On the index of minimal 2-tori in the 4-sphere

In this note, we prove that any minimal 2-torus in S-4 has Morse index at least 6, with equality if and only if it is congruent to the Clifford torus in some great S-3 subset of S-4. For a minimal 2-torus in S-n with vanishing Hopf differential, we show that its index is at least n + 3 and that this estimate is sharp: the equilateral 2-torus fully embedded in S-5 subset of S-n as a homogeneous minimal surface in S-n has index exactly n+3.