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We prove that the intrinsic geometry of compact cross-sections of an extremal horizon in four-dimensional Einstein-Maxwell theory must admit a Killing vector field or is static. This implies that any such horizon must be an extremal Kerr-Newman horizon and completes the classification of the associated near-horizon geometries. The same results hold with a cosmological constant., Comment: 7 pages

We construct all axi-symmetric non-gradient $m$-quasi-Einstein structures on a two-sphere. This includes the spatial cross-section of the extreme Kerr black hole horizon corresponding to $m=2$, as well as a family of new regular metrics with $m\neq 2$ given in terms of hypergeometric functions. We also show that in the case $m=-1$ with vanishing cosmological constant the only orientable compact solution in dimension two is the flat torus, which proves that there are no compact surfaces with a metrisable affine connection with skew Ricci tensor., Comment: v2: minor edits, reference added

We consider the classification of supersymmetric black hole solutions to five-dimensional STU gauged supergravity that admit torus symmetry. This reduces to a problem in toric K\"ahler geometry on the base space. We introduce the class of separable toric K\"ahler surfaces that unify product-toric, Calabi-toric and orthotoric K\"ahler surfaces, together with an associated class of separable 2-forms. We prove that any supersymmetric toric solution that is timelike, with a separable K\"ahler base space and Maxwell fields, outside a horizon with a compact (locally) spherical cross-section, must be locally isometric to the known black hole or its near-horizon geometry. An essential part of the proof is a near-horizon analysis which shows that the only possible separable K\"ahler base space is Calabi-toric. In particular, this also implies that our previous black hole uniqueness theorem for minimal gauged supergravity applies to the larger class of separable K\"ahler base spaces., Comment: 47 pages, 1 figure. v2: minor edits, published version

We prove that any analytic vacuum spacetime with a positive cosmological constant in four and higher dimensions, that contains a static extremal Killing horizon with a maximally symmetric compact cross-section, must be locally isometric to either the extremal Schwarzschild de Sitter solution or its near-horizon geometry (the Nariai solution). In four-dimensions, this implies these solutions are the only analytic vacuum spacetimes that contain a static extremal horizon with compact cross-sections (up to identifications). We also consider the analogous uniqueness problem for the four-dimensional extremal hyperbolic Schwarzschild anti-de Sitter solution and show that it reduces to a spectral problem for the laplacian on compact hyperbolic surfaces, if a cohomological obstruction to the uniqueness of infinitesimal transverse deformations of the horizon is absent., Comment: v2: 19 pages, 1 figure; accepted version. Corrected first order deformations for negative cosmological constant case (now Proposition 2) and added Appendix B, main results unchanged

We prove that the intrinsic geometry of compact cross-sections of any vacuum extremal horizon must admit a Killing vector field. If the cross-sections are two-dimensional spheres, this implies that the most general solution is the extremal Kerr horizon and completes the classification of the associated near-horizon geometries. The same results hold with a cosmological constant. Furthermore, we also deduce that any non-trivial vacuum near-horizon geometry, with a non-positive cosmological constant, must have an SO(2,1) isometry in all dimensions under no symmetry assumptions. We also show that, if the cross-sections are two-dimensional, the horizon Einstein equation is equivalent to a single fourth order PDE for the K\"ahler potential, and that this equation is explicitly solvable on the sphere if the corresponding metric admits a Killing vector., Comment: 13 pages

We consider the classification of supersymmetric AdS$_5$ black hole solutions to minimal gauged supergravity that admit a torus symmetry. This problem reduces to finding a class of toric K\"ahler metrics on the base space, which in symplectic coordinates are determined by a symplectic potential. We derive the general form of the symplectic potential near any component of the horizon or axis of symmetry, which determines its singular part for any black hole solution in this class, including possible new solutions such as black lenses and multi-black holes. We find that the most general known black hole solution in this context, found by Chong, Cvetic, L\"u and Pope (CCLP), is described by a remarkably simple symplectic potential. We prove that any supersymmetric and toric solution that is timelike outside a smooth horizon, with a K\"ahler base metric of Calabi type, must be the CCLP black hole solution or its near-horizon geometry., Comment: 41 pages, 3 figures. v2: typos fixed, published version

