Figure 6

Figure 6: Phase space of doubly-spinning black rings (j1 ≡ jψ, j2 ≡ jφ), restricted to the representative region j1 > j2 ≥ 0. The dashed line j1 + j2 = 1 corresponds to extremal MP black holes (see Figure 2(a)). The (upper) thin black curve corresponds to regular extremal black rings with degenerate horizons at maximal S2 spin j 2, for given S1 rotation j 1. It ends on the extremal MP curve at (3∕4,1∕4). The (lower) thick black curve corresponds to regular nonextremal black rings with minimal spin j1 along 1 S for given j2 on 2 S. It ends on the extremal MP curve at (4∕5,1∕5). Black rings exist in the gray-shaded parameter regions bounded by the black curves, the segment j1 ∈ [3∕4,4∕5] of the extremal MP dashed line, and the j1 axis with j ≥ ∘27--∕32 1. In the light-gray region there exist only thin black rings. In the dark-gray spandrel between the dashed MP line, the thick black curve, and the axis ∘ ------ 27∕32 < j1 < 1, there exist thin and fat black rings and MP black holes: there is discrete three-fold nonuniqueness.