We study the classification of immersed constant mean curvature spheres in the homogeneous 3-manifold $Sol(3)$, i.e., the only Thurston 3-dimensional geometry where this problem remains open. Our main result states that, for every $H>\sqrt{1/3}$, there exists a unique (up to translations) immersed CMC $H$ sphere $S_H$ in $Sol(3)$. Moreover, this sphere $S_H$ is embedded, and is therefore the unique compact embedded CMC $H$ surface in $Sol(3)$.
The same results are obtained for all real numbers $H$ such that there exists a solution to the isoperimetric problem with mean curvature $H$.
Joint work with Pablo Mira.