We present a classification of asymptotically flat, supersymmetric black hole and soliton solutions of five-dimensional minimal supergravity that admit a single axial symmetry which `commutes' with the supersymmetry. This includes the first examples of five-dimensional black hole solutions with exactly one axial Killing field that are smooth on and outside the horizon. The solutions have similar properties to the previously studied class with biaxial symmetry, in particular, they have a Gibbons-Hawking base and the harmonic functions must be of multi-centred type with the centres corresponding to the connected components of the horizon or fixed points of the axial symmetry. We find a large moduli space of black hole and soliton spacetimes with non-contractible 2-cycles and the horizon topologies are $S^3$, $S^1\times S^2$ and lens spaces $L(p,1)$., Comment: v2: 53 pages, 1 table; accepted version. Minor clarifications in assumptions, typos corrected, references added, table of numerical results added, results unchanged. v3: Corrected remarks after Corollary 2, minor changes in Theorem 2 and proof of Theorem 3, results unchanged

We prove uniqueness and existence theorems for four-dimensional asymptotically flat, Ricci-flat, gravitational instantons with a torus symmetry. In particular, we prove that such instantons are uniquely characterised by their rod structure, which is data that encodes the fixed point sets of the torus action. Furthermore, we establish that for every admissible rod structure there exists an instanton that is smooth up to possible conical singularities at the axes of symmetry. The proofs involve adapting the methods that are used to establish black hole uniqueness theorems, to a harmonic map formulation of Ricci-flat metrics with torus symmetry, where the target space is directly related to the metric (rather than auxiliary potentials). We also give an elementary proof of the nonexistence of asymptotically flat toric half-flat instantons. Finally, we derive a general set of identities that relate asymptotic invariants such as the mass to the rod structure., Comment: 29 pages, 3 figures. v2: remark on static case added, references added. v3: minor clarifications, published version. v4: minor correction and clarification in Introduction and Section 2.3, updated since published version (results unchanged)

We prove that any supersymmetric solution to five-dimensional minimal gauged supergravity with $SU(2)$ symmetry, that is timelike outside an analytic horizon, is a Gutowski-Reall black hole or its near-horizon geometry. The proof combines a delicate near-horizon analysis with the general form for a K\"ahler metric with cohomogeneity-1 $SU(2)$ symmetry. We also prove that any timelike supersymmetric soliton solution to this theory, with $SU(2)$ symmetry and a nut or a complex bolt, has a K\"ahler base with enhanced $U(1)\times SU(2)$ symmetry, and we exhibit a family of asymptotically AdS$_5/\mathbb{Z}_p$ solitons for $p \geq 3$ with a bolt in this class., Comment: 33 pages. v2: minor edits, published version

We demonstrate that five-dimensional, asymptotically flat, stationary and biaxisymmetric, vacuum black holes with lens space $L(n, 1)$ topology, possessing the simplest rod structure, do not exist. In particular, we show that the general solution on the axes and horizon, which we recently constructed by exploiting the integrability of this system, must suffer from a conical singularity on the inner axis component. We give a proof of this for two distinct singly spinning configurations and numerical evidence for the generic doubly spinning solution., Comment: v2: minor changes, published version; 23 pages, 3 figures

We prove that the only asymptotically flat spacetimes with a suitably regular event horizon, in a generalised Majumdar-Papapetrou class of solutions to higher-dimensional Einstein-Maxwell theory, are the standard multi-black holes. The proof involves a careful analysis of the near-horizon geometry and an extension of the positive mass theorem to Riemannian manifolds with conical singularities. This completes the classification of asymptotically flat, static, extreme black hole solutions in this theory., Comment: 10 pages. v2: minor correction, assumption added, references added. v3: main result stated as a theorem, published version

We consider the classification of asymptotically flat, stationary, vacuum black hole spacetimes in four and five dimensions, that admit one and two commuting axial Killing fields respectively. It is well known that the Einstein equations reduce to a harmonic map on the two-dimensional orbit space, which itself arises as the integrability condition for a linear system of spectral equations. We integrate the Belinski-Zakharov spectral equations along the boundary of the orbit space and use this to fully determine the metric and associated Ernst and twist potentials on the axes and horizons. This is sufficient to derive the moduli space of solutions that are free of conical singularities on the axes, for any given rod structure. As an illustration of this method we obtain constructive uniqueness proofs for the Kerr and Myers-Perry black holes and the known doubly spinning black rings., Comment: 58 pages. v2: typos fixed, minor clarifications, to appear in Adv. Theor. Math. Phys

