  Rapidly rotating Myers-Perry black holes in d>5 dimensions were conjectured
to be unstable by Emparan and Myers. In a previous publication, we found
numerically the onset of the axisymmetric ultraspinning instability in the
singly-spinning Myers-Perry black hole in d=7,8,9. This threshold signals also
a bifurcation to new branches of axisymmetric solutions with pinched horizons
that are conjectured to connect to the black ring, black Saturn and other
families in the phase diagram of stationary solutions. We firmly establish that
this instability is also present in d=6 and in d=10,11. The boundary conditions
of the perturbations are discussed in detail for the first time and we prove
that they preserve the angular velocity and temperature of the original
Myers-Perry black hole. This property is fundamental to establish a
thermodynamic necessary condition for the existence of this instability in
general rotating backgrounds. We also prove a previous claim that the
ultraspinning modes cannot be pure gauge modes. Finally we find new
ultraspinning Gregory-Laflamme instabilities of rotating black strings and
branes that appear exactly at the critical rotation predicted by the
aforementioned thermodynamic criterium. The latter is a refinement of the
Gubser-Mitra conjecture.
