Figure 1:
Horizon area vs. angular momentum for Myers–Perry black holes with a single spin in (black), (dark gray), and (light gray). 

Figure 2:
Phase space of (a) fivedimensional and (b) sixdimensional MP rotating black holes: black holes exist for parameters within the shaded regions. The boundaries of the phase space correspond to extremal black holes with regular horizons, except at the corners of the square in five dimensions, where they become naked singularities. The sixdimensional phase space extends along the axes to arbitrarily large values of each of the two angular momenta (ultraspinning regimes). 

Figure 3:
Phase space of (a) sevendimensional, and (b) eightdimensional MP rotating black holes (in a representative quadrant ). The surfaces for extremal black holes are represented: black holes exist in the region bounded by these surfaces. (a) : the hyperbolas at which the surface intersects the planes (which are , i.e., and ) correspond to naked singularities with zero area; otherwise, the extremal solutions are nonsingular. The three prongs extend to infinity; these are the ultraspinning regimes in which one spin is much larger than the other two. The prong along becomes asymptotically of the form , i.e., the same shape as the fivedimensional diagram in Figure 2(a). (b) : ultraspinning regimes exist in which two spins are much larger than the third one. The sections at large constant asymptotically approach the same shape as the sixdimensional phase space Figure 2(b). 

Figure 4:
Horizon area of fivedimensional MP black holes. We only display a representative quadrant of the full phase space of Figure 2(a), the rest of the surface being obtained by reflection along the planes and . 

Figure 5:
Curve of horizon area vs. spin for fivedimensional black rings rotating along their (solid). The dashed curve corresponds to fivedimensional MP black holes (see Figure 1). The solid curve for black rings has two branches that meet at a regular, nonextremal minimallyrotating black ring at : an upper branch of thin black rings, and a lower branch of fat black rings. Fat black rings always have a smaller area than MP black holes. Their curves meet at the same zeroarea naked singularity at . 

Figure 6:
Phase space of doublyspinning black rings (, ), restricted to the representative region . The dashed line 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 spin , for given rotation . It ends on the extremal MP curve at . The (lower) thick black curve corresponds to regular nonextremal black rings with minimal spin along for given on . It ends on the extremal MP curve at . Black rings exist in the grayshaded parameter regions bounded by the black curves, the segment of the extremal MP dashed line, and the axis with . In the lightgray region there exist only thin black rings. In the darkgray spandrel between the dashed MP line, the thick black curve, and the axis , there exist thin and fat black rings and MP black holes: there is discrete threefold nonuniqueness. 

Figure 7:
The phase space covered by doublyspinning MP black holes and black rings, obtained by replicating Figure 6 taking . The square corresponds to MP black holes (see Figure 2(a)). The lightgray zones contain thin black rings only, and the mediumgray zone contains MP black holes only. At each point in the darkgray spandrels near the corners of the square there exist one thin and one fat black ring, and one MP black hole. 

Figure 8:
Rod structures for the (a) 4D Schwarzschild and (b) 5D Tangherlini black holes. From top to bottom, the lines represent the sources for the time, and (in 5D) potentials , , . 

Figure 9:
Rod structures for (a) the seed used to generate (b) the rotating black ring. The seed metric is diagonal, and the dotted rod has negative density . In the final solution the parameters can be adjusted so that the metric at on the axis is completely smooth. The (upper) horizon rod in the final solution has mixed direction , while the other rods are aligned purely along the or directions. 

Figure 10:
Curves for phases of fivedimensional black holes with a single angular momentum: MP black hole (black), black ring (dark gray), black Saturn (light gray). We only include those black Saturns, where the central black hole and the black ring have equal surface gravities and angular velocities. The three curves meet tangentially at a naked singularity at , . The cusp of the blackring curve occurs at , . The cusp of the blackSaturn curve is at , with area . 

Figure 11:
Correspondence between phases of black membranes wrapped on a twotorus of side (left) and quicklyrotating MP black holes with rotation parameter (right: must be rotated along a vertical axis): (i) uniform black membrane and MP black hole; (ii) nonuniform black membrane and pinched black hole; (iii) pinchedoff membrane and black hole; (iv) localized black string and black ring (reproduced from [80]). 

Figure 12:
Proposal of [80] for the phase curves of thermal equilibrium phases in . The solid lines and figures have significant arguments in their favor, while the dashed lines and figures might not exist and admit conceivable, but more complicated, alternatives. Some features have been drawn arbitrarily; at any given bifurcation, and in any dimension, smooth connections are possible instead of swallowtails with cusps; also, the bifurcation into two blackSaturn phases may happen before, after, or right at the merger with the pinched black hole. Mergers to dirings or multiplering configurations that extend to asymptotically large seem unlikely. If thermal equilibrium is not imposed, the whole semiinfinite strip , is covered, and multiple rings are possible. 

Figure 13:
against for KerrAdS black holes. The thick curve corresponds to extremal black holes. Blackhole solutions lie on, or above, this curve (which was determined using results in [20]). The thin line is the BPS bound . 

Figure 14:
(vertical) against for Myers–PerryAdS black holes. Nonextremal black holes fill the region above the surface. The surface corresponds to extremal black holes, except when one of the angular momenta vanishes (in which case there is not a regular horizon, just as in the asymptoticallyflat case). This “extremal surface” lies inside the squarebased pyramid (with vertex at the origin) defined by the BPS relation , so none of the black holes are BPS. 
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