We present a systematic study of the moduli space of asymptotically flat, supersymmetric and biaxisymmetric black hole solutions to five-dimensional minimal supergravity. Previously, it has been shown that such solutions must be multi-centred solutions with a Gibbons-Hawking base. In this paper we perform a full analysis of three-centred solutions with a single black hole, for which there are seven regular black hole solutions. We find that four of these can have the same conserved charges as the BMPV black hole. These consist of a black lens with L(3,1) horizon topology and three distinct families of spherical black holes with nontrivial topology outside the horizon. The former provides the first example of a nonspherical black hole with the same conserved charges as the BMPV black hole. Moreover, of these four solutions, three can have a greater entropy than the BMPV black hole near the BMPV upper spin bound. One of these is a previously known spherical black hole with nontrivial topology and the other two are new examples of a spherical black hole with nontrivial topology and an L(3,1) black lens., Comment: version 2: typos corrected, minor clarifications added; 26 pages, 10 figures

We derive new identities for the thermodynamic variables of five-dimensional, asymptotically flat, stationary and biaxisymmetric vacuum black holes. These identities depend on the topology of the solution and include contributions arising from certain topological charges. The proof employs the harmonic map formulation of the vacuum Einstein equations for solutions with these symmetries., Comment: 5 pages. v2: minor edits

We determine all infinitesimal transverse deformations of extreme horizons in Einstein-Maxwell theory that preserve axisymmetry. In particular, we show that the general static transverse deformation of the AdS(2) X S(2) near-horizon geometry is a two-parameter family, which contains the known extreme charged, accelerating, static black hole solution held in equilibrium by an external electric or magnetic field (Ernst solution) and a special case of the extreme Kerr-Newman-Melvin solution. More generally, we find a three-parameter family of deformations of the extreme Kerr-Newman horizon, which contains the extreme Kerr-Newman-Melvin solution and a rotating generalisation of the Ernst solution. We also consider vacuum gravity with a cosmological constant and prove uniqueness of axisymmetric transverse deformations of the extreme Kerr-AdS horizon. Finally, we completely classify transverse deformations of extreme horizons in three-dimensional Einstein-Maxwell theory with a negative cosmological constant., Comment: 36 pages

We prove that five-dimensional extreme anti-de Sitter black ring solutions to the vacuum Einstein equations that admit biaxial symmetry do not exist. This is established by demonstrating the nonexistence of five-dimensional, biaxisymmetric, vacuum near-horizon geometries of extreme horizons with a nonzero cosmological constant and ring topology., Comment: 6 pages

We determine all asymptotically flat, supersymmetric and biaxisymmetric soliton and black hole solutions to five dimensional minimal supergravity. In particular, we show that the solution must be a multi-centred solution with a Gibbons-Hawking base. The proof involves combining local constraints from supersymmetry with global constraints for stationary and biaxisymmetric spacetimes. This reveals that the horizon topology must be one of S^3, S^1 x S^2 or a lens space L(p,1), thereby providing a refinement of the allowed horizon topologies. We construct the general smooth solution for each possible rod structure. We find a large moduli space of black hole spacetimes with noncontractible 2-cycles for each of the allowed horizon topologies. In the absence of a black hole we obtain a classification of the known `bubbling' soliton spacetimes., Comment: v2: 43 pages, 5 figures, references added, typos fixed, minor clarifications, expanded discussion of physical parameters. v3: clarified statement of theorem 4 (results unchanged, update made after publication)

We prove that any asymptotically flat static spacetime in higher dimensional Einstein-Maxwell theory must have no magnetic field. This implies that there are no static soliton spacetimes and completes the classification of static non-extremal black holes in this theory. In particular, these results establish that there are no asymptotically flat static spacetimes with non-trivial topology, with or without a black hole, in Einstein-Maxwell theory., Comment: 9 pages. v2: minor edits, references added

In five-dimensional minimal supergravity, there are spherical black holes with nontrivial topology outside the horizon which have the same conserved charges at infinity as the BMPV solution. We show that some of these black holes have greater entropy than the BMPV solution. These spacetimes are all asymptotically flat, stationary, and supersymmetric. We also show that there is a limit in which the black hole shrinks to zero size and the solution becomes a nonsingular "bubbling" geometry. Thus, these solutions provide explicit analytic examples of placing black holes inside solitons., Comment: 17 pages, 5 figures; v2: references added

