Some constructions of constant mean curvature surfaces in $\mathbb{S}^2\times\mathbb{R}$

This talk will consist of three videos. In the first of them, we will recall the definition of constant mean curvature surface, as well as the classical Lawson correspondence between constant mean curvature surfaces in Euclidean space $\mathbb{R}^3$ and minimal surfaces in the round sphere $\mathbb{S}^3$. The second video will explain how this setting can be extended to other homogeneous spaces, and how it leads to a fruitful technique in the construction of constant mean curvature surfaces: starting from a minimal solution of a Plateau problem, one considers the corresponding constant mean curvature surface and extends it across the boundary by means of ambient isometries. Although neither such a solution of the Plateau problem nor the correspondence are explicit, we will discuss how to analyze some properties of the target surfaces, specially those concerning the symmetries. In the last video, we will apply these ideas to obtain compact constant mean curvature surfaces in $\mathbb{S}^2\times\mathbb{R}$ focusing on the possible genera and on whether they are embedded or not. This is part of a joint project with F. Torralbo (University of Granada, Spain), e.g., see arXiv items 2007.06882, 1802.04070, 1311.2500 and 1104.1259.