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An explicit 3-dimensional Riemannian metric is constructed which can beinterpreted as the (conformal) sum of two Kerr black holes with aligned angularmomentum. When the separation distance between them is large we prove that thismetric has positive Ricci scalar and hence positive Yamabe invariant. Thismetric can be used to construct axially symmetric initial data for two Kerrblack holes with large angular momentum.

The Yamabe invariant for axially symmetric two Kerr black holes initial data