Constant mean curvature spheres in ${\rm Sol}_3$

Based on work of Daniel and Mira, we prove that for each $H>0$, there exists a unique (up to ambient isometry) immersed constant mean curvature $H$ sphere $S_H$ in ${\rm Sol}_3$, where ${\rm Sol}_3$ is the non-abelian solvable 3-dimensional Lie group equipped with its usual left-invariant metric that makes it into a model space for one of the eight Thurston geometries; see Bonahon or Scott for a discussion of the Thurston geometries that can occur for compact 3-manifolds. Since this proof also demonstrates that $S_H$ has index one, the results in~Daniel-Mira imply $S_H$ is embedded and the left-invariant Gauss map of $S_H$ is a diffeomorphism. One key new result that we prove in order to obtain these results for constant mean curvature spheres is the existence of height estimates for certain constant mean curvature $H$ graphs in ${\rm Sol}_3$, which depend only on any fixed positive lower bound for $H$. By the work in~Daniel-Mira, the existence of the spheres $S_H$ for all $H>0$ implies the existence for all $H\geq 0$ of a complex (holomorphic only when $H=0$) quadratic differential $Q_H$ defined on all constant mean curvature $H$ surfaces $M$ in ${\rm Sol}_3$ and which, for $H>0$, vanishes on an open set of $M$ if and only if $M$ is contained in some left translation of $S_H$; furthermore, when $Q_H$ is not identically zero on $M$, it has isolated zeros on $M$ and negative index at these zeros.