We present a new supersymmetric, asymptotically flat, black hole solution to five-dimensional U(1)^3-supergravity which is regular on and outside an event horizon of lens space topology L(2,1). The solution has seven independent parameters and uplifts to a family of 1/8-supersymmetric D1-D5-P black brane solutions to Type IIB supergravity. The decoupling limit is asymptotically AdS(3) x S^3 x T^4, with a near-horizon geometry that is a twisted product of the near-horizon geometry of the extremal BTZ black hole and L(2,1) x T^4, although it is not (locally) a product space in the bulk. We show that the decoupling limit of a special case of the black lens is related to that of a black ring by spectral flow, thereby supplying an account of its entropy. Analogous solutions of U(1)^N-supergravity are also presented., Comment: v2. 11 pages, 1 figure, substantial additions: new result which relates a special case of the black lens to a black ring by spectral flow; section on domain of outer communication; minor edits and additions in the main text; included Appendix on black lens solution to U(1)^N supergravity with detailed regularity analysis; references added

We consider the inverse problem of determining all extreme black hole solutions to the Einstein equations with a prescribed near-horizon geometry. We investigate this problem by considering infinitesimal deformations of the near-horizon geometry along transverse null geodesics. We show that, up to a gauge transformation, the linearised Einstein equations reduce to an elliptic PDE for the extrinsic curvature of a cross-section of the horizon. We deduce that for a given near-horizon geometry there exists a finite dimensional moduli space of infinitesimal transverse deformations. We then establish a uniqueness theorem for transverse deformations of the extreme Kerr horizon. In particular, we prove that the only smooth axisymmetric transverse deformation of the near-horizon geometry of the extreme Kerr black hole, such that cross-sections of the horizon are marginally trapped surfaces, corresponds to that of the extreme Kerr black hole. Furthermore, we determine all smooth and biaxisymmetric transverse deformations of the near-horizon geometry of the five-dimensional extreme Myers-Perry black hole with equal angular momenta. We find a three parameter family of solutions such that cross-sections of the horizon are marginally trapped, which is more general than the known black hole solutions. We discuss the possibility that they correspond to new five dimensional vacuum black holes., Comment: 37 pages. v2: typos fixed, reference added, published version

We present a new supersymmetric, asymptotically flat, black hole solution to five-dimensional supergravity. It is regular on and outside an event horizon of lens space topology L(2,1). It is the first example of an asymptotically flat black hole with lens space topology. The solution is characterised by a charge, two angular momenta and a magnetic flux through a non-contractible disc region ending on the horizon, with one constraint relating these., Comment: 4 pages. v2: minor changes, title of published version differs

We consider AdS gravity duals to CFT on background spacetimes with a null infinity. Null infinity on the conformal boundary may extend to an extremal horizon in the bulk. For example it does so for Poincare-AdS, although does not for planar Schwarzschild-AdS. If null infinity does extend into an extremal horizon in the bulk, we show that the bulk near-horizon geometry is determined by the geometry of the boundary null infinity. Hence the `infra-red' geometry of the bulk is fixed by the large scale behaviour of the CFT spacetime. In addition the boundary stress tensor must have a particular decay at null infinity. As an application, we argue that for CFT on asymptotically flat backgrounds, any static bulk dual containing an extremal horizon extending from the boundary null infinity, must have the near-horizon geometry of Poincare-AdS. We also discuss a class of boundary null infinity that cannot extend to a bulk extremal horizon, although we give evidence that they can extend to an analogous null surface in the bulk which possesses an associated scale-invariant `near-geometry'., Comment: 35 pages

The domain of outer communication of five-dimensional asymptotically flat stationary spacetimes may possess non-trivial 2-cycles. We discuss how this may lead to a gross violation of black hole uniqueness, beyond the existence of black rings, even for solutions with two commuting rotational symmetries. We illustrate this with a simple example in minimal supergravity; a four parameter family of supersymmetric black hole solutions, with spherical horizon topology and a 2-cycle in the exterior. We show there are black holes in this family with identical conserved changes to the BMPV black hole, thereby demonstrating black hole non-uniqueness in this context. We find a decoupling limit of this family of black holes that yields spacetimes asymptotic to the near-horizon geometry of a BMPV black hole which contain a black hole and an exterior 2-cycle., Comment: 24 pages. v2: minor additions, published version. v3: corrected physical quantities, results unaffected, differs from published version

