Small deformations of extreme five dimensional Myers–Perry black hole initial data

We demonstrate the existence of a one-parameter family of initial data for the vacuum Einstein equations in five dimensions representing small deformations of the extreme Myers–Perry black hole. This initial data set has ‘t-ϕi$$t-\phi ^i$$’ symmetry and preserves the angular momenta and horizon geometry of the extreme solution. Our proof is based upon an earlier result of Dain and Gabach-Clement concerning the existence of U(1)$$U(1)$$-invariant initial data sets which preserve the geometry of extreme Kerr (at least for short times). In addition, we construct a general class of transverse, traceless symmetric rank 2 tensors in these geometries.