Existence and uniqueness conjecture for constant mean curvature spheres in Sol_3

Let $\mathrm{Sol}_3$ be the three-dimensional Sol geometry, and let $H>0$. A constant mean curvature (CMC) $H$ sphere is an immersed sphere in $\mathrm{Sol}_3$ with constant mean curvature $H$; two surfaces are considered equivalent up to left translations of $\mathrm{Sol}_3$. Existence and uniqueness conjecture. For every $H>0$ there exists an embedded CMC $H$ sphere $S_H$, which is the unique, up to left translations, immersed CMC $H$ sphere and the unique, up to left translations, embedded compact CMC $H$ surface. Moreover, $S_H$ is a solution to the isoperimetric problem, and the family $(S_H)_{H>0}$ is real analytic.

The main theorem establishes the corresponding existence and uniqueness conclusions for $H>1/\sqrt{3}$; the conjecture proposes that the threshold can be removed and also includes the isoperimetric and real-analytic-family assertions.