The authors construct explicitly smooth embedded surfaces in \(\mathbb{R}^3\) of arbitrary genus whose geodesic flow is ergodic and Bernoulli with respect to the Liouville measure. The examples are obtained from embedded minimal surfaces with boundary. For a suitable choice of such a surface (any surface which is obtained from the Schwarz \(P\)-surface by a finite number of reflections in a boundary component will do) the metric can be perturbed near the boundary in such a way that a rotational symmetric focussing cap can smoothly be attached to each boundary component: it is shown that the resulting surface has the required properties.
The article is beautifully written and contains a survey of the history of the problem and earlier results as well as several very instructive pictures.