# Integrable systems methods for surfaces and new families of constant mean curvature surfaces in \(\mathbb{R}^3\)

## University College Cork

THIS SEMINAR HAS BEEN POSTPONED BECAUSE OF THE UNIVERSITY POLICY CONCERNING THE CORONAVIRUS OUTBREAK. A NEW DATE WILL BE ANNOUNCED AS SOON AS POSSIBLE.

In this lecture I will outline the integrable systems technique for CMC surfaces, but with a view at some other cases. Then I will explain some recent developments in the construction of certain families of CMC surfaces using this setup. In particular, we start with a \(2\times 2\) Cauchy problem to which we associate a scalar second order differential equation. The singularities in this ODE correspond to the ends in the resulting surface. Particularly, regular singularities produce asymptotically Delaunay ends while irregular singularities produce irregular ends. Our aim is to discuss global issues such as period problems and asymptotic behavior involved in the construction of CMC surfaces in \(\mathbb{R}^3\) arising from the family of Heun's differential equations.