Minimal surfaces with arbitrary topology in \(\mathbb{H}^2\times\mathbb{R}\) (Q2059712)

Answering to a conjecture of A. Ros, the author proves that any open orientable surface can be properly embedded in \(\mathbb{H}^2\times\mathbb{R}\) as an area-minimizing surface. The proof uses a vertical bridge principle for tall curves in \(\mathbb{S}^1_\infty\times\mathbb{R}\) and the existence of positive solutions of the asymptotic Plateau problem. This allows to construct complete, properly embedded minimal surfaces in \(\mathbb{H}^2\times\mathbb{R}\) with any (finite or infinite) number of genus and ends.