| Summary: | We study the horizon geometry of Kerr black holes (BHs) with scalar synchronized hair [ 1], 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, 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 (sic) and the horizon linear velocity v(H) are smaller than critical values, j((S)), (sic)((S)), v(H)((S)) respectively. For the hairy BHs, we find that j(H) < j((S)) is a sufficient, but not necessary, condition for being embeddable; vH < v(H)((S)) is a necessary, but not sufficient, condition for being embeddable; whereas (sic) < (sic)((S)) is a necessary and sufficient condition for being embeddable in E-3. Thus, the latter quantity provides the most faithful diagnosis for the existence of an 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.
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