature metric in a given conformal class.
In particular, the mass endomorphism mentioned above is a spinorial
version of the ADM mass that appears in the solution of Yamabe's problem.