Embedded tori in $\mathbb S^2\times\mathbb R$ with constant mean curvature

We will describe a 1-parameter family of surfaces in the Riemannian product space $\mathbb S^2\times\mathbb R$ with constant mean curvature $H>0$. Among the constructed surfaces, we find the first non-trivial examples of embedded constant mean curvature tori in $\mathbb S^2\times\mathbb R$. They form a continuous deformations from a stack of tangent spheres to a horizontal equivariant cylinder, in the same fashion as Delaunay’s unduloids in Euclidean space $\mathbb R^3$ form a continuous deformation from a stack of round spheres to a round cylinder. These non-equivariant tori have constant mean curvature $H>\frac{1}{2}$. This is part of a recent joint work with F. Torralbo (see arXiv:2007.06882).