Black Hole Uniqueness
In four dimensions, Hawking proved that the topology of the event horizon of a non-rotating black hole must be spherical. Because the proof uses the Gauss-Bonnet theorem, it does not generalize to higher dimensions. The discovery of black ring solutions in five dimensions shows that other topologies are allowed in higher dimensions, but it is unclear precisely which topologies are allowed. It has been shown that the horizon must be of positive Yamabe type, meaning that it must admit a metric of positive scalar curvature.
Read more about this topic: Higher-dimensional Einstein Gravity
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