A complete classification is possible for . Recall from Section 3.2.1, the staticity conditions for a near-horizon geometry are and .
Theorem 4.1 (). The only vacuum static near-horizon geometry for and compact is given by , and . For this result is also valid for .
In four dimensions one can solve Eq. (66) in general without assuming compactness of . For non-constant and any one obtains the near-horizon geometry (sending )
The classification of near-horizon geometries in vacuum gravity with a cosmological constant can be completely solved. Although very simple, to the best of our knowledge this has not been presented before, so for completeness we include it here.
The main simplification comes from the fact that cross sections of the horizon are one-dimensional, so the horizon equations are automatically ODEs. Furthermore, there is no intrinsic geometry on and so the only choice concerns its global topology, which must be either or .
Theorem 4.2. Consider a near-horizon geometry with compact cross section , which satisfies the vacuum Einstein equations including cosmological constant . If the near-horizon geometry is given by the quotient of AdS3 in Eq. (73). For the only solution is the trivial flat geometry . There are no solutions for .
Proof: We may choose a periodic coordinate on so the horizon metric is simply and the 1-form . Observe that must be a globally-defined function and hence must be a periodic function of . Since the curvature and metric connection trivially vanish, the horizon equations (66) and (67) simplify to71) over to deduce that and , which gives the trivial flat near-horizon geometry . For we argue as follows. Multiply Eq. (71) by and integrate over to find . Hence must be a constant and substituting into the horizon equations gives and (without loss of generality we have chosen a sign for ). The near-horizon geometry is then 3 and in the second equality we have written it as a fibration over AdS2.
It is worth remarking that the ODE (71) can be completely integrated without assuming compactness. For this reveals a second solution and , where we have set the integration constant to zero by translating the coordinate . Upon changing the resulting near-horizon geometry is:3. Unlike the previous case though, cannot be taken to be compact and hence we must have . For there is also a second solution given by and , although this is singular. For there is a unique solution given by and , although this is also singular.18
The general solution to Eq. (66) is not known in this case. In view of the rigidity theorem it is natural to assume axisymmetry. If one assumes such a symmetry, the problem becomes of ODE type and it is possible to completely solve it. The result is summarised by the following theorem, first proved in [122, 163] for and in  for .
Theorem 4.3 ([122, 163, 154]). Consider a spacetime containing a degenerate horizon, invariant under an isometry, satisfying the vacuum Einstein equations including a cosmological constant. Any non-static near-horizon geometry, with compact cross section, is given by the near-horizon limit of the extremal Kerr or Kerr-(A)dS black hole.
Now assume is compact, so by axisymmetry one must have either or . Integrating Eq. (77) over then shows that if then and so the metric in square brackets is AdS2. The horizon metric extends to a smooth metric on if and only if . It can then be checked the near-horizon geometry is isometric to that of extremal Kerr for or Kerr-AdS for . It is also easy to check that for it corresponds to extremal Kerr-dS. In the non-static case, the horizon topology theorem excludes the possibility of for . If the non-static possibility with can also be excluded .
It would be interesting to remove the assumption of axisymmetry in the above theorem. In  it is shown that regular non-axisymmetric linearised solutions of Eq. (66) about the extremal Kerr near-horizon geometry do not exist. This supports the conjecture that any smooth solution of Eq. (66) on must be axisymmetric and hence given by the above theorem.
In this case there are several different symmetry assumptions one could make. Classifications are known for homogeneous horizons and horizons invariant under a -rotational symmetry.
We may define a homogeneous near-horizon geometry to be one for which the Riemannian manifold is a homogeneous space whose transitive isometry group also leaves the rest of the near-horizon data invariant. Since any near-horizon geometry (7) possesses the 2D symmetry generated by and where , it is clear that this definition guarantees the near-horizon geometry itself is a homogeneous spacetime. Conversely, if the near-horizon geometry is a homogeneous spacetime, then any cross section must be a homogeneous space under a subgroup of the spacetime isometry group, which commutes with the 2D symmetry in the plane (since is a constant submanifold). It follows that must also be invariant under the isometry , showing that our original definition is indeed equivalent to the near-horizon geometry being a homogeneous spacetime.