We determine the most general three-dimensional vacuum spacetime with a negative cosmological constant containing a non-singular Killing horizon. We show that the general solution with a spatially compact horizon possesses a second commuting Killing field and deduce that it must be related to the BTZ black hole (or its near-horizon geometry) by a diffeomorphism. We show there is a general class of asymptotically AdS$_3$ extreme black holes with arbitrary charges with respect to one of the asymptotic-symmetry Virasoro algebras and vanishing charges with respect to the other. We interpret these as descendants of the extreme BTZ black hole., Comment: 10 pages. v2: general solution has second Killing field, changes and additions to reflect this, references added. v3: included analysis of non-extreme solution and discussion

We derive a mass formula and a mass variation law for asymptotically flat, stationary spacetimes, invariant under two commuting rotational symmetries, in a general five dimensional theory of gravity coupled to an arbitrary set of Maxwell fields and uncharged scalar fields. If the spacetime is everywhere regular, these mass formulas reduce to a sum of magnetic flux terms defined on its non-trivial 2-cycles. If there is a black hole, we obtain a mass variation law more general than previously obtained, which also has contributions from the 2-cycles exterior to the black hole. This can be interpreted as the first law of black hole mechanics in a background soliton containing bubbles., Comment: 13 pages. v2: minor changes, references added, published version

Any spacetime containing a degenerate Killing horizon, such as an extremal black hole, possesses a well-defined notion of a near-horizon geometry. We review such near-horizon geometry solutions in a variety of dimensions and theories in a unified manner. We discuss various general results including horizon topology and near-horizon symmetry enhancement. We also discuss the status of the classification of near-horizon geometries in theories ranging from vacuum gravity to Einstein-Maxwell theory and supergravity theories. Finally, we discuss applications to the classification of extremal black holes and various related topics. Several new results are presented and open problems are highlighted throughout., Comment: 70 pages; invited review article for Living Reviews in Relativity; v2 some improvements and references added

We consider stationary extreme black hole solutions to the Einstein-Yang-Mills equations in four dimensions, allowing for a negative cosmological constant. We prove that any axisymmetric black hole of this kind possesses a near-horizon AdS(2) symmetry and deduce its near-horizon geometry must be that of the abelian embedded extreme Kerr-Newman (AdS) black hole. We also show that the near-horizon geometry of any static black hole is a direct product of AdS(2) and a constant curvature space., Comment: 12 pages

Aretakis has proved that a massless scalar field has an instability at the horizon of an extreme Reissner-Nordstr\"om black hole. We show that a similar instability occurs also for a massive scalar field and for coupled linearized gravitational and electromagnetic perturbations. We present numerical results for the late time behaviour of massless and massive scalar fields in the extreme RN background and show that instabilities are present for initial perturbations supported outside the horizon, e.g.\ an ingoing wavepacket. For a massless scalar we show that the numerical results for the late time behaviour are reproduced by an analytic calculation in the near-horizon geometry. We relate Aretakis' conserved quantities at the future horizon to the Newman-Penrose conserved quantities at future null infinity., Comment: 44 pages, 19 figures

We present a new infinite class of near-horizon geometries of degenerate horizons, satisfying Einstein's equations for all odd dimensions greater than five. The symmetry and topology of these solutions is compatible with those of black holes. The simplest examples give horizons of spatial topology S^3xS^2 or the non-trivial S^3-bundle over S^2. More generally, the horizons are Lens space bundles associated to certain principal torus-bundles over Fano Kaehler-Einstein manifolds. We also consider the classification problem for Einstein metrics on such Lens space bundles and derive a family which unifies all the known examples (Sasakian and non-Sasakian)., Comment: 42 pages. v2: minor changes, published version

We show that any extreme black hole with an orthogonally transitive abelian isometry group has a near-horizon geometry with enhanced symmetry. We also point out a simple proof of the horizon topology theorem of Galloway and Schoen for degenerate horizons., Comment: 5 pages

