Navigating the space of symmetric CMC surfaces
Abstract of Talk:
We consider compact surfaces of constant mean curvature (CMC) in 3-dimensional space forms.
While the only CMC spheres are round spheres and CMC tori can be explicitly parametrized via
integrable systems methods, only very little is known about higher genus CMC surfaces.
In this talk we first give a brief introduction to the spectral curve approach to CMC tori due to
Hitchin. In general the constant mean curvature condition of a surface can be translated into the
flatness condition of an associated family of SL(2,C)-connections. For tori, the abelian fundamental
group allows to reduce flat SL(2,C)-connections to flat line bundle connections, and the associated
family can be parametrized in terms of certain algebraic geometric objects - the spectral data - from
which the conformal immersion can be reconstructed.
Under the assumption of certain discrete symmetries, irreducible connections on higher genus
surfaces can also be parametrized by flat line bundle connections. This enables us to define a
generalization of the spectral curve theory for higher genus CMC surfaces. Due to the non-abelian
nature, it is hard to construct spectral data for higher genus CMC surfaces explicitly.
In a recent preprint (joint work with L. Heller and N. Schmitt) we have introduced a flow on the
spectral data from CMC tori towards higher genus CMC surfaces. We explain how this flow can be
used to construct spectral data for higher genus CMC surfaces and to study the moduli space of
symmetric CMC surfaces of higher genus.