Homogeneous geometries can be straightforwardly classified without assuming compactness of as follows.
Theorem 4.4 (). Any vacuum, homogeneous, non-static near-horizon geometry is locally isometric towhere is a -connection over a 2D base space satisfying with . The curvature of the connection is , where is the volume form of the 2D base, and .
The proof uses the fact that homogeneity implies must be a Killing field and then one reduces the problem onto the 2D orbit space. Observe that for one recovers the static near-horizon geometries. For and we see that so that the 2D metric is a round and the horizon metric is locally isometric to a homogeneously squashed . Hence we have:
Corollary 4.1. Any vacuum, homogeneous, non-static near-horizon geometry is locally isometric to the near-horizon limit of the extremal Myers–Perry black hole with rotational symmetry (i.e., equal angular momenta). For one gets the same result with the Myers–Perry black hole replaced by its generalisation with a cosmological constant .
For we see that there are more possibilities depending on the sign of . If we again have a horizon geometry locally isometric to a homogeneous . If , we can write and the -connection is non-trivial, so the cross sections are the Nil group manifold with its standard homogeneous metric. For , we can write and the connection , so the cross sections are the group manifold with its standard homogeneous metric, unless , which gives . Hence we have:
Corollary 4.2. For any vacuum, homogeneous, non-static near-horizon geometry is locally isometric to either the near-horizon limit of the extremal rotating black hole  with rotational symmetry, or a near-horizon geometry with: (i) and its standard homogenous metric, (ii) and its standard homogeneous metric or (iii) .
This is analogous to a classification first obtained for supersymmetric near-horizon geometries in gauged supergravity, see Proposition (5.3).
We now consider a weaker symmetry assumption, which allows for inhomogeneous horizons. A -rotational isometry is natural in five dimensions and all known explicit black-hole solutions have this symmetry. The following classification theorem has been derived:
Theorem 4.5 (). Consider a vacuum non-static near-horizon geometry with a -rotational isometry and a compact cross section . It must be globally isometric to the near-horizon geometry of one of the following families of black-hole solutions:
- : the 3-parameter boosted extremal Kerr string.
- : the 2-parameter extremal Myers–Perry black hole or the 3-parameter ‘fast’ rotating extremal KK black hole .
- : the Lens space quotients of the above solutions.
- The near-horizon geometry of the vacuum extremal black ring  is a 2-parameter subfamily of case 1, corresponding to a Kerr string with vanishing tension .
- The near-horizon geometry of the ‘slowly’ rotating extremal KK black hole  is identical to that of the 2-parameter extremal Myers–Perry in case 2.
- The cases can be written as a single 3-parameter family of near-horizon geometries .
- The case has been ruled out .
For , the analogous problem has not been solved. The only known solution in this case is the near-horizon geometry of the rotating black hole with a cosmological constant , which generalises the Myers–Perry black hole. It would be interesting to classify near-horizon geometries with in this case since this would capture the near-horizon geometry of the yet-to-be-found asymptotically-AdS5 black ring. A perturbative attempt at constructing such a solution is discussed in .
For spacetime dimension , so the horizon cross section dim, the horizon equation (66) is far less constraining than in lower dimensions. Few general classification results are known, although several large families of vacuum near-horizon geometries have been constructed.
The only known classification result for vacuum near-horizon geometries is for solutions with -rotational symmetry. These generalise the axisymmetric solutions and solutions with -symmetry discussed in Sections 4.3 and 4.4 respectively. By performing a detailed study of the orbit spaces it has been shown that the only possible topologies for are: , , , and .
An explicit classification of the possible near-horizon geometries (for the non-toroidal case) was derived in , see their Theorem 1. Using their theorem it is easy to show that the most general solution with is in fact isometric to the near-horizon geometry of a boosted extremal Kerr-membrane (i.e., perform a general boost of Kerr along the coordinates and then compactify ). Non-static near-horizon geometries with have been ruled out  (including a cosmological constant).
