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"Classification of Near-Horizon Geometries of Extremal Black Holes"

Hari K. Kunduri and James Lucietti

	Abstract
1	Introduction
	1.1	Black holes in string theory
	1.2	Gauge/gravity duality
	1.3	Black hole classification
	1.4	This review
2	Degenerate Horizons and Near-Horizon Geometry
	2.1	Coordinate systems and near-horizon limit
	2.2	Curvature of near-horizon geometry
	2.3	Einstein equations and energy conditions
	2.4	Physical charges
3	General Results
	3.1	Horizon topology theorem
	3.2	AdS₂-structure theorems
4	Vacuum Solutions
	4.1	Static: all dimensions
	4.2	Three dimensions
	4.3	Four dimensions
	4.4	Five dimensions
	4.5	Higher dimensions
5	Supersymmetric Solutions
	5.1	Four dimensions
	5.2	Five dimensions
	5.3	Six dimensions
	5.4	Ten dimensions
	5.5	Eleven dimensions
6	Solutions with Gauge Fields
	6.1	Three dimensional Einstein–Maxwell–Chern–Simons theory
	6.2	Four dimensional Einstein–Maxwell theory
	6.3	Five dimensional Einstein–Maxwell–Chern–Simons theory
	6.4	Theories with hidden symmetry
	6.5	Non-Abelian gauge fields
7	Applications and Related Topics
	7.1	Black-hole uniqueness theorems
	7.2	Stability of near-horizon geometries and extremal black holes
	7.3	Geometric inequalities
	7.4	Analytic continuation
	7.5	Extremal branes
	Acknowledgements
	References
	Footnotes

Although spacetimes correspond to Lorentzian metrics, one can often analytically continue these to complete Riemannian metrics. Indeed, the first example of an inhomogeneous Einstein metric on a compact manifold was found by Page, by taking a certain limit of the Kerr–de Sitter metrics [185

], giving a metric on $--- ℂ ℙ2#ℂ ℙ2$ .