The second variation of 2-spheres in the 2n-sphere

We prove sharp upper and lower bounds for the Morse index and nullity of linearly full branched minimal 2-spheres in the round 2n-sphere. In particular, we show that the Morse index of a linearly full minimal 2-sphere in the round 2n-sphere is at least n(n-1)(2n+1), with equality if and only if the 2-sphere is twistor-equivalent to the Boruvka sphere. Our techniques also apply more generally to give bounds for the Morse index and nullity for totally isotropic surfaces in the 2n-sphere.