Aretakis has proved the existence of an instability of a massless scalar field at the horizon of an extreme Kerr or Reissner-Nordstrom black hole: for generic initial data, a transverse derivative of the scalar field at the horizon does not decay, and higher transverse derivatives blow up. We show that a similar instability occurs for linearized gravitational, and electromagnetic, perturbations of an extreme Kerr black hole. We show also that the massless scalar field instability occurs for extreme black hole solutions of a large class of theories in various spacetime dimensions., Comment: 13 pages. v2: minor clarifications. v3: minor changes, published version

We point out a simple construction of an infinite class of Einstein near-horizon geometries in all odd dimensions greater than five. Cross-sections of the horizons are inhomogeneous Sasakian metrics (but not Einstein) on S^3xS^2 and more generally on Lens space bundles over any compact positive Kaehler-Einstein manifold. They are all consistent with the known topology and symmetry constraints for asymptotically flat or globally AdS black holes., Comment: 10 pages, Latex. v2: corrected minor error, results unaffected; minor improvements. v3: improved and clarified global analysis and topology (updated since published version)

The elliptic Einstein-DeTurck equation may be used to numerically find Einstein metrics on Riemannian manifolds. Static Lorentzian Einstein metrics are considered by analytically continuing to Euclidean time. Ricci-DeTurck flow is a constructive algorithm to solve this equation, and is simple to implement when the solution is a stable fixed point, the only complication being that Ricci solitons may exist which are not Einstein. Here we extend previous work to consider the Einstein-DeTurck equation for Riemannian manifolds with boundaries, and those that continue to static Lorentzian spacetimes which are asymptotically flat, Kaluza-Klein, locally AdS or have extremal horizons. Using a maximum principle we prove that Ricci solitons do not exist in these cases and so any solution is Einstein. We also argue that Ricci-DeTurck flow preserves these classes of manifolds. As an example we simulate Ricci-DeTurck flow for a manifold with asymptotics relevant for AdS_5/CFT_4. Our maximum principle dictates there are no soliton solutions, and we give strong numerical evidence that there exists a stable fixed point of the flow which continues to a smooth static Lorentzian Einstein metric. Our asymptotics are such that this describes the classical gravity dual relevant for the CFT on a Schwarzschild background in either the Unruh or Boulware vacua. It determines the leading O(N^2) part of the CFT stress tensor, which interestingly is regular on both the future and past Schwarzschild horizons., Comment: 48 pages, 7 figures; Version 2 - section 2.2.1 on manifolds with boundaries substantially modified, corrected and extended. Discussion in section 3.1 amended. References added and minor changes

We consider the classification of near-horizon geometries in a general two-derivative theory of gravity coupled to abelian gauge fields and uncharged scalars in four and five dimensions, with one and two commuting rotational symmetries respectively. Assuming that the theory of gravity reduces to a 3d non-linear sigma model (as is typically the case for ungauged supergravities), we show that the functional form of any such near-horizon geometry may be determined. As an example we apply this to five dimensional minimal supergravity. We also construct an example of a five parameter near-horizon geometry solution to this theory with S^1 X S^2 horizon topology. We discuss its relation to the near-horizon geometries of the yet to be constructed extremal black rings with both electric and dipole charges., Comment: Latex, 30 pages. v2: discussion in section 5 modified and improved, other minor changes, references added

We present a new class of near-horizon geometries which solve Einstein's vacuum equations, including a negative cosmological constant, in all even dimensions greater than four. Spatial sections of the horizon are inhomogeneous S^2-bundles over any compact Kaehler-Einstein manifold. For a given base, the solutions are parameterised by one continuous parameter (the angular momentum) and an integer which determines the topology of the horizon. In six dimensions the horizon topology is either S^2 x S^2 or CP^2 # -CP^2. In higher dimensions the S^2-bundles are always non-trivial, and for a fixed base, give an infinite number of distinct horizon topologies. Furthermore, depending on the choice of base we can get examples of near-horizon geometries with a single rotational symmetry (the minimal dimension for this is eight). All of our horizon geometries are consistent with all known topology and symmetry constraints for the horizons of asymptotically flat or globally Anti de Sitter extremal black holes., Comment: 42 pages, latex. v2: corrected section 6.1, two references added. v3: modified angular momentum and corrected area comparison, version to be published in Commun. Math. Phys

