Horizon geometry for Kerr black holes with synchronised hair

We study the horizon geometry of Kerr black holes (BHs) with scalar synchronised hair, a family of solutions of the Einstein-Klein-Gordon system that continuously connects to vacuum Kerr BHs. We identify the region in parameter space wherein a global isometric embedding in Euclidean 3-space, $\mathbb{E}^3$, is possible for the horizon geometry of the hairy BHs. For the Kerr case, such embedding is possible iff the horizon dimensionless spin $j_H$ (which equals the total dimensionless spin, $j$), the sphericity $\mathfrak{s}$ and the horizon linear velocity $v_H$ are smaller than critical values, $j^{\rm (S)},\mathfrak{s}^{\rm (S)}, v_H^{\rm (S)}$, respectively. For the hairy BHs, we find that $j_H<j^{\rm (S)}$ is a sufficient, but not necessary, condition for being embeddable; $v<v_H^{\rm (S)}$ is a necessary, but not sufficient, condition for being embeddable; whereas $\mathfrak{s}<\mathfrak{s}^{\rm (S)}$ is a necessary and sufficient condition for being embeddable in $\mathbb{E}^3$. Thus the latter quantity provides the most faithful diagnosis for the existence of an $\mathbb{E}^3$ embedding within the whole family of solutions. We also observe that sufficiently hairy BHs are always embeddable, even if $j$ -- which for hairy BHs (unlike Kerr BHs) differs from $j_H$ --, is larger than unity.