The Myers–Perry (MP) black-hole solutions  generically have isometry groups where . Observe that if then and hence these solutions fall outside the classification discussed in Section 4.5.1. They are parameterised by their mass parameter and angular momentum parameters for . The topology of the horizon cross section . A generalisation of these metrics with non-zero cosmological constant has been found . We will focus on the case, although analogous results hold for the solutions.
The location of the horizon is determined by the largest positive number such that in odd and even dimensions and , respectively, where: . It is worth noting that if subsets of the angular momentum parameters are set equal, the rotational symmetry enhances to a non-Abelian unitary group.
Since these are vacuum solutions one can trivially add flat directions to generate new solutions. For example, by adding one flat direction one can generate a boosted MP string, whose near-horizon geometries have topology. Interestingly, for odd dimensions the resulting geometry has commuting rotational isometries. For this reason, it was conjectured that a special case of this is also the near-horizon geometry of yet-to-be-found asymptotically-flat black rings (as is known to be the case in five dimensions) .
Despite the absence of explicit black-hole solutions, a number of solutions to Eq. (66) are known. It is an open problem as to whether there are corresponding black-hole solutions to these near-horizon geometries.
All the constructions given below employ the following data. Let be a compact Fano Kähler–Einstein manifold19 of complex dimension and is the indivisible class given by with (the Fano index and satisfies with equality iff ). The Kähler–Einstein metric on is normalised as and we denote its isometry group by . The simplest example occurs for , in which case with .
In even dimensions greater than four, an infinite class of near-horizon geometries is revealed by the following result.
Proposition 4.1 (). Let and be the principal -bundle over any Fano Kähler–Einstein manifold , specified by the characteristic class . For each there exists a 1-parameter family of smooth solutions to Eq. (66) on the associated -bundles .
The dim ansatz used for the near-horizon data is the invariant form
The simplest example is the , solution, for which the near-horizon data takes the explicit form
For the Fano base is higher dimensional and there are more choices available. The topology of the total space is always a non-trivial -bundle over and in fact different give different topologies, so there are an infinite number of horizon topologies allowed. Furthermore, one can choose to have no continuous isometries giving examples of near-horizon geometries with a single -rotational isometry. Hence, if there are black holes corresponding to these horizon geometries they would saturate the lower bound in the rigidity theorem.
It is worth noting that the local form of the above class of near-horizon metrics includes as a special case that of the extremal MP metrics with equal angular momenta (for ). The above class of horizon geometries are of the same form as the Einstein metrics on complex line bundles , which in four dimensions corresponds to the Page metric , although we may of course set .
Similar constructions of increasing complexity can be made in odd dimensions, again revealing an infinite class of near-horizon geometries.
Proposition 4.2. Let and be the principal -bundle over specified by the characteristic class . There exists a 1-parameter family of Sasakian solutions to Eq. (66) on .
As a simple example consider . This leads to an explicit homogeneous near-horizon geometry with
The above proposition can be generalised as follows.
Proposition 4.3 (). Given any Fano Kähler–Einstein manifold of complex dimension and coprime satisfying , there exists a 1-parameter family of solutions to Eq. (66) where is a compact Sasakian -manifold.
These examples have symmetry, although possess only one independent angular momentum along the -fibres. These are deformations of the Sasaki–Einstein manifolds .
There also exist a more general class of non-Sasakian horizons in odd dimensions.
Proposition 4.4 (). Let be the principal -bundle over any Fano Kähler–Einstein manifold , specified by the characteristic classes where . For a countably infinite set of non-zero integers , there exists a two-parameter family of smooth solutions to Eq. (66) on the associated Lens space bundles .
The dim form of the near-horizon data in the previous two propositions is the invariant form4.3 arise as a special case with and possess only one independent angular momentum. The base gives horizon topologies or depending on whether is even or odd respectively.
It is worth noting that the local form of this class of near-horizon metrics includes as a special case that of the extremal MP metrics with all but one equal angular momenta. The above class of horizon geometries are of the same form as the Einstein metrics found in [37, 157].