We consider the classification of static near-horizon geometries of stationary extremal (not necessarily BPS) black hole solutions of five dimensional Einstein-Maxwell theory coupled to a Chern-Simons term with coupling xi (with xi=1 corresponding to supergravity). Assuming the black holes have two rotational symmetries, we show that their near-horizon geometries are either the direct product AdS_3 X S^2 or a warped product of AdS_2 and compact 3d space. In the AdS_2 case we are able to classify all possible near-horizon geometries with no magnetic fields. There are two such solutions: the direct product AdS_2 X S^3 as well as a warped product of AdS_2 and an inhomogeneous S^3. The latter solution turns out to be near-horizon limit of an extremal Reissner-Nordstrom black hole in an external electric field. In the AdS_2 case with magnetic fields, we reduce the problem (in all cases) to a single non-linear ODE. We show that if there are any purely magnetic solutions of this kind they must have S^1 X S^2 horizon topology, and for xi^2 <1/4 we find examples of solutions with both electric and magnetic fields., Comment: Latex, 28 pages. v2: minor changes, reference added

We prove uniqueness theorems for asymptotically flat, stationary, extremal, vacuum black hole solutions, in four and five dimensions with one and two commuting rotational Killing fields respectively. As in the non-extremal case, these problems may be cast as boundary value problems on the two dimensional orbit space. We show that the orbit space for solutions with one extremal horizon is homeomorphic to an infinite strip, where the two boundaries correspond to the rotational axes, and the two asymptotic regions correspond to spatial infinity and the near-horizon geometry. In four dimensions this allows us to establish the uniqueness of extremal Kerr amongst asymptotically flat, stationary, rotating, vacuum black holes with a single extremal horizon. In five dimensions we show that there is at most one asymptotically flat, stationary, extremal vacuum black hole with a connected horizon, two commuting rotational symmetries and given interval structure and angular momenta. We also provide necessary and sufficient conditions for four and five dimensional asymptotically flat vacuum black holes with the above symmetries to be static (valid for extremal, non-extremal and even non-connected horizons)., Comment: Latex, 23 pages, 4 figures. v2: improved theorem 2, clarifications, references added. v3: new result added (staticity theorem), minor clarifications. v4: minor changes, published version.

We consider stationary extremal black hole solutions of the Einstein-Maxwell equations with a negative cosmological constant in four dimensions. We determine all non-static axisymmetric near-horizon geometries (with non-toroidal horizon topology) and all static near-horizon geometries for black holes of this kind. This allows us to deduce that the most general near-horizon geometry of an asymptotically globally AdS(4) rotating extremal black hole, is the near-horizon limit of extremal Kerr-Newman-AdS(4). We also identify the subset of near-horizon geometries which are supersymmetric. Finally, we show which physical quantities of extremal black holes may be computed from the near-horizon limit alone, and point out a simple formula for the entropy of the known supersymmetric AdS(4) black hole. Analogous results are presented in the case of vanishing cosmological constant., Comment: 18 pages, Latex. v2: footnote added on pg. 12. v3: assumption of non-toroidal horizon topology made explicit, minor clarifications

We consider the near-horizon geometries of extremal, rotating black hole solutions of the vacuum Einstein equations, including a negative cosmological constant, in four and five dimensions. We assume the existence of one rotational symmetry in 4d, two commuting rotational symmetries in 5d and in both cases non-toroidal horizon topology. In 4d we determine the most general near-horizon geometry of such a black hole, and prove it is the same as the near-horizon limit of the extremal Kerr-AdS(4) black hole. In 5d, without a cosmological constant, we determine all possible near-horizon geometries of such black holes. We prove that the only possibilities are one family with a topologically S^1 X S^2 horizon and two distinct families with topologically S^3 horizons. The S^1 X S^2 family contains the near-horizon limit of the boosted extremal Kerr string and the extremal vacuum black ring. The first topologically spherical case is identical to the near-horizon limit of two different black hole solutions: the extremal Myers-Perry black hole and the slowly rotating extremal Kaluza-Klein (KK) black hole. The second topologically spherical case contains the near-horizon limit of the fast rotating extremal KK black hole. Finally, in 5d with a negative cosmological constant, we reduce the problem to solving a sixth-order non-linear ODE of one function. This allows us to recover the near-horizon limit of the known, topologically S^3, extremal rotating AdS(5) black hole. Further, we construct an approximate solution corresponding to the near-horizon geometry of a small, extremal AdS(5) black ring., Comment: 45 pages, latex. v2: minor clarifications in section 3. v3: assumption of non-toroidal horizon topology made explicit, improved section 3, references added

