Ultraspinning instability: the missing link

We study linearized perturbations of Myers-Perry black holes in d = 7, with two of the three angular momenta set to be equal, and show that instabilities always appear before extremality. Analogous results are expected for all higher odd d. We determine numerically the stationary perturbations that mark the onset of instability for the modes that preserve the isometries of the background. The onset is continuously connected between the previously studied sectors of solutions with a single angular momentum and solutions with all angular momenta equal. This shows that the near-extremality instabilities are of the same nature as the ultraspinning instability of d >= 6 singly-spinning solutions, for which the angular momentum is unbounded. Our results raise the question of whether there are any extremal Myers-Perry black holes which are stable in d >= 6.