We consider extremal black hole solutions to the vacuum Einstein equations in dimensions greater than five. We prove that the near-horizon geometry of any such black hole must possess an SO(2,1) symmetry in a special case where one has an enhanced rotational symmetry group. We construct examples of vacuum near-horizon geometries using the extremal Myers-Perry black holes and boosted Myers-Perry strings. The latter lead to near-horizon geometries of black ring topology, which in odd spacetime dimensions have the correct number rotational symmetries to describe an asymptotically flat black object. We argue that a subset of these correspond to the near-horizon limit of asymptotically flat extremal black rings. Using this identification we provide a conjecture for the exact ``phase diagram'' of extremal vacuum black rings with a connected horizon in odd spacetime dimensions greater than five., Comment: 53 pages, latex, 19 eps figures. v2: added refs + one eps figure. v3: published version

We consider some aspects of counting BPS operators which are annihilated by two supercharges, in superconformal field theories. For non-zero coupling, the corresponding multi-variable partition functions can be written in terms of generating functions for vector partitions or their weighted generalisations. We derive asymptotics for the density of states for a wide class of such multi-variable partition functions. We also point out a particular factorisation property of the finite N partition functions. Finally, we discuss the concept of a limit curve arising from the large N partition functions, which is related to the notion of a typical state and discuss some implications for the holographic duals., Comment: 39 pages latex. v2: minor improvements, reference added

We provide a classification of near-horizon geometries of supersymmetric, asymptotically anti-de Sitter, black holes of five-dimensional U(1)^3-gauged supergravity which admit two rotational symmetries. We find three possibilities: a topologically spherical horizon, an S^1 \times S^2 horizon and a toroidal horizon. The near-horizon geometry of the topologically spherical case turns out to be that of the most general known supersymmetric, asymptotically anti-de Sitter, black hole of U(1)^3-gauged supergravity. The other two cases have constant scalars and only exist in particular regions of this moduli space -- in particular they do not exist within minimal gauged supergravity. We also find a solution corresponding to the near-horizon geometry of a three-charge supersymmetric black ring held in equilibrium by a conical singularity; when lifted to type IIB supergravity this solution can be made regular, resulting in a discrete family of warped AdS(3) geometries. Analogous results are presented in U(1)^n gauged supergravity., Comment: Latex, 29 pages. v2: minor improvements, references added

Recent work has demonstrated an attractor mechanism for extremal rotating black holes subject to the assumption of a near-horizon SO(2,1) symmetry. We prove the existence of this symmetry for any extremal black hole with the same number of rotational symmetries as known four and five dimensional solutions (including black rings). The result is valid for a general two-derivative theory of gravity coupled to abelian vectors and uncharged scalars, allowing for a non-trivial scalar potential. We prove that it remains valid in the presence of higher-derivative corrections. We show that SO(2,1)-symmetric near-horizon solutions can be analytically continued to give SU(2)-symmetric black hole solutions. For example, the near-horizon limit of an extremal 5D Myers-Perry black hole is related by analytic continuation to a non-extremal cohomogeneity-1 Myers-Perry solution., Comment: 21 pages, latex. v2: minor improvements v3: Corrected error in argument excluding de Sitter and Poincare-symmetric cases. Results unaffected

We determine the most general near-horizon geometry of a supersymmetric, asymptotically anti-de Sitter, black hole solution of five-dimensional minimal gauged supergravity that admits two rotational symmetries. The near-horizon geometry is that of the supersymmetric, topologically spherical, black hole solution of Chong et al. This proves that regular supersymmetric anti-de Sitter black rings with two rotational symmetries do not exist in minimal supergravity. However, we do find a solution corresponding to the near-horizon geometry of a supersymmetric black ring held in equilibrium by a conical singularity, which suggests that nonsupersymmetric anti-de Sitter black rings may exist but cannot be "balanced" in the supersymmetric limit., Comment: Latex, 18 pages, 1 figure. v2: minor changes